Peirce's Law

Strong Form: Formulation 1
The logical implication does in fact go both ways:
 * $((p \implies q) \implies p) \dashv \vdash p$

Strong Form: Formulation 2

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

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


 * Peirce's Law Strong Form 1.0 Splash Page.png

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 the tableau method of natural deduction:

Now from True Statement is Implied by Every Statement, 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 Clavius's Law.

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 $B.\!$