Logic I Notes | B.A. LL.B. (Five Year Course) Semester 1 | Mumbai University | munotes

Logic is the science of reasoning. This is the definition most Indian textbooks put first, and it is the narrowest of the three. It says that logic is concerned with one particular mental operation, inference, the passage from something known to something not yet known.

Logic is the science and art of reasoning. This third form, associated with Archbishop Whately, adds a word deliberately. A science is a systematic body of knowledge about something. An art is a practical skill, a set of rules for doing something well. Whately's point was that logic is both: it studies inference and it also trains you to infer better, in the way that grammar both describes a language and teaches you to write it.

Science, art, or normative science

Examiners ask this as a short note, so it is worth having the answer ready in three sentences.

Logic is a science because it is a systematic and reasoned body of knowledge about a definite subject matter, namely the forms of correct inference. It is an art because its rules can be applied, and applying them is a skill that improves with practice. It is called a normative science because it does not merely record how people in fact think, which is what a purely positive science would do; it lays down a standard which thinking ought to satisfy.

The contrast that makes this clear is with psychology. Both logic and psychology study thinking. Psychology asks how people actually think, including all the ways they think badly, and it treats a fallacy as a fact to be explained. Logic asks how people ought to think if their conclusions are to be worth anything, and it treats a fallacy as an error to be corrected. Ethics and aesthetics are normative in the same way: ethics sets a standard for conduct, aesthetics for beauty, logic for thought.

"Logic is the science of sciences"

Added after the past-paper check. MU sets this exact phrase as an essay, usually joined to the questions whether logic is an art and whether deductive logic is purely formal while inductive is purely material. All three are answered here.

What the phrase claims. That logic is not one science among others but the science every other science depends on, because every science reasons and none of them studies reasoning. Chemistry tells you what happens when two substances meet; it does not tell you whether a conclusion follows from what it has found. Logic supplies that to all of them and takes nothing from any of them.

It is Aristotle's point in later words. He called his logical works the Organon, the instrument, for the same reason: an instrument is used by every craft and is the property of none.

How far the claim is fair. Fair, in that no science can dispense with logic and logic borrows nothing from any of them. Overstated, in that logic supplies only the form of reasoning: it can tell a chemist that his conclusion follows and can never tell him that his premises are true, which is the limit already stated at sequence 110.

Is logic an art as well? Yes, on Whately's answer above: a science because it is an organised body of knowledge about the forms of valid inference, and an art because its rules can be applied and applying them is a skill that improves with practice.

Is deductive logic purely formal and inductive purely material? The claim is half right and the half that is wrong is the examinable half.

Deductive logic is purely formal, and this part is correct. Validity depends on the arrangement of terms and propositions alone, so any two arguments of the same form stand or fall together whatever they are about. That is why the whole of Modules II and III can be conducted without a single real fact.

Inductive logic is NOT purely material. It certainly depends on matter in a way deduction does not: whether a sample is varied, whether an analogy's resemblances are relevant, whether a cause is plausible, are all questions about the subject and not about the shape. But induction also has form. Simple enumeration has a form, analogy has a form, and the tests of strength at sequence 90 are general rules that apply whatever the instances are about.

So the accurate answer is that deduction is formal and induction is both formal and material, its form supplying the pattern and its matter supplying the strength. A student who agrees with the proposition as put has conceded more than the evidence allows.

The three parts of traditional logic, which are also this syllabus

Traditional logic divided its own subject into three, following the three operations the mind was thought to perform.

Simple apprehension, the grasping of a single idea, expressed in language by a term. "Contract" is a term.

Judgment, the putting of two ideas together and asserting or denying one of the other, expressed in language by a proposition. "Every contract is an agreement" is a proposition.

Reasoning, the passing from one or more judgments to another, expressed in language by an inference or argument. "Every contract is an agreement; this is a contract; therefore this is an agreement" is an inference.

Now look at the syllabus. Module I ends with terms. Module II is propositions. Module III is inference. MU's Modules I to III are the three parts of traditional logic in their traditional order, and the reason the order cannot be rearranged is that each part is built out of the one before it: an inference is made of propositions, and a proposition is made of terms.

A worked example

Two neighbours, Anil and Bharat, are arguing. Anil says the boundary wall belongs to him because his grandfather built it. Bharat says it belongs to him because the municipal record shows the plot boundary running on the other side of it.

Nothing about that dispute is a matter of logic yet. Whether the grandfather built the wall is a question of fact, to be answered by evidence. Whether a municipal record proves ownership is a question of law, to be answered out of the statutes and the cases.

Logic enters at the third stage, and only there. Suppose the court accepts that the grandfather built the wall, and accepts a legal rule that a person who builds a wall on their own land owns it. Then "Anil's grandfather built the wall on his own land; a person who builds a wall on their own land owns it; therefore Anil's family owns the wall" is an inference, and logic can say whether the conclusion follows. It cannot say whether the grandfather built anything.

That is the whole division of labour. Evidence supplies the facts, law supplies the rules, and logic supplies the guarantee that the conclusion drawn from them is the conclusion they actually support.

Distinctions that carry marks

What this does not mean

Logic is not about being clever in an argument. Winning a quarrel by shouting, by ridicule or by appealing to sympathy is exactly what logic exists to expose. Those are studied later as fallacies, and they are studied as faults.

Logic does not tell you whether a statement is true. It tells you whether a statement follows from other statements. Whether those other statements are themselves true is a question for evidence, for science, or for the law of the land.

"Logical" does not mean "sensible". An argument can be perfectly valid and lead to a conclusion nobody would accept, because it started from a premise nobody should have accepted. Validity is about the connection, never about the starting point.

Limits of the traditional definitions

Each of the three traditional definitions has been criticised, and the criticism is what produced the modern definition in the next chapter.

"Science of the laws of thought" is too wide and too narrow at once. Too wide, because psychology also studies the laws of thought and is not logic. Too narrow, because a great deal of thinking, remembering a face, imagining a scene, feeling regret, involves no inference at all and logic has nothing to say about it.

"Science of reasoning" is criticised for the opposite reason: it names only the third of the three operations, when traditional logic in fact studies terms and propositions as well. The answer usually given is that terms and propositions are studied only because inference is made of them, which is fair, but it makes the definition depend on a qualification that the definition itself does not state.

Quick revision

Definition to write: logic is the science which investigates the general principles of valid thought, that is, of the distinction between correct and incorrect reasoning.

Etymology: Greek logos, meaning word, thought or reason. Aristotle called his logical works the Organon, the instrument.

Three traditional definitions: science of the laws of thought (Hamilton); science of reasoning; science and art of reasoning (Whately).

Logic is a normative science: it prescribes a standard, it does not merely describe. Compare ethics for conduct and aesthetics for beauty.

Logic against psychology: ought against is; product against process; error against fact.

Three parts of traditional logic: terms, propositions, inference. They are MU's Modules I, II and III.

"Logic is the science of sciences": every science reasons and none studies reasoning, so logic serves all of them and borrows from none. Aristotle's Organon makes the same point. Overstated only in that logic supplies form and never the truth of the premises.

Deduction is purely formal; induction is NOT purely material, since it has form as well, the tests of strength being general rules. Agreeing with the proposition as put concedes too much.

Main criticisms: "laws of thought" is both too wide (psychology) and too narrow (non-inferential thinking); "science of reasoning" ignores terms and propositions.

Test yourself

1. Define logic and explain each part of your definition.

Logic is the science which investigates the general principles of valid thought. "Science" means a systematic body of reasoned knowledge, not a collection of tips. "General principles" means that logic studies the form of reasoning and not the subject matter, so one principle covers arguments about contracts and arguments about chemistry alike. "Valid" means that logic asks whether the conclusion follows from the premises, not whether the premises are true.

2. Why is logic called a normative science?

Because it lays down a norm or standard which thinking ought to satisfy, instead of merely recording how people in fact think. A positive science such as physics describes what happens. Logic tells us what ought to follow from what, and treats a piece of reasoning that departs from it as wrong rather than as merely unusual. Ethics and aesthetics are normative in the same sense, for conduct and for beauty.

3. Distinguish logic from psychology.

Both study thought, but they ask different questions. Psychology asks how the mind actually works and includes bad reasoning among the facts it explains; it studies thought as a process going on in a particular person at a particular time. Logic asks how one ought to reason if the conclusion is to be worth anything, and studies thought as a product, judged by its form and independently of who thought it. A fallacy is a fact for the psychologist and an error for the logician.

4. Is logic a science or an art?

Both, and the traditional answer is Whately's. It is a science because it is an organised body of knowledge about a definite subject, the forms of valid inference. It is an art because its principles can be applied and the application is a skill that improves with practice. The comparison usually drawn is with grammar, which both describes a language and teaches its correct use.

5. State two criticisms of the definition "logic is the science of the laws of thought".

First, it is too wide: psychology also investigates the laws of thought, and psychology is not logic, so the definition fails to mark logic off from a neighbouring subject. Second, it is too narrow: a great deal of thinking, such as remembering, imagining or wishing, involves no inference, and logic says nothing whatever about it, so the definition claims more territory than logic actually occupies.

7. Explain "logic is the science of sciences".

The phrase claims that logic is not one science beside the others but the one every other science depends on, since every science reasons and none of them studies reasoning. Chemistry can tell you what two substances do and cannot tell you whether a conclusion follows from that. Aristotle made the same point by calling his logical works the Organon, the instrument, which every craft uses and none owns. The claim is fair as far as it goes and overstated in one respect: logic supplies the form of reasoning and never the truth of the premises.

8. Is it correct to say that deductive logic is purely formal and inductive logic purely material?

Half correct. Deductive logic is purely formal, since validity depends on the arrangement of terms and propositions alone and any two arguments of the same form stand or fall together. Inductive logic is not purely material: it does depend on the matter in a way deduction does not, since the variety of a sample and the relevance of a resemblance are questions about the subject, but it also has form, simple enumeration and analogy each having a pattern and the tests of strength being general rules. Deduction is formal; induction is both formal and material.

6. What are the three parts of traditional logic and how do they relate to each other?

Terms, propositions and inference, corresponding to simple apprehension, judgment and reasoning. They are studied in that order because each is built out of the one before: a term is the smallest unit, a proposition is made by joining two terms, and an inference is made by putting propositions together. This is also the order of Modules I, II and III of this syllabus, which is why the modules cannot be read out of sequence.

It is symbolic. Because the form is what matters, the form is written down in symbols and the matter is thrown away. Ordinary language is ambiguous, long, and full of words that do no logical work; a symbol is exact. This is why modern logic is often simply called symbolic logic, and why George Boole's book of 1854 was called The Laws of Thought but did its work in algebra.

It is truth-functional. In the modern treatment of compound propositions, the truth of the whole is settled entirely by the truth of the parts. Knowing whether "p" is true and whether "q" is true is enough to settle whether "p and q" is true. Nothing else about p and q matters. Module II's truth tables are this feature at work, and it is the single largest technical difference from the traditional scheme.

Who did it

A short answer is worth having, because it is asked as a short note.

George Boole, 1854, treated logical relations as an algebra, so that reasoning could be calculated rather than argued about. Gottlob Frege, 1879, invented the modern treatment of quantifiers and generality and insisted that logic is not psychology. Bertrand Russell and A. N. Whitehead, in Principia Mathematica from 1910, built the whole of arithmetic out of logical notions and gave the subject the symbolism most books now use.

You are not asked to know their work. You are asked to know that modern logic is not a rival opinion about the same material but a rebuilding of the subject on a different foundation, and these are the names attached to the rebuilding.

A worked example

Take an ordinary legal sentence. "If the notice was validly served, the tenancy stood terminated on 31 March."

The traditional logician asks what kind of proposition this is, finds it is a hypothetical proposition, notes its antecedent and its consequent, and moves on. The apparatus stops fairly quickly.

The modern logician symbolises it. Let p stand for "the notice was validly served" and q for "the tenancy stood terminated on 31 March". The sentence is p ⊃ q.

Now something becomes visible that was not visible before. Suppose we also know ~q, that is, that the tenancy did not stand terminated on 31 March. From p ⊃ q and ~q it follows that ~p: the notice was not validly served. Suppose instead we know ~p, that the notice was not validly served. Nothing follows about q at all, because the sentence never said that valid service was the only way the tenancy could end.

That second point is a mistake lawyers make constantly, and it has a name, the fallacy of denying the antecedent. The symbolism makes it obvious in a line. The English original hides it, because "if" in English is often used where "if and only if" is meant.

Distinctions that carry marks

The last two rows are the ones examiners return to, and both come back in Module II. They are also the reason MU's topics 2.5, 2.8 and 2.9 exist at all.

What this does not mean

Modern logic did not prove traditional logic wrong. Every valid syllogism is still valid. What modern logic showed is that the traditional scheme was incomplete: there are valid inferences it cannot represent, of which relational arguments are the plainest example.

"Symbolic" does not mean "difficult". The symbols are shorthand. Anything written in them can be written in English, at greater length and with more risk of ambiguity.

Modern logic is not only about mathematics. It was built by people interested in the foundations of mathematics, which is a fact about its history and not about its scope. Its use in law is exactly the use shown in the worked example above.

Limits and criticism

The commonest criticism of modern logic is that it buys precision at the cost of realism. Real arguments, in courts especially, are not made of neat propositions with settled truth values. A witness is partly believed. A statute is ambiguous. A conclusion is reached on the balance of probabilities and not by proof. Truth-functional logic has nothing to say about any of that.

The honest answer is that this is a limit and not a refutation. Modern logic tells you what follows with certainty from what. Where certainty is not available, other tools are needed, which is why induction, probability and the assessment of evidence are separate topics on this very syllabus.

Quick revision

Definition to write: logic is the study of the methods and principles used to distinguish correct from incorrect reasoning.

The move that made it modern: logic is about propositions and their relations, not about thought. This is called anti-psychologism.

Four features: anti-psychologistic, formal, symbolic, truth-functional.

Names: Boole 1854, Frege 1879, Russell and Whitehead 1910.

Against traditional logic: whole proposition rather than term as the unit; symbols rather than words; handles relational propositions; denies existential import to universals.

Standing criticism: it assumes propositions are simply true or false, which real argument, and real litigation, often are not.

Test yourself

1. State the modern definition of logic and explain why each part of it is worded as it is.

Logic is the study of the methods and principles used to distinguish correct from incorrect reasoning. "Methods and principles" keeps the definition to things that can be taught and applied, and so keeps psychology out. "Used to distinguish" makes logic a test rather than a description. "Correct from incorrect reasoning" fixes the object of judgment as the inference, the passage from premises to conclusion, and not the truth of any individual statement.

2. What is meant by saying modern logic is formal?

That it studies the form of an argument and disregards its matter, meaning its subject. Two arguments about entirely different things have the same form if they have the same arrangement of terms and propositions, and every argument of a valid form is valid. This is why symbols can replace the content: the content was never doing any work.

3. Explain truth-functionality with one example.

A compound proposition is truth-functional when its truth value is completely determined by the truth values of its parts. "The notice was served and the rent was paid" is true when both parts are true and false otherwise, and nothing else about the two parts is relevant. Truth tables are possible only because of this feature, and they are the standard modern method of testing compound propositions.

4. Give two respects in which modern logic goes beyond traditional logic.

First, relational propositions. "A is taller than B" cannot be forced into the traditional subject-predicate scheme without distortion, and arguments turning on relations were simply outside the older system; modern logic treats them as ordinary. Second, existential import: the traditional scheme takes a universal proposition to assert that its subject exists, so "all trespassers will be prosecuted" implies there are trespassers, while modern logic takes it as a conditional that implies nothing of the kind.

5. Symbolise "If the goods were delivered then the price is payable", and state what follows if the price is not payable.

Let p stand for "the goods were delivered" and q for "the price is payable". The proposition is p ⊃ q. If the price is not payable, that is ~q, then it follows that ~p, the goods were not delivered. That inference is valid and is called denying the consequent. The reverse move, from ~p to ~q, does not follow: the proposition never said delivery was the only ground on which the price could become payable.

6. Is the traditional account of logic now worthless?

No. Every valid syllogism remains valid and every rule of eduction still holds; modern logic did not contradict traditional logic but showed that it covered less ground than had been supposed. It remains the natural way to talk about ordinary categorical statements, and this syllabus examines both, which is why Module II asks for a comparative study rather than for one scheme alone.

Standard form

To analyse an argument it is set out in standard form: each premise on its own numbered line, then a line, then the conclusion. The order of the premises never matters.

Take: "Since every agreement enforceable by law is a contract, and this agreement is enforceable by law, it is a contract."

1. Every agreement enforceable by law is a contract.

2. This agreement is enforceable by law.

Therefore, this agreement is a contract.

Rewriting an argument this way is not decoration. It forces three decisions: which statements are in the argument at all, which one is the conclusion, and what the premises actually say once the connecting words are stripped off. Most disagreements about an argument turn out to be disagreements about one of those three.

Arguments with more than one step

Not every argument is one move. In a chain argument, the conclusion of one step becomes a premise of the next, and the intermediate statement is called a sub-conclusion.

1. The notice was posted on 1 March.

2. A notice posted by registered post is deemed served on the third day.

3. Therefore the notice was served on 4 March. [sub-conclusion]

4. The tenancy ends thirty days after service.

Therefore the tenancy ended on 3 April.

Statement 3 is a conclusion in relation to 1 and 2, and a premise in relation to 4. Every judgment of any length is built this way, and a student who cannot see the joints will read the whole thing as one undifferentiated assertion.

There is a second structure worth naming. In a convergent argument, several independent premises each support the conclusion on their own, so that destroying one leaves the others standing. In a linked argument the premises work only together, so that destroying one destroys the argument. A pleader who advances four independent grounds is arguing convergently on purpose.

A worked example

Here is a passage of the kind that appears in an examination paper.

"The defendant company cannot escape liability. It employed the driver, and the driver was on his employer's business when the collision happened. An employer is answerable for a wrong committed by an employee in the course of employment."

Step one, find the conclusion. There is no indicator word, which is common. Ask which statement the others are there to support. The first sentence is the point being urged; the rest is offered as reason for it. The conclusion is "the defendant company cannot escape liability", which restated positively is "the defendant company is liable".

Step two, list the premises. That the company employed the driver; that the driver was acting in the course of his employment; that an employer is answerable for an employee's wrong committed in the course of employment.

Step three, standard form.

1. An employer is answerable for a wrong committed by an employee in the course of employment.

2. The defendant company employed the driver.

3. The driver was acting in the course of his employment when the collision happened.

Therefore the defendant company is liable.

Step four, notice what the form shows. The premises are linked and not convergent: knock out premise 3 and the argument collapses entirely, which is exactly why the whole of the litigation in such a case is fought over premise 3. Setting the argument out in standard form does not tell you who wins. It tells you where the fight is.

Distinctions that carry marks

What this does not mean

An argument is not a heated exchange. In ordinary speech "they had an argument" means a quarrel. In logic it means a piece of reasoning, and the two have almost nothing to do with each other. A perfectly polite paragraph in a textbook is an argument; two people shouting is not.

A long passage is not automatically an argument. Much writing states facts, tells a story or describes a scene without claiming that anything follows from anything. The next chapter is about telling the difference.

The conclusion is not the last sentence. It is very often the first, particularly in legal writing, where the practice is to state the proposition and then support it.

Quick revision

Definition: an argument is a group of propositions in which one, the conclusion, is claimed to follow from the others, the premises.

Only statements can be premises or conclusions. Questions and commands cannot.

One conclusion, any number of premises. Truth of the premises is not required for something to be an argument.

Conclusion indicators: therefore, hence, thus, so, consequently, accordingly, it follows that.

Premise indicators: because, since, for, as, given that, inasmuch as, on the ground that.

Standard form: premises numbered, line, conclusion. Order of premises is irrelevant.

Chain argument: a sub-conclusion serves as a premise for the next step.

Linked against convergent: linked premises work only together; convergent ones work independently.

Test yourself

1. Define an argument and say what an argument must contain.

An argument is a group of propositions in which one proposition, the conclusion, is asserted on the strength of the others, the premises, which are put forward as reasons for it. It must contain at least one premise and exactly one conclusion, and every one of its members must be a proposition, that is, a sentence capable of being true or false. It need not contain true premises and its conclusion need not actually follow.

2. How do you find the conclusion of a passage that contains no indicator word?

Ask which single statement the rest of the passage is there to support. Test a candidate by putting "therefore" in front of it and "because" in front of the others and seeing whether the passage still reads as the author intended. In legal writing the conclusion is very often the opening sentence, since the practice is to state the proposition first and then give reasons.

3. Put this into standard form: "He cannot be convicted, for the prosecution has not proved the last link in the chain, and a case on circumstantial evidence fails unless every link is proved."

1. A case resting on circumstantial evidence fails unless every link in the chain is proved.

2. The prosecution has not proved the last link in the chain.

Therefore he cannot be convicted.

The word "for" marks the premises and the conclusion stands first, which is the usual legal order.

4. Distinguish linked premises from convergent premises, and say why the difference matters to a lawyer.

Linked premises support the conclusion only in combination, so that if one is destroyed the argument fails altogether. Convergent premises each support the conclusion independently, so that destroying one leaves the others intact. It matters because it tells a pleader where to attack and how to draft: a case built on linked premises has a single weakest point, while independent grounds of appeal are deliberately convergent so that losing one ground does not lose the appeal.

5. Is "Please serve the notice today, because the limitation period expires tomorrow" an argument?

No, or at least not as it stands. "Please serve the notice today" is a request and not a proposition, so it cannot be a conclusion; it is neither true nor false. The passage does contain a reason, and it can be turned into an argument by restating the request as the proposition "the notice ought to be served today", which is capable of truth and falsity and can then be supported.

6. What is a sub-conclusion?

A statement that is the conclusion of one step of an argument and a premise of the next. It appears wherever an argument takes more than one move, which is the normal shape of a judgment. Identifying sub-conclusions is what separates reading a judgment as reasoning from reading it as a block of assertion.

An argument asserts its premises and asserts its conclusion. A conditional asserts neither. That is the whole distinction, and it can be tested in one move: ask whether the speaker has committed themselves to anything being true. In a conditional they have not.

A conditional can of course be a part of an argument, and usually is. "If the notice was not served, the suit is premature. The notice was not served. Therefore the suit is premature." That is an argument with three statements, of which the first is a conditional serving as a premise.

The explanation trap

Compare two sentences, both containing "because".

A. "The suit was dismissed because it was filed after the period of limitation had expired."

B. "The suit must have been dismissed, because no execution proceedings were ever taken out."

They look identical in structure and they are doing completely different work. In A, everybody already accepts that the suit was dismissed; the sentence tells you why it happened. In B, nobody yet knows whether the suit was dismissed; the sentence gives a reason for believing that it was.

A is an explanation. B is an argument. The test is the one implicit in that comparison, and it is the standard test:

Is the statement in question already accepted as a fact? If it is, the passage is telling you the cause of it, and is an explanation. If it is in doubt, and the other statement is offered to remove the doubt, the passage is an argument.

The two are worth separating carefully, because law is full of both. A judgment that says "the accused was acquitted because the sole witness turned hostile" is explaining. A judgment that says "the witness must have been won over, because his statement in court contradicts his statement to the police in every material particular" is arguing.

Enthymemes: the argument with a piece missing

Real arguments almost never state everything. An enthymeme is an argument in which a premise, or occasionally the conclusion, is left unstated because it is thought too obvious to be worth saying.

"He is a partner in the firm, so he is liable for its debts."

The stated premise is that he is a partner. The stated conclusion is that he is liable. The unstated premise is the whole of the law being relied on: that every partner is liable for the debts of the firm. Supplying it is called completing the enthymeme, and it is the single most useful thing a law student can learn to do, for two reasons.

First, the suppressed premise is usually where the argument is weakest, precisely because nobody said it out loud. Second, in legal argument the suppressed premise is usually the proposition of law, and identifying it is identifying what the case is actually about.

There is a rule for supplying it. Supply the weakest premise that makes the argument work. If the missing premise can be stated modestly or sweepingly, choose the modest form, because attributing a sweeping claim to someone who did not make it is a way of losing an argument you should have won.

A worked example

Read this passage and decide what it is.

"The plaintiff has produced no receipt. In a suit for recovery of money lent, the burden of proving the loan lies on the plaintiff. Where a person who could easily produce the best evidence of a transaction does not produce it, a court may presume that it would not have helped him. The claim therefore deserves to be dismissed."

Is anything offered in support of anything? Yes, plainly. The last sentence carries the indicator "therefore".

Is it an explanation? No. It is not telling us why the claim was dismissed, as a known past event. It is urging that it ought to be dismissed, which is in doubt.

Are there any conditionals? The third sentence has the shape of one but is in fact a general rule, asserted. It is a premise.

Any enthymeme? Yes, and finding it is the point of the exercise. The argument moves from "no receipt was produced" to "the claim deserves to be dismissed", and the step is missing that a receipt is the best evidence of a money loan and could easily have been produced. Without that step the third premise does not connect to the first at all. Once supplied, the weakness is visible: whether a receipt could easily have been produced is a question of fact that this passage never establishes.

That is what identification is for. The passage looked like four confident sentences. Setting it out showed that it stands on an assumption nobody stated.

Distinctions that carry marks

What this does not mean

"Because" does not mark an argument. It marks a reason, and reasons come in two kinds: reasons why something happened and reasons for believing something is so. Only the second makes an argument.

A passage without indicator words may still be an argument. Legal writing frequently omits them entirely and relies on the reader to see the structure.

An enthymeme is not a defective argument. It is the ordinary form of argument in speech and in writing. Only the completed version can be assessed, which is why completing it is a step and not a criticism.

Quick revision

The test: is any statement offered as evidence or reason for accepting another? If not, no argument.

Not arguments: reports, descriptions, illustrations, statements of opinion standing alone, loosely associated statements, warnings, advice, conditionals, explanations.

Conditional: asserts neither its antecedent nor its consequent, so it cannot be an argument by itself, though it can be a premise.

Explanation against argument: is the statement already accepted, or in doubt? Accepted means explanation.

Enthymeme: an argument with a premise or conclusion left unstated. Complete it before assessing it, and supply the weakest premise that makes it work.

Test yourself

1. State the test for whether a passage contains an argument.

Ask whether any statement in the passage is being offered as evidence or as a reason for accepting another statement, that is, whether there is an inferential claim. If there is, the passage is an argument, however badly expressed. If nothing supports anything, the passage is not an argument, however long, learned or persuasive it may be.

2. Distinguish an argument from an explanation and give one example of each.

In an argument the statement in question is in doubt and the other statements are offered as grounds for believing it. In an explanation the statement is already accepted and the other statements give its cause. "The decree must be a nullity, since the court had no pecuniary jurisdiction" is an argument. "The decree was set aside because the court had no pecuniary jurisdiction" is an explanation of something already known to have happened.

3. Why is a conditional statement not an argument?

Because it asserts neither of its two component statements. An argument puts its premises forward as true and claims its conclusion is true; a conditional claims only that the one would follow from the other, and commits the speaker to nothing about whether either is actually so. It can, however, serve as a premise in an argument, and in legal reasoning it usually does.

4. Complete this enthymeme and identify the suppressed premise: "The agreement was made with a minor, so it is void."

1. An agreement with a person who is not competent to contract is void.

2. A minor is not competent to contract.

3. The agreement was made with a minor.

Therefore the agreement is void.

The suppressed material is the whole of the law relied on, statements 1 and 2, which is the usual position: what is left unsaid in a legal enthymeme is the legal proposition, and that is exactly what the case turns on.

5. Is the following an argument? "Trespass to land is actionable without proof of damage. So is libel. So is assault."

No. It is an illustration. The three sentences do not support one another; the second and third are further examples of the same kind of tort, offered to make the class clear rather than to establish that any of them is actionable without damage. Nothing is in doubt and nothing is being proved.

6. What rule governs the supplying of a missing premise?

Supply the weakest premise that makes the argument work. A missing premise can usually be stated either modestly or sweepingly, and only the modest version is a fair reconstruction of what the speaker meant. Attributing an extravagant claim to an opponent in order to refute it easily is itself a fault of reasoning, and one that a court will notice.

Three questions do most of the work here. What is asserted as fact? What is asserted as law or as general rule? What is assumed and never asserted at all? The third is the enthymeme of the last chapter, and it belongs to content analysis because a suppressed premise is a suppressed assertion.

In a legal argument the content divides cleanly into the two kinds. Facts are established by evidence and can be disputed by evidence. Rules are established out of statutes and cases and can be disputed by argument about statutes and cases. Confusing the two is the commonest failure in first-year answers: a student who tries to prove a rule by producing more facts, or to prove a fact by citing a case, has not analysed the content.

Language

Ask how the words are being used, and whether they are doing more than stating.

Four things are looked for here, and each of them will be met again as a fallacy or as a defect of definition later in this book.

Ambiguity. A word with two settled meanings, used so that the argument slides from one to the other. "Sanction" means both a penalty and an approval. An argument that uses it in one sense in the premise and the other in the conclusion proves nothing at all.

Vagueness. A word with one meaning but no clear boundary. "Reasonable", "prompt", "material" are vague, and vagueness is not always a fault: statutes use vague words deliberately when the range of future cases cannot be foreseen. It becomes a fault when the argument treats the vague word as though it were precise.

Emotive language. Words chosen for the attitude they carry rather than for what they describe. "The defendant admitted" and "the defendant confessed" report the same event and invite different verdicts. Emotive language is not automatically illegitimate, but an argument that does its work by it has not done its work.

Rhetorical form. Questions asked to make an assertion, repetition, and the studied understatement that legal writing is fond of. These affect how the argument lands and change nothing about whether it is sound.

Structure or form

Ask how the statements are connected: which are premises, which is the conclusion, and what depends on what.

This is the work of the previous two chapters, applied. Set the argument in standard form. Identify sub-conclusions. Decide whether the premises are linked or convergent. Note whether the argument claims to be deductive, in which case the conclusion is said to follow necessarily, or inductive, in which case it is said to follow with some degree of probability.

Structure is the dimension on which logic proper has most to say, and it is the only one of the four that can be assessed without knowing anything about the subject matter. That is why it is the dimension the rest of this syllabus is about.

Form and content, taken as a pair

Added after the past-paper check, because MU sets "form and content" as a short note in its own right, and the two are worth putting side by side rather than leaving at opposite ends of the list.

The content of an argument is what it is about: the propositions themselves, the facts alleged, the rules relied on. It is also called the matter of the argument.

The form of an argument is how those propositions are arranged: which is premise, which is conclusion, how they connect.

The whole of logic is on the form side. This is the point sequence 20 made about modern logic and it is worth stating plainly here: two arguments about entirely different subjects have the same form if they have the same arrangement, and every argument of a valid form is valid. "All contracts are agreements; this is a contract; therefore this is an agreement" and "All whales are mammals; this is a whale; therefore this is a mammal" differ completely in content and not at all in form.

Which is why the two are assessed by different people. The form is assessed by logic and needs no knowledge of the subject at all. The content is assessed by evidence, by science or by the law of the land, and logic has nothing to contribute to it.

