Peirce's law

Inlogic,Peirce's lawis named after thephilosopherandlogician Charles Sanders Peirce.It was taken as anaxiomin his first axiomatisation ofpropositional logic.It can be thought of as thelaw of excluded middlewritten in a form that involves only one sort of connective, namely implication.

Inpropositional calculus,Peirce's lawsays that ((P→Q)→P)→P.Written out, this means thatPmust be true if there is a propositionQsuch that the truth ofPfollows fromthe truth of "ifPthenQ".

Peirce's law does not hold inintuitionistic logicorintermediate logicsand cannot be deduced from thededuction theoremalone.

Under theCurry–Howard isomorphism,Peirce's law is the type ofcontinuationoperators, e.g.call/ccinScheme.^[1]

History

Here is Peirce's own statement of the law:

Afifth iconis required for the principle ofexcluded middleand 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 consequentxbeing false while its antecedent (x⤙y) ⤙xis true. If this is true, either its consequent,x,is true, when the whole formula would be true, or its antecedentx⤙yis false. But in the last case the antecedent ofx⤙y,that isx,must be true. (Peirce, theCollected Papers3.384).

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 theais used in such a sense that (x⤙y) ⤙ameans 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 ofxfollows the truth ofx.(Peirce, theCollected Papers3.384).

Warning:As explained in the text, "a"here does not denote a propositional atom, but something like thequantified propositional formula $\forall p\,p$ .The formula ((x→y) →a) →xwould not be atautologyifawere interpreted as an atom.

Relations between principles

In intuitionistic logic, if $P$ is proven or rejected, or if $Q$ is provenly valid, then Piece's law for the two propositions holds. But the law's special case when $Q$ is rejected, calledconsequentia mirabilis,is equivalent to excluded middle already overminimal logic.This also means that Piece's law entails classical logic over intuitionistic logic, as also shown below. Intuitionistically, not even the constraint $\neg Q\to P$ implies the law for two propositions. Postulating the latter to be valid results inSmetanich's intermediate logic.

It is helpful to consider Pierce's law in the equivalent form $((P\to Q)\to (P\land Q))\to P$ .Indeed, from $P\to Q$ follows $P\leftrightarrow (P\land Q)$ ,and so $(P\to Q)\to P$ is equivalent to $(P\to Q)\to (P\land Q)$ .The case $Q=\bot$ now directly shows how double-negation elimination $\neg \neg P\to P$ implies consequentia mirabilis over minimal logic.

In intuitionistic logic, explosion can be used for $\bot \to (P\land \bot )$ ,and so here the law's special case consequentia mirabilis also implies double-negation elimination. As the double-negated excluded middle is always already valid even in minimal logic, this also intuitionistically proves excluded middle. In the other direction, one can intuitionistically also show that excluded middle implies Piece's law directly. To this end, note that using theprinciple of explosion,excluded middle may be expressed as $P\lor (P\to Q)$ .In words, this may be expressed as: "Every proposition $P$ either holds or implies any other proposition. " Now to prove the law, note that $(P\lor R)\to ((R\to P)\to P)$ is derivable from just implication introduction on the one hand andmodus ponenson the other. Finally, in place of $R$ consider $P\to Q$ .

Another proof of the law in classical logic proceeds by passing through the classically valid reversedisjunctive syllogismtwice: First note that $\neg \neg P$ is implied by $(\neg \neg P\land \neg Q)\lor P$ ,which is intuitionistically equivalent to $\neg (\neg P\lor Q)\lor P$ .Now explosion entails that $\neg A\lor B$ implies $A\to B$ ,and using excluded middle for $A$ entails that these two are in fact equivalent. Taken together, this means that in classical logic $P$ is equivalent to $(P\to Q)\to P$ .

Using Peirce's law with the deduction theorem

Peirce's law allows one to enhance the technique of using thededuction theoremto prove theorems. Suppose one is given a set of premises Γ and one wants to deduce a propositionZfrom them. With Peirce's law, one can add (at no cost) additional premises of the formZ→Pto Γ. For example, suppose we are givenP→Zand (P→Q)→Zand we wish to deduceZso that we can use the deduction theorem to conclude that (P→Z)→(((P→Q)→Z)→Z) is a theorem. Then we can add another premiseZ→Q.From that andP→Z,we getP→Q.Then we apply modus ponens with (P→Q)→Zas the major premise to getZ.Applying the deduction theorem, we get that (Z→Q)→Zfollows from the original premises. Then we use Peirce's law in the form ((Z→Q)→Z)→Zand modus ponens to deriveZfrom 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)→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)→Z	12. deduction from 2 to 11
(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
fromPandP→QinferQ

(whereP,Q,Rcontain only "→" as a connective) are all thetautologieswhich use only "→" as a connective.

Failure in non-classical models of intuitionistic logic

Since Peirce's law implies the law of the excluded middle, it must always fail in non-classical intuitionistic logics. A simple explicit counterexample is that ofGödel many valued logics,which are afuzzy logicwhere truth values are real numbers between 0 and 1, with material implication defined by:

{\begin{aligned}u\mathrel {\xrightarrow[{G}]{}} v&={\begin{cases}1,&{\text{if }}u\leq v\\v,&{\text{if }}u>v\end{cases}}\end{aligned}}

and where Peirce's law as a formula can be simplified to:

{\begin{aligned}((u\mathrel {\xrightarrow[{G}]{}} v)\mathrel {\xrightarrow[{G}]{}} u)\mathrel {\xrightarrow[{G}]{}} u&={\begin{cases}1,&{\text{if }}u\leq v\\u,&{\text{if }}u>v\end{cases}}\end{aligned}}

where it always being true would be equivalent to the statement that u > v implies u = 1, which is true only if 0 and 1 are the only allowed values. At the same time however, the expression cannot ever be equal to the bottom truth value of the logic and its double negation is always true.

Notes

^Timothy G. Griffin, A Formulae-as-Types Notion of Control, 1990- Griffin defines K on page 3 as an equivalent to Scheme's call/cc and then discusses its type being the equivalent of Peirce's law at the end of section 5 on page 9.