The Laws of Thought
Chapter Sixteen
Syllabus topic 1.6, "Laws of Thought."
Pages 77 to 81 of 334
In one line
The three laws of thought are the most basic principles any piece of coherent reasoning must obey: a thing is what it is; nothing can both be and not be; and everything either is or is not.
In the wording a student can write in an examination: the traditional laws of thought are the law of identity, the law of contradiction and the law of excluded middle. They are not descriptions of how people happen to think but necessary conditions of any thinking at all, since abandoning them makes assertion and denial meaningless.
Why they are called laws of thought
The name is old and slightly misleading, and the misleading part is examinable.
They were called laws of thought because it was supposed that they state how the mind must operate. On that reading they belong to psychology, and the difficulty of chapter 10 returns: psychology describes how people actually think, and people in fact contradict themselves constantly.
The modern reading is different and better. They are not laws about minds at all. They are laws about propositions and about the world. The law of contradiction does not say that nobody can believe a contradiction; plainly people do. It says that no proposition can be both true and false, and that no thing can both have and lack the same property at the same time in the same respect.
Both readings should be mentioned in an answer, because the shift from one to the other is exactly the shift from traditional to modern logic described at sequence 20.
The law of identity
Whatever is, is. A thing is identical with itself. If a proposition is true, then it is true.
In symbols, A is A, or p if and only if p.
It looks empty, and its content shows itself only when it is broken. The law is what forbids a term from changing its meaning in the middle of an argument. If "possession" means physical control in the first premise, the law of identity requires that it mean physical control in the conclusion. An argument that lets the word shift has broken the law of identity, and the resulting fallacy has a name, equivocation, which Module IV meets again as a fault of definition.
That is the practical content of the law: one word, one meaning, throughout one argument. It is not a triviality when applied to a document.
The law of contradiction
Nothing can both be and not be. No proposition can be both true and false at the same time and in the same respect.
In symbols, not both p and not p.
The qualifications are the whole of the law's usefulness, and students omit them.
The Laws of Thought
"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.
The Laws of Thought
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
| Law | Statement | Symbolic form | What it forbids |
|---|---|---|---|
| Identity | Whatever is, is | A is A | A term changing meaning within an argument |
| Contradiction | Nothing both is and is not | Not both p and not p | Both members of a contradictory pair being true |
| Excluded middle | Everything either is or is not | Either p or not p | Both members of a contradictory pair being false |
| Law of contradiction | Law of excluded middle | |
|---|---|---|
| Says | They cannot both be true | They cannot both be false |
| Applies to | Contradictories and contraries | Contradictories only |
| Seriously disputed | No | Yes, for future and vague propositions |
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 Laws of Thought
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?
The Laws of Thought
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.
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.