Formal truth and material truth. The pair has a second use, applied to a whole argument rather than to a proposition. An argument has formal truth when its conclusion follows from its premises, whatever they are; it has material truth when its premises are in fact true. An argument may have either without the other, which is the four-line table at sequence 110 in different words.

The trap. An argument can be formally impeccable and materially worthless, and a beginner impressed by the neatness of a structure has been persuaded by form alone. The remedy is the two-question discipline of sequence 60, kept in that order.

A worked example: all four dimensions on one passage

"No responsible court could take a different view. The appellant, a man of means who had every opportunity to pay, kept the workman waiting for his wages for four years. The Act was passed precisely to prevent such delay. It follows that the appellant must pay compensation in addition to the arrears."

Purpose. Persuasive and justificatory together. The opening sentence is addressed to an audience and asks it to feel that only one view is open; the rest supports an order. It is not primarily probative.

Content. Assertions of fact: that the appellant is a man of means, that he had every opportunity to pay, that the delay was four years. Assertion about law: that the Act was passed to prevent such delay. Assumed and never asserted: that a purpose of the Act by itself authorises the award of compensation. That is the enthymeme, and it is the whole case.

Language. "No responsible court" is emotive, and it works by making disagreement a criticism of the reader. "A man of means" is emotive and also vague. "Precisely" adds nothing but confidence. Strip all four and the passage is materially shorter.

Structure. One conclusion, "the appellant must pay compensation in addition to the arrears", marked by "it follows that". The premises are linked. There is a suppressed premise supplying the legal power to award compensation, and it is doing more work than any of the stated premises.

What the analysis produces. Not a verdict. It produces a list of exactly four things to check: whether the purpose licenses the omissions, whether the three facts are proved, whether the emotive words are covering a gap, and whether the missing legal premise exists. That list is what evaluation, in the next chapter, works through.

Distinctions that carry marks

What this does not mean

Analysis is not criticism. A perfectly good argument can be analysed at length, and the analysis will find nothing wrong. Students often assume that being asked to analyse is being invited to attack, and then manufacture objections.

Emotive language is not automatically a fallacy. It becomes one when it replaces a reason. A judgment that describes conduct as callous, having found facts that justify the word, is not reasoning badly.

Vagueness is not always a defect. A statute that said a notice must be served within 17 days rather than within a reasonable time would be precise and much worse. The fault lies in treating a vague standard as if it had a sharp edge.

Quick revision

Four dimensions: purpose, content, language, structure or form. MU prints them in that order and an answer should keep it.

Purpose: probative, persuasive, justificatory or explanatory. It settles what counts as success and what omissions are legitimate.

Content: what is asserted as fact, what as rule, and what is assumed and never asserted.

Language: ambiguity, vagueness, emotive loading, rhetorical form.

Structure: standard form, premises and conclusion, linked or convergent, deductive or inductive.

Form and content as a pair: content, also called matter, is what the argument is about; form is how its propositions are arranged. Logic is entirely on the form side. Formal truth is that the conclusion follows; material truth is that the premises are in fact true, and an argument may have either without the other.

Analysis precedes evaluation. Analysis describes; evaluation judges.

Test yourself

1. Name the four dimensions of the analysis of an argument and state the question each asks.

Purpose asks what the argument is for and whom it addresses. Content asks what it asserts as fact, what it asserts as rule, and what it assumes without asserting. Language asks how the words are being used, in particular whether they are ambiguous, vague or emotive. Structure or form asks how the statements are connected, which is the premise, which the conclusion, and whether the premises are linked or convergent.

2. Why does purpose have to be settled first?

Because it decides what counts as a defect. An argument in a pleading that omits the best point for the other side is doing what pleadings are for, while a textbook that did the same would be misleading its reader. The same feature is a fault in one setting and a virtue in another, and until the purpose is known there is no standard to measure the argument against.

3. Distinguish ambiguity from vagueness with one legal example of each.

Ambiguity is having more than one settled meaning: "sanction" means both a penalty and an approval, and an argument that shifts between them proves nothing. Vagueness is having one meaning with no clear boundary: "reasonable time" has a single sense but no sharp edge, so that some periods are clearly reasonable, some clearly not, and a range in between about which the word gives no answer.

4. What does analysing the content of an argument chiefly look for?

The assertions, separated into those of fact and those of law or general rule, and above all the assumptions that are never asserted. A suppressed premise is a suppressed assertion, and in legal argument it is usually the proposition of law being relied on, which means it is usually the thing actually in dispute.

5. Analyse the language of this sentence: "Even the most indulgent tribunal could hardly overlook so flagrant a breach."

"Even the most indulgent" is emotive and pre-empts disagreement by suggesting that anyone who disagrees is more indulgent than the most indulgent tribunal. "Hardly overlook" is a studied understatement of the kind legal writing favours, and it asserts a conclusion while appearing to concede. "Flagrant" is emotive and vague at once: it carries strong disapproval and states no test. Nothing in the sentence is a reason.

7. Write a short note on form and content.

The content, or matter, of an argument is what it is about: the propositions themselves, the facts alleged and the rules relied on. The form is how those propositions are arranged, which is premise and which conclusion and how they connect. Logic is entirely concerned with the form, so that two arguments about different subjects with the same arrangement stand or fall together and neither needs to be understood in order to be judged. Content is assessed by evidence, by science or by law. The pair extends to whole arguments as formal truth, meaning that the conclusion follows, and material truth, meaning that the premises are in fact true, and an argument may have either without the other.

6. Is analysis the same as evaluation?

No. Analysis takes an argument apart and describes what is there: its purpose, its assertions, its wording and its structure. Evaluation passes judgment on whether the premises are acceptable and whether the conclusion follows. Analysis comes first, and its product is not a verdict but a list of the things that have to be checked before a verdict is possible.

Strong means that if the premises were true, the conclusion would probably be true. Strength is a matter of degree: an inductive argument can be very strong, moderately strong or weak, where validity admits of no degrees at all.

Cogent is the inductive counterpart of sound: strong, with true premises, and with no relevant evidence left out. That last requirement has no deductive counterpart and it is where most inductive arguments fail.

An example makes the difference plain. "Ninety-eight of the hundred consignments delivered under this contract were defective, so the ninety-ninth was probably defective too" is strong. Add the information that the ninety-ninth came from a different factory, and the argument is no longer cogent, though nothing in it has become false. Suppressing relevant evidence destroys an inductive argument in a way it cannot destroy a deductive one.

Interpreting an argument: the principle of charity

Before an argument can be evaluated it has to be interpreted, because arguments in the world are compressed, ambiguous and enthymematic. The rule for doing this fairly has a name.

The principle of charity: interpret an argument in the strongest form its words will reasonably bear. Where a sentence has two readings, one of which makes the argument obviously silly and the other of which makes it worth taking seriously, take the second. Where a premise is missing, supply the weakest one that does the job.

Two reasons, one honest and one practical. The honest one is that refuting an argument nobody made proves nothing; that error has its own name, the straw man. The practical one is that in law the opponent's argument will be restated by the court in its strongest form, and a pleader who has only prepared to meet the weak version will be answering the wrong case.

The limit of charity is that it is not invention. If the argument as made contains no premise capable of supporting the conclusion, the answer is that the argument fails, not that a better one could be constructed.

A worked example

"The witness cannot be believed. He gave a different account to the police. Anyone who changes his story is lying."

Interpret first. The third sentence, read literally, is a sweeping claim that is plainly false: people change accounts through error, through fear and through the passage of time. Charity asks whether a weaker reading is available. It is: "a witness who materially contradicts his earlier statement is not reliable on that point." That is a defensible proposition and it is what the speaker probably meant.

Now test the form. On the charitable reading the argument is:

1. A witness who materially contradicts his earlier statement is unreliable on that point.

2. This witness materially contradicted his earlier statement to the police.

Therefore this witness is unreliable on that point.

That is valid. Notice what has happened to the conclusion, though: the original said the witness "cannot be believed" at all, and the charitable reconstruction supports only unreliability on the point of contradiction. The argument, at its strongest, proves less than it claimed.

Now test the premises. Premise 1 is a proposition about how evidence is weighed and it is broadly correct. Premise 2 is a question of fact, to be established by putting the earlier statement to the witness. If it is not established, the argument is unsound.

Verdict. Valid on the charitable reading, but for a narrower conclusion than the one asserted, and sound only if premise 2 is proved. That three-part answer is what evaluation looks like when it is done properly, and no part of it is an opinion about whether the witness was in fact truthful.

Distinctions that carry marks

What this does not mean

A valid argument is not a true one. Validity is a relation between premises and conclusion. An argument from false premises can be perfectly valid, and its conclusion may be nonsense.

Showing that an argument is bad does not show its conclusion is false. It shows the conclusion is unproved by this argument. Treating a refuted argument as a disproved conclusion is itself a fallacy.

Inductive arguments are not defective deductive ones. They are a different kind of reasoning with their own standard. Calling an inductive argument invalid is like calling a chair badly written.

Quick revision

Two independent questions: are the premises true, and does the conclusion follow?

Deductive: valid or invalid; sound means valid plus true premises. Only a sound argument establishes its conclusion.

Inductive: strong or weak; cogent means strong plus true premises plus no relevant evidence omitted.

Four combinations: only true premises with a valid form gives a guaranteed conclusion. The other three tell you nothing.

Principle of charity: read the argument in the strongest form its words will bear. The opposite fault is the straw man.

A bad argument for a true conclusion is still a bad argument, and the conclusion may be established some other way.

Test yourself

1. What are the two questions asked in evaluating an argument, and why are they independent?

Whether the premises are true, and whether the conclusion follows from them. They are independent because the first is a question about the world, answered by evidence or by law, while the second is a question about the argument's form, answered by logic and answerable without knowing whether any premise is true. An argument can pass either test and fail the other.

2. Define valid and sound, and say why both words are needed.

An argument is valid when the conclusion cannot be false if the premises are true, which is a matter of form alone. It is sound when it is valid and all its premises are in fact true. Both words are needed because validity alone establishes nothing: an argument from false premises may be perfectly valid, and only soundness guarantees the conclusion.

3. Why may "valid" not be used of an inductive argument?

Because validity means that the conclusion cannot be false if the premises are true, and an inductive argument never claims that. It claims only that the conclusion is probable, so the proper words are strong and weak, which admit of degrees as validity does not. Calling an inductive argument invalid criticises it for failing to do something it never attempted.

4. What does cogency require that soundness does not?

That no relevant evidence has been left out. A deductive argument is unaffected by additional information: adding premises to a valid argument cannot make it invalid. An inductive argument can be destroyed by a single further fact, so cogency requires strength, true premises and completeness of the evidence relied on.

5. State the principle of charity and give the name of the fault it prevents.

Interpret an argument in the strongest form its words will reasonably bear, choosing the reading that makes it worth answering and supplying the weakest missing premise that makes it work. The fault it prevents is the straw man, which is refuting a weaker argument than the one actually made and then claiming to have answered the real one.

6. An argument is shown to be invalid. What follows about its conclusion?

Nothing at all. Invalidity shows that this argument does not establish the conclusion; it does not show that the conclusion is false. The proposition may be true and may be provable by some other argument. Treating the refutation of an argument as the disproof of its conclusion is a distinct error and is common in first-year answers.

Claim no more than your premises give you. Overstating a conclusion is the commonest self-inflicted wound in advocacy. If the evidence shows the notice was posted, conclude that it was posted, not that it was received. A modest conclusion that follows is worth more than a bold one that does not.

Use each word in one sense throughout. If "possession" means physical control in your first premise, it must mean physical control in the third. This is the requirement that the fallacy of equivocation breaks.

Make every premise relevant. A premise that could be deleted without weakening the argument is not helping it; it is inviting the reader to wonder why it was included.

Anticipate the strongest objection and answer it. Not the weakest. An argument that disposes of a trivial objection and ignores the real one has drawn attention to the real one.

Do not assume what you are trying to prove. An argument whose premise is a restatement of its conclusion is circular. It is a fault that is easy to see in someone else's writing and almost invisible in one's own, which is why the standard form of the last three chapters is worth the trouble: a circular argument is obvious the moment it is written out with the premises above the line.

A worked example: building an argument from nothing

The problem. Your client, Meena, lent Rs 2,00,000 to Suresh in cash in January 2023. He has not repaid. You are asked whether a suit will succeed.

Step one, the conclusion. "Suresh is liable to repay Rs 2,00,000 to Meena." Notice how much sharper this is than "Meena has a case."

Step two, what would have to be true.

1. A person who receives money as a loan is bound to repay it.

2. Suresh received Rs 2,00,000 from Meena.

3. The money was received as a loan and not as a gift or in discharge of some other obligation.

4. The claim is within the period of limitation.

Therefore Suresh is liable to repay Rs 2,00,000 to Meena.

Step three, test each premise. Premise 1 is law and is not in doubt. Premise 2 is fact, and here the argument meets its first real difficulty: the payment was in cash, so what proves it? A bank withdrawal on the same day, a witness, a message. Premise 3 is fact, and it is the premise the defence will attack, because "it was a gift" is the standard answer to a cash loan between people who know each other. Premise 4 is law applied to fact and requires the date to be fixed.

Step four, what the construction has produced. Not an argument yet, but something more useful: a list of exactly two things to prove and one date to check, arrived at before a single document was collected. That is the practical value of constructing backwards, and it is why the method is taught in a first-year logic paper rather than in a procedure paper.

A note on modesty. The conclusion as drafted claims liability for the whole sum. If the evidence supports only Rs 1,50,000, the argument as built fails entirely, and a conclusion drafted to the evidence would have succeeded in part. Claim what your premises give you.

Distinctions that carry marks

What this does not mean

Constructing an argument is not deciding what you want to be true and then finding support for it. The test of a constructed argument is the same as the test of anybody else's: are the premises supportable, and does the conclusion follow? An argument built backwards from a conclusion you cannot support is not an argument; it is a wish.

Anticipating an objection is not conceding it. Stating the strongest point against you and answering it is what a court expects, and it costs nothing where the answer is good.

A structured argument need not read like a list. The four steps are the skeleton. A well written paragraph has the same skeleton and does not show it.

Quick revision

Method: fix the conclusion, find the premises that would yield it, test whether each can be supported, arrange rule then facts then application then conclusion.

Backwards, not forwards. Forwards produces true statements that prove nothing.

Rules: state your real premises; claim no more than they give; one sense per word; every premise relevant; answer the strongest objection; never assume the conclusion.

Circularity is invisible in prose and obvious in standard form, which is a reason to write the standard form out.

Legal shape: one premise of law, one or more of fact. That shape tells you what evidence to collect.

Test yourself

1. Why is an argument constructed backwards from its conclusion?

Because the conclusion is what fixes which premises are needed, and asking what would have to be true for the conclusion to follow produces a short and definite list. Building forwards from what happens to be known produces a collection of true statements with no guarantee that anything follows from them, and the gaps are discovered only at the end, when the work of collecting evidence has already been done.

2. State four rules of good construction.

State the premises actually relied on rather than leaving the load-bearing assumption unstated; claim no more in the conclusion than the premises will support; use every key word in a single sense throughout; and make sure each premise is relevant, so that none could be deleted without weakening the argument. To these may be added answering the strongest objection rather than the weakest, and never assuming what is to be proved.

3. What is the usual shape of a legal argument's premises?

One premise of law and one or more premises of fact, with the conclusion following from their combination. The legal premise is supported out of statutes and decided cases and is contested by argument; the factual premises are supported by evidence and are contested by evidence. Separating them tells a lawyer what has to be researched and what has to be proved.

4. Why does overstating a conclusion weaken an argument?

Because validity is destroyed the moment the conclusion goes beyond what the premises support, so an overstated conclusion turns a good argument into a bad one without adding anything. Where the premises establish that a notice was posted, a conclusion that it was received does not follow, and the whole argument fails, whereas the modest conclusion would have succeeded and might well have been enough.

5. What is a circular argument, and how is it detected?

An argument in which a premise is a restatement of the conclusion, so that the argument assumes what it was supposed to prove. It is difficult to see in flowing prose, where the same claim in two different forms of words reads like two claims. It is detected by writing the argument in standard form, with the premises listed above the line and the conclusion below it, where the repetition becomes obvious.

6. Your client wants a declaration that a gift deed is void. State the first step and give the conclusion in proper form.

The first step is to state the conclusion as a single proposition capable of being true or false, not as a subject of dispute. "The gift deed dated 4 April 2024 executed by X in favour of Y is void." From that sentence the necessary premises can be read off: the rule under which such a deed is void, and the facts that bring this deed within the rule.

7. The premises need not be true. Validity is a relation and not a certificate. "All lawyers can fly; she is a lawyer; therefore she can fly" is perfectly valid and thoroughly absurd, and both of those things are true at once.

Demonstrating feature 1

The claim that denial produces a contradiction is not a slogan and can be shown.

Take: every agreement enforceable by law is a contract; this agreement is enforceable by law; therefore this agreement is a contract.

Now try to deny the conclusion while keeping the premises. You must say: every agreement enforceable by law is a contract, and this agreement is enforceable by law, and this agreement is not a contract. The second and third statements together give you an agreement which is enforceable by law and is not a contract, and the first statement says there is no such thing. You have asserted and denied the same proposition. That is what "necessity" means here, and it is not a claim about how confident anybody feels.

The forms this syllabus will teach

Deductive inference is divided by how many premises it uses, and the division is MU's own, at topic 3.1.

Immediate inference draws a conclusion from a single premise. From "all contracts are agreements" it follows immediately that "some agreements are contracts". The whole of Module III, opposition and eduction, is immediate inference.

Mediate inference draws a conclusion from two or more premises taken together, and its classical form is the syllogism, three propositions and three terms. Every example in this chapter is a syllogism.

Note carefully: the syllogism itself is Logic II's subject and is not on this paper. It appears here only because the basic features of deduction cannot be shown without an example, and the syllogism is the natural one. What this paper examines is immediate inference.

A worked example

Section 11 of the Indian Contract Act 1872 provides that every person is competent to contract who is of the age of majority according to the law to which he is subject, who is of sound mind, and who is not disqualified from contracting by any law to which he is subject.

Ravi, aged seventeen, signs an agreement to sell his motorcycle.

The deduction.

1. Only a person who is of the age of majority is competent to contract.

2. Ravi is not of the age of majority.

Therefore Ravi is not competent to contract.

Test each feature against it. Is it truth preserving? If both premises are true the conclusion cannot be false. Is it ampliative? No: premise 1 already says that non-majors are not competent, and premise 2 puts Ravi among them; nothing new has entered. Does form alone decide? Yes: replace "competent to contract" with any predicate and "Ravi" with any name and the argument stands. Would further facts destroy it? No: learning that Ravi is unusually mature, or that the price was fair, or that he has a bank account, changes nothing whatever, because none of it makes him of the age of majority.

That last point is the one worth dwelling on. In an inductive argument, every one of those additional facts would matter. In a deductive one they are simply irrelevant, and a student who begins arguing about Ravi's maturity has not understood which kind of reasoning is in play.

Distinctions that carry marks

That last row is a tendency and not a definition, and the reason is under "What this does not mean" below.

What this does not mean

Deduction is not "general to particular". This is the definition most students arrive with and it is wrong. "All contracts are agreements; all agreements are promises; therefore all contracts are promises" moves from general to general and is deductive. "Ravi is a minor; Ravi signed this agreement; therefore some minor signed this agreement" moves from particular to particular and is deductive. The definition is necessity, not direction. The direction is only a common pattern.

Deduction does not give certainty about the world. It gives certainty about the connection. The conclusion of a valid argument is only as good as its premises, and the premises come from outside logic.

"Not ampliative" is not a criticism. That deduction adds no information is the price of its reliability, and it is a bargain: a lawyer who can show that a conclusion is already contained in a statute and an admitted fact does not need any further information.

Limits

The great limitation of deduction is that it cannot establish its own premises. Every deductive argument has to be given something to work on, and where the premises come from is not a deductive question. In law they come from statutes, from precedent and from findings of fact, and findings of fact are reached inductively, which is the subject of the next chapter and the reason the two are taught together.

Quick revision

Definition: an argument whose premises are claimed to provide conclusive grounds for its conclusion, so that if they are true it cannot be false.

Seven features: necessity; truth preserving; not ampliative; judged by validity, which has no degrees; validity depends on form alone; adding premises cannot destroy it; premises need not be true.

Necessity means contradiction: asserting the premises and denying the conclusion is self-contradictory.

Two kinds: immediate inference, one premise, which is Module III; mediate inference, the syllogism, which is Logic II.

Not "general to particular", which is the standard student error. General to general and particular to particular deductions both exist.

Its limit: it cannot establish its own premises.

Test yourself

1. Define deductive reasoning and explain the word "necessity" in your definition.

Deductive reasoning is reasoning in which the premises are claimed to provide conclusive grounds for the conclusion, so that if the premises are true the conclusion cannot be false. "Necessity" is not a description of the arguer's confidence. It means that asserting the premises while denying the conclusion produces a contradiction, that is, requires the same proposition to be both asserted and denied.

2. List the basic features of deductive reasoning.

The conclusion follows necessarily; the reasoning is truth preserving, so true premises cannot yield a false conclusion; it is not ampliative, adding no information beyond what the premises already contain; it is judged by validity, which admits of no degrees; validity depends on the form of the argument and not on its subject matter; adding further premises cannot make a valid argument invalid; and the premises need not in fact be true for the argument to be valid.

3. Why is it wrong to define deduction as reasoning from the general to the particular?

Because the direction of movement is a common pattern and not the defining feature. Deductions run from general premises to general conclusions, as in a chain of universal propositions, and from particular premises to particular conclusions. What makes an argument deductive is the claim that the conclusion follows necessarily, and arguments of every direction can make that claim.

4. What does it mean to say that deduction is not ampliative, and is that a defect?

It means the conclusion contains no information that was not already present, at least implicitly, in the premises: deduction makes explicit what was implicit rather than adding to our knowledge of the world. It is not a defect but the reverse side of reliability. The reason deduction can guarantee its conclusion is precisely that it never ventures beyond what it was given.

5. What is meant by saying deduction is monotonic, and why does it matter in litigation?

That adding further premises to a valid argument cannot make it invalid. It matters because it tells a lawyer which additional facts are worth investigating. Where a conclusion rests on a valid deduction from a rule and an admitted fact, no amount of further evidence about surrounding circumstances can touch it, and time spent gathering that evidence is wasted. Where the reasoning is inductive, the position is exactly the opposite.

6. Give a valid deductive argument with false premises and explain what it shows.

"Every document registered under the Registration Act is a contract; this sale deed is registered under that Act; therefore this sale deed is a contract." The first premise is false, since many registered documents are not contracts, so the argument is unsound. It is nevertheless valid, because if the premises were true the conclusion would have to be. It shows that validity concerns the connection between premises and conclusion and says nothing about truth.

Induction by simple enumeration. From "these observed instances of A were B" to "all A are B", or to "the next A will be B". The weakest form, and MU's topic 1.8.

Analogy. From "these two things resemble each other in several respects" to "they resemble each other in a further respect". MU's topic 1.9, and the form of reasoning that precedent is built on.

Causal inference. From an observed regularity to the conclusion that one thing produces another. The methods for testing it were set out by John Stuart Mill and are the ordinary equipment of accident investigation.

Statistical inference. From the proportion in a sample to the proportion in the whole, or from a proportion in a class to a probability about an individual member of it.

Hypothesis, also called inference to the best explanation. From a set of facts to the explanation that accounts for them better than any rival. This is the form of reasoning a court uses on circumstantial evidence, and it is why that topic sits where it does.

What makes an inductive argument strong

The tests differ by kind, and the detailed ones come in the chapters on simple enumeration and analogy. Four apply generally.

Number of instances. More observations, more strength, though with rapidly diminishing returns.

Variety of instances. Ten observations under ten different conditions are worth far more than a hundred under one. A drug tested only on healthy young men supports a very narrow conclusion however many times it is tested.

Modesty of the conclusion. The less the conclusion claims, the stronger the argument for it. "The next consignment will probably be defective" is far better supported than "all consignments from this supplier are defective".

Absence of counter-instances, and honest search for them. An argument that has looked for contrary evidence and not found it is worth more than one that never looked, even if the evidence in hand is identical.

The problem of induction

Every syllabus expects this to be named, and it can be stated in four sentences.

Induction assumes that what has been observed is a guide to what has not been observed, which is to say that the future will resemble the past and the unexamined resembles the examined. What justifies that assumption? It cannot be justified deductively, because there is no contradiction in supposing that nature changes tomorrow. It cannot be justified inductively either, because that would be to argue that induction has worked before and will therefore work again, which is itself an induction and assumes the very thing in question.

This is David Hume's problem, and no answer to it commands general agreement. The practical position everybody in fact adopts, courts included, is that induction is indispensable and is used with care rather than defended. What matters for this paper is being able to state the problem, not to solve it.

A worked example

A workman sues, alleging that a defective machine caused his injury. There is no direct evidence of the defect, because the machine was repaired the same evening.

The following facts are proved: three other workmen on the same machine suffered similar injuries in the preceding six months; each injury happened when the guard was in the raised position; the machine's maintenance register shows no inspection for eleven months; and the employer repaired the machine on the evening of the accident without recording what was repaired.

The inductive argument.

1. Three previous injuries on this machine happened with the guard raised.

2. The machine had not been inspected for eleven months.

3. The machine was repaired immediately after this accident, with no record of what was done.

Therefore the machine was probably defective and the defect probably caused this injury.

Assess it by the four tests. Number: four incidents including the present one, which is a small but not negligible number. Variety: weak, because all four involve the same machine and the same guard, so the conclusion is properly limited to this machine. Modesty: the conclusion says "probably" and confines itself to this machine, which is correct. Counter-instances: unknown, and the employer is in the best position to produce them.

Now destroy it with one fact. Suppose the employer proves that the three earlier injuries were caused by workmen deliberately raising the guard to work faster, and that the repair on the evening in question was to a light fitting. Nothing already established has become false. The argument is nevertheless worthless. That is feature 4, and no deductive argument could ever be treated this way.

Distinctions that carry marks

What this does not mean

Induction is not "particular to general". That is the common pattern, not the definition. "Every consignment so far has been defective, so the next one will be" runs from particular to particular and is inductive. What makes it inductive is that the conclusion goes beyond the premises.

A weak inductive argument is not invalid. The word does not apply. It is weak, and it may be strengthened by more or better evidence, which is something no invalid deductive argument can be.

Probability here is not always a number. Courts reason inductively all day without assigning figures to anything. "Probable" means better supported than the alternatives, not calculated at some percentage.

Quick revision

Definition: an argument whose premises are claimed to give some but not conclusive support, so the conclusion may be false though the premises are true.

Six features: probable not certain; ampliative; judged by strength, which has degrees; can be destroyed by new premises; good version is cogency; never described as valid or invalid.

Kinds: simple enumeration, analogy, causal, statistical, hypothesis or inference to the best explanation.

Tests of strength: number of instances, variety of instances, modesty of the conclusion, absence of counter-instances after an honest search.

The problem of induction: the assumption that the unobserved resembles the observed cannot be justified deductively, and justifying it inductively is circular. Hume.

Not "particular to general", which is a pattern and not a definition.

Test yourself

1. Define inductive reasoning and state how it differs from deductive reasoning in the claim it makes.

Inductive reasoning is reasoning whose premises are claimed to provide some but not conclusive support for the conclusion, so that the conclusion may be false even though all the premises are true. A deductive argument claims that its conclusion follows with necessity, so that it cannot be false if the premises are true. The difference is in the strength of the claim made, and it is what makes induction ampliative and deduction safe.

2. Give the basic features of inductive reasoning.

The conclusion is probable and not certain; the reasoning is ampliative, the conclusion asserting more than the premises; it is judged by strength, which comes in degrees; further premises may destroy a strong argument without falsifying anything already asserted; its good version is cogency, meaning strength together with true premises and no relevant evidence suppressed; and the words valid and invalid have no application to it.

3. Name the kinds of inductive argument and give a legal example of one.

Simple enumeration, analogy, causal inference, statistical inference, and hypothesis or inference to the best explanation. A court convicting on circumstantial evidence uses the last of these: it takes a set of proved facts and concludes that guilt is the only explanation that accounts for all of them, which is why the Supreme Court's rule requires every other hypothesis to be excluded.

4. What tests determine the strength of an inductive argument?

The number of instances relied on; their variety, since instances gathered under differing conditions support a wider conclusion than the same number gathered under one; the modesty of the conclusion, a narrower conclusion always being better supported than a sweeping one; and whether contrary instances have been honestly looked for and not found, as against merely not encountered.

5. State the problem of induction.

Induction assumes that the unobserved resembles the observed. That assumption cannot be established deductively, because there is no contradiction in supposing that the pattern breaks. It cannot be established inductively either, since arguing that induction has succeeded before and will succeed again is itself an induction and assumes precisely what is in question. The problem is Hume's, and no solution is generally accepted.

6. Why is cogency a stricter standard than soundness?

Because it contains a requirement that soundness has no need of: that no relevant evidence has been left out. A valid deductive argument cannot be spoiled by additional information, so soundness need only demand validity and true premises. An inductive argument can be destroyed by a single further fact, so a strong argument from true premises is still not cogent if the arguer has kept back evidence pointing the other way.

The two standards of proof are two settings of the same inductive dial. In a civil case the court asks which version is more probable. In a criminal case it asks whether guilt has been established beyond reasonable doubt, which is a demand for a much stronger inductive argument, not for a deductive one.

Where the deduction is

Once the facts are found, the application of the rule to them is deductive and admits of no discretion. If the rule is that a minor's agreement is void, and the finding is that this party was a minor, the agreement is void. A judge who accepted both and held otherwise would not be exercising a judgment; he would be contradicting himself.

This is why the conclusion of a judgment is usually a single sentence after twenty pages. The deductive step is the cheapest part of the whole process, and the entire cost lies in establishing the two premises.

Three devices that convert one into the other

The law of evidence contains machinery for turning an inductive inference into a deductive one, which is a striking thing for a logic student to notice, and MU's topic invites the noticing.

"May presume", section 2(1)(h): the Court may either regard the fact as proved unless and until it is disproved, or may call for proof of it. This leaves the inference inductive and merely permits it.

"Shall presume", section 2(1)(l): the Court shall regard the fact as proved unless and until it is disproved. The inference is now compulsory, but rebuttable: the inductive conclusion is imposed and may still be displaced by evidence.

"Conclusive proof", section 2(1)(b): on proof of one fact the Court shall regard the other as proved, and shall not allow evidence to be given for the purpose of disproving it.

That last one is the interesting case. By forbidding contrary evidence, the statute makes the step from the first fact to the second unchallengeable, and an unchallengeable step is a deductive one. Parliament has converted an inductive inference into a deductive rule by legislation. It does this where the value of certainty outweighs the risk of occasionally being wrong, and a student who understands the trade-off understands both the logic and the policy.

Where precedent fits

Following a precedent is not deduction. The earlier case is not a rule that contains this one; it is another dispute that resembles this one, and the reasoning runs: the earlier case resembles this one in the respects that mattered to the decision, so it should be decided the same way. That is argument by analogy, and analogy is inductive.

