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?