mathematics

Peirce's law

Peirce's law is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Peirce's law rather than just read about it. In short: In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce. It was taken as an axiom in his first axiomatisation of propositional logic.

Key takeaways

Peirce's law belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Peirce's law to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Peirce's law from memory before moving on to harder problems.

Reference excerpt

In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce. It was taken as an axiom in his first axiomatisation of propositional logic. It can be thought of as the law of excluded middle written in a form that involves only one sort of connective, namely implication. In propositional calculus, Peirce's law says that ((P→Q)→P)→P. Written out, this means that P must be true if there is a proposition Q such that the truth of P follows from the truth of "if P then Q". Peirce's law does not hold in intuitionistic logic or intermediate logics and cannot be deduced from the deduction theorem alone. Under the Curry–Howard isomorphism, Peirce's law is the type of continuation operators, e.g. call/cc in Scheme.

History Here is Peirce's own statement of the law:

A fifth icon is required for the principle of excluded middle and other propositions connected with it. One of the simplest formulae of this kind is:

This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent x being false while its antecedent (x ⤙ y) ⤙ x is true. If this is true, either its consequent, x, is true, when the whole formula would be true, or its antecedent x ⤙ y is false. But in the last case the antecedent of x ⤙ y, that is x, must be true. (Peirce, the Collected Papers 3.384). Peirce goes on to point out an immediate application of the law:

From the formula just given, we at once get:

where the a is used in such a sense that (x ⤙ y) ⤙ a means that from (x ⤙ y) every proposition follows. With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of x follows the truth of x. (Peirce, the Collected Papers 3.384). Warning: As explained in the text, "a" here does not denote a propositional atom, but something like the quantified propositional formula ∀ p p {\displaystyle \forall p\,p} . The formula ((x → y) → a) → x would not be a tautology if a were interpreted as an atom.

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Peirce's law

Start with the simplest possible case. Write down what Peirce's law claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Peirce's law before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Peirce's law ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Peirce's law

In research: Peirce's law appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Peirce's law in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Peirce's law is common in secondary-school and first-year university syllabi. It links to neighbouring topics Charles Sanders Peirce, Intuitionism, Mathematical logic, so understanding it makes those chapters shorter.
In everyday life: Look for Peirce's law outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Peirce's law in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Peirce's law means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Peirce's law out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Peirce's law in simple terms?

In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce. It was taken as an axiom in his first axiomatisation of propositional logic.

Why does Peirce's law matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Peirce's law?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Peirce's law.