This is why distinguishing a case is a real argument and not an evasion. To distinguish is to say that a resemblance relied on is not relevant, or that a difference is. Those are exactly the tests for a good analogy, which are MU's own topic 1.9 and are worked in this module's last chapter.

It also explains why a line of cases is stronger than one case. More instances and greater variety strengthen an induction, and a proposition applied across many different fact situations is better supported than one applied once.

Where interpretation fits

Statutory interpretation is neither cleanly deductive nor cleanly inductive, and pretending otherwise is a common error.

Working out whether a bicycle is a "vehicle" within an Act is not a deduction from the definition, because if the definition settled it there would be no question. Nor is it an induction from instances. It is closer to what Module IV calls a précising definition: a decision, made for the purposes of this Act, about where to draw a boundary the ordinary word leaves fuzzy. That is why interpretation is taught in this course as its own subject and why the topic sits in Module IV of this one.

A worked example: one judgment, both kinds of reasoning

A shopkeeper is prosecuted for selling adulterated milk. The evidence is that a food inspector took a sample, that the sample was sealed in his presence, that the public analyst's report says the fat content was below the prescribed standard, and that the shopkeeper's signature appears on the seizure memo.

The inductive part. From the memo and the inspector's testimony the court concludes that this sample came from this shop, which no witness saw in its entirety and which is an inference from documents and testimony. From the analyst's report it concludes that the milk in the sealed container was below standard. From those two together it concludes that the milk sold was below standard. Each step goes beyond its evidence and each is judged by the prudent-person standard of section 2(1)(j).

The place where an inductive step is made compulsory. Where a statute provides that the analyst's report shall be evidence of the facts stated in it, an inference that the court would otherwise have to reason its way to is supplied by the legislature. That is the machinery of the previous section at work.

The deductive part. The rule is that selling an article of food below the prescribed standard is an offence. The finding is that the accused sold an article of food below the prescribed standard. The conclusion that he committed the offence follows necessarily, and nothing further is argued about it.

Where the appeal will go. An appeal against the finding that the sample came from this shop is an attack on an inductive conclusion, and it will be argued by pointing at gaps in the chain. An appeal against the holding that this substance is "food" within the Act is an attack on the major premise, and it will be argued out of the statute. The two appeals have nothing in common except the case they arise from, and confusing them is the surest way to lose both.

Distinctions that carry marks

What this does not mean

The syllogism is not a description of how judges reach decisions psychologically. It is a description of how a decision is justified. A judge may reach a view long before articulating the premises, as writers in the realist tradition have argued at length. The reply is that the justification is what is published, what binds later courts and what an appeal is against, so the syllogism remains the right account of the reasoning even if it is a poor account of the thinking.

Not every judgment is one syllogism. Most are chains of them, with the conclusion of one becoming a premise of the next, which is the chain argument of sequence 30.

Induction in a court is not guesswork. It is governed by rules of evidence, standards of proof and, in circumstantial cases, by a rule requiring every other hypothesis to be excluded. Those rules are the discipline that a bare inductive argument lacks.

Quick revision

Facts inductively, law deductively. That one line is the answer to the question as MU prints it.

Section 2(1)(j), BSA 2023 defines "proved" by a standard of probability and by what a prudent person would act on. An inductive standard in a statute.

Standards of proof: preponderance of probability in civil cases, beyond reasonable doubt in criminal ones. Two settings of one inductive dial, not a change of logical kind.

Presumptions: may presume, 2(1)(h); shall presume, 2(1)(l); conclusive proof, 2(1)(b), which forbids contrary evidence and thereby makes the step deductive.

Precedent is analogy, therefore inductive, therefore judged by the relevance of the resemblances. Distinguishing is an attack on the analogy.

The judicial syllogism describes justification, not psychology. That is the realists' criticism and its answer.

Test yourself

1. Explain the use of deductive and inductive reasoning in a court, with the stage of the case each belongs to.

Finding facts from evidence is inductive: no judge witnessed the events, so the court reasons from testimony, documents and circumstances to a conclusion that goes beyond them and is reached on a standard of probability. Applying the legal rule to the facts once found is deductive: given the rule and the finding, the result follows necessarily and no further evidence is relevant. Almost every argument in litigation is an argument about one premise or the other.

2. How does the Bharatiya Sakshya Adhiniyam 2023 define "proved", and why is the definition inductive?

Section 2(1)(j) provides that a fact is proved when, after considering the matters before it, the Court either believes it to exist or considers its existence so probable that a prudent man ought, in the circumstances of the particular case, to act on the supposition that it exists. It is inductive because it asks for probability and not for necessity, and because it fixes the required degree by reference to what a prudent person would act on rather than to any demonstration.

3. Distinguish "may presume", "shall presume" and "conclusive proof".

Under section 2(1)(h) the court may either regard the fact as proved unless disproved or call for proof of it, so the inference is permitted only. Under section 2(1)(l) the court must regard it as proved unless and until it is disproved, so the inference is compelled but remains rebuttable. Under section 2(1)(b) the court must regard it as proved and must not allow evidence to disprove it, so the inference cannot be challenged at all, which makes it deductive in effect.

4. Why is following a precedent an inductive process?

Because the earlier decision is not a rule that logically contains the present case; it is another dispute said to resemble this one in the respects that mattered. The reasoning runs from resemblance to like treatment, which is argument by analogy, and analogy is a form of induction. This is why distinguishing is a genuine argument: it denies the relevance of a resemblance or asserts the relevance of a difference, which are the standard tests of a good analogy.

5. Is statutory interpretation deductive?

No. If the words of the definition settled the question there would be no question to argue, so interpretation is not a deduction from the definition. Nor is it an induction from instances. It is a decision, made for the purposes of the particular Act, about where to draw a boundary that the ordinary word leaves indeterminate, which is what Module IV calls a précising definition.

6. What is the realist criticism of the judicial syllogism, and what is the answer to it?

That the syllogism describes how a decision is presented rather than how it is reached, judges often forming a view first and constructing the reasoning afterwards. The answer is that the syllogism is an account of justification, not of psychology, and justification is what matters institutionally: it is what is published, what binds later courts, what the parties can attack, and what an appellate court reviews. How the judge came to the view is not reviewable; the stated reasons are.

Validity depends on form alone. Two arguments with the same arrangement of terms and propositions are both valid or both invalid, whatever they are about. This is why symbols work and why one rule covers every subject.

Validity has no degrees. An argument is valid or invalid. "Fairly valid" and "more valid" are not expressions in this subject.

The eight combinations, and the one that cannot happen

Take a deductive argument. Its premises are all true or not, its conclusion is true or not, and it is valid or not. That is eight cases. Seven of them occur, and one is impossible, and knowing which is the examinable point.

The third line is the definition of validity, stated as an impossibility. Everything a student needs about truth and validity is contained in the fact that exactly one of these eight rows is empty.

Examples of the ones people find hard to believe:

True premises, true conclusion, invalid. "All lawyers are graduates; she is a graduate; therefore she is a lawyer." Both premises may be true and the conclusion may be true, and the argument is still worthless, because being a graduate does not put her among the lawyers. Had she been a doctor, the same argument with the same true premises would have produced a false conclusion.

False premises, true conclusion, valid. "All Acts of Parliament are printed in green ink; the Contract Act is an Act of Parliament; therefore the Contract Act is printed in green ink." Change the conclusion to something true and adjust: "All statutes have sections; the Contract Act is a statute; therefore the Contract Act has sections", where the first premise is not quite true of every statute, is valid and reaches a true conclusion.

False premises, false conclusion, valid. "All contracts are in writing; this contract is a contract; therefore this contract is in writing." Valid, false throughout.

What validity is worth

If a valid argument can have false premises and a false conclusion, what is validity for?

It is worth exactly this: a valid argument transmits truth. It guarantees that you will never move from truth to falsehood. Put true premises in and you must get a true conclusion out. Put anything else in and the guarantee is off, but the machine is not at fault.

That is why the two questions of the evaluation chapter are asked in the order they are. Test validity first, because it is a question logic can answer on its own. Then test the premises, which is a question for evidence and for law. An argument that passes both is sound, and a sound argument's conclusion is established.

A worked example

In cross-examination, counsel puts the following to a witness.

"You said you were at the shop at six o'clock. The shop closes at five thirty. So you were not there at all."

Set it out.

1. The witness says he was at the shop at six o'clock.

2. The shop closes at five thirty.

Therefore the witness was not at the shop at all.

Is it valid? No, and the invalidity is gross. Even granting both premises, all that follows is that he was not there at six, or that the shop was open late, or that he was mistaken about the time. That he was never there at all does not follow from anything.

Are the premises true? Quite possibly both.

What is the verdict? True premises, invalid argument, and a conclusion that may be true or false for reasons this argument has not touched. The witness may indeed have been lying about everything. This question does not show it, and if the point is left there the record contains an unanswered assertion rather than a proved contradiction.

Why it matters practically. Counsel who understands the distinction asks the next question, which is what the witness was doing at five thirty. Counsel who does not sits down satisfied.

Distinctions that carry marks

What this does not mean

An invalid argument is not one with false premises. That is unsoundness. Invalidity is a defect in the connection, and it is present or absent whatever the premises are.

A valid argument is not necessarily a good argument. It may be valid and start from premises nobody accepts, in which case it establishes nothing and wastes everybody's time.

Truth is not "what the court finds". What the court finds is what is proved. The two usually coincide and sometimes do not, which is why appeals and reviews exist.

Quick revision

Truth belongs to propositions, validity to arguments. Never say a proposition is valid or an argument is true.

Valid: impossible for the premises to be true and the conclusion false.

Sound: valid and all premises true. Only a sound argument establishes its conclusion.

Of the eight combinations, only true premises with a false conclusion in a valid argument is impossible. That single impossibility is the definition of validity.

Validity transmits truth: true in, true out. Nothing else is guaranteed.

Validity depends on form alone and has no degrees.

Test yourself

1. Define truth and validity and state what each belongs to.

A proposition is true when what it asserts is the case and false when it is not; truth belongs to propositions. An argument is valid when it is impossible for all its premises to be true while its conclusion is false; validity belongs to arguments. The two notions apply to different kinds of thing, so calling a proposition valid or an argument true is not a mistake of fact but a misuse of the words.

2. Which combination of premise truth, conclusion truth and validity is impossible, and why?

An argument cannot be valid, have all true premises, and have a false conclusion. That combination is impossible because it is precisely what validity is defined to exclude: to say an argument is valid is to say that this situation cannot arise. Every other combination of the three can and does occur.

3. Give an example of a valid argument with false premises and a true conclusion, and say what it shows.

"Every statute is divided into sections; the Bharatiya Nyaya Sanhita is a statute; therefore the Bharatiya Nyaya Sanhita is divided into sections." The first premise is not true of every statute, so the argument is unsound, yet it is valid and its conclusion is true. It shows that validity concerns the connection alone, and that a true conclusion is no evidence that the argument which produced it was any good.

4. Why must validity be tested before the truth of the premises?

Because validity is a question logic can settle by itself, needing no facts, while the truth of the premises requires evidence, investigation or research into the law. If the argument is invalid the enquiry stops there and the premises need not be examined at all, since even true premises would establish nothing. Testing in the other order can waste all the work.

5. What does a valid argument guarantee?

That it will never carry you from truth to falsehood. If the premises are all true the conclusion must be true. It guarantees nothing when the premises are false: the conclusion may then be true or false, and the argument gives no indication which. This is what is meant by saying that validity is truth preserving.

6. Counsel says: "That is not a valid argument, because I do not accept your second premise." What is wrong with the sentence?

It confuses unsoundness with invalidity. Refusing to accept a premise is a challenge to the truth of the premise, which if made out shows the argument to be unsound while leaving it perfectly valid. Validity concerns whether the conclusion would follow if the premises were granted, and that question is untouched by whether anyone grants them.

The four kinds of implication

MU asks for implication as a basic concept, and the standard treatment distinguishes four senses in which one proposition is said to imply another. Only the last is the one modern logic works with, and knowing why is worth marks.

Logical implication. "This is a square" implies "this has four sides". The connection holds because of the meanings and the logic alone, and denying the second while asserting the first is a contradiction.

Definitional implication. "He is a bachelor" implies "he is unmarried". The connection comes from the definition of a word rather than from the structure of the propositions.

Causal implication. "The glass was heated to a thousand degrees" implies "the glass melted". The connection is a matter of physical fact discovered by observation, not of meaning at all.

Decisional implication. "This is a public nuisance" implies "an ordinary member of the public cannot sue for it without leave". The connection is neither logical nor definitional nor causal; somebody has decided that it shall hold. Legal implications are overwhelmingly of this kind, which is why they can be altered by legislation and the others cannot.

Material implication

Modern logic uses a fifth and deliberately thin notion, the one written with the horseshoe. p ⊃ q is defined to be false in exactly one case, where p is true and q is false, and true in the other three. Nothing else is required: p and q need have no connection of meaning, cause or decision whatsoever.

This produces results that look wrong on first meeting, and they are called the paradoxes of material implication. A false proposition materially implies anything: "the Contract Act was passed in 1900 ⊃ the moon is made of cheese" is true, because its antecedent is false. A true proposition is materially implied by anything.

The answer is that material implication is not an analysis of the English "if". It is a truth function, adopted because it is the weakest relation strong enough to make deduction work, and because a truth-functional connective is one a truth table can handle. Ordinary conditionals carry more than material implication does, and Module II's chapter on truth tables returns to this.

The legal sense of "implication", which is different

A law student meets the word "implication" every day and almost never in the logician's sense, so the two must be separated deliberately.

An implied term in a contract is a term the parties did not write and which the law reads in. An implied repeal is a repeal Parliament did not enact in words and which a court finds because two Acts cannot stand together. A necessary implication in a statute is a meaning not stated but forced by the words used.

In all three the word describes something that has been read in by a court because it was not expressed. That is close to the logician's decisional implication and quite unlike logical implication, and a student who explains the legal doctrine of implied terms in an answer about the basic concepts of logic has answered a different question.

A worked example

An inspector finds a shop closed at ten in the morning on a working day, the shutters locked, and the licence displayed in the name of the accused. He records: "the accused was absent from the shop, therefore the shop was being run in contravention of the licence condition requiring the licensee to be present during business hours."

Find the inference. The act performed by the inspector: from the closed shutters and the displayed licence he passed to the conclusion that the licensee was absent, and from that to a contravention.

Find the implication, if any. Does "the shutters were down at ten" imply "the licensee was absent"? No. It is consistent with the licensee being inside. The inspector performed an inference for which no implication existed, which is precisely what an invalid argument is.

Now the second step. Does "the licensee was absent during business hours" imply "the licence condition was contravened"? Yes, and it is a decisional implication: it holds because the licensing authority laid down the condition, and it could be repealed tomorrow. A student who called it a logical implication would have said something false about the source of the connection.

What the analysis gives the defence. Two distinct answers, and they are answers of different kinds. Against the first step, evidence: the licensee was inside doing stock. Against the second, argument: the condition on its true construction requires presence only when the shop is open for business. Only separating inference from implication makes it obvious that these are two answers and not one.

Distinctions that carry marks

What this does not mean

To imply is not to hint. In ordinary English "he implied that the witness was lying" means he suggested it without saying so. In logic implication is a relation between propositions and nobody performs it. The word for hinting is insinuating.

Inferring is not implying. The listener infers; the speaker implies, in the loose sense. In the strict sense the speaker does neither, because implication holds between propositions.

Material implication is not the English "if". It is a deliberately weakened substitute that a truth table can handle, and its oddities are a known price rather than a discovery about conditionals.

Quick revision

Inference: the act of passing from premises to a conclusion. Personal, dated, may be performed badly.

Implication: the relation in virtue of which the conclusion follows. Impersonal, timeless, either holds or does not.

The link: an inference is correct when there is a corresponding implication; validity is the name of an argument that exhibits one.

Four kinds: logical, definitional, causal, decisional. Legal implications are decisional, which is why statute can change them.

Material implication: p ⊃ q, false only when p is true and q false. Its oddities are the paradoxes of material implication.

The legal word "implication", as in implied terms and implied repeal, means something read in because it was not expressed. Do not answer with it.

Test yourself

1. Distinguish inference from implication.

Inference is an act performed by a person at a particular time, by which they pass from one or more propositions to another; it can be done well or badly and it is studied by psychology as well as by logic. Implication is a relation between propositions, holding independently of whether anybody has ever thought of them, and it is timeless and impersonal. An inference is correct exactly when there is an implication for it to follow.

2. How are implication and validity connected?

Validity is the property an argument has when its premises imply its conclusion. So implication is the relation, validity is the name for an argument that exhibits it, and inference is the act of drawing the conclusion by relying on it. To say that an argument is valid and to say that its premises imply its conclusion are two ways of saying one thing.

3. Name the four kinds of implication with an example of each.

Logical: "this is a square" implies "this has four sides", the connection coming from logic and meaning. Definitional: "he is a bachelor" implies "he is unmarried", the connection coming from the definition of the word. Causal: "the water was cooled below zero" implies "the water froze", the connection being a discovered fact about the world. Decisional: "this agreement was made by a minor" implies "this agreement is void", the connection existing because the law has so decided.

4. What is material implication and what are its paradoxes?

Material implication is the truth-functional connective written p ⊃ q, defined to be false only when p is true and q false and true in every other case, so that no connection of meaning between p and q is needed. Its paradoxes are that a false proposition materially implies every proposition, and that a true proposition is materially implied by every proposition, both of which are absurd if the horseshoe is read as the English "if".

5. Why does modern logic use material implication despite the paradoxes?

Because it is truth functional, so the truth value of the compound is fixed entirely by the truth values of its parts, and only such a connective can be handled by a truth table. It is also the weakest relation that still supports valid deduction: any stronger reading would import claims about meaning or cause that logic has no way to test. The paradoxes are accepted as the price of a workable calculus, not defended as an account of ordinary conditionals.

6. A statute is said to repeal an earlier one by necessary implication. Is this implication in the logician's sense?

Not in the sense of logical implication. It is closest to decisional implication: a court decides that the later Act cannot stand together with the earlier one and reads the repeal in although Parliament did not enact it in words. The connection exists because an authority has held that it does, and it could be displaced by an express saving in a later statute, which no logical implication could be.

Wittgenstein's picture answer. In the Tractatus Logico-Philosophicus a proposition is a picture of a possible state of affairs, in the way a diagram of a road accident is a picture of the accident: not a resemblance in colour or size, but a mapping of parts onto parts. A true proposition is a picture that fits.

Neither answer has satisfied everybody, and Wittgenstein himself abandoned his later on. That is not a small point: it means the theory that is easiest to believe is the hardest to make precise.

The criticisms

What is a fact? The theory needs facts to be items in the world, and it is not obvious that they are. The desk is a thing you can point to. The fact that the desk is brown is not a further thing standing beside it. A theory whose central term is this obscure has explained truth by something less clear than truth.

Negative propositions. "There is no cheque in the file" is true. What fact does it correspond to? There is no such item as an absence sitting in the file. Either the theory needs negative facts, which most philosophers find intolerable, or it needs some other account of these propositions, and either way the simple formula does not hold.

General propositions. "All contracts require consideration" is true, and there is no single fact for it to correspond to, because it is about every contract that ever was and ever will be.

We cannot step outside our beliefs to check. To see whether a proposition corresponds to a fact we would have to compare the proposition with the world directly. But every comparison we actually make is between a proposition and other propositions, our observations included. This is the objection that motivates the coherence theory in the next chapter.

A worked example

A suit is filed alleging that a partnership was dissolved on 30 June 2023. The defendant denies it.

On the correspondence theory, the plaintiff's assertion is true if the partnership was in fact dissolved on that date, and false otherwise. Whether the plaintiff can prove it, whether it fits comfortably with the rest of the story, and whether it would be convenient for anybody, are all irrelevant to whether it is true. They are relevant to whether it will be found true, which is a different matter.

That gap between being true and being found true is what the whole law of evidence is about. A fact may be true and unprovable, in which case the party asserting it loses. Section 2(1)(j) of the Bharatiya Sakshya Adhiniyam 2023 is careful about this: it defines when a fact is "proved", not when it is true, and it does so by a standard of probability that a prudent person would act on. A statute cannot make something so; it can only say when a court shall proceed as though it were.

The lesson for a student. A judgment that a fact is proved is a finding, and it can be wrong. If truth were defined as what the court finds, no judgment could ever be mistaken and there would be no reason to have appeals, reviews or the power to set aside a decree obtained by fraud. The existence of those remedies is the legal system stating, in its own way, that it holds the correspondence theory.

Distinctions that carry marks

What this does not mean

Correspondence is not resemblance. Words do not look like the things they are about, and no theory has ever claimed they do. What is claimed is a mapping of structure.

The theory is not about how we find out. It says what truth is, not how truth is discovered. Discovery is the business of evidence, observation and inquiry, and the theory is silent on it.

"True" is not "certain". A proposition is true or false whether or not anyone can establish which, and the whole apparatus of standards of proof exists because certainty is unavailable.

Quick revision

Statement: p is true if and only if p corresponds to a fact.

Origin: Plato and Aristotle; the classical formulation is to say of what is that it is.

Realist: truth depends on the world, not on belief, usefulness or agreement.

Accounts of correspondence: Russell's shared structure; Wittgenstein's picture theory in the Tractatus.

Four criticisms: what is a fact; negative propositions; general propositions; and the impossibility of comparing a proposition with the world except through other propositions.

Legal significance: it is the theory the law of evidence assumes. "Proved" under section 2(1)(j) BSA 2023 is a different notion from "true", and appeal and review exist because the two can come apart.

Test yourself

1. State the correspondence theory of truth and explain each element of the statement.

A proposition is true if and only if it corresponds to a fact. "Proposition" is used because only propositions bear truth values, not sentences or beliefs. "Fact" means a part of the world, the way things are arranged, and not another proposition. "Corresponds" names a relation, not a resemblance: the words need not look like what they describe, and what is claimed is that the structure of the proposition maps onto the structure of the fact.

2. Why is the correspondence theory called realist?

Because it makes the truth of a proposition depend entirely on how the world is, independently of what anyone believes, agrees or finds useful. On this theory everybody may believe a proposition and it may still be false, and nobody may believe it and it may still be true. Theories that make truth depend on beliefs, coherence among beliefs or usefulness are not realist in this sense.

3. State two criticisms of the theory.

First, the notion of a fact is obscure: a fact is not an object one can point to, so the theory explains truth by something less clear than what it is explaining. Second, negative and general propositions fit badly: "there is no receipt in the file" is true but corresponds to no positive item in the world, and "all contracts require consideration" is true but there is no single fact for it to correspond to.

4. What accounts have been given of the correspondence relation?

Russell proposed that a true proposition shares a structure with the fact that makes it true, its constituents standing in the same arrangement as the constituents of the fact. Wittgenstein, in the Tractatus Logico-Philosophicus, proposed that a proposition pictures a state of affairs, mapping part onto part as a diagram maps a scene. Neither account has been generally accepted, and Wittgenstein later abandoned his.

5. Distinguish being true from being proved, with the statutory definition.

A proposition is true when the world is as it says, which does not depend on any tribunal. A fact is proved, under section 2(1)(j) of the Bharatiya Sakshya Adhiniyam 2023, when the Court believes it to exist or considers its existence so probable that a prudent person ought in the circumstances to act upon the supposition that it exists. The one is a relation to the world, the other a relation to a tribunal's assessment, and they can come apart in either direction.

6. What does the existence of appeals show about the theory of truth the law assumes?

That the law assumes the correspondence theory. An appeal, a review, or the setting aside of a decree obtained by fraud all presuppose that a court's finding can be wrong, that is, that the finding and the truth are two different things. If truth were simply what the court found, no finding could be mistaken and none of those remedies would have anything to correct.

The same reasoning drives circumstantial evidence, corroboration, and the treatment of a defence as an afterthought because it was raised late and does not sit with the earlier conduct.

But notice how far the law refuses to go. A story that hangs together perfectly is not thereby true, and every court knows it, because a fabricated account is constructed precisely so as to hang together. Coherence is treated as evidence of truth and never as the definition of it. That refusal is the coherence theory's chief difficulty, met in a courtroom every day.

The criticisms

Two rival systems may each be coherent. The plaintiff's case and the defendant's case are each, at their best, complete and internally consistent stories. If coherence is truth, both are true, and they contradict each other. The theory then collides with the law of contradiction, which is a very high price.

Truth becomes relative to a system. A proposition may cohere with my beliefs and not with yours, so it is true for me and false for you. If "true for me" means true, this is again a contradiction. Most philosophers prefer to keep the law of contradiction and drop the theory. As the previous chapter's distinction suggests, someone who says "that is true for me" is almost always saying only "I believe it".

Coherence with what? If the system is the beliefs of a society, then a society that agreed the earth was flat would have made it flat, which nobody accepts. If the system is the ideal set of all true propositions, the theory has explained truth by using the notion of truth, which is circular.

A consistent fiction is still fiction. A well constructed novel is internally coherent throughout and no part of it is true. This is the objection in its shortest form and it is the one to write down.

A worked example

Two accounts are given of the same evening.

The plaintiff's account. He handed over Rs 2,00,000 in cash at the defendant's shop at about seven in the evening; the defendant counted it and promised to repay in six months; the plaintiff then went home and told his wife.

The defendant's account. No money was handed over at all; the plaintiff came to the shop about an unrelated matter; the suit is an afterthought following a family quarrel.

Each account is internally consistent. Each explains why the plaintiff was at the shop. Neither contains a contradiction.

How the court proceeds. It looks for the points at which each account has to meet something independently fixed. Was Rs 2,00,000 withdrawn from any account that week? Does the shop's own record show the plaintiff's visit? Was the family quarrel before the alleged loan or after it? What did the plaintiff say in his notice, and does the notice fit his evidence?

What this shows about the theory. The court is testing coherence, and testing it against a wider and wider body of material until one account fails to fit. That is the coherence theory as a method. But the court's justification for the method is a correspondence assumption: it believes the wider body of material to be connected to what actually happened, and that is why fitting with it is worth anything. Take the correspondence assumption away and the exercise is only a comparison of two fictions.

The honest summary, and the one that earns marks: coherence is an excellent test of truth and a poor definition of it.

Distinctions that carry marks

What this does not mean

Coherence is not mere consistency. Two unrelated propositions that do not contradict each other are consistent and not coherent. Coherence requires positive support among the members of the system.

The theory is not "truth is what people agree on". That is a cruder view. The coherence theorist demands a logically organised system, not a show of hands, although the version that fills the blank with the beliefs of a society comes close to it.

Rejecting the theory does not mean rejecting the method. Courts, historians and detectives all use coherence to decide what to believe, and are right to. What they do not do is define truth by it.

Quick revision

Statement: a proposition is true if and only if it coheres with a specified system of propositions.

Motivation: we can never compare a proposition with the world directly, only with other propositions.

Two blanks to fill: which system, and what coherence requires. Coherence needs at least consistency and usually mutual support.

Advocates: Leibniz, Spinoza, Hegel, Bradley, Blanshard and others in the rationalist tradition.

Criticisms: rival coherent systems; truth becomes relative and collides with the law of contradiction; coherence with what; a consistent fiction comes out true.

Legal use: the assessment of a witness is a coherence test. The law treats coherence as evidence of truth, never as its definition.

The line to write: coherence is a good test of truth and a bad definition of it.

Test yourself

1. State the coherence theory and identify what has to be specified before it says anything definite.

A proposition is true if and only if it coheres with a system of other propositions. Two things must be specified. First, which system: one's own beliefs, the beliefs of one's society, the beliefs of the informed, or the ideal complete set of true propositions. Second, what coherence requires: at a minimum logical consistency, on most versions mutual support, and on the strongest versions mutual entailment.

2. What objection to the correspondence theory motivates the coherence theory?

That we cannot compare a proposition with the world directly. To do so we would have to step outside all our beliefs and inspect reality unmediated, which nobody can do. Every check that is actually carried out compares one proposition with others, our records of observation included. The coherence theorist concludes that if comparison with other propositions is all we ever do, that is what truth must consist in.

3. Give the strongest criticism of the theory and explain it.

A well constructed fiction is internally coherent throughout, and every part of it is false. More sharply, in litigation each side's case at its best is a complete and consistent story, so if coherence were truth both would be true and they contradict one another. The theory then violates the law of contradiction, which most philosophers are unwilling to give up for any theory of truth.

4. How do courts use coherence, and where do they stop?

They use it constantly to assess evidence: a witness is disbelieved because the account is internally inconsistent, or conflicts with a document, or cannot be fitted into the admitted movements of the parties or the time available. They stop at the point of definition. No court holds that a coherent story is thereby true, precisely because a fabricated account is built to be coherent, so coherence is treated as evidence of truth and never as truth itself.

5. Distinguish a test of truth from a definition of truth.

A test tells you how to find out whether a proposition is true; a definition tells you what its being true consists in. The two need not be the same, and here they are not: coherence is an excellent test, because a proposition that will not fit with anything else we know is almost always false, and a poor definition, because fitting with other propositions is not what we mean when we call something true.

6. Why does the coherence theory tend towards relativism?

Because a proposition may cohere with one person's or one group's system and not with another's, so the theory yields that it is true for the first and false for the second. If "true for me" is taken to mean true, the same proposition is both true and false, which the law of contradiction forbids. The usual conclusion is that the phrase "true for me" is a loose way of saying "believed by me", and that truth itself is not relative in this way.

Where the law looks pragmatist

Not in its account of facts. No court decides what happened by asking which version would be more useful to believe, and any court that did would be corrupt.

But in its account of legal doctrine, courts talk like pragmatists constantly. A rule is said to be unworkable. A test is said to have produced uncertainty and to require reformulation. A construction is preferred because the alternative would lead to absurd consequences. A precedent is departed from because it has proved impossible to apply consistently.

None of those is an argument that the rule corresponds to anything. They are arguments about what works, and they are perfectly respectable arguments in their place. The place is the making and reform of legal doctrine, and not the finding of facts.

The criticisms

Useful to whom? A proposition may be useful for one person to believe and useless or damaging for another. The theory then makes the same proposition true for the one and false for the other, which collides with the law of contradiction exactly as the coherence theory did.

Useful beliefs are often false. People are sustained by beliefs about themselves and their circumstances which are demonstrably untrue, and being sustained is a real benefit. The theory overstates the link between truth and utility.

True beliefs are often useless. The number of grains of sand on a particular beach is a definite figure, and the proposition stating it is true, and believing it would do nobody any good at all.

It confuses truth with its value. That true beliefs are generally useful is agreed by everybody. It does not follow that being useful is what being true consists in, any more than the fact that healthy food is generally nourishing makes nourishing the definition of healthy. This is the objection to state last, because it is the one that survives the pragmatist's replies.

Peirce's version postpones the problem. Truth as the ideal end of inquiry avoids the crude objections, but nobody knows what inquiry would settle on, so the theory tells us nothing about any proposition now. It is a promise rather than a definition.

A worked example

A court is asked to decide whether the manager of a company branch had authority to sign a particular guarantee. There is no rule directly in point.

A correspondence question would be: did the board in fact confer that authority? That is a question of historical fact, decided on the minutes and the evidence, and the pragmatic theory has nothing to contribute to it.

