How is the law of contraposition a tautology? I recently started the study of Aristotelian logic in Math class. I wanted to ask (as the title suggests) why the law of contraposition is a tautology. My book states that a tautology is a statement which is true for the all the values of the variables included. I ask this because I constructed the truth table for the law, but the statement didn't satisfy the definition of a tautology. It would be much appreciated if anyone would help me understand this, thanks.

Just fill in the truth table like this:


P | Q | P → Q | ~Q → ~P | (P → Q) ↔ (~Q → ~P)
––|–––|–––––––|––––––––––|––––––––––––––––––––
T | T | T | T | T
T | F | F | F | T
F | T | T | T | T
F | F | T | T | T


As you see, these two have identical truth values, and they are _logically equivalent_. It means that `(P → Q) ↔ (~Q → ~P)` is a tautology, not `~Q → ~P`. If two things are logically equivalent, they create a tautology when in the form of a biconditional.

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