Exhaustive Paradox


John asks Sally if she is tired.
Sally responds: "I'm not tired, I'm exhausted."
John responds: "Well, I'm glad to hear you aren't exhausted!"

Sally is perplexed. But so is John. Explain both perplexities.


Clearly, what Sally means is that she is not just tired. That is, she is both exhausted and tired. So John suggesting she is not exhausted is perplexing to Sally.

Still, claiming to not be tired and to be exhausted is an odd way to claim to be both tired and exhausted. At least, John thinks so. He reasons that for any agent S, if S is exhausted then S is tired, since being exhausted is an extreme form of being tired. Hence, to claim:

  1. S is exhausted


       2. S is tired

So claiming, in addition:

       3. S is not tired

Is inconsistent. But John is trying to be charitable, and so’d rather not attribute an inconsistency to Sally. Rather, John takes Sally at her word when she claims "I'm not tired." Moreover, John believes, as is plausible:

        4. S is not tired


        5. S is not exhausted

And since John is a nice guy, he's glad to hear Sally isn't exhausted given that she isn't tired. Of course, that still leaves Sally as speaking inconsistently, since she explicitly says she is exhausted, and this conflicts with (5). In other words, if John assumes Sally speaks consistently then either Sally is not tired and not exhausted or tired and exhausted, either of which conflicts with Sally’s expressed claim. John is perplexed because he seems forced to conclude his friend Sally is inconsistent.