A pragmatic question would be: which rule about the apparent authority of branch managers should the law adopt? Here the court reasons about consequences. A rule that no branch manager can bind the company would make commercial dealing impossible, since no counterparty could ever be safe. A rule that any employee can bind the company would make companies uninsurable. The rule chosen is the one that works: the company is bound where the counterparty dealt in good faith with someone the company held out as having that authority.

What the example shows. The two questions live in the same judgment and are answered by different standards. Mixing them is a serious error in either direction: deciding what happened by what would be convenient is corrupt, and choosing a rule by asking what "corresponds to the facts" is meaningless, because a rule is not the sort of thing that corresponds to facts at all.

The three theories compared

How to write the assessment. The three are not simply rivals; they answer three different questions that the one word "truth" runs together. Correspondence answers what truth is. Coherence answers how we test for it. Pragmatism answers why it is worth having. The dominant view is that correspondence gives the right definition, that coherence gives the best available test, and that the pragmatic account describes the value of truth rather than its nature. An answer that says this, and defends it in a paragraph, has done what the topic asks.

What this does not mean

Pragmatism is not "believe whatever suits you". The test is survival under sustained experience and by many people, not immediate convenience, and beliefs that flatter are exactly the ones that fail it.

It is not the same as the coherence theory. Both deny that truth is a relation to the world, but coherence looks inward at the system of beliefs and pragmatism looks outward at consequences in action.

Judicial talk of workability is not a theory of truth. It is an argument about which rule to adopt, and rules are made rather than discovered. Nothing follows from it about how facts are found.

Quick revision

Statement: a proposition is true if believing it works, that is, if acting on it succeeds and it withstands the test of experience.

Advocates: Peirce, James, Dewey. Peirce's version: truth is what inquiry would settle on in the long run.

"Works" means reliable over the long run under testing, not convenient.

Criticisms: useful to whom; useful falsehoods; useless truths; it confuses truth with the value of truth; Peirce's version tells us nothing about any proposition now.

Legal use: the making and reform of doctrine, where courts argue from workability and consequences. Never the finding of facts.

The comparison to write: correspondence says what truth is, coherence says how we test it, pragmatism says why it matters.

Test yourself

1. State the pragmatic theory of truth and name its advocates.

A proposition is true if it is useful to believe, in the sense that acting on it succeeds, it predicts correctly and it survives sustained testing by experience. Its principal advocates are Charles Sanders Peirce and William James, with John Dewey developing it later. Peirce's refinement, which is the more defensible version, is that truth is the opinion which inquiry, pursued far enough, would eventually settle upon.

2. In what sense does a pragmatist mean "useful"?

Not convenient, comforting or agreeable. The sense is reliability in practice over the long run: the belief guides action successfully, produces the consequences it led one to expect, continues to do so when others act on it, and survives everything experience puts against it. Beliefs held because they are comforting are precisely the ones that fail this test.

3. Give three criticisms of the theory.

First, a belief may be useful to one person and damaging to another, so the same proposition comes out both true and false, which violates the law of contradiction. Second, many demonstrably false beliefs are useful and many true ones are useless, so the connection between truth and utility is weaker than the theory needs. Third, and most fundamentally, the theory confuses truth with the value of truth: that true beliefs are generally useful is agreed, but it does not follow that usefulness is what truth consists in.

4. Where do courts reason pragmatically, and where do they refuse to?

They reason pragmatically about legal doctrine: a rule is rejected as unworkable, a test is reformulated because it has produced uncertainty, a construction is preferred because the alternative leads to absurdity, a precedent is departed from because it cannot be applied consistently. They refuse to reason this way about facts. No court asks which version of events would be more useful to believe, and one that did would not be administering justice.

5. Compare the three theories in one paragraph.

The correspondence theory holds that truth is a relation between a proposition and the world, which is what the word ordinarily means but is hard to make precise. The coherence theory holds that truth is a relation between a proposition and a system of other propositions, which describes how we actually check but wrongly makes a consistent fiction true. The pragmatic theory holds that truth is a property a belief earns by working, which explains why truth is worth having but is defeated by useful falsehoods and useless truths.

6. Why is it said that the three theories answer different questions?

Because the single word "truth" runs three questions together: what truth is, how we find out what is true, and why truth matters. Correspondence answers the first, coherence answers the second and pragmatism answers the third. Read that way they are less in competition than they appear, and the dominant position is to take correspondence as the definition, coherence as the working test, and the pragmatic account as a description of the value of truth rather than of its nature.

"At the same time." A person may be a minor in 2022 and a major in 2026. There is no contradiction, because the two propositions are about different times, and the law only bites when the time is the same.

"In the same respect." A document may be admissible for one purpose and inadmissible for another. A person may be an employee for one Act and an independent contractor for another. There is no contradiction, because the respect differs.

Once those two qualifications are stated, the law is unanswerable. The reason is that it is presupposed by the act of asserting anything at all. To assert p is to exclude not p; if asserting p left not p open, assertion would have no content, and every sentence would be compatible with every other. This is why the law cannot be proved: any proof would have to use it.

Where a lawyer meets it. The commonest use is the demonstration that an opponent's case is self-contradictory, which is a complete answer without needing any evidence at all. A defence that pleads both that the money was never received and that it was received as a gift is not offering two alternatives; it is offering something that cannot be true. Pleading in the alternative is different, and the difference is that alternative pleas are expressly put forward as alternatives, so neither is asserted outright.

The law of excluded middle

Everything either is or is not. Every proposition is either true or false; there is no third possibility.

In symbols, either p or not p.

This is not the same as the law of contradiction, and the difference is worth a mark. The law of contradiction says the two cannot both be true. The law of excluded middle says they cannot both be false. Together they say that exactly one of p and not p is true.

The demonstration is short. "This agreement is void" and "this agreement is not void" cannot both be true, by the law of contradiction, and cannot both be false, by the law of excluded middle. So exactly one holds. Now take "this agreement is void" and "this agreement is valid": they cannot both be true, but they can both be false, because the agreement may be voidable. The first pair are contradictories; the second are contraries. That distinction is topic 1.7, three chapters ahead, and the two laws are what it rests on.

Where it is disputed. The law of excluded middle is the only one of the three that has been seriously challenged. It is doubted for propositions about the future, since it is not obvious that "there will be a sea battle tomorrow" is already true or already false; and for propositions using vague words, since a man with a certain number of hairs may be neither definitely bald nor definitely not bald. Intuitionist mathematicians reject it outright.

Where a lawyer meets it. The whole of division by dichotomy, at topic 4.5, depends on it: divide anything into A and not A and you have divided it exhaustively and exclusively, because excluded middle guarantees nothing falls outside and contradiction guarantees nothing falls into both. This is why statutory drafting uses the form so often.

A worked example

A written statement pleads: "The defendant denies that any notice was received. In any event, the notice received by the defendant was defective in form."

Apply the law of contradiction. The first sentence asserts that no notice was received. The second asserts that a notice was received and was defective. Both cannot be true at the same time and in the same respect. As drafted, the pleading is self-contradictory, and the contradiction can be pointed out without a single witness.

Is there a way to save it? Yes, and it is instructive. Recast the second plea expressly in the alternative: "without prejudice to the above, if the court finds that a notice was received, the defendant says that it was defective in form." Now nothing is asserted twice over. The first plea is asserted; the second is put forward conditionally on a finding against the first. A conditional asserts neither of its parts, which is the point made at sequence 40, and so no contradiction arises.

The lesson. The law of contradiction does not forbid pleading alternative cases. It forbids asserting both members of a contradictory pair, and the whole art of drafting alternative pleas is the art of not asserting them.

Distinctions that carry marks

What this does not mean

They are not rules people always follow. People contradict themselves daily. The laws say that when they do, they have asserted nothing, not that they were incapable of the words.

The law of contradiction is not broken by change. A statement true in 2022 and false in 2026 is two statements about two times, and the law expressly requires the same time.

The law of excluded middle does not apply to contraries. "Guilty" and "innocent" behave like contradictories in a criminal trial only because the law makes them so by the presumption of innocence, which is a legal device and not a law of logic. "Valid" and "void" are contraries, and both can be false.

Quick revision

Three laws: identity, contradiction, excluded middle.

Identity: A is A. Its practical content is one word, one meaning, throughout an argument. Breaking it is equivocation.

Contradiction: not both p and not p, at the same time and in the same respect. Both qualifications are essential.

Excluded middle: either p or not p. Cannot both be false. Doubted for future events and vague terms.

Together: exactly one of a contradictory pair is true.

Modern reading: they are laws about propositions and the world, not about minds. That shift is the shift from traditional to modern logic.

Legal uses: identity forbids equivocation in a document; contradiction defeats a self-contradictory plea; excluded middle underwrites division by dichotomy.

Test yourself

1. State the three laws of thought.

The law of identity: whatever is, is, so that a thing is identical with itself and a true proposition is true. The law of contradiction: nothing can both be and not be, so that no proposition is both true and false at the same time and in the same respect. The law of excluded middle: everything either is or is not, so that of a proposition and its denial one must be true and there is no third possibility.

2. Why is the name "laws of thought" said to be misleading?

Because it suggests that they describe how minds work, which would make them part of psychology, and people manifestly do contradict themselves. On the modern reading they are not about minds at all but about propositions and about the world: no proposition can be both true and false, and no thing can both have and lack the same property at the same time and in the same respect.

3. Distinguish the law of contradiction from the law of excluded middle.

The law of contradiction says that a proposition and its denial cannot both be true. The law of excluded middle says that they cannot both be false. Taken together they yield that exactly one of them is true. The difference shows itself with contraries: "this agreement is void" and "this agreement is valid" cannot both be true, so contradiction applies, but both can be false if the agreement is voidable, so excluded middle does not.

4. Why are the qualifications "at the same time" and "in the same respect" essential to the law of contradiction?

Because without them the law would forbid ordinary and correct statements. A person may be a minor at one date and a major at another, and a document may be admissible for one purpose and inadmissible for another, without any contradiction at all. The law bites only when the very same property is affirmed and denied of the very same thing at the very same moment.

5. Why can the law of contradiction not be proved?

Because any proof would have to assume it. Asserting a proposition means excluding its denial; if asserting p left not p open, assertion would carry no content whatever and no proof of anything could get started. The law is therefore presupposed by the activity of proving, and a demand for its proof misunderstands what kind of principle it is.

6. A written statement denies that any notice was received and also says the notice received was defective. What is wrong, and how is it cured?

As drafted the two pleas assert contradictory propositions at the same time and in the same respect, so the pleading is self-contradictory and can be attacked without evidence. The cure is to plead the second expressly in the alternative and conditionally, so that it is not asserted but advanced only if the court rejects the first. A conditional asserts neither of its parts, so no contradiction arises, and this is the whole basis of alternative pleading.

Collective and distributive use. A collective term names a group taken as a whole: "the Bench", "the jury", "the committee". The distinction is really about use rather than about words, and this is where students are trapped. "The judges of the Supreme Court are twenty six" is true of the judges taken together and absurd of any judge individually. "The judges of the Supreme Court are appointed by the President" is true of each one taken separately. The same words, used collectively in the first and distributively in the second. Arguing from one use to the other is the fallacy of composition or of division, met again at sequence 640.

Concrete and abstract. A concrete term names a thing possessing a quality: "honest man", "registered document". An abstract term names the quality itself: "honesty", "registration". Legal writing is full of abstract terms and they are the harder ones to define, because there is nothing to point at.

Positive and negative. MU's own next division but one, taken at sequence 190.

Relative and absolute. A relative term cannot be understood without another: "creditor" makes no sense without a debtor, "landlord" without a tenant, "mortgagee" without a mortgagor. An absolute term stands alone: "tree", "document". Almost the whole vocabulary of the law of obligations is relative, which is why MU sets inference by converse relation at topic 3.4 and why that chapter is the most legally useful in Module III.

Terms in law

A statute is a document in which a small number of terms are given fixed meanings and everything else is left to ordinary usage. That is what a definition clause is, and it is worth seeing it as a piece of logic rather than as a piece of drafting.

Three things follow, and they are the whole practical content of this chapter for a lawyer.

A defined term means what the Act says and nothing else, within that Act. The ordinary meaning is displaced. This is the law of identity of the last chapter applied to a document.

A term defined in one Act is not thereby defined in another. The same word may have three statutory meanings and one ordinary one, and choosing between them is the beginning of most arguments about construction.

An undefined term carries its ordinary meaning, and the ordinary meaning of a general term has no sharp boundary, which is why litigation about whether something falls within it is possible at all. That indeterminacy is the subject of Module IV.

A worked example

An Act penalises "any person who keeps a dangerous animal without a licence". A prosecution is brought against a company that keeps a large dog at its warehouse.

Identify the terms. "Person", "dangerous animal", "licence", and the compound "person who keeps a dangerous animal without a licence". "Any" is the quantifier, "who" and "without" are syncategorematic.

Classify each. "Person" is a general term, and it is defined in the Act or by the General Clauses Act, so its meaning is fixed rather than ordinary; on the usual definition it includes a company. "Dangerous animal" is a general term left undefined, so it carries its ordinary meaning and has no sharp edge. "Licence" is a general term whose meaning is fixed by the Act's own scheme.

Notice where the case is. Not in "person": that is settled by a definition. It is entirely in "dangerous animal", an undefined general term with a vague boundary, and the argument will be about whether this dog falls inside it.

Notice also the collective trap. If the Act were about "the occupiers of the estate", a submission that the occupiers as a body kept the dog would not establish that any individual occupier kept it. Slipping between the collective and the distributive use of a term is a real and frequent error in pleadings, and the analysis of the term is what catches it.

Distinctions that carry marks

What this does not mean

A term is not necessarily one word. "A person of unsound mind who is disqualified by law from contracting" is a single term, because it occupies a single term position.

Collective and distributive are not two classes of word. They are two ways of using a term, and most terms admit of both.

Abstract does not mean vague. "Registration" is abstract and perfectly precise. "Reasonable" is concrete in its usual employment and thoroughly vague. The two distinctions are independent.

Quick revision

Term: a word or group of words capable of being the subject or predicate of a proposition.

Categorematic words can be terms; syncategorematic words, such as quantifiers, the copula and prepositions, cannot.

Quantifier, copula, quality: "all", "are", "not". Module II is built on them.

Divisions: singular and general; collective and distributive use; concrete and abstract; positive and negative; relative and absolute.

Composition and division are the fallacies of arguing between the collective and distributive uses.

In law: a definition clause fixes a term's meaning for that Act only; an undefined term carries its ordinary and therefore fuzzy meaning, which is where litigation lives.

Test yourself

1. Define a term and distinguish categorematic from syncategorematic words.

A term is a word or group of words capable of standing as the subject or the predicate of a proposition. Categorematic words have a meaning by themselves and can occupy those positions, as "contract" and "void" can. Syncategorematic words have no meaning standing alone and cannot be terms, though they contribute to the meaning of the proposition: quantifiers such as "all", the copula "is", conjunctions and prepositions are of this kind.

2. Distinguish singular from general terms and say why the distinction matters.

A singular term applies to exactly one individual, such as "the Indian Contract Act 1872". A general term applies to each member of a class taken separately, such as "statute". The distinction matters because only general terms can be quantified: one can speak of all or some statutes, but "all the Indian Contract Act 1872" is nonsense. It also matters in Module III, where singular propositions have only one opposite.

3. Explain the collective and distributive use of a term with an example.

A term is used collectively when the predicate is asserted of the group taken as a whole, and distributively when it is asserted of each member separately. "The committee consists of seven persons" is collective and is absurd of any individual member; "every member of the committee is a graduate" is distributive. Arguing from one use to the other is the fallacy of composition or of division, and it appears in pleadings more often than it ought.

4. What is a relative term, and why do they matter to lawyers?

A relative term is one that cannot be understood without a correlative: creditor and debtor, landlord and tenant, mortgagee and mortgagor, vendor and purchaser. They matter because the vocabulary of obligations is built almost entirely out of them, so that establishing one half of the relation establishes the other. This is what makes inference by converse relation, at topic 3.4, immediately useful in practice.

5. What does a definition clause in a statute do, in logical terms?

It fixes the meaning of a term for the purposes of that Act, displacing the ordinary meaning within that Act and only within it. Logically it is a précising definition: it takes a word whose ordinary boundaries are fuzzy and draws a sharp line for a particular purpose. The same word may therefore bear different meanings in different Acts, and the ordinary meaning survives everywhere the Act does not reach.

6. In "All contracts made by minors are void", list the terms and the words that are not terms.

The terms are "contracts", "minors", the compound "contracts made by minors", and "void". "All" is a quantifier, "by" is a preposition and "are" is the copula, and none of the three is a term, since none can stand as a subject or predicate and none has a meaning by itself.

Connotation without denotation: empty terms. "A married bachelor" has a connotation, which is why we can see at once that nothing satisfies it, and denotes nothing. In law, "an agreement with a minor which is enforceable against him" is such a term.

This asymmetry matters for legal drafting. A definition which is perfectly clear may nevertheless describe an empty class, and a schedule of names may pick out a class no definition captures. Statutes use both devices, and the choice between them is a choice between defining by connotation and defining by denotation.

The kinds of connotation

Some textbooks divide connotation three ways and examiners occasionally ask for it. The division belongs to the psychology of meaning rather than to logic proper, but it is short.

Subjective connotation is the set of attributes a particular person associates with the term. It varies from person to person and is what makes people talk past one another.

Objective connotation is the whole set of attributes actually common to everything the term denotes. It may include attributes nobody has yet noticed.

Conventional connotation is the set of attributes the term is agreed in a language to imply, which is what a dictionary records and what logic works with. When "connotation" is used without qualification, this is what is meant.

Where the law fixes the one and the other

This is the practically useful part, and it is why the chapter is here.

A definition clause fixes the conventional connotation of a term for one Act. Section 2 of almost any statute is a list of attributes attached to words.

A schedule or a list fixes a denotation. Where an Act applies to the establishments named in a Schedule, the class is fixed by enumeration, not by meaning, and nothing outside the list is in however similar it may be.

Illustrations and examples show denotation. The illustrations to a section do not define the term; they display members of its class, so that the reader can work out where the boundary runs.

Inclusive definitions do something in between. A definition saying that a word "includes" certain things is not stating the full connotation; it is adding named members to the denotation while leaving the ordinary meaning otherwise intact. That is why an inclusive definition enlarges a term and an exhaustive one confines it, which is one of the most heavily litigated points in statutory construction.

A worked example

An Act applies to any "factory", defined as "any premises where ten or more workers are working and where a manufacturing process is being carried on with the aid of power".

Read the connotation off the definition. Four attributes: premises; ten or more workers; a manufacturing process being carried on; the aid of power. Every one must be satisfied, because the attributes are joined by "and".

Predict the litigation. Since each attribute is necessary, an occupier who wants to be outside the Act needs to defeat only one of them, and the argument will be about whichever is easiest to defeat on the facts. That is a direct consequence of the connotation being a conjunction of four attributes, and it can be predicted before any case arises.

Now apply the law of inverse variation. Suppose the legislature deletes "with the aid of power". The connotation loses an attribute, so the denotation grows: premises using no power now fall inside, and the Act covers more establishments. Suppose instead it adds "and where the process is carried on for twelve months in the year". The connotation gains an attribute, so the denotation shrinks and seasonal establishments drop out.

A drafter can therefore widen or narrow the reach of an Act by adding or removing a single word in the definition, and the direction of the change is settled in advance by the law of inverse variation. That is not a metaphor; it is how amendment of definition clauses actually works.

Distinctions that carry marks

What this does not mean

Connotation here is not the ordinary English "connotation". In everyday use the word means an overtone or an association, as when "cheap" is said to have negative connotations. In logic it means the defining attributes and nothing else. This is a false friend and it catches students every year.

Inverse variation is not a formula. It states a direction, not a ratio, and it fails where a term already denotes one thing or none.

A term with no denotation is not meaningless. It is meaningful precisely because it has a connotation; that is how we know nothing satisfies it.

Quick revision

Connotation, or intension: the attributes a thing must have to be called by the term. Denotation, or extension: the things that have them.

Law of inverse variation: more connotation, less denotation. A direction, not a ratio, and it fails at the ends.

Proper names: denotation without connotation, on Mill's view.

Empty terms: connotation without denotation, such as "an enforceable agreement with a minor".

Three kinds of connotation: subjective, objective, conventional. Logic works with the conventional.

In statutes: a definition clause fixes connotation; a schedule or list fixes denotation; illustrations display denotation; "includes" enlarges and "means" confines.

Forward link: a definition states a connotation and a division sorts a denotation, which is the whole of Module IV in one line.

Test yourself

1. Define connotation and denotation, and give an example of each for one term.

The connotation, or intension, of a term is the set of attributes a thing must possess to be called by it; the denotation, or extension, is the set of things that possess them. For "contract", the connotation is agreement, competent parties, free consent, lawful consideration, lawful object and not being expressly declared void; the denotation is every actual contract, this lease and that sale deed among them.

2. State the law of inverse variation and give its two limits.

As the connotation of a term increases the denotation decreases, and as the connotation decreases the denotation increases. Its first limit is that the relation is one of direction and not of proportion: there is no ratio between the number of attributes and the number of objects. Its second is that it breaks down at the ends, since a term denoting a single thing cannot denote fewer without denoting none, and a term may have a full connotation and an empty denotation.

3. Can a term have denotation without connotation, or connotation without denotation?

Both, on the traditional account. A proper name such as "Ravi" denotes an individual and, on Mill's view, connotes nothing, being a mark rather than a description. An empty term such as "an agreement with a minor enforceable against him" has a clear connotation and denotes nothing at all, and it is precisely because the connotation is clear that we can see the class is empty.

4. How does a statute fix the connotation of a term, and how does it fix the denotation?

It fixes connotation by a definition clause, which attaches a list of attributes to the word for the purposes of that Act. It fixes denotation by enumeration, in a schedule or a list of establishments, persons or articles to which the Act applies. Illustrations to a section display members of the denotation without defining the term, which is why they guide but do not control construction.

5. What is the difference between a definition that says "means" and one that says "includes"?

A definition that says "means" is exhaustive: it states the whole connotation and the term bears that meaning and no other within the Act, which confines it. A definition that says "includes" is inclusive: it leaves the ordinary meaning standing and adds named members to the denotation, which enlarges the term. The distinction is heavily litigated, since a great deal can turn on whether a class is closed or open.

6. An Act defines "factory" as premises where ten or more workers work and a manufacturing process is carried on with the aid of power. What happens to its reach if "with the aid of power" is deleted?

The connotation loses one attribute, so by the law of inverse variation the denotation grows: premises using no power now satisfy the remaining three attributes and fall within the Act. The reach of the statute is widened, and it is widened by deleting words rather than by adding them, which is a direct and predictable consequence of the relation between connotation and denotation.

To shift the practical burden. A rule that a thing may be done only if a condition is satisfied throws the work onto whoever asserts the condition; a rule that it may be done unless a condition exists throws the work the other way. The difference is a positive or a negative formulation of the same test.

There is a matching danger, and it is a drafting fault rather than a logical one. A double negative in a statute is very hard to read. "No person shall be disqualified unless he has not complied with sub-section (2)" takes several readings to unpick, and provisions of this shape have generated real litigation.

A worked example

Section 11 of the Indian Contract Act 1872 provides that every person is competent to contract who is of the age of majority according to the law to which he is subject, who is of sound mind, and who is not disqualified from contracting by any law to which he is subject.

Notice the three attributes. Two are stated positively, majority and soundness of mind. The third is stated negatively, not disqualified by any law.

Why the third is negative. Because the disqualifications are scattered across other statutes and cannot be listed in this one. A negative term is the only way to capture a class whose members are fixed elsewhere and may change.

Now form the negative of the whole. "Incompetent to contract" is the negative of "competent to contract", and it applies within the universe of persons. It is satisfied by failing any one of the three attributes, because the attributes are joined conjunctively. This is a small illustration of a general rule: the negative of a conjunction is a disjunction of negatives, which Module II states formally at sequence 350.

And notice the privative point. A minor is not "of unsound mind"; he is incompetent for a different reason. Using "unsound mind" for a minor would be using a privative term outside its proper universe, and it would misdescribe the ground of incompetence.

Distinctions that carry marks

What this does not mean

A prefix does not make a term negative. "Blind" is negative without one and "immortal" is positive with one. Look at what is asserted.

The negative term is not everything else in existence. It is everything else in the universe of discourse, and the universe must be fixed first.

Negative wording in a statute is not a drafting failure. It is the standard way of expressing a prohibition and of catching an unlistable residue.

Quick revision

Positive: presence of an attribute. Negative: absence of it.

The prefix is not the test. "Blind" is negative; "immortal" is positive.

Privative: absence of an attribute normally possessed. Not true of things that never had it.

Universe of discourse must be fixed before a negative term has definite members.

Together they exhaust the universe, by excluded middle, and overlap nowhere, by contradiction. This is the basis of dichotomy.

In statutes: negatives catch the unlistable residue, express prohibitions, and shift the practical burden. Double negatives are a known source of litigation.

Forward link: obversion at sequence 460 turns a positive proposition into a negative one by negating the predicate term, and it cannot be done without this chapter.

Test yourself

1. Distinguish positive from negative terms and state the correct test.

A positive term signifies that a thing possesses an attribute; a negative term signifies that it lacks one. The test is what the term asserts and not whether it carries a negative prefix. "Blind" has no prefix and is negative because it means wanting sight, while "immortal" has one and is best treated as positive because it names a definite condition rather than a bare absence.

2. What is a privative term?

A term signifying the absence of an attribute which the thing in question would in the ordinary course possess. "Blind", "deaf" and "illiterate" are privative: they can be predicated only of things capable of sight, hearing or literacy. A stone is non-seeing but not blind. Privative terms therefore carry an implication that pure negatives lack, and legal drafting relies on that implication.

3. Why must the universe of discourse be fixed before a negative term is used?

Because the negative of a term is everything within the field under discussion that lacks the attribute, not everything in existence. If the discussion is about persons, "non-adult" covers persons who are not adults and not stones or ideas. A statute performs this fixing when it states the establishments, persons or transactions to which it applies, and within that field its negative expressions have a definite membership.

4. Give three reasons a statute uses negative wording.

To catch a residue that cannot be enumerated, as with "any person other than a member". To impose a prohibition, negative language such as "no suit shall lie" being the standard signal of a mandatory provision. And to place the practical burden on one party rather than the other, since a rule permitting an act only if a condition is met operates differently from one permitting it unless a condition exists.

5. Why do positive and negative terms together exhaust their universe?

Because of the two laws of thought. The law of excluded middle guarantees that everything in the universe of discourse either has the attribute or lacks it, so nothing falls outside the pair. The law of contradiction guarantees that nothing both has and lacks it, so nothing falls into both. Division by dichotomy relies on exactly this and is therefore always exhaustive and always exclusive.

6. Form the negative of "competent to contract" under section 11 of the Indian Contract Act 1872, and say what satisfies it.

The negative is "incompetent to contract", within the universe of persons. Since section 11 requires majority, soundness of mind and freedom from disqualification, all three together, the negative is satisfied by the failure of any one of them: a person may be incompetent by being a minor, or by being of unsound mind, or by being disqualified by some other law. The negative of a conjunction of conditions is a disjunction of their negatives.

"Valid" and "void" are contraries because "voidable" sits between them.

"Legal" and "illegal" are contraries in most contexts, because a great deal of conduct is neither commanded nor forbidden, which is why the law recognises acts that are merely permitted.

Why it matters in argument

Treating contraries as contradictories is a standard error and it produces a standard bad argument: disprove one and claim the other.

"The prosecution has not proved that the accused was at the scene, therefore he was elsewhere." No. "At the scene" and "elsewhere" would be contradictories if the universe were fixed, but the argument does not reach that far: failure to prove presence establishes only that presence is not proved, which under section 2(1)(i) is the middle state and not the opposite one.

"The agreement is not valid, therefore it is void." No. It may be voidable, in which case it is valid until avoided.

"The witness has not been shown to be truthful, therefore he is lying." No. He may be mistaken, which is neither.

The rule in one line: from the falsity of one contrary, nothing follows about the other. From the falsity of one contradictory, the truth of the other follows at once. That single asymmetry is what the whole distinction is for.

A worked example

An Act provides that a licence shall be granted to any applicant who is a "fit person", and that no licence shall be granted to a person who is "unfit".

Are "fit" and "unfit" contradictories or contraries here? If they are contradictories, every applicant is one or the other and the Act is complete. If they are contraries, there is a middle class of applicants who are neither fit nor unfit, and the Act says nothing about them.

The answer turns on the wording and it is not a matter of taste. "Unfit" formed by simple negation of "fit" would give contradictories. But the Act uses "unfit" as a positive characterisation, a finding to be made against a person, and a finding of unfitness would need material. An applicant about whom nothing is known is not thereby unfit; he is simply not shown to be fit.

The practical consequence. On the contrary reading, an authority that refuses a licence must find the applicant unfit and cannot rest on the absence of proof of fitness, unless the Act places the burden on the applicant. On the contradictory reading it may. The whole administrative law of the provision follows from a distinction taught in a first-year logic paper, which is why the topic is on a law syllabus at all.

Distinctions that carry marks

What this does not mean

Contrary is not the same as merely different. "Red" and "chair" are different and are not contraries. Contraries are extremes within one class, so they must be comparable to begin with.

A negative prefix does not settle it. "Unfit" may be used as a simple negation or as a positive finding, and the Act's own scheme decides which. This is the point of the worked example.

"Contradictory" here is not the law of contradiction. The law says a proposition cannot be both true and false; contradictory terms are a relation between two terms. They are connected but they are not the same thing.

Quick revision

Contradictory terms: one is the simple negative of the other; nothing falls outside the pair and nothing falls in both.

Contrary terms: extremes of one scale; both can be false, and the middle ground often has its own name.

The test: can both be false? Yes means contrary; no means contradictory.

Legal examples: registered and unregistered are contradictory; valid and void are contrary, with voidable between; proved and disproved are contrary, and section 2(1)(i) BSA 2023 names the middle as "not proved".

Guilty and innocent are contraries made to behave as contradictories by the presumption of innocence, which is a legal rule and not a logical one.

The standard bad argument: disproving one contrary and claiming the other.

Terms here, propositions at sequence 410. Same words, different subject.

Test yourself

1. Distinguish contradictory from contrary terms and state the test.

Two terms are contradictory when one is the simple negation of the other, so that within the universe of discourse everything falls under exactly one of them. They are contrary when they are the extremes of a single scale, so that a thing may fall under neither. The test is whether both can be false of the same thing at once: if they can, the terms are contrary; if they cannot, they are contradictory. Both can never be true, whichever relation holds.

2. Give two legal examples of contrary terms and identify the middle ground in each.

"Valid" and "void" are contrary, the middle ground being "voidable", an agreement which is valid until it is avoided. "Proved" and "disproved" are contrary, and here the statute names the middle itself: section 2(1)(i) of the Bharatiya Sakshya Adhiniyam 2023 provides that a fact is not proved when it is neither proved nor disproved.

