Concept
Peirce's law
Overview
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.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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One of the most successfully cultivated branches of philosophy in our time is what is called inductive logic, the study of the conditions under which our sciences have evolved. Writers on this subject have begun to show a singular unanimity as to what the laws of nature and elements of fact mean, when formulated by mathematicians, physicists and chemists.
William James, Pragmatism
States what Peirce's law is.
Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.
Alfred North Whitehead, Science and the modern world
States what Peirce's law is.
LII. Let this suffice as to the respective dignity of prerogatives of instances. But it must be noted, that in this our organ, we treat of logic, and not of philosophy. Seeing, however, that our logic instructs and informs the understanding, in order that it may not, with the small hooks, as it were, of the mind, catch at, and grasp mere abstractions, but rather actually penetrate…
Francis Bacon, Novum organum
States what Peirce's law is.
Related ideas
- Philosophy
- Intuitionistic logic
- Law of excluded middle
- Propositional logic
- Modus ponens
- Minimal logic
- Logical consequence
- Many-valued logic
- Tautology (logic)
- Logic
- Deduction theorem
- Fuzzy logic
- Principle of explosion
- Charles Sanders Peirce
- Intermediate logic
- Disjunctive syllogism
- Axiom
- Consequentia mirabilis
Place in the tradition
Part of Intermediate logic.