The simplest rule in logic may break when Hobbes wrote Hamlet
If P implies Q, and P holds, then Q holds. Modus ponens is about as basic as reasoning gets, and logicians have leaned on it since antiquity. Yet the philosopher Vann McGee built a puzzle about who wrote Hamlet in which following the rule step by step seems to lead somewhere absurd.
The Latin name means roughly the mode that affirms by affirming, and the rule also goes by affirming the antecedent or conditional elimination. It takes two premises, a conditional claim and the truth of its first half, and delivers the second half. Theophrastus, Aristotle's successor, was the first to spell it out explicitly. Its sibling, modus tollens, runs the other way, from a false consequence back to a false premise, and both have look-alike fallacies: affirming the consequent and denying the antecedent.
Validity is not truth. Take the argument that if today is Tuesday John goes to work, and today is Tuesday, so John goes to work. The form is valid every day of the week, but it is only sound when the premises are actually true. If John also works Wednesdays, the reasoning offers no explanation for why he is at his desk that day. Because the rule lets a proof drop a conditional once it has been used, it is also called the rule of detachment; Bertrand Russell described an inference as the dissolving of an implication, leaving just the conclusion behind.
The pattern turns up far from philosophy. Under the Curry-Howard correspondence, which pairs proofs with programs, it is simply function application: feed a function expecting type P an input of type P and you get a result of type Q. In artificial intelligence the same move is known as forward chaining.
McGee's challenge goes like this. Either Shakespeare or Hobbes wrote Hamlet. If one of them wrote it, then if Shakespeare did not, Hobbes did. Applying the rule, we conclude that if Shakespeare did not write Hamlet, Hobbes did, a statement few would accept. McGee argued the rule can fail when the conclusion is itself a conditional, and in some non-classical logics its validity cannot be taken for granted.
Source: Modus ponens