3. Are "guilty" and "innocent" contradictory terms?

Not as a matter of logic. Logically there is a middle state, namely that guilt has not been established and innocence has not been established either. What makes the pair behave as contradictories in a criminal trial is the presumption of innocence, a legal rule directing that a person not proved guilty be treated as innocent. The gap is closed by decision rather than by necessity, and that is worth saying in an answer.

4. What follows from the falsity of one of a pair of contraries?

Nothing about the other. If it is false that the agreement is void, it does not follow that it is valid, because it may be voidable. This is the asymmetry that makes the distinction worth learning: from the falsity of one contradictory the truth of the other follows immediately, while from the falsity of one contrary nothing whatever follows.

5. Identify the fault: "The prosecution has failed to prove that the accused was at the scene, so he was elsewhere."

The argument treats a pair of contraries as if they were contradictories. Failure to prove presence establishes only that presence is not proved, which under section 2(1)(i) of the Bharatiya Sakshya Adhiniyam 2023 is the middle state of "not proved", and it says nothing about where the accused actually was. The conclusion asserted requires affirmative material of its own.

6. Why must this distinction be kept separate from the one at topic 3.2?

Because topic 3.2 concerns contradictory and contrary propositions, not terms. "All A are B" and "No A is B" are contrary propositions and both can be false; "All A are B" and "Some A are not B" are contradictory propositions and exactly one is true. The test is the same question, whether both can be false, but the things related are different, and answers that mix them up describe the square of opposition when asked about terms.

Why it is called the weakest form

Because a single counter-instance destroys it completely, and because nothing in the argument protects against one.

The classical illustration is the swan. Europeans observed swans for centuries, every one of them white, and concluded that all swans are white. The conclusion was overturned in 1697, when black swans were found in Western Australia. The number of confirming instances had been enormous and it counted for nothing.

No number of confirming instances makes the conclusion certain, and one contrary instance makes it false. That asymmetry is the whole case against relying on simple enumeration where anything better is available.

Two further weaknesses are examinable.

It gives no reason. A causal induction tells you why the connection holds and can be tested by intervening. Simple enumeration says only that it has held so far, so it cannot distinguish a real connection from a coincidence.

The instances are usually not varied. People observe what is nearest, so the sample is drawn from one place and one period. The swans were all European.

Why it survives anyway

If it is so weak, why is it the commonest form of reasoning in ordinary life and in practice?

Because it is the starting point. All the better forms of induction begin with somebody noticing a regularity, and noticing a regularity is simple enumeration. A causal hypothesis has to be suggested before it can be tested, and the suggestion comes from the enumeration.

Because it is often all there is. Where the subject matter does not permit experiment, and human affairs mostly do not, counting instances may be the only method available.

Because it is strong enough for many purposes. Nobody demands a causal explanation before accepting that a particular registry office is always closed on the second Saturday. For practical action, a well supported enumeration is frequently enough, and the pragmatists of chapter 150 would say that is all anyone should ask.

Improving it

An enumeration is strengthened by exactly the tests given at sequence 90, and the improvements are worth stating in the form of instructions.

Increase the number of instances, remembering that the returns diminish quickly.

Vary the circumstances. Twenty instances gathered in twenty different conditions are worth far more than two hundred gathered in one. This is the single most effective improvement available.

Look for the exception rather than waiting for it. An enumeration that has survived a genuine search for counter-instances is worth much more than one that has merely not tripped over any.

Weaken the conclusion. "All crows are black" is destroyed by one white crow. "Crows are generally black" and "the next crow will probably be black" survive it. Since the conclusion is what the argument has to support, claiming less is the cheapest way of claiming something defensible.

Look for a connection. The moment a reason can be given why the property should attach to the class, the argument stops being simple enumeration and becomes something better.

Where the law uses it, and where it distrusts it

Custom. A custom is proved by showing a practice observed uniformly and continuously over a long period, which is an enumeration of instances, and the law then treats the practice as a rule. The requirements the law imposes, antiquity, continuity, uniformity, are precisely the tests for strengthening an enumeration.

Course of dealing. Where parties have dealt on the same terms many times, a court will infer that the current transaction was on those terms too. Again an enumeration, and again the law asks about the number and the consistency of the earlier dealings.

Business practice and market usage are established the same way.

And where the law distrusts it. Evidence that a person has behaved in a certain way on other occasions is generally not admissible to show that he behaved that way on this occasion. The reasoning would be an enumeration, and the law rejects it, not because the inference is worthless but because it is weak and highly prejudicial. That is a legal judgment about the strength of a form of induction, made by a rule of evidence, and it is exactly the kind of thing this syllabus exists to let a student notice.

A worked example

A tenant claims that rent has always been payable on the tenth of each month, though the lease is silent. He proves that on each of the previous thirty six months the landlord accepted rent on the tenth without objection.

The argument.

1. On each of the last thirty six occasions rent was accepted on the tenth without objection.

Therefore rent is payable on the tenth.

Assess it. Number: thirty six, which is substantial. Variety: poor, because the instances are all of the same kind and all between the same parties, though that limitation matters less here, since the conclusion is only about these parties. Modesty: the conclusion asserts a term of the tenancy, which is a good deal more than the premises establish; a modest version, that the parties treated the tenth as acceptable, follows much more comfortably. Search for exceptions: unknown, and the landlord is the person who can supply them.

How the landlord attacks it. Not by disputing the thirty six instances, which are admitted. By producing a counter-instance, a single month in which he objected, or by breaking the connection: proving that the tenth was accepted as an indulgence and not as of right. A counter-instance destroys the enumeration; showing the instances have another explanation destroys the inference. Those are the two lines of attack on any argument of this form, and they are always available.

Distinctions that carry marks

What this does not mean

It is not the same as deduction from a general rule. Enumeration produces the general rule; deduction uses one that is already given.

A large number is not a substitute for variety. Two hundred observations of one kind support a narrower conclusion than twenty of twenty kinds.

Rejecting it in evidence is not calling it worthless. The rule against character evidence reflects a judgment that the inference is weak relative to its prejudicial effect, not that no inference is possible.

Quick revision

Definition: a general conclusion drawn from particular instances on the sole ground that no contrary instance has been observed. Also called induction per enumerationem simplicem.

Primary induction makes the inductive leap to instances nobody has observed, and simple enumeration is one. Secondary induction does not: it applies an established generalisation, or moves particular to particular, as analogy does.

Weakest form of induction: one counter-instance destroys it, and no number of instances makes it certain. The black swan is the standard illustration.

Three weaknesses: no reason is given; the sample is usually unvaried; nothing guards against the exception.

Why it survives: it is where every better induction starts, it is often the only method available, and it is frequently strong enough to act on.

Improvements: more instances, greater variety, an honest search for exceptions, a weaker conclusion, and a connection if one can be found.

Legal uses: custom, course of dealing, market usage. Legal distrust: the exclusion of character and similar-fact evidence.

Test yourself

1. Define induction by simple enumeration and set out its form.

It is the form of induction in which a general conclusion about a class is drawn from a number of observed instances, on the sole ground that no contrary instance has been observed. Its form is: this A is B, that A is B, every other observed A has been B, therefore all A are B. Nothing beyond the counting of instances is offered in support, which is what makes the enumeration simple.

2. Why is it called the weakest form of induction?

Because a single counter-instance falsifies its conclusion outright, while no number of confirming instances can make the conclusion certain, and nothing in the argument guards against the exception. It also gives no reason for the connection it asserts, so it cannot tell a real regularity from a coincidence, and its instances are usually drawn from a narrow range of circumstances.

3. Give the classical illustration of its failure.

The swan. For centuries every swan observed in Europe was white, and the generalisation that all swans are white was supported by an immense number of instances. Black swans were found in Western Australia in 1697 and the conclusion was destroyed at once. The number of confirming observations counted for nothing against one contrary observation.

4. How can an enumeration be strengthened?

By increasing the number of instances, though the returns diminish quickly; by varying the circumstances under which they are gathered, which is the most effective single improvement; by searching honestly for counter-instances rather than merely not encountering any; by weakening the conclusion, since a claim that crows are generally black survives a white crow while a claim that all crows are black does not; and by finding a reason for the connection, at which point the argument ceases to be simple enumeration.

5. Give two legal doctrines that rest on simple enumeration.

Custom, which is established by proving a practice observed uniformly and continuously over a long period, the law's requirements of antiquity, continuity and uniformity being the standard tests for strengthening an enumeration. And course of dealing, where a court infers that the present transaction was on the terms the parties have used many times before, the strength of the inference depending on the number and consistency of the earlier dealings.

7. Distinguish primary from secondary induction.

Primary induction infers a general conclusion from particular instances, covering instances nobody has observed, and so makes the inductive leap; simple enumeration and causal induction are of this kind. Secondary induction does not make that leap: it applies a generalisation already established, or reasons from one particular to another without asserting anything general, as analogy does. The test is whether the conclusion reaches unobserved instances.

6. How is an argument of this form attacked?

In two ways, and both are always available. First, by producing a counter-instance, which destroys the generalisation outright however many confirming instances there were. Second, by showing that the instances have some other explanation, so that the observed regularity is a coincidence or the product of a cause that will not operate in the present case. The first attacks the conclusion, the second attacks the inference.

6. The modesty of the conclusion. The less the conclusion claims relative to the premises, the stronger the argument. Concluding that B is probably liable is far better supported than concluding that B is certainly liable to the same extent and on the same terms.

Putting the tests in order

An examination answer that lists six tests as equals has missed the point. They are not equal.

Relevance is the master test. Numbers and variety matter because they make relevance more likely to be real rather than accidental; they have no independent value. An analogy with two relevant resemblances beats one with fifty irrelevant ones, always.

Modesty is free. It costs nothing to weaken the conclusion and it improves every analogy.

Disanalogies are the opponent's tool and the arguer's blind spot. Anyone constructing an analogy will have thought about the resemblances and will not have looked for the differences, which is why the differences are where an argument by analogy is attacked.

A worked example: one analogy, six tests

The situation. A court has held that a shopkeeper who left a loose floor tile unrepaired for a week is liable to a customer injured by it. A different case now arises: a bank has left a loose floor tile unrepaired for a week and a customer has been injured.

The analogy. The two occupiers resemble each other in inviting the public onto premises for the purposes of their business, in controlling the premises, in the nature of the hazard and in the period for which it was left. The shopkeeper was liable, so the bank is probably liable.

Now run the tests.

Number. One prior case, which is the minimum. A line of decisions covering shops, restaurants and cinemas would be much stronger.

Variety. Nil, since there is only one instance. If the line existed and covered shops, restaurants and cinemas, the variety would be high and would show that the liability does not depend on being a shop.

Number of resemblances. Four, listed above.

Relevance. Strong, and this is the whole argument. Inviting the public, controlling the premises, the hazard, the delay: every one of them bears on why an occupier is liable at all. Add that both premises had white walls and the number of resemblances rises to five and the strength of the argument does not move at all.

Disanalogies. Look for them deliberately. The bank's floor may be cleaned by a contractor rather than by staff; the customer may have been in a part of the premises not open to customers; the tile may have been loosened that morning by another customer. The first is probably irrelevant, because control is what matters and not who does the cleaning. The second and third are highly relevant, and either would defeat the analogy.

Modesty. "The bank is probably liable" follows. "The bank is liable in the same amount" does not, because nothing in the analogy concerns quantum.

The result. A reasonably strong analogy, resting almost entirely on the relevance of four resemblances, and vulnerable to exactly two facts. That is a usable answer, and it is what the six tests are for.

Bad analogies and how they go wrong

The false analogy is the standard fallacy of this form: an argument whose resemblances are numerous but irrelevant. "This agreement, like the one in that case, was made on stamp paper of the same value, in the same city, before the same sub-registrar, so it must be enforceable in the same way." Nothing in the list bears on enforceability.

The analogy stretched too far takes a comparison that holds for one purpose and carries it into another. A company is like a person for the purposes of owning property and suing; it is not like a person for the purposes of imprisonment.

The analogy that ignores the crucial difference is the commonest in practice, because the arguer is looking at resemblances by construction.

The analogy used as proof rather than as support. An analogy can never establish its conclusion with certainty, and an argument that treats it as if it had has mistaken an induction for a deduction.

Distinctions that carry marks

What this does not mean

More resemblances is not automatically better. Adding irrelevant resemblances adds nothing, and an answer that says "the more points of similarity the stronger the analogy" without the qualification about relevance has stated the common error.

An analogy is not a metaphor. A metaphor illustrates; an analogy argues. Calling a company a person may be either, and only one of them supports a conclusion.

A weak analogy is not invalid. The word does not apply to inductive arguments at all, as chapter 90 established.

Quick revision

Form: A and B resemble each other in p, q, r; A has s; therefore B probably has s.

Six tests: number of instances; variety of instances; number of resemblances; relevance of the resemblances; number and importance of disanalogies; modesty of the conclusion.

Relevance is the master test. One relevant resemblance beats a hundred irrelevant ones.

Modesty is free and improves every analogy.

Disanalogies are where an analogy is attacked, because the arguer will not have looked for them.

False analogy: many resemblances, none relevant. The standard fallacy of this form.

It moves particular to particular and never states a general rule, which is why one prior case is enough to argue from.

Test yourself

1. Define argument by analogy and give its form.

An argument by analogy infers, from the resemblance of two or more things in certain respects, that they resemble one another in a further respect. Its form is: A and B resemble each other in respects p, q and r; A has the further property s; therefore B probably has s. Its conclusion is probable only, and it moves from one particular to another without asserting any general rule.

2. State the characteristics of a good analogy.

A greater number of instances; greater variety among them, so that the resemblance relied on is not accidental; a greater number of respects in which the things resemble each other; relevance of those resemblances to the property inferred; few and unimportant disanalogies; and a modest conclusion that claims no more than the premises support.

3. Which of the tests is decisive, and why?

Relevance. A resemblance counts only in so far as it is connected with the property being inferred, so a single relevant resemblance outweighs any number of irrelevant ones. Two documents may be alike in paper, typeface, length and place of execution without any of it bearing on enforceability, while one resemblance in the capacity of the parties bears on it completely. The other tests matter because they make a relevance claim more credible, not on their own account.

4. What is a false analogy?

An argument by analogy in which the resemblances relied on, however numerous, have no connection with the property inferred. It is the characteristic fallacy of this form and it is persuasive precisely because the list of similarities can be made long. The answer to it is not to deny the similarities, which are usually admitted, but to show that none of them bears on the conclusion.

5. Why are disanalogies the natural place to attack an argument by analogy?

Because whoever constructed the argument was looking for resemblances and will not have searched for differences. The tests for a difference are the same as for a resemblance: what matters is whether it bears on the property inferred. A difference in some feature unconnected with the conclusion is harmless, while a single relevant difference may destroy the argument by itself.

6. Why does weakening the conclusion strengthen an analogy?

Because the strength of an inductive argument is the relation between what the premises support and what the conclusion claims. Reducing the claim leaves the same support carrying a lighter load. Concluding that a party is probably liable is far better supported than concluding that he is certainly liable in a stated amount, and since the modest conclusion is usually all that is needed, the improvement costs nothing.

The reasoning involved is inductive and it is the form named at sequence 90 as inference to the best explanation. A number of facts are proved. Several explanations are conceivable. The court concludes that one explanation accounts for all of them and the others do not.

The danger in that reasoning is obvious once it is stated. It is that the court may fail to think of the explanation nobody suggested, and having eliminated the two it thought of, treat the third as eliminated too. Indian law meets this danger with a rule.

The five golden principles

Sharad Birdhichand Sarda v. State of Maharashtra, AIR 1984 SC 1622, decided on 17 July 1984, is where the rule is stated in the form now always cited.

Facts. The appellant's wife Manju died of potassium cyanide poisoning at her matrimonial home about four months after the marriage. The prosecution case was that the appellant had administered the poison. There was no direct evidence of administration at all: the case rested entirely on circumstances, namely that the marriage was unhappy, that the appellant was alleged to be involved with another woman, that cyanide was present, and that he had the opportunity. The defence was that Manju, who was deeply unhappy, had taken the poison herself. The trial court convicted and the High Court confirmed.

Held. The appeal was allowed and the conviction set aside. At paragraph 152 the Court set out five conditions which must be fulfilled before a case against an accused resting on circumstantial evidence can be said to be fully established:

(1) the circumstances from which the conclusion of guilt is to be drawn should be fully established;

(2) the facts so established should be consistent only with the hypothesis of the guilt of the accused;

(3) the circumstances should be of a conclusive nature and tendency;

(4) they should exclude every possible hypothesis except the one to be proved; and

(5) there must be a chain of evidence so complete as not to leave any reasonable ground for the conclusion consistent with the innocence of the accused, and must show that in all human probability the act must have been done by the accused.

At paragraph 153 the Court called these five golden principles "the panchsheel of the proof of a case based on circumstantial evidence". On the facts, the possibility that Manju had taken the poison herself had not been excluded, so conditions four and five failed and the conviction could not stand.

Why it matters here. Because every one of the five conditions is a rule of logic given the force of law, and the fourth is eliminative induction stated in a sentence. The Court also drew a distinction that belongs to this syllabus rather than to the law of evidence: it insisted, at paragraph 152, that the circumstances "must or should" be established and not "may be" established, citing the observation that the mental distance between "may be" and "must be" is long and divides vague conjectures from sure conclusions.

The rule did not begin in 1984. Hanumant v. State of Madhya Pradesh, AIR 1952 SC 343, is where Mahajan J first expounded it, and the Court in Sarda called it the locus classicus and reproduced the passage twice.

Facts. Not read, and so not stated here. What was read is the Supreme Court's own reproduction of the passage in Sarda, and Hanumant is cited for that passage alone.

Held. That where the evidence is of a circumstantial nature, the circumstances from which the conclusion of guilt is to be drawn should in the first instance be fully established, and all the facts so established should be consistent only with the hypothesis of the guilt of the accused; that the circumstances should be of a conclusive nature and tendency and should exclude every hypothesis but the one proposed to be proved; and that there must be a chain of evidence so far complete as not to leave any reasonable ground for a conclusion consistent with the innocence of the accused, and such as to show that within all human probability the act must have been done by the accused.

Why it matters here. It is the source of the five conditions, and it shows that the logical structure was in Indian law from the Supreme Court's earliest years.

Reading the five conditions as logic

This is the part of the chapter that a logic paper is actually asking for, and it is worth writing out.

Condition 1, the circumstances must be fully established. The premises of an inductive argument must themselves be proved. An argument whose premises are only suspected proves nothing, and section 2(1)(j) of the Bharatiya Sakshya Adhiniyam 2023 fixes what proof of a fact means.

Condition 2, consistency only with guilt. No rival hypothesis may be consistent with the whole body of proved facts. Note "only": consistency with guilt is not enough, since a set of facts consistent with guilt may be equally consistent with innocence.

Condition 3, conclusive nature and tendency. The resemblance test of analogy, in another dress: the circumstances must be relevant to the conclusion and must point towards it, not merely accompany it.

Condition 4, exclusion of every other hypothesis. Eliminative induction. This is the condition that converts an ordinary inference to the best explanation into something a court may act on, and it is the one prosecutions most often fail, as Sarda itself failed it.

Condition 5, a chain complete beyond a reasonable ground for innocence. The strength requirement, fixed at the criminal standard. It also supplies the image the courts use constantly: a chain is as strong as its weakest link, so a single unproved circumstance breaks the whole.

The one thing the conditions cannot supply. They require every other hypothesis to be excluded, and no rule can make a court think of a hypothesis nobody put forward. That is why a defence has real work to do even where it bears no burden: suggesting the alternative explanation is what makes condition four bite.

A worked example

A watchman is found dead in a locked godown. Proved: the accused had the only other key; he was seen entering at 9.10 p.m.; the watchman was last heard alive at 9.15 p.m.; the accused left at 9.40 p.m. carrying a bag; the deceased had been struck with a heavy object; a bloodstained iron bar was recovered from the accused's house; and the accused had quarrelled with the deceased a week earlier over pilferage.

Condition 1. Each circumstance must be proved to the standard of section 2(1)(j). The recovery is the one most likely to be attacked, because a recovery proves possession and not use.

Condition 2. Are the facts consistent only with guilt? Almost, but the bag is not: a man may leave a godown carrying a bag for many reasons, and it is consistent with everything.

Condition 3. The quarrel is relevant to motive but weakly, being a week old and about pilferage. The key and the timings are strongly relevant.

Condition 4. What other hypotheses exist? That a third person entered with the accused's key before 9.10 and remained; that the deceased was struck after 9.40 by someone with access; that the injury was accidental. Each has to be excluded, and the exclusion has to come from the proved facts, not from the improbability felt by the court.

Condition 5. The chain runs: only key, entry, last heard alive, exit, injury, weapon, motive. Break any link and the chain fails. If the bar cannot be connected to the injury by medical evidence, the chain has a hole where its most important link should be.

The verdict as a logician would report it. A strong inductive argument, conditions one to three broadly satisfied, condition four unsatisfied on the present material because the third-person hypothesis has not been excluded, and condition five accordingly not met. That is the shape of the answer Sarda requires, and it is the shape of an assessment of an inductive argument generally.

Distinctions that carry marks

What this does not mean

Circumstantial evidence is not weaker evidence. Indian courts have said repeatedly that it may be stronger than direct testimony, since circumstances do not lie although witnesses may. What it requires is a stricter method, which is what the five conditions supply.

The five conditions are not a formula for acquittal. They are the conditions on which a conviction may safely rest, and convictions on circumstantial evidence are recorded constantly.

Distinguishing is not evasion. It is the standard test of an analogy, and a court that distinguishes has given a reason that can itself be examined.

Quick revision

Precedent is analogy: the earlier case resembles this one in the respects the earlier decision turned on. Distinguishing attacks the relevance of a resemblance or asserts the relevance of a difference. The ratio is the relevant resemblance.

Circumstantial evidence: evidence of facts from which the fact in issue is inferred. The reasoning is inference to the best explanation.

Sharad Birdhichand Sarda v. State of Maharashtra, AIR 1984 SC 1622, 17 July 1984: the five golden principles at paragraph 152, called the panchsheel at paragraph 153. Conviction set aside because suicide had not been excluded.

Hanumant v. State of Madhya Pradesh, AIR 1952 SC 343: the source of the rule, per Mahajan J.

The five conditions: fully established; consistent only with guilt; conclusive in nature and tendency; exclude every other hypothesis; a chain complete beyond any reasonable ground consistent with innocence.

"May be" against "must be": the mental distance between them divides vague conjectures from sure conclusions.

The limit: no rule can make a court think of an unsuggested hypothesis, which is why the defence has work to do under condition four.

Test yourself

1. Explain why following a precedent is an argument by analogy.

Because the earlier decision is not a rule that contains the present case but another dispute on other facts, and the reasoning runs from the resemblance between the two to the conclusion that they should be decided alike. The respects relied on must be the ones the earlier decision turned upon, which is the relevance test of any analogy. It follows that distinguishing a case is an attack on the analogy, and that a line of decisions across varied facts is stronger than a single decision.

2. State the five golden principles and name the case.

They are stated in Sharad Birdhichand Sarda v. State of Maharashtra, AIR 1984 SC 1622, at paragraph 152, drawn from Hanumant v. State of Madhya Pradesh, AIR 1952 SC 343. The circumstances from which guilt is drawn must be fully established; the facts so established must be consistent only with the hypothesis of guilt; the circumstances must be of a conclusive nature and tendency; they must exclude every possible hypothesis except the one to be proved; and there must be a chain of evidence so complete as to leave no reasonable ground for a conclusion consistent with innocence, showing that in all human probability the act was done by the accused.

3. Which of the five conditions is eliminative induction, and what does it require?

The fourth, that the circumstances exclude every possible hypothesis except the one to be proved. It requires the court not merely to find that guilt explains the facts but to satisfy itself that nothing else does, so that the conclusion is reached by eliminating rivals rather than by selecting the most striking candidate. In Sarda itself the condition failed, because the possibility that the deceased had taken the poison herself had not been excluded.

4. Distinguish direct from circumstantial evidence.

Direct evidence establishes the fact in issue itself, as where a witness says he saw the shot fired. Circumstantial evidence establishes other facts from which the fact in issue must be inferred, as where a witness says he saw a man running from the house with a pistol. In the first, the inference is made by the witness and asserted; in the second, the inference is the court's own work, which is why it is governed by the stricter method laid down in Sarda.

5. What was the significance of the distinction between "may be" and "must be" in Sarda?

The Court insisted that the circumstances must or should be established, not merely may be, and adopted the observation that the mental distance between "may be" and "must be" is long and divides vague conjectures from sure conclusions. In logical terms, a proposition that may be true is one whose falsity has not been excluded, and an argument built on such propositions cannot satisfy the fourth and fifth conditions, however plausible its conclusion appears.

6. What is the limit of the five conditions?

That no rule can require a court to consider a hypothesis nobody has raised. Condition four demands the exclusion of every other explanation, but the court can only exclude explanations it has thought of, so the risk remains that the true explanation was never before it. This is why the defence has real work under this head even though it bears no burden of proof: suggesting the alternative is what gives the fourth condition something to operate on.

The safe course in an examination is to say that a proposition is what is capable of truth or falsity, and to note that some writers call it a statement. That is accurate and costs one line.

A worked example

A written statement contains the following sentence: "It is denied that the defendant received any notice, and in any event, was the notice not sent to an address the plaintiff knew to be wrong?"

How many propositions? Two, and the second is disguised.

The first is asserted by the opening clause: that the defendant received no notice.

The second is put in interrogative form, but it is a rhetorical question and it asserts that the notice was sent to an address the plaintiff knew to be wrong. Reduced to a proposition, it is capable of truth or falsity, and it can be traversed, put to a witness and found upon.

Why the distinction matters here practically. A pleading is a set of propositions, and every one of them must be capable of being admitted or denied. Anything in a pleading that is not a proposition, an exclamation, a question, an expression of indignation, is unanswerable and therefore has no place there. Rewriting a rhetorical question as the proposition it asserts is not pedantry: it is what allows the other side to plead to it.

Distinctions that carry marks

What this does not mean

A proposition is not a sentence written carefully. It is not a form of words at all. Two entirely different forms of words express one proposition when they assert the same thing.

A proposition is not a fact. That distinction is the next chapter's, and it is the one most often confused.

Not every declarative sentence expresses a determinate proposition. "He came fairly soon after that" is declarative and, without more, asserts nothing definite enough to be true or false. Making such sentences definite is the work of sequence 300.

Quick revision

Proposition: what a declarative sentence asserts; the bearer of truth and falsity.

Sentence: a form of words in a particular language.

One proposition, many sentences: active and passive, different word orders, different languages.

One sentence, many propositions: "I was there yesterday", said by different people on different days.

Only declarative sentences express propositions. Interrogative, imperative and exclamatory sentences do not, though a rhetorical question asserts what it appears to ask.

"Statement" is used by some writers for the same thing; where distinguished, the statement is the act and the proposition is what is stated.

Test yourself

1. Distinguish a proposition from a sentence.

A sentence is a form of words belonging to a particular language; a proposition is what a declarative sentence asserts, and it is the thing that is true or false. The two are distinct because one proposition can be expressed by many sentences, in different voices, different word orders and different languages, while one sentence can express different propositions on different occasions of its use.

2. Give an example of one proposition expressed by three sentences.

"Ravi paid the money", "The money was paid by Ravi" and "It was Ravi who paid the money". The three differ in voice and emphasis and are identical in what they assert: each is true in exactly the circumstances in which the others are true. Since truth and falsity attach to the proposition and not to the wording, there is one proposition here and three sentences.

3. Give an example of one sentence expressing different propositions.

"I was there yesterday." Spoken by the plaintiff on 5 April it asserts that the plaintiff was at the place on 4 April; spoken by the defendant on 9 April it asserts something entirely different, about a different person and a different day. The words are the same throughout, so what varies must be the proposition and not the sentence.

4. Which kinds of sentence express propositions, and why does it matter in a pleading?

Only declarative sentences, since only they assert something capable of truth or falsity. It matters in a pleading because a pleading is a set of propositions each of which must be capable of being admitted or denied. A question, a command or an exclamation cannot be traversed, cannot be put to a witness and cannot be found upon, so anything in a pleading that is not a proposition has no work to do there.

5. What is a rhetorical question, and how is it treated?

A sentence in interrogative form that in fact asserts something, being asked for effect rather than for an answer. It is treated as the proposition it asserts, since what matters logically is what is being claimed and not the punctuation. In a pleading or an argument it should be rewritten as the assertion it is, so that it can be answered.

6. Why can Module II not classify sentences instead of propositions?

Because sentences differ in ways that make no logical difference, such as voice, word order, emphasis and language, so a classification of sentences would be a classification of grammar. The classifications in this module, into categorical and conditional, into A, E, I and O, and into simple and compound, all turn on what is asserted, which is the proposition. Two sentences asserting the same thing must fall in the same class, and they do only if the proposition is the unit.

Proposition against fact

This is the harder half, and the trap is that a true proposition and the fact that makes it true are described in the same words.

"The notice was served" is a proposition. The fact that the notice was served is a fact. The English words are almost identical and the things are not.

The proposition is what is asserted; the fact is what is the case. The proposition may be true or false. The fact cannot be either, because a fact is not a claim about anything. It makes no sense to say a fact is false; if it were false it would not be a fact.

Facts correspond to true propositions only. For a false proposition there is no corresponding fact, which is the difficulty the correspondence theory met at sequence 130 with negative propositions.

In law the word "fact" is defined, and not in the ordinary sense. Section 2(1)(f) of the Bharatiya Sakshya Adhiniyam 2023 provides that "fact" means and includes any thing, state of things, or relation of things, capable of being perceived by the senses, and any mental condition of which any person is conscious. The second half is what surprises a beginner: a person's intention, good faith or knowledge is a fact in law, provable by evidence, and not an opinion or a legal conclusion.

And "fact in issue" is defined separately, in section 2(1)(g): any fact from which, either by itself or in connection with other facts, the existence, non-existence, nature or extent of any right, liability or disability asserted or denied in any suit or proceeding, necessarily follows. This is a logician's definition in a statute: it defines the important facts by what follows from them.

A worked example

At a trial the plaintiff pleads that the defendant knew the goods were defective when he sold them. The defendant denies it. Evidence is led, and the court finds that he knew.

Identify the three.

The fact is the defendant's state of mind at the time of sale. Under section 2(1)(f) it is a fact, because it is a mental condition of which a person is conscious, and it existed whether or not anybody ever proved it.

The proposition is "the defendant knew the goods were defective when he sold them". It was true from the moment of the sale, if it was true at all, and it was true before the suit was filed and while the defendant was denying it.

The judgment is the court's act of finding, on a particular day, that the proposition is proved. It is dated, it is attributable to a named judge, and it may be wrong.

Now see what the distinctions buy. An appeal is against the judgment, not against the fact and not against the proposition. Fresh evidence, if allowed, does not alter the fact and does not alter the truth of the proposition; it alters what the tribunal ought to find. A decree obtained by fraud is set aside because the judgment was made on false material while the fact remained as it always was.

