Peirce's law
Encyclopedia
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". In particular, when Q is taken to be a false formula, the law says that if P must be true whenever it implies the false, then P is true. In this way Peirce's law implies the law of excluded middle
.
Peirce's law does not hold in intuitionistic logic
or intermediate logic
s 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.
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, 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
Law of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
written in a form that involves only one sort of connective, namely implication.
In propositional calculus
Propositional calculus
In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...
, 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". In particular, when Q is taken to be a false formula, the law says that if P must be true whenever it implies the false, then P is true. In this way Peirce's law implies the law of excluded middle
Law of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
.
Peirce's law does not hold in intuitionistic logic
Intuitionistic logic
Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
or intermediate logic
Intermediate logic
In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic, thus consistent superintuitionistic logics are called intermediate logics .-Definition:A superintuitionistic logic is a...
s and cannot be deduced from the deduction theorem
Deduction theorem
In mathematical logic, the deduction theorem is a metatheorem of first-order logic. It is a formalization of the common proof technique in which an implication A → B is proved by assuming A and then proving B from this assumption. The deduction theorem explains why proofs of conditional...
alone.
Under the Curry–Howard isomorphism, Peirce's law is the type of continuation
Continuation
In computer science and programming, a continuation is an abstract representation of the control state of a computer program. A continuation reifies the program control state, i.e...
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:
| {(x → y) → x} → x.>- 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:
| {(x → y) → a} → x,>- 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: ((x→y)→a)→x is not a tautology. However, [a→x]→[((x→y)→a)→x] is a tautology.
Other proofs of Peirce's law
Showing Peirce's Law applies does not mean that P→Q or Q is true, we have that P is true but only (P→Q)→P, not P→(P→Q) (see affirming the consequentAffirming the consequentAffirming the consequent, sometimes called converse error, is a formal fallacy, committed by reasoning in the form:#If P, then Q.#Q.#Therefore, P....
).
simple proof:
Using Peirce's law with the deduction theorem
Peirce's law allows one to enhance the technique of using the deduction theoremDeduction theoremIn mathematical logic, the deduction theorem is a metatheorem of first-order logic. It is a formalization of the common proof technique in which an implication A → B is proved by assuming A and then proving B from this assumption. The deduction theorem explains why proofs of conditional...
to prove theorems. Suppose one is given a set of premises Γ and one wants to deduce a proposition Z from them. With Peirce's law, one can add (at no cost) additional premises of the form Z→P to Γ. For example, suppose we are given P→Z and (P→Q)→Z and we wish to deduce Z so that we can use the deduction theorem to conclude that (P→Z)→(((P→Q)→Z)→Z) is a theorem. Then we can add another premise Z→Q. From that and P→Z, we get P→Q. Then we apply modus ponens with (P→Q)→Z as the major premise to get Z. Applying the deduction theorem, we get that (Z→Q)→Z follows from the original premises. Then we use Peirce's law in the form ((Z→Q)→Z)→Z and modus ponens to derive Z from the original premises. Then we can finish off proving the theorem as we originally intended.-
- P→Z 1. hypothesis
- (P→Q)→Z 2. hypothesis
- Z→Q 3. hypothesis
- P 4. hypothesis
- Z 5. modus ponens using steps 4 and 1
- Q 6. modus ponens using steps 5 and 3
- P→Q 7. deduction from 4 to 6
- Z 8. modus ponens using steps 7 and 2
- Z→Q 3. hypothesis
- (Z→Q)→Z 9. deduction from 3 to 8
- ((Z→Q)→Z)→Z 10. Peirce's law
- Z 11. modus ponens using steps 9 and 10
- (P→Q)→Z 2. hypothesis
- ((P→Q)→Z)→Z 12. deduction from 2 to 11
- P→Z 1. hypothesis
- (P→Z)→((P→Q)→Z)→Z) 13. deduction from 1 to 12 QED
Completeness of the implicational propositional calculus
One reason that Peirce's law is important is that it can substitute for the law of excluded middle in the logic which only uses implication. The sentences which can be deduced from the axiom schemas:- P→(Q→P)
- (P→(Q→R))→((P→Q)→(P→R))
- ((P→Q)→P)→P
- from P and P→Q infer Q
(where P,Q,R contain only "→" as a connective) are all the tautologiesTautology (logic)In logic, a tautology is a formula which is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; it had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense...
which use only "→" as a connective.