Peirce's Law

Peirce's law is a formula in propositional calculus that is commonly expressed in the following form:

$$((p \implies q) \implies p) \implies p$$

Peirce's law holds in classical propositional calculus, but not in intuitionistic propositional calculus. The precise axiom system that one chooses for classical propositional calculus determines whether Peirce's law is taken as an axiom or proven as a theorem.

History
Here is Peirce's own statement and proof 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:

$$\{ (x \prec y) \prec x \} \prec x.$$

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 $$(x \prec y) \prec x$$ is true. If this is true, either its consequent, $$x,\!$$ is true, when the whole formula would be true, or its antecedent $$x \prec y$$ is false. But in the last case the antecedent of $$x \prec y,$$ that is $$x,\!$$ must be true. (Peirce, CP 3.384).

Peirce goes on to point out an immediate application of the law:

From the formula just given, we at once get:

$$\{ (x \prec y) \prec a \} \prec x,$$

where the $$a\!$$ is used in such a sense that $$(x \prec y) \prec a$$ means that from $$(x \prec y)$$ 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, CP 3.384).

Note. The above transcription uses the "precedes sign" ($$\prec$$) for the "sign of illation" that Peirce customarily wrote as a cursive symbol somewhat like a gamma ($$\gamma\!$$) turned on its side or else typed as a bigram consisting of a dash and a "less than" sign.

Context : Classical Propositional Calculus
Classical Propositional Calculus

Statement
$$((p \implies q) \implies p) \implies p$$

Proof
See below.

Context : Logical Graphs
Logical Graphs

Statement
Under the existential interpretation of logical graphs, Peirce's law is represented by means of the following formal equivalence or logical equation.

Proof by Logical Graphs
Using the axiom set given in the entry for logical graphs, Peirce's law may be proved in the following manner.

Proof by Natural Deduction
This is proved by the tableau method:

Comment
A non-obvious result that has the same strength as the Law of the Excluded Middle.

Statement
A stronger form of Peirce's law also holds, in which the final implication is observed to be reversible:

$$((p \implies q) \implies p) \iff p$$

Proof 1
Given what precedes, it remains to show that:

$$p \implies ((p \implies q) \implies p)$$

But this is immediate, since $$p \implies (r \implies p)$$ for any proposition $$r.\!$$

Proof 2
Representing propositions as logical graphs under the existential interpretation, the strong form of Peirce's law is expressed by the following equation:

Using the axioms and theorems listed in the entry for logical graphs, the equational form of Peirce's law may be proved in the following manner:

Proof by Truth Table
Let $$v: \left\{{p, q}\right\} \to \left\{{T, F}\right\}$$ be an interpretation for a logical formula $$\phi$$ of two variables $$p, q$$.

We see that $$v \left({\left({p \implies q}\right) \implies p}\right) = v \left({p}\right)$$ for all interpretations $$v$$.

By the definition of interderivable, we have that $$\left({\left({p \implies q}\right) \implies p}\right) \dashv \vdash p$$.

Hence, as Equivalences are Interderivable, the result follows.