Collapse the three and none of that is expressible. If the fact simply were what the court found, a wrong finding would be a contradiction in terms, and there would be nothing for an appeal to correct. The whole architecture of review presupposes that the three are separate.

Distinctions that carry marks

What this does not mean

A fact is not a true proposition. They correspond; they are not identical. A true proposition is something asserted; a fact is something that is the case.

"Fact" in law is wider than a beginner expects. A state of mind is a fact under section 2(1)(f), so intention, good faith and knowledge are proved by evidence like any other fact.

A judgment in this chapter is not a judgment of a court. The logical sense is the mental act of asserting. A court's judgment is a document recording, among other things, many such acts. The words coincide and the meanings do not.

Quick revision

Three positions on one line: judgment at the mental end, proposition in the middle, fact at the world end.

A judgment asserts a proposition; a fact makes a proposition true. Neither relation runs backwards.

A proposition exists unjudged, which is why it cannot be a judgment.

A fact is neither true nor false, which is why it cannot be a proposition.

Section 2(1)(f) BSA 2023: "fact" includes a mental condition of which a person is conscious. A state of mind is a fact.

Section 2(1)(g) BSA 2023: a "fact in issue" is one from which the existence or extent of a right or liability necessarily follows.

Why it matters: appeal, review and setting aside a fraudulent decree all presuppose that a finding can be wrong, which requires the three to be distinct.

Test yourself

1. Distinguish a judgment from a proposition.

A judgment is a mental act performed by a particular person at a particular time, in which something is asserted or denied. A proposition is the content of that act, what is asserted, and it has no date and no owner. The distinction is proved by the existence of unjudged propositions: a proposition about the number of blades of grass in a field in 1800 is true or false although nobody has ever judged it, which would be impossible if propositions were judgments.

2. Distinguish a proposition from a fact.

A proposition is what is asserted and it is capable of being true or false. A fact is a state of affairs in the world and is neither true nor false, since it makes no claim about anything. A true proposition corresponds to a fact; a false proposition corresponds to none, which is why the correspondence theory of truth has difficulty with negative and false propositions.

3. How does the Bharatiya Sakshya Adhiniyam 2023 define "fact", and what is surprising about it?

Section 2(1)(f) provides that "fact" means and includes any thing, state of things or relation of things capable of being perceived by the senses, and any mental condition of which any person is conscious. What surprises a beginner is the second limb: a person's intention, knowledge or good faith is a fact in law, to be proved by evidence like any other, and not a matter of opinion or a conclusion of law.

4. What is a fact in issue, and why is its definition of interest to a logician?

Under section 2(1)(g) it is any fact from which, either by itself or with other facts, the existence, non-existence, nature or extent of a right, liability or disability asserted or denied in a proceeding necessarily follows. It is of interest because the statute picks out the important facts by what follows from them, that is, by an implication relation. The definition is logical in form, not evidentiary.

5. Why must the three be kept apart if the system of appeals is to make sense?

Because an appeal presupposes that a finding can be wrong, and a finding can be wrong only if what was found and what is the case are distinct. If a fact simply were whatever the court found, no finding could be mistaken and there would be nothing to correct. The same is true of review and of setting aside a decree obtained by fraud, both of which assume the fact remained unchanged while the judgment about it was defective.

6. Is a judgment in the logical sense the same as the judgment of a court?

No. In logic a judgment is the mental act of asserting or denying, and every such act has a proposition as its content. A court's judgment is a document, which records a great many such acts together with the reasons for them and the order made. The word is the same and the things are different, and an answer that runs them together will misdescribe both.

Why the distinction matters

It is what "simple" and "compound" mean. A simple proposition has constituents only. A compound proposition has at least one component. That is the definition modern logic uses, and it is why MU sets this distinction at 2.1 and the kinds of simple and compound propositions at 2.6: the second cannot be stated without the first.

It tells you what can be symbolised as a unit. In the notation of sequence 350, a whole simple proposition is written with a single letter, p, because it has no propositional parts to expose. A compound proposition must be broken into its components before it can be symbolised at all.

It tells you what can be separately proved or disproved. A component can be admitted or denied on its own. A constituent cannot: denying "the notice" is not a possible move in a pleading, whereas denying "the notice was served" is.

A worked example

A section provides: "Where a tenant has been in arrears for three months and has failed to comply with a notice of demand, the landlord may apply for possession, unless the tenant deposits the arrears within thirty days."

Break it into components. Four parts can be true or false on their own:

p: the tenant has been in arrears for three months

q: the tenant has failed to comply with a notice of demand

r: the landlord may apply for possession

s: the tenant deposits the arrears within thirty days

Break one of them into constituents. In p, the parts are "the tenant", "in arrears" and "for three months". None is a proposition. They are constituents.

Why the exercise is worth doing. The section's structure is now visible: if p and q, then r, unless s. Four components, three connectives, and the whole is a compound proposition. Every dispute under the section will be about one of the four components, and the drafting question of what "unless" does to r is a question about a connective and not about any component.

And the trap. "A notice of demand" contains "demand", and "of demand" is not a proposition; it qualifies the notice. A reader who treated it as a component would produce a fifth item to prove that the section never required.

Distinctions that carry marks

What this does not mean

A component need not be asserted. In a conditional, both components are present and neither is asserted. Presence and assertion are different questions.

A long constituent is still a constituent. "The notice served on 4 April at the registered office by a person authorised in that behalf" is one term however long, because it asserts nothing standing alone.

Component is not the same as clause. Grammar counts clauses; logic counts parts capable of truth and falsity, and the two lists differ, as the qualifying phrase in the worked example shows.

Quick revision

Component: a part of a proposition which is itself a proposition, and so can be true or false alone.

Constituent: a part which is not a proposition, such as a term.

The test: can this part be true or false by itself?

Simple proposition: constituents only. Compound proposition: at least one component. That is the whole basis of topic 2.6.

A component need not be asserted, as the parts of a conditional show.

Propositions about belief embed a component and are not truth-functional, which is why truth tables do not reach them.

Test yourself

1. Distinguish a constituent from a component and state the test.

A component of a proposition is a part which is itself a proposition, and which is therefore capable of being true or false on its own. A constituent is a part which is not a proposition, such as a term or a qualifying phrase, and is neither true nor false standing alone. The test is to take the part and ask whether it can be true or false by itself: if it can, it is a component, and if it cannot, it is a constituent.

2. Identify the components and constituents in "The notice was served and the rent was unpaid".

The components are "the notice was served" and "the rent was unpaid", each being a proposition capable of truth or falsity on its own, and either may be true while the other is false. The constituents are the parts of those components: "the notice", "was served", "the rent" and "was unpaid", none of which asserts anything standing alone.

3. How does the distinction define simple and compound propositions?

A simple proposition has constituents only and no components, so it cannot be broken into smaller propositions. A compound proposition has at least one component, and it is built out of propositions joined by connectives. The definition matters for symbolisation, since a simple proposition is written with a single letter while a compound one must be broken into its components before it can be written at all.

4. Is "served on 4 April" in "the notice served on 4 April was defective" a component?

No, it is a constituent. As it stands it qualifies "the notice" and forms part of the subject term; it asserts nothing separately and is neither true nor false standing alone. The whole proposition asserts one thing, that a particular notice was defective, and the reference to 4 April merely identifies which notice is meant.

5. Must a component be asserted?

No. In the conditional "if the notice was served, the rent was payable", both parts are components because each can be true or false alone, and neither is asserted: the proposition claims only that the second would follow from the first. Being a component is a question about whether the part is a proposition, not about whether the speaker has committed to it.

6. Why do truth tables not reach "Ravi believes the notice was served"?

Because the truth of the whole is not fixed by the truth of the embedded component. Whether the notice was served may be true or false without changing whether Ravi believes it, so the whole is not a truth function of its parts. Truth tables work only for compounds whose truth value is completely determined by the truth values of their components, which is the truth-functionality feature described at sequence 20.

By modality: how the predicate is said to belong to the subject. This is the basis the other three chapters do not touch, and it has three members.

Apodictic, or necessary: "every contract must have consideration". The predicate is asserted to belong necessarily.

Assertoric, or actual: "this contract has consideration". The predicate is asserted simply to belong, as a matter of fact.

Problematic, or possible: "this agreement may be a contract". The predicate is asserted only to be possible.

Why modality was dropped by modern logic, and where it went. The three modal forms are not truth-functional, since knowing whether p is true does not settle whether p is necessary, so no truth table can reach them. Modern logic accordingly left them out of the propositional calculus and they became the subject of a separate modal logic, developed much later. That is the same fate as the tenses at sequence 390, and for the same reason.

Why it matters to a lawyer. Because legal language is saturated with modality and the three forms have entirely different effects: "shall" is apodictic, "is" is assertoric, and "may" is problematic. A provision saying the Court shall admit the document, one saying it admits the document and one saying it may admit it are three different rules, and the difference is a difference of modality and of nothing else.

Analytic and synthetic propositions

Added after the past-paper check, since MU asks for these by name with an example.

An analytic proposition is one whose predicate is already contained in the connotation of its subject, so that denying it produces a contradiction and no observation is needed to establish it. "A contract is an agreement" is analytic, because being an agreement is part of what "contract" means. So is "a bachelor is unmarried".

A synthetic proposition is one whose predicate adds something the connotation of the subject does not contain, so that denying it produces no contradiction and observation or evidence is needed. "This contract is in writing" is synthetic, and so is "most contracts are performed".

The test. Ask whether the proposition can be shown true by examining the meaning of the subject term alone. If it can, it is analytic; if the world has to be consulted, it is synthetic. The distinction is Kant's and it is the ancestor of the modern distinction between what is true by definition and what is true as a matter of fact.

Where a lawyer meets it, and this is the useful part. A definition clause makes propositions analytic within its own Act. Once an Act provides that "vehicle" means a conveyance used for the carriage of persons or goods, the proposition "every vehicle under this Act is used for the carriage of persons or goods" is analytic: it cannot be disproved by evidence, because it follows from the definition. Whether a particular object is a vehicle remains synthetic and is proved by evidence.

The practical rule that comes out of it. Evidence is relevant to synthetic propositions and irrelevant to analytic ones. A party who leads evidence to show that some vehicles are not used for carriage has misunderstood what a definition clause does, and this is a real and frequent error.

Categorical propositions

Every categorical proposition has four parts, and naming them is examinable in its own right.

The quantity word, or quantifier: "all", "no", "some". It says how much of the subject is being spoken about.

The subject term: what is being spoken about.

The copula: "is", "are", "is not", "are not". It joins subject to predicate and carries the quality, affirmative or negative. MU sets it as a topic of its own at 2.10.

The predicate term: what is being said about the subject.

Every categorical proposition can be forced into this shape, and forcing it there is the exercise called reduction to logical form, at sequence 300.

Conditional propositions, named here and taken at sequence 290

Traditional logic divides conditionals into two.

Hypothetical, of the form "if p then q". The first part is the antecedent and the second the consequent.

Disjunctive, of the form "either p or q". Each part is called an alternative.

Some books add a third, the conjunctive proposition of the form "not both p and q", though most treat it as a disjunctive in disguise. MU's syllabus does not name it separately and neither does this book, beyond this line.

Where the law puts each

This is worth a paragraph, because it explains the shape of every statute a student will read.

Definition clauses are categorical. "'Immoveable property' includes land, benefits to arise out of land, and things attached to the earth." The assertion is flat.

Operative provisions are conditional, almost without exception. "Where a person, being under no obligation to do so, lawfully does anything for another person, and such other person enjoys the benefit thereof, the latter is bound to make compensation." That is "if p and q then r", and the section asserts nothing about whether anybody has done anything.

A statute is therefore mostly a set of conditionals whose antecedents are questions of fact. The trial establishes the antecedents; the section supplies the consequents. That is the judicial syllogism of sequence 100 seen from the drafting side, and it is why the conditional is the characteristic form of legal language.

A worked example

Classify each of the following.

"Every agreement enforceable by law is a contract." Categorical. Quantity "every", subject "agreement enforceable by law", copula "is", predicate "a contract". The qualification "enforceable by law" is a constituent of the subject term and does not make the proposition conditional, which is the trap in this example.

"If an agreement is enforceable by law, it is a contract." Conditional, hypothetical. Antecedent "an agreement is enforceable by law", consequent "it is a contract". Nothing is asserted about any agreement.

"No agreement made by a minor is enforceable." Categorical and negative. Quantity "no", and the copula carries the negative quality.

"Either the notice was served or the suit is premature." Conditional, disjunctive. Two alternatives, neither asserted by itself.

Compare the first two. They look almost identical and behave differently. The categorical version, combined with "this agreement is enforceable by law", yields "this agreement is a contract" immediately. The conditional version yields the same conclusion, but only after the antecedent has been affirmed, and the affirming is a separate step that somebody has to take and prove. In an examination the difference between a proposition whose qualification sits inside the subject term and one whose qualification is an antecedent is exactly what is being tested.

Distinctions that carry marks

What this does not mean

A qualified subject does not make a proposition conditional. "All agreements enforceable by law are contracts" is categorical. The qualification is inside the subject term.

"Categorical" does not mean emphatic. In ordinary speech a categorical denial is a firm one. In logic it means unconditional, and a hesitant unconditional assertion is still categorical.

A conditional is not a weaker categorical. It is a different kind of proposition, with two components rather than two terms.

Quick revision

Basis of the division: the manner of assertion. MU also calls the traditional scheme Aristotle's classification.

Categorical: asserts absolutely. Four parts: quantifier, subject, copula, predicate.

Conditional: asserts a relation between propositions and asserts neither of them. Two kinds: hypothetical, if p then q, with antecedent and consequent; disjunctive, either p or q, with alternatives.

A conditional cannot support an inference by itself until the condition is settled, which is why it cannot be an argument on its own.

In law: definitions are categorical, operative provisions are conditional. A statute is largely a set of conditionals whose antecedents are questions of fact.

Four bases of classification: quantity, quality, relation and modality. The categorical and conditional division is the division by RELATION.

Modality: apodictic or necessary, assertoric or actual, problematic or possible. Not truth-functional, so modern logic left it to a separate modal logic. In law, "shall", "is" and "may" are the three modalities.

Analytic: the predicate is contained in the connotation of the subject, so denying it contradicts. Synthetic: the predicate adds something, so evidence is needed. A definition clause makes propositions analytic within its own Act, and evidence is irrelevant to them.

Trap: a qualification inside the subject term does not make a categorical proposition conditional.

Test yourself

1. On what basis does traditional logic divide propositions into categorical and conditional, and what does each mean?

The basis is the manner of assertion. A categorical proposition asserts its predicate of its subject absolutely, without attaching any condition, as in "all contracts are agreements". A conditional proposition asserts only a relation between two propositions and asserts neither of them, as in "if an agreement is enforceable by law it is a contract", which says nothing about whether any agreement is enforceable.

2. Name the four parts of a categorical proposition and identify them in an example.

The quantifier, the subject term, the copula and the predicate term. In "All contracts made by minors are void", "all" is the quantifier, "contracts made by minors" is the subject term, "are" is the copula, carrying affirmative quality, and "void" is the predicate term. Every categorical proposition can be put into this shape, and doing so is what reduction to logical form means.

3. Name the kinds of conditional proposition and their parts.

Hypothetical propositions, of the form "if p then q", whose parts are the antecedent and the consequent; and disjunctive propositions, of the form "either p or q", whose parts are called alternatives. Some writers add the conjunctive proposition, "not both p and q", though it is usually treated as a disjunctive in another form and MU's syllabus does not name it separately.

4. Why is the conditional the characteristic form of legal language?

Because an operative provision has to state what shall follow if certain facts exist, without asserting that they do. A section is therefore a conditional whose antecedent is a question of fact and whose consequent is a legal result, and the trial exists to establish the antecedent. This is the drafting side of the judicial syllogism: the statute supplies the major premise and the evidence supplies the minor.

5. Classify: "All agreements made without consideration are void" and "If an agreement is made without consideration, it is void". What difference does the classification make?

The first is categorical, the qualification "made without consideration" being part of the subject term. The second is hypothetical, with that qualification as the antecedent. The difference is what is needed to draw a conclusion about a particular agreement: from the categorical one, the additional premise that this agreement is of the described kind; from the conditional one, the affirmation of the antecedent as a separate step. In practice the two come to the same result, and only the categorical form asserts anything about a class of agreements outright.

7. On what four bases does traditional logic classify a proposition?

By quantity, universal or particular; by quality, affirmative or negative; by relation, categorical or conditional; and by modality, apodictic or necessary, assertoric or actual, and problematic or possible. The four are independent, so every proposition has an answer under each of them, and the division into categorical and conditional made in this chapter is the division by relation.

8. What is modality, and why did modern logic drop it?

Modality is the manner in which the predicate is said to belong to the subject: necessarily in an apodictic proposition, actually in an assertoric one, and possibly in a problematic one. Modern logic dropped it from the propositional calculus because modal propositions are not truth-functional: knowing whether a proposition is true does not settle whether it is necessary, so no truth table can reach it. Modality became the subject of a separate modal logic developed much later.

9. Distinguish analytic from synthetic propositions, and give the legal application.

An analytic proposition has a predicate already contained in the connotation of its subject, so that denying it is a contradiction and no evidence is needed: "a contract is an agreement". A synthetic proposition has a predicate that adds something the subject does not contain, so evidence is needed: "this contract is in writing". The legal application is that a definition clause makes propositions analytic within its own Act, so evidence directed against them is irrelevant, while whether a particular thing falls within the definition remains synthetic and is proved by evidence.

6. Does a qualification in the subject make a proposition conditional?

No. "All agreements enforceable by law are contracts" is categorical, because the qualification is a constituent of the subject term and nothing is held back. A proposition is conditional only when what it asserts is a relation between two propositions, neither of which is asserted on its own. Confusing a qualified subject with an antecedent is the commonest error in classifying propositions.

Singular propositions count as universal. "Ravi is a minor" speaks of a single individual, and the whole of the subject class, which has one member, is covered. Traditional logic therefore treats singular propositions as A, and singular negatives as E. This matters at sequence 430, where MU sets the opposition of singular propositions as a topic of its own.

The four forms are about categoricals only. A conditional proposition is not an A, E, I or O proposition at all, and forcing it into one is an error.

Where the four forms appear in statutes

Reading the four forms off legislative language is the practical skill, and it is worth doing once here.

A form. "Every person is competent to contract who is of the age of majority." All S is P.

E form. "No suit shall lie against the Government except after notice." No S is P.

I form. "Some of the provisions of this Act shall come into force at once." Some S is P.

O form. The rarest in drafting, because a legislature that means "some S is not P" usually says it as an exception to a universal. "Nothing in this section shall apply to a transfer by operation of law" creates an O proposition indirectly, by excepting part of the class from a rule stated universally.

That last observation is the useful one. Legislative drafting overwhelmingly uses A and E forms with provisos and exceptions, rather than I and O forms directly. A proviso carves a part out of a universal, and the result is that the universal now holds of the remainder. Recognising this shape saves a great deal of confusion when reading a section whose main limb and proviso appear to contradict each other.

A worked example

Classify each and give its letter.

"Every agreement made without consideration is void." Quantity universal, quality affirmative. A.

"No agreement made by a minor is enforceable." Quantity universal, quality negative. E.

"Some agreements without consideration are valid." Quantity particular, quality affirmative. I. This is the proposition section 25 of the Indian Contract Act 1872 makes true, since it saves natural-love-and-affection promises in writing and registered, promises to compensate for past voluntary services, and promises to pay a time-barred debt.

"Some registered documents are not compulsorily registrable." Quantity particular, quality negative. O.

Now notice a relation. The A proposition and the I proposition just given appear to conflict, and they do: if every agreement without consideration is void, no such agreement can be valid. What resolves it is that the statute states the universal in section 25 and then carves out the exceptions in the same section, so the true A proposition is about agreements without consideration other than the three excepted classes. The apparent contradiction was produced by taking the main limb without the exception, which is exactly the drafting shape described above.

The logical lesson. An A proposition and an I proposition with the same subject and predicate can both be true, and an A proposition and an O proposition cannot. Working out which pairs can hold together is the square of opposition, at sequence 410, and this example is a preview of it.

Distinctions that carry marks

What this does not mean

"Some" does not mean "not all". It means at least one, and it leaves open whether all are.

"No S is P" is not a particular proposition. It is universal in quantity and negative in quality.

A singular proposition is not a particular one. "Ravi is a minor" is treated as universal, because it covers the whole of a one-membered subject class. Students confuse "singular" with "particular" every year, and the words are not synonyms in this subject.

The classification is not about sentences. "Not a single agreement made by a minor is enforceable" is an E proposition however it is worded.

Quick revision

Quantity: universal or particular, settled by the quantifier. Quality: affirmative or negative, settled by the copula.

Four forms: A, all S is P; E, no S is P; I, some S is P; O, some S is not P.

Letters from AffIrmo and nEgO, first vowel universal, second particular.

"Some" means at least one, and does not exclude all.

Singular propositions are treated as universal, A if affirmative and E if negative.

In statutes: A and E forms with provisos and exceptions are the normal drafting shape; I and O forms appear indirectly, by exception.

Test yourself

1. On what two grounds are categorical propositions classified, and what are the four resulting forms?

On quantity, which is settled by the quantifier and is either universal or particular, and on quality, which is settled by the copula and is either affirmative or negative. Crossing the two gives four forms: the universal affirmative or A, "all S is P"; the universal negative or E, "no S is P"; the particular affirmative or I, "some S is P"; and the particular negative or O, "some S is not P".

2. Where do the letters A, E, I and O come from?

From two Latin words. AffIrmo, meaning "I affirm", gives A and I to the two affirmative forms, the first vowel to the universal and the second to the particular. NEgO, meaning "I deny", gives E and O to the two negative forms in the same order. The mnemonic is traditional and is worth stating in an answer, since it also fixes which letter goes with which quantity.

3. What does "some" mean in logic, and how does that differ from ordinary speech?

It means "at least one", and it leaves entirely open whether all are. In ordinary speech saying "some" carries a suggestion that not all are, so that "some agreements are void" would be heard as denying that all are. In logic that suggestion forms no part of what is asserted, so an I proposition is true even where the corresponding A proposition is also true.

4. Is a singular proposition particular?

No. A singular proposition such as "Ravi is a minor" speaks of a subject class with one member and speaks about the whole of it, so traditional logic treats it as universal: an A proposition if affirmative and an E proposition if negative. The word "singular" describes the subject, while "particular" describes the quantity, and confusing the two produces errors in the square of opposition.

5. Classify "No suit shall lie against the Government except after notice" and explain the exception.

The main limb is an E proposition, universal in quantity and negative in quality: no S is P. The exception carves a part out of that universal, so the proposition actually enacted is that no suit for which notice has not been given shall lie. This is the standard drafting shape: a universal stated in the main limb with a proviso or exception restricting the subject class, and the universal then holds of the remainder.

6. Can an A proposition and an I proposition with the same subject and predicate both be true?

Yes. "All contracts are agreements" and "some contracts are agreements" are both true, since "some" means at least one and does not exclude all. What cannot both be true is an A proposition and the corresponding O proposition, since one says the predicate belongs to every member of the subject class and the other says it fails of at least one. The full set of such relations is the square of opposition.

When would the two fallacies become valid? Only if the conditional were replaced by a biconditional, "p if and only if q", which asserts that each is necessary for the other. English "if" is very often used where "if and only if" is meant, which is why the fallacies feel right.

Disjunctive propositions

"Either the notice was defective or the tenant waived the defect."

Each part is an alternative. The proposition asserts that at least one of them is true, and it does not say which.

The inclusive and exclusive senses. In its inclusive sense, "or" allows both alternatives to be true; in its exclusive sense, it means one but not both. Logic takes "or" as inclusive unless something signals otherwise, and modern notation writes the inclusive sense as p ∨ q.

The same pair under MU's own names: weak and strong disjuncts. A weak disjunction is the inclusive one, "p or q or both", and it is weak because it claims less: it is satisfied by any of three situations. A strong disjunction is the exclusive one, "p or q but not both", and it is strong because it claims more, excluding the case where both hold. "The applicant shall be a citizen of India or a person of Indian origin" is strong, the two being mutually exclusive; "a person who obstructs or resists" is weak, since doing both is certainly not outside the section. Weak is the default in logic and in the reading of a penal provision, because it is the wider.

The two senses give different inferences and the difference is practical.

From an inclusive disjunction, denying one alternative establishes the other. Either p or q; not p; therefore q. This is valid on both readings.

From an inclusive disjunction, affirming one alternative establishes nothing about the other. Either p or q; p; therefore not q, is invalid, because both may be true.

On the exclusive reading that last inference is valid, which is why the reading matters and why careful drafting says "or both" or "but not both" instead of leaving it to be guessed.

In statutes. "A person who obstructs or resists" is inclusive: doing both is worse and certainly not outside the section. "The applicant shall be either a citizen of India or a person of Indian origin" is exclusive in effect, since the categories do not overlap. The safest reading of a penal provision is the inclusive one, because it is the wider.

A worked example

Section 91 of the Code of Civil Procedure 1908 provides that in the case of a public nuisance or other wrongful act affecting, or likely to affect, the public, a suit for a declaration and injunction or for such other relief as may be appropriate may be instituted by the Advocate-General, or with the leave of the Court by two or more persons even though no special damage has been caused to them.

Find the conditional. The provision is a hypothetical: if there is a public nuisance or other wrongful act affecting or likely to affect the public, then such a suit may be instituted in the stated ways. The antecedent is the state of affairs and the consequent is the permission.

Find the disjunctions, and there are three. "A public nuisance or other wrongful act": inclusive, since an act may be both. "Affecting or likely to affect": inclusive, since an act already affecting the public is certainly likely to. "By the Advocate-General, or with the leave of the Court by two or more persons": inclusive, since nothing prevents both routes being taken in different proceedings.

Now test an inference. "The Advocate-General has not sued. Therefore two or more persons may sue with leave." Is this valid? It is not, and the reason is instructive: the disjunction is about who may sue, not about who has sued. Denying that one permitted person has acted says nothing at all, because the proposition never asserted that anybody had. Confusing a permissive disjunction with an assertive one is a real error in reading statutes.

And test a fallacy. "Two or more persons have obtained leave to sue. Therefore there is a public nuisance." That is affirming the consequent: leave may have been granted wrongly, or for an "other wrongful act" which is not a nuisance. The section supplies no such inference.

Distinctions that carry marks

What this does not mean

A conditional does not assert its antecedent. A section beginning "where a person fails to appear" does not say that anybody has failed to appear.

"If" in English often means "if and only if". Logic does not read it that way, and the difference is what makes denying the antecedent feel valid when it is not.

A disjunction is not a choice offered. "Either the notice was defective or the tenant waived the defect" does not invite anyone to pick; it asserts that at least one is so.

Quick revision

Hypothetical: if p then q. Antecedent, consequent. Neither asserted.

Valid: affirming the antecedent, and denying the consequent.

Fallacies: denying the antecedent, and affirming the consequent. Both would be valid only for a biconditional.

Affirming the consequent is the standard error in circumstantial reasoning, and Sarda's fourth condition exists to prevent it.

Disjunctive: either p or q. Alternatives. Asserts at least one.

Inclusive by default, written p ∨ q. Denying one alternative establishes the other; affirming one establishes nothing. MU calls the two WEAK and STRONG disjuncts: weak is the inclusive one, claiming less, and strong is the exclusive one, claiming more.

In statutes: "where" and "provided that" introduce antecedents; "shall" often marks a consequent; "or" is normally read inclusively.

Test yourself

1. Define a hypothetical proposition and name its parts.

A hypothetical proposition asserts that one proposition follows from another, taking the form "if p then q". The part introduced by "if" is the antecedent and the part that follows from it is the consequent. It asserts neither part on its own: it does not say that p is true and does not say that q is true, but only that q would hold if p did.

2. State the two valid inferences from a hypothetical proposition and the two fallacies.

Valid: affirming the antecedent, from "if p then q" and p to q; and denying the consequent, from "if p then q" and not q to not p. Fallacious: denying the antecedent, from "if p then q" and not p to not q; and affirming the consequent, from "if p then q" and q to p. Both fallacies assume that p is necessary for q when the conditional says only that it is sufficient.

3. Demonstrate the fallacy of affirming the consequent with a legal example.

"If the accused was at the scene, his fingerprints would be on the door. His fingerprints are on the door. Therefore he was at the scene." The conclusion does not follow, because the prints may have been left at another time or in another way. This is the characteristic error in reasoning from circumstantial evidence, and it is what the fourth of the five golden principles guards against by requiring every other hypothesis to be excluded.

4. Distinguish the inclusive from the exclusive sense of "or", and give the inference each supports.

The inclusive sense allows both alternatives to be true and is written p ∨ q; the exclusive sense means one but not the other. On both readings, denying one alternative establishes the other. On the inclusive reading, affirming one alternative establishes nothing about the other, since both may hold; on the exclusive reading, affirming one establishes that the other is false. Logic takes "or" as inclusive unless something signals otherwise.

5. What words introduce an antecedent in legislative drafting?

"Where" and "provided that" far more often than "if", together with "in case", "on condition that" and "in the event that". The consequent is frequently marked by "shall". Recognising these is necessary because a section written as "where a person does X, he shall be liable to Y" is a hypothetical proposition, and reading it as an assertion that somebody has done X is a plain misreading of the statute.

7. What is meant by weak and strong disjuncts?

They are the two senses of "or". A weak disjunct is the inclusive one, asserting p or q or both, and it is called weak because it claims less, being satisfied in three of the four possible situations. A strong disjunct is the exclusive one, asserting p or q but not both, and it is called strong because it excludes the case in which both hold. Logic takes a disjunction as weak unless something signals otherwise, and a penal provision is read the same way because the weak sense is the wider.

6. Why can a hypothetical proposition not by itself be an argument?

Because an argument asserts its premises and asserts its conclusion, while a hypothetical asserts neither of its parts. It claims only a connection, so nobody has committed to anything being true and nothing has been established. It can be a premise in an argument, and in legal reasoning it usually is: the section supplies the conditional and the evidence supplies the affirmation of the antecedent.

The rule in one line: "only S is P" becomes "all P is S". The same holds for "none but", "no one except" and "alone".

"All except" and "all but". "All the defendants except the third were served." Two propositions are being asserted at once, and both must be written: All the defendants other than the third are persons who were served, and The third defendant is not a person who was served. An answer that gives only the first has dropped half of what the sentence says.

Exclusive propositions using "the only". "The only remedy is an injunction." Reduce: All remedies available here are injunctions.

Numerical and quasi-numerical quantity. "Most of the witnesses were unreliable", "a few sections apply", "several documents were produced". All of these are particular in logic: they say something about part of the class and not the whole. Reduce with "some". "Most witnesses were unreliable" becomes Some witnesses are persons who were unreliable. Note what is lost: logic's "some" cannot express "most", and the traditional scheme has no way to record it. That loss is one of the failures of the traditional classification, at sequence 320.

Singular propositions. "Ravi is a minor." The subject class has one member and the whole of it is spoken about, so the proposition is universal. Reduce: All persons identical with Ravi are minors, in strict form, or simply treat it as an A proposition and say so. The strict form looks laboured and is needed only when an operation such as conversion has to be carried out on it.

