I recently encountered Peirce's law, which states for a given proposition P, if there exists a proposition Q such that P follows from "if P then Q", then P is true. Alternatively: ((P→Q)→P)→P. Below, I've described a scenario that I believe illustrates a serious problem with this law. Could someone please defend Peirce's law? I have seen its symbolic proof, but I found it difficult to build an argument from it in English without resorting to arcane philosophical terminology. So I would be grateful if someone could attack this question using plain English.

There is a room where a button is affixed to a candy dispenser. You and your friend enter the room. Your friend says, "I will push the button if doing so will result in candy being released from the dispenser." Your friend then walks over to the dispenser and reads a placard that informs her as to whether pressing the button would result in candy being released. You leave the room. Later, you see your friend and ask her whether she pushed the button. She says "no".

Here are two propositions:

P=Your friend pushed the button

Q=Candy was released from the dispenser.

If your friend's initial statement was true, then (P→Q)→P. By Peirce's Law, it follows that your friend pushed the button. And yet she claims she did not. Is your friend a liar?


Here is Peirce's own defence:

"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, Collected Papers 3.384]

Shorter: if P implying random something implies the P itself then P better be true in the first place.

But the intuitive meaning of conditional promises such as "I will do X if condition C is met" is by no means expressed by the material conditional C→X. It expresses that not X is incompatible with C, there is no "freedom" to choose as in the promise. As truth tables or Boolean algebra readily show, (P→Q)→P and P always have identical truth values. So if we insist on treating promises this way then "I will push the button if doing so will result in candy being released from the dispenser" is just a long way of saying "I will push the button". So either she "lied" in her long-winded declaration or in her "no".

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As a matter of observation, it’s worth pointing out that all the arrows and variables here mean the same thing. Pierce’s law is not ‘(P implies Q) implies P entails P’.

Our intuition around implication is always to understand the outermost statement as an argument, and the inner ones as hypotheticals. But you can’t try to interpret the statement by switching contexts, because that equivocates on the meaning of statements inside and outside the hypothetical.

It’s better to understand the statement as an algebraic sentence and to appeal directly to the semantics or the sound proof methods of the logic. This makes sure you’re meaning the same thing each time you mention P.

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