The Modern Definition of Logic
Chapter Two
Syllabus topic 1.1, "Traditional and Modern definitions of Logic."
Pages 8 to 12 of 334
In one line
Modern logic is the study of the methods and principles used to distinguish correct reasoning from incorrect reasoning.
In the wording a student can write in an examination: modern logic sets aside the question how the mind works and studies instead the relations that hold between propositions, expressing those relations in symbols so that the correctness of an inference can be tested by its form alone.
Why the definition changed
The traditional definitions were not abandoned because they were false. They were abandoned because they were unusable, and the previous chapter shows why: "the science of the laws of thought" cannot be told apart from psychology, and "the science of reasoning" quietly ignores two of the three things traditional logic actually studies.
There was a deeper objection, and it is the one that produced modern logic. If logic is about thought, then logic is about something that goes on inside a person's head, and different heads work differently. A rule about thinking is a rule about people. But "if all contracts are agreements and this is a contract, then this is an agreement" is not a fact about anybody's head. It would hold if nobody had ever thought it.
So the modern logician moves the subject matter. Logic is not about thought. It is about propositions and the relations between them. Whether anyone ever entertains the propositions is irrelevant, in the way that the truth of a theorem in geometry does not depend on anybody drawing the triangle.
The definition, taken apart
The definition most Indian syllabuses quote is Copi's, and MU's own reading list carries his book. It runs: logic is the study of the methods and principles used to distinguish correct from incorrect reasoning.
Methods and principles. Not habits, not tendencies. A method is something you can be taught and can apply deliberately, such as constructing a truth table. A principle is a general rule the method rests on. Both words are chosen to keep psychology out.
Used to distinguish. Logic is a test. It is not a description of what reasoning is like; it is an instrument for sorting reasoning into two heaps.
Correct from incorrect reasoning. Not true from false statements. The subject of the verdict is the reasoning, that is, the passage from premises to conclusion, and never the premises by themselves.
The four features of modern logic
It is anti-psychologistic. This is the word for the move described above: the refusal to explain logical relations as facts about how minds work. Its most famous statement is Gottlob Frege's, in the 1880s. Nothing in modern logic depends on how anybody actually thinks.
It is formal. Modern logic studies the form of an inference and disregards its matter, that is, what the inference is about. "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" are the same argument in different clothes. Every argument of that shape is valid, whatever the subject.
The Modern Definition of Logic
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.
The Modern Definition of Logic
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
| Traditional logic | Modern logic | |
|---|---|---|
| Founder | Aristotle, fourth century BC | Boole, Frege, Russell, from 1854 |
| Subject matter | Thought, and its laws | Propositions, and the relations between them |
| Basic unit | The term, inside a subject-predicate proposition | The whole proposition, taken as a unit |
| Method | Classification of propositions and rules of inference stated in words | Symbolisation and calculation |
| Handles relations? | Badly; "A is taller than B" does not fit the scheme | Yes; relational propositions are ordinary members of it |
| Existential import | A universal proposition asserts its subject exists | It does not |
| Verdict on an argument | Valid or invalid, by rules | Valid or invalid, by a mechanical test |
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.
The Modern Definition of Logic
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.
The Modern Definition of Logic
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.
The rest of this subject
These notes are cut from the University's printed syllabus. Open the syllabus itself, or the past papers, for the same subject.