Negative words other than in the copula. "No document was produced" is already E in shape. But "Documents were not produced" may be E or O depending on the sense, and "Not all documents were produced" is O and not E: it denies the universal and asserts only that at least one was not. "Not all S is P" reduces to "Some S is not P". This is the second most examined trap in the topic.

Sentences whose grammatical subject is not the logical subject. "There are agreements which are void." Reduce: Some agreements are void. I form. The word "there" is doing grammatical work only.

Interrogative and exclamatory forms. A rhetorical question is reduced to the proposition it asserts, as at sequence 240. A genuine question is not a proposition and cannot be reduced at all, and saying so is the correct answer.

Sentences containing "few" and "a few". "Few sections apply" suggests a negative: it means that not many do, and it is usually reduced as Some sections are not sections which apply. "A few sections apply" is affirmative: Some sections are sections which apply. The article changes the quality, which is a real feature of English and a favourite of examiners.

Adverbs of time and place. "He always pays late" reduces to All occasions of his paying are occasions of late payment. "He never appeared" reduces to No occasion of the hearing is an occasion on which he appeared. The trick is to make the class a class of occasions, times or places, and it is the standard method for any sentence about frequency.

A worked example: a statutory sentence

Section 91(1)(b) of the Code of Civil Procedure 1908 permits a suit in the case of a public nuisance to be instituted, with the leave of the Court, by two or more persons, even though no special damage has been caused to such persons by reason of such public nuisance.

Find the propositions. There are two, and the second is the one carried by "even though".

First: with leave, two or more persons may institute such a suit. Reduce: All cases of a suit under this clause by two or more persons with leave are cases of a permitted suit. A form.

Second: the absence of special damage is no bar. This is a negative and it must be got into the copula. Reduce: No case of such a suit is a case barred by the absence of special damage to the plaintiffs. E form.

What reduction reveals. The words "even though" carry a whole proposition, and it is a negative universal excluding a defence. A reader who did not reduce would very likely read "even though" as an aside rather than as an enactment, and would miss that the clause abolishes a requirement which the general law of tort otherwise imposes.

That is the practical value of the topic. Reduction is not an exercise in tidiness. It makes a reader count the propositions in a sentence, and legislative sentences routinely carry more than one.

Distinctions that carry marks

What this does not mean

Reduction does not improve the sentence. If the original asserts something doubtful, the reduced form asserts the same doubtful thing.

"Only" does not simply mean "all". It reverses the terms, and reading it as an ordinary universal is the single commonest error in this topic.

"Not all" is not "none". It is an O proposition, and treating it as E turns a modest denial into a sweeping one.

Reduction is not always possible. A genuine question, a command or an exclamation expresses no proposition and cannot be reduced. Saying so is the right answer, not a failure.

Quick revision

Target form: quantifier, subject, copula, predicate. Quality lives in the copula.

Procedure: predicate, subject, quantity, quality, write out.

"Only S is P" becomes "all P is S". Same for "none but", "no one except", "alone".

"All except" carries two propositions and both must be written.

"Not all S is P" is O, not E.

"Few" is negative, "a few" is affirmative.

Most, many, several, a majority: all reduce to "some", and the extra information is simply lost.

Singular propositions are universal, A or E.

Frequency and place are handled by making the class a class of occasions.

Test yourself

1. What is reduction to logical form, and what are its three governing rules?

It is the rewriting of an ordinary sentence as a standard categorical proposition in the order quantifier, subject, copula, predicate. The rules are: change nothing that is asserted, since reduction is translation and not improvement; make the quantifier explicit, since ordinary sentences often omit it; and put the quality into the copula rather than leaving a negative floating in the predicate.

2. Reduce "Only registered documents are admissible" and explain the trap.

It reduces to "All admissible documents are registered documents", an A proposition with the terms reversed from where the sentence places them. The trap is to read it as "all registered documents are admissible", which asserts something quite different and much stronger. The rule is that "only S is P" becomes "all P is S", and the same applies to "none but", "no one except" and "alone".

3. Reduce "All the defendants except the second were served."

Two propositions are asserted and both must be written. First, an A proposition: "All defendants other than the second are persons who were served." Second, an E proposition about the exception: "The second defendant is not a person who was served." An answer that gives only the first has dropped half of what the sentence says, and in a pleading that half is usually the important one.

4. Distinguish the reduction of "Few witnesses were reliable" from "A few witnesses were reliable".

"Few" carries a negative sense, meaning that not many were, and it reduces to the O proposition "Some witnesses are not reliable witnesses". "A few" is affirmative, meaning that some were, and it reduces to the I proposition "Some witnesses are reliable witnesses". The article changes the quality of the proposition, which is a genuine feature of English usage and a favourite examination point.

5. How are propositions about frequency reduced?

By making the subject a class of occasions, times or places. "He always pays late" becomes "All occasions of his paying are occasions of late payment", an A proposition. "He never appeared" becomes "No occasion of the hearing is an occasion on which he appeared", an E proposition. Without this device such sentences have no obvious subject term and cannot be fitted into the standard forms at all.

6. Why does reduction matter when reading a statute?

Because it forces a reader to count the propositions in a sentence, and legislative sentences routinely carry more than one. Words such as "even though", "except" and "provided that" each introduce a further proposition, often a negative universal excluding a defence or a requirement. A reader who does not reduce is likely to treat such words as asides, and so to miss part of what the section enacts.

Do not memorise the table. Memorise two rules and derive it.

Rule of quantity: universal propositions distribute their subject. A and E are universal, so their subjects are distributed. I and O are particular, so theirs are not.

Rule of quality: negative propositions distribute their predicate. E and O are negative, so their predicates are distributed. A and I are affirmative, so theirs are not.

Two rules, four propositions, table complete. It is also worth noting what comes out of it: A distributes one term, E distributes two, I distributes none, O distributes one.

Why this matters

Because of one rule that governs every inference in Module III and every syllogism in Logic II.

No term may be distributed in a conclusion unless it was distributed in a premise.

The reason is not a convention. A term distributed in the conclusion is a term about the whole of which the conclusion makes a claim. If the premises spoke only of part of that class, the conclusion has asserted something the premises never gave it, which is precisely what an invalid inference is. Distribution is the bookkeeping that catches it.

Two immediate consequences, both of which are proved in their own chapters and are stated here so the reader knows what the chapter is for.

A proposition cannot be converted if conversion would distribute a term that was undistributed. "All contracts are agreements" cannot become "all agreements are contracts", because "agreements" is undistributed in the first and distributed in the second. This is the whole of the rule of conversion at sequence 450.

O propositions cannot be converted at all. In "Some agreements are not contracts", the subject is undistributed and the predicate distributed. Converting would put the distributed term in the subject place, where it would have to be undistributed, and the undistributed term in the predicate place of a negative, where it would have to be distributed. Both fail at once.

A worked example

A section provides: "No document required by law to be registered shall be received in evidence unless it has been registered."

Reduce it. The main proposition is E: No document required by law to be registered and not registered is a document receivable in evidence.

Distribution. Subject distributed, predicate distributed, because E distributes both.

Now test two proposed inferences.

"Therefore no document receivable in evidence is a document required to be registered and unregistered." This simply converts the E proposition. Both terms were distributed in the original and both are distributed in the conclusion, so nothing has been claimed that was not given. The inference is valid, and it is the conversion of E at sequence 450.

"Therefore all registered documents are receivable in evidence." Now look at the terms. "Registered documents" appears in the conclusion as a distributed subject. Was it distributed in the premise? The premise never spoke about registered documents at all; it spoke about unregistered ones. The conclusion has claimed something about a class the premise never addressed, and it is invalid.

And it is invalid for a reason that matters practically. Registration removes one bar to admissibility. It does not confer admissibility, since a registered document may still be inadmissible for want of stamp, for irrelevance, or for want of proof of execution. The distribution rule caught a real legal error, and it caught it mechanically, without knowing anything about the law of evidence.

Distinctions that carry marks

What this does not mean

Distribution is not about the number of things. It is about whether the proposition speaks of the whole class or part of it. A class with one member is spoken of in its entirety when that member is spoken of.

Distribution is not a property of a term. It is a property of a term as it occurs in a particular proposition. "Agreements" is undistributed in "all contracts are agreements" and distributed in "no agreements are void promises". The same word, two occurrences, two answers.

An affirmative proposition never distributes its predicate, however emphatic. "All contracts, without exception, are agreements" still says nothing about agreements generally.

Quick revision

Distributed: the proposition refers to the whole of the class the term names. Undistributed: to part of it only.

Two rules: universals distribute their subject; negatives distribute their predicate.

Table: A distributes S only; E distributes both; I distributes neither; O distributes P only.

The governing rule of inference: no term distributed in the conclusion unless distributed in a premise.

Consequences: A converts only by limitation; O cannot be converted at all.

Distribution belongs to an occurrence, not to a word.

Test yourself

1. Define distribution and state the test.

A term is distributed in a proposition when the proposition refers to the whole of the class the term denotes, that is, when it says something about every member of that class; it is undistributed when it refers to part of the class only. The test is to take the term and ask whether the proposition tells you something about every member of the class it names.

2. State the distribution of terms in each of A, E, I and O.

In A, "all S is P", the subject is distributed and the predicate undistributed. In E, "no S is P", both are distributed. In I, "some S is P", neither is distributed. In O, "some S is not P", the subject is undistributed and the predicate distributed.

3. Give the two rules from which the table follows.

Universal propositions distribute their subject, which covers A and E; particular propositions do not, which covers I and O. Negative propositions distribute their predicate, which covers E and O; affirmative propositions do not, which covers A and I. From these two rules the whole table can be reconstructed without memorising it.

4. Why is the predicate of a negative proposition distributed?

Because exclusion is total in a way that inclusion is not. To say that no minor's agreement is enforceable is to exclude every minor's agreement from the whole class of enforceable things, so something is being said about every member of the predicate class, namely that none of them is a minor's agreement. The same holds of O: to say that some agreement is not a contract is to exclude it from every contract there is.

5. State the rule of inference that distribution exists to enforce, and explain why it holds.

No term may be distributed in a conclusion unless it was distributed in a premise. It holds because a term distributed in the conclusion is one about the whole of which the conclusion makes a claim, and if the premises spoke only of part of that class then the conclusion has asserted something the premises did not supply. That is exactly what invalidity is, and distribution is the bookkeeping that detects it.

6. Is "agreements" a distributed term?

The question cannot be answered as it stands, because distribution belongs to an occurrence of a term in a particular proposition and not to the word itself. In "all contracts are agreements" it is the predicate of an affirmative proposition and is undistributed. In "no agreements are enforceable without consideration" it is the subject of a universal and is distributed. The same word can be either, depending on where and in what proposition it occurs.

The scheme cannot distinguish between a universal that carries a commitment to the existence of its subject and one that does not, because it treats them all alike. That is the failure, and the modern classification's answer is to deny existential import to universals altogether, which is topic 2.9.

Failure three: compound propositions

The traditional scheme puts hypothetical and disjunctive propositions into a separate class called conditional and then does very little with them. It has no systematic account of how their truth depends on the truth of their parts, and therefore no way of testing an argument built out of them.

That is a large gap. Statutes are written in compounds, as sequence 290 showed, and the traditional scheme offers no equivalent of a truth table.

Conjunction is worse off still. "The notice was served and the rent was unpaid" is a compound proposition with two components, and traditional logic has no class for it at all: it is not categorical, since it is not one subject and one predicate, and it is not conditional. In practice the scheme simply treats it as two propositions, which is right in this case and cannot be generalised.

Failure four: multiple quantifiers

"Every creditor has some remedy."

There are two quantifiers here and the traditional form allows only one. Worse, the sentence is ambiguous in a way the scheme cannot even express: it may mean that for each creditor there is some remedy or other, or that there is one remedy available to every creditor. Those are different propositions with different truth conditions, and the difference matters enormously in law.

The traditional scheme has no apparatus for the distinction. The modern one has, and its ability to handle it is one of the main reasons the modern scheme exists.

Failure five: existence and the copula

"Ghosts do not exist." What is being denied of what? If "ghosts" is the subject, the proposition says of ghosts that they lack existence, which requires there to be ghosts for the denial to be about. The subject-predicate form treats existence as a quality on a level with being white or being enforceable, and it does not behave like one.

"There is no contract between the parties" raises the same difficulty in a legal setting, and it is not academic: whether such a plea denies a quality of a thing or denies that there is any such thing decides what has to be pleaded and proved.

Failure six: what "some" loses

As sequence 300 showed, "most witnesses were unreliable", "a majority of the sections apply" and "two of the three defendants were served" all reduce to "some". The scheme has one particular quantifier and it means "at least one". Everything more precise is discarded in the reduction.

This is a genuine loss for a lawyer, since proportions and numbers are the ordinary currency of proof.

Failure seven: tense and modality

The copula is "is" or "are", and it is timeless. "The notice was served", "the notice will be served" and "the notice must be served" differ in ways the scheme cannot record, and the third of them, which is the ordinary form of a legal obligation, is not a matter of quantity or quality at all.

Traditional logic handles tense by the device of sequence 300, turning the proposition into one about occasions. It has no comparable device for "must", "may" and "shall", which is why the logic of obligation and permission had to be developed separately and much later.

A worked example

Take a single sentence from the law of contract and see how many of the failures it triggers.

"A minor's agreement is void, and every person who has received a benefit under it must restore it, unless the court otherwise directs."

Relational. "Received a benefit under it" relates a person to an agreement. Two ends, one place to put them.

Compound. The sentence has three components joined by "and" and "unless". The traditional scheme has no class for the whole.

Multiple quantifiers. "Every person who has received a benefit" quantifies over persons; "a benefit" quantifies over benefits. Read one way it is every person and every benefit; read another, every person and at least one benefit.

Modal. "Must restore" is an obligation, and neither quantity nor quality captures it.

Existential import. Does the sentence assert that anybody has received a benefit? Traditionally yes; obviously not.

The verdict. One ordinary legislative sentence, five failures. That is why the modern classification exists, and it is a better answer to the examination question than a list of abstract objections with no example attached.

Distinctions that carry marks

What this does not mean

The traditional scheme is not useless. It handles ordinary categorical statements well, and every valid inference it licenses is still valid. It is incomplete, not incorrect.

"Failure" does not mean the scheme produces wrong answers. It means it produces no answer where an answer is available, and this is worth saying explicitly, because "failure" sounds stronger than the criticism actually is.

The examination question is not an invitation to abuse the older logic. The marks are for naming the specific failures and illustrating them, not for saying that traditional logic is outdated.

Quick revision

The assumption attacked: every proposition is a subject, a copula and a predicate, and quantity and quality exhaust its form.

Seven failures: relational propositions; existential import; compound propositions; multiple quantifiers; existence as a predicate; the loss of numerical quantity in "some"; tense and modality.

The most examined: existential import, because it produces MU's own topic 2.9. "All trespassers will be prosecuted" does not assert that there are trespassers.

Relational failure matters most to a lawyer, since the vocabulary of obligations is relational throughout.

The right characterisation: incomplete, not incorrect. Every inference it licenses remains valid.

Test yourself

1. What assumption of the traditional classification is under attack, and why does it fail?

The assumption that every proposition consists of a subject term joined by a copula to a predicate term, with quantity and quality exhausting its logical form. It fails because a great many propositions do not have that shape: relational propositions have two or more ends, compound propositions are built out of whole propositions rather than terms, and propositions with more than one quantifier cannot be written in a form that has room for only one.

2. Explain the failure over relational propositions.

"Bombay is larger than Pune" can be forced into the scheme only by treating "larger than Pune" as a quality of Bombay, which conceals the fact that a relation between two things is involved. The consequence is that the obvious inference to "Pune is smaller than Bombay" becomes invisible, since the two propositions then share no term. Relations with three ends, such as "A paid B on behalf of C", cannot be represented at all.

3. What is the difficulty about existential import?

Traditional logic takes a universal affirmative to assert that its subject exists, allowing the inference from "all S is P" to "some S is P". But "all trespassers will be prosecuted" plainly does not assert that there are trespassers, and a section providing for all persons convicted under a repealed provision does not assert that any such persons exist. The scheme cannot distinguish universals that carry an existential commitment from those that do not, because it treats them all alike.

4. Why are compound propositions a problem for the traditional scheme?

Because it classes hypothetical and disjunctive propositions together as conditional and then supplies no systematic account of how their truth depends on the truth of their parts, so it offers no method of testing an argument built out of them. Conjunctions fare worse still: "p and q" is neither categorical nor conditional, and the scheme has no class for it, treating it in practice as two separate propositions.

5. Illustrate the difficulty with multiple quantifiers.

"Every creditor has some remedy" contains two quantifiers, and the traditional form has room for one. The sentence is also ambiguous between meaning that each creditor has some remedy or other and meaning that there is one remedy available to all creditors, which are different propositions with different truth conditions. Traditional logic cannot express the ambiguity, let alone resolve it, whereas the modern scheme does both by giving quantifiers a scope.

6. Is it right to say the traditional classification is wrong?

No, and the distinction matters for the answer. Every inference the traditional scheme licenses remains valid, and it handles ordinary categorical statements well. What it does is fail to reach a large class of propositions, so that it returns no answer where an answer is available. The correct description is that it is incomplete rather than incorrect, and the modern classification was built to cover what it leaves out.

Why the modern cut is the better one

Because the members of each modern class behave alike and the members of each traditional class do not.

All compounds are governed by their connectives and can be tested by truth tables. All simples must be analysed internally, into subject and predicate or into a relation and its terms, and quantified where necessary.

The traditional class of "conditional" propositions contained hypotheticals and disjunctions, which is right, and excluded conjunctions, which is arbitrary: "p and q" is exactly as compound as "if p then q". The traditional class of "categorical" propositions contained singular, universal and relational propositions together, though they behave quite differently.

A classification is good when knowing which class a thing is in tells you how it behaves. That is a criterion from Module IV, at sequence 630, applied here, and it is the honest answer to why the modern classification replaced the older one.

What the modern scheme does with quantity

It does not throw quantity away; it relocates it.

In the modern treatment, quantity is not a feature of the proposition as a whole. It is a quantifier attached to a variable inside a simple proposition, and it can occur more than once, with a scope. That is what answers failure four of the last chapter.

"All contracts are agreements" becomes, in the modern reading, a general proposition: for anything whatever, if it is a contract then it is an agreement. Written that way, it says nothing about whether any contracts exist, which answers failure two, and it makes the universal into a conditional, which is where existential import goes.

This is MU's topic 2.7, at sequence 360, and it is the deepest single difference between the two schemes.

A worked example

Take three propositions and classify each in both schemes.

"No agreement made by a minor is enforceable."

Traditional: categorical, universal negative, E.

Modern: simple, and generally quantified. For anything whatever, if it is an agreement made by a minor then it is not enforceable.

"If the notice was served, the tenancy ended."

Traditional: conditional, hypothetical.

Modern: compound, an implication with two components.

"The notice was served and the rent was unpaid."

Traditional: no class. The scheme has nowhere to put it and treats it as two propositions.

Modern: compound, a conjunction with two components.

What the comparison shows. The second and third are treated as quite different things by the traditional scheme and as the same kind of thing by the modern one, and the modern grouping is the useful one, because both can be tested by the same method. Meanwhile the first is treated as a single indivisible form by the traditional scheme and is opened up by the modern one into a quantifier, a variable and a conditional, which is what allows the modern scheme to say what the older one could not about existence.

Distinctions that carry marks

What this does not mean

The modern scheme is not a rival account of the same material. It is a different division of a wider field. Every A, E, I and O proposition is a simple proposition in the modern scheme, so the two classifications intersect rather than compete.

"Simple" does not mean short or easy. "Every creditor of an insolvent company incorporated before 1956 has a remedy against the directors personally" is a simple proposition: long, difficult, and containing no component.

Compound does not mean complicated. "p and q" is compound and trivial.

Quick revision

The aim, four parts: classify by what actually decides inference; reach the propositions the older scheme could not; make the test mechanical; separate form from content completely.

First cut: simple, containing no component, against compound, containing at least one.

A simple proposition becomes one letter. A compound becomes components joined by connectives.

Quantity is relocated, from a feature of the proposition to a quantifier with a scope inside it.

"All S is P" becomes a conditional: for anything whatever, if it is S then it is P. This is where existential import goes.

Why the modern cut is better: the members of each class behave alike, which is the test of a good division.

Test yourself

1. State the aim of the modern classification.

To divide propositions according to the features on which their logical behaviour actually depends, rather than according to quantity and quality alone. That means classifying by structure, so that propositions which behave alike fall together; reaching the propositions the traditional scheme could not handle, such as relational and compound propositions; making the testing of inferences mechanical, which truth tables allow; and separating form from content completely, so that a whole simple proposition can be replaced by a single letter.

2. What is the first cut the modern classification makes, and how does it differ from the traditional first cut?

The modern scheme first asks whether a proposition contains another proposition as a component, dividing propositions into simple and compound. The traditional scheme first asks whether the assertion is absolute or conditional, dividing them into categorical and conditional. The difference matters because the traditional cut separates hypotheticals from conjunctions, which behave alike, and groups singular, universal and relational propositions together, which do not.

3. Why is the modern division said to be a better one?

Because knowing which class a proposition belongs to tells you how it behaves, which is the test of a good division. Every compound is governed by its connectives and can be tested by a truth table; every simple proposition has to be analysed internally into predicates, relations and quantifiers. The traditional classes do not have this property, since conjunctions have no class at all and the categorical class contains propositions of quite different logical character.

4. What happens to quantity in the modern scheme?

It is relocated rather than discarded. Instead of being a feature of the proposition as a whole, occurring once, it becomes a quantifier attached to a variable inside the proposition, and it may occur more than once with a scope. This is what allows the modern scheme to represent propositions such as "every creditor has some remedy" and to distinguish the two readings of it, which the traditional form cannot express.

5. How does the modern scheme read "All contracts are agreements", and what follows about existence?

As a general proposition: for anything whatever, if it is a contract then it is an agreement. The universal has become a conditional, and a conditional asserts neither of its parts. It follows that the proposition asserts nothing about whether any contracts exist, which is precisely the denial of existential import and the answer to the difficulty raised by "all trespassers will be prosecuted".

6. Is "Every creditor of an insolvent company incorporated before 1956 has a remedy against the directors personally" simple or compound?

Simple. Length and difficulty are irrelevant: what matters is whether any part of it is itself a proposition capable of being true or false on its own, and none is. "Of an insolvent company incorporated before 1956" qualifies the subject term and asserts nothing separately. The proposition is a single quantified statement, however complicated its terms may be.

Kinds of compound proposition

Negative propositions, ~p. The denial of a proposition is itself a compound in the modern scheme, because it has a component. "It is not the case that the notice was served." ~p is true exactly when p is false.

Conjunctive propositions, p • q. Two or more components asserted together, each of them asserted. "The notice was served and the rent was unpaid." Both are claimed, and the whole is true only if both are true.

Disjunctive propositions, p ∨ q. At least one component is asserted, without saying which. "Either the notice was defective or the tenant waived the defect." Read inclusively, as sequence 290 established.

Implicative propositions, p ⊃ q. One component is asserted to follow from another, and neither is asserted by itself. "If the notice was served, the tenancy ended."

Equivalent propositions, p ≡ q. Each component is asserted to follow from the other. "A person is a major if and only if he has completed eighteen years." Statutory definitions are frequently equivalences, and reading them as mere implications is a common error: "means" in a definition clause is usually an equivalence, while "includes" is not.

Which components are asserted

This is the point of the classification and it is worth a table of its own, because it decides what an opponent has to disprove.

Read that table as a litigator. A pleading that asserts a conjunction has given the other side two targets and needs both to survive. A pleading that asserts a disjunction has given two targets and needs only one to survive. That asymmetry is the whole reason alternative pleading exists, and it is a consequence of the truth conditions of the connectives rather than of any rule of procedure.

A worked example: symbolising a section

Section 14 of the Indian Contract Act 1872 provides that consent is said to be free when it is not caused by coercion as defined in section 15, or undue influence as defined in section 16, or fraud as defined in section 17, or misrepresentation as defined in section 18, or mistake, subject to the provisions of sections 20, 21 and 22.

Assign letters.

f: the consent is free

c: the consent is caused by coercion

u: the consent is caused by undue influence

d: the consent is caused by fraud

m: the consent is caused by misrepresentation

k: the consent is caused by mistake within sections 20, 21 and 22

Symbolise. The section states an equivalence, because "is said to be free when" in a definitional provision fixes both directions:

f ≡ ~(c ∨ u ∨ d ∨ m ∨ k)

Read what the symbolism shows. Consent is free exactly when none of the five vitiating factors is present. The section is a negated disjunction, and a negated disjunction is true only when every alternative is false. So a party alleging that consent was not free needs to establish only one of the five, while a party asserting that it was free must be able to meet all of them.

And a second thing, which the words hide. Because the five are joined by "or" and then denied, the section is equivalent to a conjunction of five denials:

f ≡ (~c • ~u • ~d • ~m • ~k)

That transformation is one of De Morgan's rules and it is worked at sequence 350. The practical point is that the section can be read either as "not any of these" or as "none of these, and none of these, and none of these", and the second reading is the one that tells a drafter what has to be pleaded.

Distinctions that carry marks

What this does not mean

A negative proposition is not the same as a negative categorical. "No S is P" is an E proposition and is simple in the modern scheme, because it contains no component; "it is not the case that all S is P" is compound, because it denies a whole proposition. The English is close and the analysis is not.

"Or" in a statute is not automatically exclusive. Sequence 290 dealt with this, and it is worth repeating because it changes what a party has to prove.

Brackets are not decoration. ~(p • q) and ~p • q are different propositions with different truth conditions, and dropping the brackets changes the meaning rather than the appearance.

Quick revision

Notation: ~ negation, • conjunction, ∨ disjunction, ⊃ implication, ≡ equivalence. Simple propositions are single letters. Brackets group.

Simple propositions: subject-predicate, relational, class-membership, class-inclusion. In a relational proposition the term from which the relation proceeds is the referent and the term to which it proceeds is the relatum, and the order is part of what is asserted.

Compound propositions: negative, conjunctive, disjunctive, implicative, equivalent.

Which components are asserted: both in a conjunction, neither in a disjunction, an implication or an equivalence.

Litigation consequence: a conjunction gives the opponent two targets and needs both; a disjunction gives two targets and needs one.

Definitions using "means" are usually equivalences; those using "includes" are not.

Test yourself

1. Name the kinds of simple proposition with an example of each.

Subject-predicate propositions, asserting a quality of a thing, as in "the document is registered". Relational propositions, asserting a relation among two or more things, as in "Ravi owes Meena two lakh rupees". Class-membership propositions, placing an individual in a class, as in "Ravi is a lawyer". And class-inclusion propositions, placing one class within another, as in "all lawyers are graduates".

2. Name the kinds of compound proposition with their symbols.

Negative, ~p; conjunctive, p • q; disjunctive, p ∨ q; implicative, p ⊃ q; and equivalent, p ≡ q. Each is classified by its connective, and the connective is what determines the truth of the whole from the truth of the components.

3. Which components of a compound proposition are asserted?

Both components of a conjunction are asserted, so both must be true for the whole to be true. Neither component of a disjunction is asserted individually; what is asserted is that at least one holds. Neither component of an implication is asserted; only the connection is. Neither component of an equivalence is asserted; only that they stand or fall together.

4. Why does the difference between a conjunction and a disjunction matter to a pleader?

Because it decides how many of the assertions must survive. A case pleaded as a conjunction fails if the other side disproves any one of its parts, so every part is a target and all must hold. A case pleaded in the alternative is a disjunction and survives if any one part holds, so the other side must defeat every alternative. Alternative pleading is a direct application of the truth conditions of the connectives.

5. Symbolise "Consent is free when it is not caused by coercion, undue influence, fraud, misrepresentation or mistake", and say what a party must prove.

Writing f for free consent and c, u, d, m and k for the five factors, the section is f ≡ ~(c ∨ u ∨ d ∨ m ∨ k). A party alleging that consent was not free needs to establish any one of the five, since a disjunction is true if any alternative is. A party asserting that consent was free must be in a position to meet all five, since a negated disjunction is true only when every alternative is false.

7. In a relational proposition, what are the referent and the relatum?

The referent is the term from which the relation proceeds and the relatum is the term to which it proceeds. In "Ravi owes Meena two lakh rupees", Ravi is the referent and Meena the relatum; in "the tenant holds under the landlord", the tenant is the referent. The two cannot be exchanged without changing the proposition, and inference by converse relation is the operation that exchanges them while replacing the relation by its converse.

6. What is the difference between ~(p • q) and ~p • q?

The first denies the conjunction, and is true whenever at least one of p and q is false. The second asserts the conjunction of ~p with q, and is true only when p is false and q is true. They differ in truth value in two of the four possible cases, so the brackets are not a matter of style: removing them states a different proposition.

The first answer is what the table is for. The connective is defined to be false in exactly one case, where the antecedent is true and the consequent false, because that is the only case in which a promise of the form "if p then q" has been broken. Consider an undertaking to the court: "if the appeal is dismissed, my client will vacate within thirty days." When has the undertaking been violated? Only if the appeal is dismissed and the client does not vacate. If the appeal is allowed, the undertaking has not been violated, whatever the client does. Rows three and four record exactly that.

The second answer is that any other table would be worse. Suppose rows three and four were both made false. Then p ⊃ q would mean the same as p • q, and the connective would be useless. Suppose row three were true and row four false. Then the truth of a conditional would depend on the truth of its consequent when the antecedent is false, which would make "if p then q" behave differently from "if p then q" with the same parts in a different context, and truth-functionality would be lost.

The third answer is the honest one. Material implication is not an analysis of the English "if". It is a deliberately weakened substitute, chosen because it is truth-functional and because it validates the inferences deduction needs, namely affirming the antecedent and denying the consequent. Ordinary conditionals carry more, usually a connection of meaning or cause, and modern logic simply declines to represent that surplus. Saying this in an examination is worth more than reciting the table.

Constructing a table for a longer compound

Method. Count the distinct simple components. Write 2 to that power rows. Fill the component columns in the standard order. Then build the compound up in stages, one connective at a time, giving each stage its own column. The last column is the answer.

Example: ~(p • q) ≡ (~p ∨ ~q), which is one of De Morgan's rules.

The last column is true in every row.

Tautology, contradiction and contingency

A tautology is a compound proposition true in every row of its table. The example above is one. So is p ∨ ~p, which is the law of excluded middle written as a formula.

A contradiction is false in every row. p • ~p is the standard case, and it is the law of contradiction written as a formula.

A contingent proposition is true in some rows and false in others. Almost every proposition anybody actually asserts is contingent.

Why this matters. A tautology says nothing about the world, since it is true whatever the facts are. That is not a criticism: the rules of logic are tautologies, and they are useful precisely because they hold whatever happens. But an argument whose conclusion is a tautology has told you nothing, and a pleading whose case is a tautology has pleaded nothing.

De Morgan's rules

Two transformations, both provable by table, and both used constantly in reading statutes.

