Peirce's Law

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


 * $$((p \implies q) \implies p) \vdash p$$

or, which is equivalent:
 * $$((p \implies q) \implies p) \implies p$$

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.

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

Strong Form of Peirce's Law
The logical implication does in fact go both ways:
 * $$((p \implies q) \implies p) \dashv \vdash p$$

or equivalently:
 * $$((p \implies q) \implies p) \iff p$$

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

Proof by Logical Graphs
First we show $$((p \implies q) \implies p) \implies p$$:

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

The steps of the proof are replayed in the following animation.

Similarly, the strong form may be proved in the following manner:

The following animation replays the steps of the proof.

Proof by Natural deduction
By the Tableau method:

Now from Implication Properties, we have $$p \vdash r \implies p$$ which is equivalent to $$p \implies (r \implies p)$$.

Now we put a substitution instance of $$p \implies q$$ for $$r$$, and:
 * $$p \implies ((p \implies q) \implies p)$$

follows immediately.

Proof by Truth Table
We can directly prove $$((p \implies q) \implies p) \dashv \vdash p$$.

We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, the truth values under the main connectives match for all models.

$$\begin{array}{|ccccc||c|}\hline ((p & \implies & q) & \implies & p) & p \\ \hline F & T & F & F & F & F \\ F & T & T & F & F & F \\ T & F & F & T & T & T \\ T & T & T & T & T & T \\ \hline \end{array}$$

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

Consider $$\left({\left({p \implies q}\right) \implies p}\right) \implies p$$.

Now let $$q = \bot$$.

So we have $$((p\implies \bot)\implies p)\implies p$$.

What means $$(\neg p\implies p)\implies p$$. That is Consequentia Mirabilis.

Now let $$\,\!q=p$$

So we have $$((p\implies p)\implies p)\implies p$$.

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

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

Note. Peirce uses the sign of illation “$$-\!\!\!<$$” for implication. In one place he explains “$$-\!\!\!<$$” as a variant of the sign “$$\le$$” for less than or equal to; in another place he suggests that $$A \,-\!\!\!< B$$ is an iconic way of representing a state of affairs where $$A,\!$$ in every way that it can be, is $$B.\!$$