~(p • q) is equivalent to ~p ∨ ~q. The denial of a conjunction is the disjunction of the denials. To deny that both conditions were satisfied is to say that at least one was not.

~(p ∨ q) is equivalent to ~p • ~q. The denial of a disjunction is the conjunction of the denials. To deny that either happened is to say that neither did.

Where a lawyer meets them. Section 14 of the Indian Contract Act 1872 says consent is free when it is not caused by coercion, undue influence, fraud, misrepresentation or mistake. That is ~(c ∨ u ∨ d ∨ m ∨ k), and by the second rule it is equivalent to ~c • ~u • ~d • ~m • ~k. The section can therefore be read as "none of these five", which is how a court states it, or as "not this, and not this, and not this, and not this, and not this", which is how a pleading has to meet it.

Testing an argument by table

An argument is valid when there is no row in which every premise is true and the conclusion is false. That is the definition of validity from sequence 110, applied mechanically.

Test: affirming the antecedent. Premises p ⊃ q and p; conclusion q.

Look for a row where both premises are true. Only row one: p ⊃ q is true and p is true. In that row the conclusion q is true. There is no row with true premises and a false conclusion, so the argument is valid.

Test: affirming the consequent. Premises p ⊃ q and q; conclusion p.

Rows where both premises are true: row one, where p ⊃ q is T and q is T; and row three, where p ⊃ q is T and q is T. In row three the conclusion p is false. There is a row with true premises and a false conclusion, so the argument is invalid, which is what sequence 290 asserted and this proves.

That is the whole method, and it is mechanical: no judgment is exercised anywhere in it, which is what "the test can be computed" meant at sequence 330.

A worked example

A section provides: "A licence shall be cancelled if the holder has been convicted of an offence under this Act and has failed to pay the penalty within thirty days."

Symbolise. c: the holder has been convicted. f: the holder has failed to pay within thirty days. x: the licence shall be cancelled. The section is (c • f) ⊃ x.

Question one. The holder was convicted but paid within thirty days. May the licence be cancelled under this section?

Build the antecedent. c is true, f is false, so c • f is false. With a false antecedent the implication is true whatever x is, which means the section is satisfied and says nothing about x. The section supplies no power to cancel. That is rows three and four of the implication table doing real work: a false antecedent leaves the consequent entirely open.

Question two. The licence was cancelled. Does it follow that the holder was convicted?

This is affirming the consequent, and it is invalid. The licence may have been cancelled under some other provision.

Question three. The licence was not cancelled. What follows?

Denying the consequent, which is valid: ~x gives ~(c • f), and by De Morgan that is ~c ∨ ~f. Either he was not convicted or he did not fail to pay, and the section does not tell us which. A conclusion in the alternative is a real conclusion, and it is often all a provision will yield.

Distinctions that carry marks

What this does not mean

A true conditional is not a good argument. p ⊃ q being true in a row where p is false does not mean anything follows; it means the conditional has not been falsified.

Truth tables do not test the premises. They test the connection. Whether p is in fact true is not a question a table can answer.

Not every "if" is material implication. Conditionals about what would have happened, and conditionals expressing a causal connection, are not truth-functional, and a table cannot represent them.

Quick revision

Truth-functional: the truth of the whole is fixed by the truth of the parts. This is why tables are possible.

Rows: 2 to the power n, in the standard order TT, TF, FT, FF.

Tables: ~p true when p false; p • q true only when both true; p ∨ q false only when both false; p ⊃ q false only when p true and q false; p ≡ q true when they agree.

The two odd rows: a false antecedent makes an implication true, because the only way to break "if p then q" is to have p and not q.

Tautology true in every row; contradiction false in every row; contingent neither.

De Morgan: ~(p • q) is ~p ∨ ~q; ~(p ∨ q) is ~p • ~q.

Validity test: no row where every premise is true and the conclusion is false.

Test yourself

1. What makes a truth table possible, and how many rows does one have?

That the connectives of modern logic are truth-functional, meaning that the truth value of the compound is completely determined by the truth values of its components and by nothing else. Since each simple component is either true or false, a compound containing n distinct simple components has 2 to the power n rows, so two components give four rows, three give eight and four give sixteen.

2. Give the table for implication and explain the last two rows.

p ⊃ q is true when p and q are both true, false when p is true and q false, and true in both cases where p is false. The last two rows record that a conditional is not broken when its antecedent fails: an undertaking that "if the appeal is dismissed my client will vacate" is violated only if the appeal is dismissed and the client does not vacate, and it is not violated at all if the appeal is allowed, whatever the client then does.

3. Why is material implication not an analysis of the English "if"?

Because ordinary conditionals usually assert a connection of meaning, cause or law between the parts, and material implication asserts nothing of the kind: it requires only that the antecedent not be true while the consequent is false. It is adopted because it is truth-functional, so a table can handle it, and because it validates the two inferences deduction needs. The oddities that result are accepted as the price, not defended as a discovery about conditionals.

4. Define tautology, contradiction and contingent proposition, with an example of each.

A tautology is true in every row of its truth table, such as p ∨ ~p. A contradiction is false in every row, such as p • ~p. A contingent proposition is true in some rows and false in others, such as p ⊃ q. A tautology and a contradiction each say nothing about the world, since their truth value does not depend on the facts, while a contingent proposition does.

5. State De Morgan's rules and give a statutory application.

The denial of a conjunction is the disjunction of the denials, ~(p • q) being equivalent to ~p ∨ ~q; and the denial of a disjunction is the conjunction of the denials, ~(p ∨ q) being equivalent to ~p • ~q. Section 14 of the Indian Contract Act 1872, which makes consent free when it is not caused by any of five factors, is a negated disjunction and is therefore equivalent to the conjunction of five denials, which is the form a pleading has to meet.

6. How is an argument tested for validity by a truth table, and what does the test show about affirming the consequent?

By constructing the table and looking for a row in which every premise is true and the conclusion is false; if there is no such row the argument is valid, and if there is one it is invalid. For affirming the consequent, with premises p ⊃ q and q and conclusion p, there is a row in which p is false and q is true: both premises are true there and the conclusion is false, so the argument is invalid.

Why the distinction is worth a mark. Because it says what symbolising a general proposition actually consists in. The examiner's instruction, "symbolise by using propositional functions and quantifiers", is an instruction to do two things in order: write the propositional functions, then bind their variables with the right quantifier. A student who writes Sx and Px and stops has done the first half.

The four forms in modern dress

Notice the pattern and it is the whole chapter. The two universals use the horseshoe. The two existentials use the dot. That is not a convention chosen for tidiness; it is forced, and the next two sections show why.

Why a universal takes a conditional

Try writing "all contracts are agreements" as a universal conjunction: ∀x(Sx • Px). That says: for anything whatever, it is a contract and it is an agreement. That is, everything in the universe is a contract, which is absurd and is plainly not what was meant.

The reason is that a universal proposition is not about everything. It is about the members of the subject class, and it says nothing at all about anything else. The conditional captures exactly that: for anything whatever, if it happens to be a contract, then it is an agreement. Where the antecedent fails, the conditional is true and nothing has been claimed, which is rows three and four of the implication table doing precisely the work they were defended for at sequence 350.

And now the consequence that MU sets as topic 2.9. Suppose there are no contracts at all. Then Sx is false for everything, so Sx ⊃ Px is true for everything, so ∀x(Sx ⊃ Px) is true. A universal proposition with an empty subject class comes out true.

That is the modern denial of existential import, and it is why "all trespassers will be prosecuted" no longer asserts that there are trespassers.

Why an existential takes a conjunction

Try writing "some contracts are agreements" as an existential conditional: ∃x(Sx ⊃ Px). That says: there is at least one thing such that, if it is a contract, it is an agreement. Take any object that is not a contract, a chair for example. For the chair, the antecedent is false, so the conditional is true, so such a thing exists. The proposition would be true no matter what, and would tell us nothing.

The conjunction avoids that: ∃x(Sx • Px) requires there to be a thing that really is a contract and really is an agreement. It asserts existence and it asserts the predicate, which is what "some" is for.

So the asymmetry is not arbitrary. A universal conjunction would say far too much; an existential conditional would say almost nothing. The only workable pairing is universal with conditional and existential with conjunction.

Multiple quantifiers, and why order matters

This is what answers failure four of sequence 320, and it is the most legally useful part of the chapter.

"Every creditor has some remedy." Two quantifiers, and two readings.

Reading one: ∀x(Cx ⊃ ∃y(Ry • Hxy)). For every creditor, there is some remedy or other that he has. Different creditors may have different remedies.

Reading two: ∃y(Ry • ∀x(Cx ⊃ Hxy)). There is one remedy such that every creditor has that one.

These are different propositions. The second implies the first and the first does not imply the second. Reversing the order of the quantifiers changes what is asserted, and traditional logic could not even record the difference.

Where this decides a case. A statute providing that "every workman shall be paid a wage fixed by the appropriate Government" reads naturally in the first way: each workman has a wage fixed for him. Read the second way it would mean there is a single wage fixed for all workmen, which is a completely different scheme. Provisions of this shape are litigated, and the whole argument is about quantifier order.

A worked example

Section 11 of the Indian Contract Act 1872 provides that every person is competent to contract who is of the age of majority according to the law to which he is subject, who is of sound mind, and who is not disqualified from contracting by any law to which he is subject.

Symbolise. Write Px for "x is a person", Mx for "x is of the age of majority", Sx for "x is of sound mind", Dx for "x is disqualified by law", Cx for "x is competent to contract".

∀x((Px • Mx • Sx • ~Dx) ⊃ Cx)

Read what this does and does not say.

It does not say that anybody is competent. It is a conditional under a universal quantifier, so it asserts nothing about whether the antecedent is ever satisfied. If there were no persons of full age at all, the section would still be true. That is the denial of existential import, in a statute.

It does not say that only such persons are competent. The section as symbolised is an implication and not an equivalence. Whether the three conditions are also necessary is a further question, and it is answered by the section's context rather than by its form. A student who symbolises it with the triple bar has asserted something the words do not.

And it gives the negation directly. ~Cx follows if the antecedent fails, only on the equivalence reading. On the implication reading, the failure of the antecedent yields nothing, which is denying the antecedent from sequence 290. The section's real effect, that a minor's agreement is void, comes from elsewhere and not from section 11 alone, and the symbolism makes that visible in a line.

Distinctions that carry marks

What this does not mean

"For anything whatever" is not "for everything in the subject class". The quantifier ranges over everything there is, and the antecedent does the work of confining attention to the subject class.

Denying existential import is not saying the subject does not exist. It is saying the proposition does not assert that it does. "All contracts are agreements" leaves the question of whether there are contracts entirely open.

A quantifier is not a term. It is a syncategorematic device, as sequence 170 said, and it cannot be a subject or a predicate.

Quick revision

Apparatus: variable x, predicate letters Sx and Px, universal quantifier ∀x or (x), existential quantifier ∃x.

Propositional function: an expression with a variable, such as Sx, which is neither true nor false until the variable is replaced by a name or bound by a quantifier. Quantification is asserting a propositional function of all or of some of the values of the variable. A variable is bound inside the scope of its quantifier and free outside it, and an expression with a free variable is never a proposition.

The four forms: A is ∀x(Sx ⊃ Px); E is ∀x(Sx ⊃ ~Px); I is ∃x(Sx • Px); O is ∃x(Sx • ~Px).

Universals take the horseshoe, existentials take the dot, and the pairing is forced: a universal conjunction would assert that everything is S, and an existential conditional would be true whatever the facts.

Existential import: a universal with an empty subject class is true; an existential with an empty subject class is false. This is topic 2.9.

Quantifier order matters: "every creditor has some remedy" has two readings, and the second implies the first but not conversely.

Statutes are universals over conditionals, which is why a section asserts nothing about whether its conditions are ever satisfied.

Test yourself

1. Write the four traditional forms in modern notation.

A becomes ∀x(Sx ⊃ Px), read as: for anything whatever, if it is S then it is P. E becomes ∀x(Sx ⊃ ~Px). I becomes ∃x(Sx • Px), read as: there is at least one thing which is S and is P. O becomes ∃x(Sx • ~Px). The two universals use a conditional and the two existentials a conjunction.

2. Why can a universal proposition not be written as a universal conjunction?

Because ∀x(Sx • Px) says that for anything whatever, it is S and it is P, that is, that everything in the universe belongs to the subject class and has the predicate. "All contracts are agreements" would then assert that everything is a contract, which is absurd. A universal proposition is not about everything; it is about the members of the subject class only, and the conditional is what confines it to them.

3. Why can an existential proposition not be written with a conditional?

Because ∃x(Sx ⊃ Px) would be satisfied by any object at all for which the antecedent is false. Take anything that is not a contract: the conditional is true of it, so such a thing exists, so the whole proposition is true whatever the facts. It would therefore assert nothing. The conjunction is required because "some" is meant to assert that a thing exists which really does have both properties.

4. Explain existential import in the modern scheme.

A universal proposition asserts nothing about whether its subject class has any members, because it is a conditional and a conditional with a false antecedent is true. If there are no trespassers at all, "all trespassers will be prosecuted" comes out true. An existential proposition does assert existence, since it is a conjunction requiring a thing that actually is S and is P. So universals carry no existential import and existentials carry it by definition.

5. Show, with a legal example, why the order of quantifiers matters.

"Every workman shall be paid a wage fixed by the appropriate Government" has two readings. Written with the universal outermost, it says that for each workman there is some wage fixed for him, which permits different wages for different workmen. Written with the existential outermost, it says there is one wage such that every workman is paid it. The second implies the first but not the reverse, and provisions of this shape are litigated precisely because the words do not settle which was meant.

7. What is a propositional function, and how does it become a proposition?

A propositional function is an expression containing a variable, such as "x is a contract", written Sx. It is neither true nor false, because nothing has been said about which x is meant, so it is not a proposition. It becomes one in either of two ways: by substitution, putting a name in place of the variable, which yields a singular proposition; or by quantification, binding the variable with a universal or existential quantifier, which yields a general proposition.

8. Distinguish free and bound variables.

A variable is bound when it falls within the scope of a quantifier that governs it, and free when it does not. In Sx the variable is free and the expression is a propositional function; in ∀x(Sx ⊃ Px) every occurrence of x is bound and the expression is a proposition. An expression containing at least one free variable is always a propositional function and never a proposition, which is why symbolising a general proposition means writing the functions and then binding them.

6. What does the symbolisation of section 11 of the Indian Contract Act 1872 show about what the section does not say?

Symbolised as a universally quantified conditional, it shows that the section asserts nothing about whether any person satisfies its conditions, since a conditional with a false antecedent is true. It also shows that the section is an implication and not an equivalence, so it does not say that only such persons are competent, and nothing about incompetence follows from the failure of its conditions. That result comes from other provisions, and treating the section as a definition of incompetence is a symbolisation error.

Both classifications intersect rather than compete. Every A, E, I and O proposition is a simple proposition in the modern scheme. Every traditional hypothetical is a modern implicative compound. The two schemes are two grids over one field, and the modern grid extends beyond the edges of the older one.

Where the modern scheme is better, and where it costs something

Better: coverage, mechanical testing, the handling of relations and quantifier scope, and honesty about existence.

The cost, and it is real. The modern scheme is further from ordinary language. Reading "all contracts are agreements" as "for anything whatever, if it is a contract then it is an agreement" is exact and unnatural, and material implication's odd rows are the price of truth-functionality, as sequence 350 admitted. A student who says the modern scheme is simply superior has not noticed that it buys its precision with a currency.

And a practical cost for a lawyer. Legal reasoning is largely conducted in the traditional vocabulary. Judgments speak of universals, particulars, exceptions and provisos, not of quantifier scope. A lawyer who can only think in the modern notation will find nothing to talk to.

A worked example

Take one proposition and put it through both schemes completely.

"No agreement made by a person of unsound mind is enforceable."

Traditional analysis. Categorical, since it asserts absolutely. Quantity universal, quality negative, so it is an E proposition. Subject term "agreement made by a person of unsound mind", predicate term "enforceable". Both terms distributed, by the rules at sequence 310. It converts simply to "no enforceable agreement is one made by a person of unsound mind", and it obverts to "all agreements made by a person of unsound mind are unenforceable".

Modern analysis. Simple, since it contains no component proposition. Written ∀x(Ux ⊃ ~Ex), where Ux is "x is an agreement made by a person of unsound mind" and Ex is "x is enforceable". It asserts nothing about whether any such agreements exist. It contains one quantifier and one conditional, and its negation is ∃x(Ux • Ex), which asserts that there is at least one such agreement which is enforceable.

What each analysis gives you. The traditional one gives you the immediate inferences: what else follows from this proposition alone, which is Module III's subject. The modern one gives you the truth conditions and the exact form of the denial, which is what a pleader needs when drafting a traverse. Both are useful and neither replaces the other, which is the fair conclusion to a comparison question.

Distinctions that carry marks

What this does not mean

The modern scheme did not refute the traditional one. It extended the field and revised one doctrine, existential import. Nothing else was withdrawn.

A comparison question is not an invitation to praise one scheme. The marks are for points of comparison, and an answer that runs both past six named points beats an answer that argues for a winner.

"Traditional" does not mean obsolete. The vocabulary of legal reasoning is traditional, and the whole of Module III is traditional logic.

Quick revision

Six points of comparison: basis of division, unit of analysis, classes produced, method of testing, coverage, treatment of existence.

Traditional: manner of assertion; the term; categorical and conditional; rules in words; ordinary categoricals only; universals carry existential import.

Modern: structure; the whole proposition; simple and compound; truth tables; relations, conjunctions and quantifier scope included; universals carry no existential import.

Three agreements: every traditionally valid inference remains valid; both are formal; the classifications intersect, since every A, E, I and O proposition is simple.

The cost of the modern scheme: distance from ordinary language, and material implication's odd rows.

Test yourself

1. On what points should the two classifications be compared?

On the basis of division, the unit of analysis, the classes each produces, the method by which each tests an inference, the range of propositions each can represent, and the treatment of existential import. An answer built on these six points, taking each scheme past each of them, is a comparison; an answer that describes one scheme and then the other is two descriptions.

2. State the basis of division and the unit of analysis in each scheme.

The traditional scheme divides by the manner of assertion, absolute or conditional, and then by quantity and quality, and its unit is the term, since distribution, conversion and obversion all operate on subjects and predicates. The modern scheme divides by structure, asking whether a proposition contains another proposition as a component, and its unit is the whole proposition, which is why a simple proposition can be replaced by a single letter.

3. Give three respects in which the modern scheme has wider coverage.

It represents relational propositions, treating the relation as a predicate with two or more arguments, where the traditional subject-predicate form has only one place. It gives conjunctions a class, which the traditional scheme does not have at all. And it handles propositions with more than one quantifier by giving quantifiers a scope, so that the two readings of "every creditor has some remedy" can be distinguished and written down.

4. In what respects do the two schemes agree?

Every inference the traditional scheme licenses remains valid under the modern one, so there is no conflict about results. Both are formal, studying the shape of propositions and disregarding their subject matter. And the two classifications intersect rather than compete: every A, E, I and O proposition is a simple proposition in the modern scheme, and every traditional hypothetical is a modern implicative compound.

5. What does the modern scheme cost?

Distance from ordinary language. Reading "all contracts are agreements" as a quantified conditional is exact and unnatural, and material implication's treatment of a false antecedent is accepted as the price of truth-functionality rather than defended as an account of "if". There is also a practical cost for a lawyer, since judgments and statutes are written in the traditional vocabulary of universals, exceptions and provisos, and not in quantifier notation.

6. Analyse "No agreement made by a person of unsound mind is enforceable" in both schemes.

Traditionally it is a categorical proposition, universal in quantity and negative in quality, so an E proposition, with both terms distributed; it converts simply and obverts to a universal affirmative with a negative predicate. In the modern scheme it is a simple proposition, written ∀x(Ux ⊃ ~Ex), asserting nothing about whether any such agreements exist, and its denial is ∃x(Ux • Ex). The first analysis yields the immediate inferences, the second the truth conditions and the exact form of the traverse.

What the disagreement costs each side

What the traditional scheme cannot do. It cannot make a true universal statement about an empty class, and there are many worth making. "All trespassers will be prosecuted" is not falsified by the absence of trespassers. "All persons convicted under the repealed section shall receive a refund" is not an assertion that anybody was convicted. A scheme that makes these assert existence has misdescribed them.

What the modern scheme loses. Three of the traditional square's relations. Subalternation disappears, since A no longer implies I. Contraries disappear, since A and E can now both be true when the subject class is empty. Subcontraries disappear, since I and O can now both be false for the same reason. Only the contradictories survive, A against O and E against I, and the square of opposition is reduced to its diagonals.

That is a real loss, and it is why the square is taught in the traditional form at sequence 410, with this qualification attached to it.

The empty-class test

The quickest way to see the difference, and a good way to answer an examination question, is to take an empty class and run the four propositions past it.

Let S be "unicorns in this courtroom". There are none.

A, "all unicorns in this courtroom are white." Traditional: false, because it asserts there are unicorns here. Modern: true, vacuously, because there is nothing to be a counterexample.

E, "no unicorn in this courtroom is white." Traditional: false, same reason. Modern: true, vacuously.

Both A and E true at once, on the modern reading, which is exactly why contraries fail: two propositions that traditional logic says can never both be true are both true here.

I, "some unicorn in this courtroom is white." False on both readings.

O, "some unicorn in this courtroom is not white." False on both readings.

Both I and O false at once, which is why subcontraries fail.

A worked example

A section provides: "Every person who was in occupation of the premises on 1 January 1990 shall be deemed to be a tenant."

Read it traditionally. The section asserts that there were persons in occupation on that date. If it turned out that the premises were vacant, the section would be false, which is an odd thing to say about a piece of legislation: a statute is not false, it simply does not apply.

Read it modernly. For anything whatever, if it is a person who was in occupation on 1 January 1990, then it shall be deemed a tenant. If nobody was in occupation, the provision is true and applies to nobody. That is the correct account of what a legislature does, and it matches how courts treat such provisions.

Now the practical question. A party asserts rights under the section. What must he prove?

On either reading, he must prove that he was in occupation on the date, which is a particular proposition and carries existential import on both schemes. The section itself never proved existence for him. The distinction therefore has a practical edge: a party cannot rely on the enacting words to establish that anybody was in occupation, because the section, properly read, asserts nothing of the kind.

And the drafting lesson. A legislature that wishes to assert existence must do it separately, which is why Acts contain recitals and declarations. The operative words do not carry it.

Distinctions that carry marks

What this does not mean

The modern scheme does not say the subject class is empty. It says the proposition does not assert that it has members. The question is left open, exactly as it should be.

The dispute does not affect particular propositions. Both schemes agree that I and O assert existence.

Denying existential import is not a verbal trick. It falls out of reading a universal as a conditional, which is itself forced, as sequence 360 showed.

Quick revision

The one difference: traditional universals assert that their subject class has members; modern universals do not.

Traditional reason: subalternation, the inference from A to I, requires it.

Modern reason: ∀x(Sx ⊃ Px) is true when nothing is S, because a conditional with a false antecedent is true.

Cost to the traditional scheme: it cannot state a true universal about an empty class, and legal provisions are full of them.

Cost to the modern scheme: subalternation, contraries and subcontraries all fail, and only the contradictories of the square survive.

Empty-class test: with no S, both A and E come out true and both I and O come out false on the modern reading.

No dispute about I and O: both schemes agree that particular propositions assert existence.

Test yourself

1. State the distinction between traditional and modern general propositions.

On the traditional view a universal proposition asserts that its subject class has at least one member, so "all S is P" implies "some S is P". On the modern view a universal proposition is a conditional about anything whatever, ∀x(Sx ⊃ Px), which is true when nothing is S and therefore asserts nothing about existence. The two schemes agree entirely about particular propositions, which assert existence on both.

2. Why does the traditional scheme need existential import?

Because it accepts subalternation, the inference from A to I. If "all S is P" did not assert that some S exists, that inference would carry one from a true universal to a false particular whenever the subject class was empty, which no valid inference may do. Existential import is therefore not an add-on but a consequence of an inference the traditional square relies on.

3. Why does the modern scheme deny existential import?

Because it symbolises "all S is P" as ∀x(Sx ⊃ Px), and a conditional with a false antecedent is true. If nothing is S, the conditional holds of everything, so the universal is true. The symbolisation is itself forced, since writing the universal as a conjunction would assert that everything in the universe belongs to the subject class, so the denial of existential import follows rather than being chosen.

4. What does the modern scheme lose from the square of opposition?

Three of its four relations. Subalternation fails, since A no longer implies I. The contrary relation fails, since A and E can both be true when the subject class is empty. The subcontrary relation fails, since I and O can both be false for the same reason. Only the contradictory relations survive, A against O and E against I, so the square is reduced to its diagonals.

5. Apply the empty-class test to the four forms.

Take a subject class with no members, such as unicorns in this courtroom. On the modern reading "all unicorns here are white" and "no unicorn here is white" are both true, since there is nothing to falsify either, which shows the contrary relation failing. "Some unicorn here is white" and "some unicorn here is not white" are both false, since neither can find an instance, which shows the subcontrary relation failing. On the traditional reading the two universals are both false, since each asserts that unicorns are present.

6. What practical difference does the distinction make to reading a statute?

A provision such as "every person in occupation on a stated date shall be deemed a tenant" does not, properly read, assert that anybody was in occupation. It states what shall follow if anyone was. A party claiming under it must therefore prove his own occupation as a particular proposition; he cannot rely on the enacting words to establish that occupation existed. A legislature that wishes to assert existence has to do so separately, which is one reason Acts carry recitals and declarations.

"Aristotle is a Greek philosopher" does not assert that Aristotle is alive. "The accused is a person who struck the deceased on 4 April 2019" does not assert that the striking is going on now. The present tense of the copula is a grammatical convention, and what it expresses is a relation between the subject and the predicate, not a moment.

How time is expressed instead. Time goes into the terms, not into the copula. "The notice was served in March" becomes, in logical form, "the notice is a thing served in March", where the whole of the temporal information sits inside the predicate term. This is the device already used at sequence 300 for propositions about frequency, and it is the same device applied to tense.

Why it matters. Because inference in Module III operates on the copula, and every operation there, conversion, obversion, contraposition, assumes that the copula is a plain timeless link. A student who thinks the copula carries tense will produce nonsense at the first obversion.

The four things the English "is" does

English uses one word for four different jobs, and telling them apart is a standard examination question in this topic. Only the first is the copula.

The "is" of predication, the copula proper. "The document is registered." A quality is asserted of a thing.

The "is" of class membership. "Ravi is a lawyer." An individual is placed in a class. Traditional logic treats this as predication too, which is one of the places the modern scheme separates what the older one ran together, at sequence 340.

The "is" of identity. "The Managing Director is the person who signed the guarantee." Here the two sides name the same individual, and the proposition can be read in either direction, which no ordinary predication can. Identity statements are the reason some logics need a special sign for identity.

The "is" of existence. "There is a contract between the parties." Nothing is being predicated of anything; the claim is that something exists. This is failure five at sequence 320, and it is why the modern scheme uses a quantifier for existence rather than a predicate.

A worked distinction. "The tenant is the occupier" may be any of three of these. If it means that the tenant has the quality of occupying, it is predication. If it means that the tenant belongs to the class of occupiers, it is class membership. If it means that the person called the tenant and the person called the occupier are the same person, it is identity. In a dispute about who is liable to be evicted, the three readings lead to different results, and only the third supports the inference that anything true of the occupier is true of the tenant.

A worked example

A section provides: "A person is said to be an insider if he is or has been connected with the company."

Find the copulas. There are three "is" or "has been" constructions and they are not doing the same work.

"A person is said to be an insider" is a stipulative definition: the statute is laying down that the word "insider" shall apply. It is not an ordinary predication at all but an act of definition, which is Module IV's subject at sequence 570.

"if he is connected with the company" is a predication with a relational predicate, and the "is" here is the copula proper.

"or has been connected" is the same predication with the time shifted, and the shift is carried by the verb and not by the copula. In logical form the two together become one predicate: "x is a person who is or has at any time been connected with the company". The whole of the temporal range sits inside the term.

What the analysis shows. The definition covers past connection as well as present, and once the temporal words are inside the term, the proposition is timeless and the ordinary operations of Module III can be performed on it. Anyone who left "has been" outside the term would find that obversion produced something meaningless.

Distinctions that carry marks

What this does not mean

The copula is not the verb of the sentence. "The tenant occupies the premises" has no copula on its face; reduced to logical form it becomes "the tenant is a person who occupies the premises", and the copula appears.

The copula does not assert existence, and reading it as though it did is what produces the traditional doctrine of existential import.

Predication is not the same as description. A description names; a predication asserts. The difference is the presence of a copula doing its third job.

Quick revision

Predication: asserting a predicate of a subject. Copula: the sign that joins them.

Three functions of the copula: joining, carrying quality, asserting.

The copula is timeless. Its present tense is a convention, and time is expressed by putting temporal words inside the predicate term.

The copula is syncategorematic: not a term, and not to be counted among the terms of a proposition.

Four uses of the English "is": predication, class membership, identity, existence. Only the first is the copula.

Why it matters: every operation in Module III works on the copula, and treating it as tensed or as existential breaks all of them.

Test yourself

1. What is predication, and what is the copula?

Predication is the act of asserting a predicate of a subject, that is, of saying something about something. The copula is the sign of that joining, expressed in English by "is", "are", "is not" and "are not". Remove the copula and no assertion remains, only two terms placed side by side, which is why a term is not a proposition.

2. State the three functions of the copula.

It joins the subject term to the predicate term, making one proposition out of two terms. It carries the quality of the proposition, affirming or denying, which is why reduction to logical form requires any negative to be brought into it. And it asserts, signalling that the speaker commits to the proposition, which is why a conditional, whose parts contain copulas, asserts neither of them.

3. Why is the copula said to be timeless, and where does time go instead?

Because its present tense is a grammatical convention and expresses a relation rather than a moment: "Aristotle is a Greek philosopher" does not assert that Aristotle is alive. Time is expressed by absorbing the temporal words into the predicate term, so that "the notice was served in March" becomes "the notice is a thing served in March". This is necessary because every operation in Module III assumes a plain, timeless link.

4. Distinguish the four uses of the English "is".

The "is" of predication asserts a quality of a thing and is the copula proper. The "is" of class membership places an individual in a class. The "is" of identity says that the expressions on either side name the same individual, and is reversible in a way no ordinary predication is. The "is" of existence asserts that something exists and predicates nothing of anything, which is why modern logic uses a quantifier for it.

5. Why does the copula not assert existence?

Because its function is to link a subject to a predicate, not to claim that the subject has instances. Reading it as an assertion of existence is exactly what gives the traditional universal its existential import, and produces the result that "all trespassers will be prosecuted" asserts that there are trespassers. The modern scheme keeps the two apart by using a separate quantifier for existence.

6. Reduce "The tenant occupies the premises" to logical form and identify the copula.

It becomes "The tenant is a person who occupies the premises". The copula is "is", and it did not appear in the original sentence, whose verb was "occupies". This shows that the copula is not the verb of the sentence: it appears when the sentence is put into standard form, with the whole of the assertion about occupation absorbed into the predicate term.