The Premise Paradox

This paper argues that if our talk of “premises” is not handled carefully, a classical language will contain sentences that are both true and false. This is so, even if a “premise” is defined in non-semantic terms, viz., as a sentence that is underived in the context of a proof. As a preliminary, I first explain that in classical logic, expressions must be seen as linguistic types rather than tokens. (Otherwise, ‘this very term = this very term’ is a false instance of the Law of Identity.) Yet, if expressions are types, one can then derive a contradiction from the following premises: (1) Socrates is mortal; (2) (5) is not a premise, In the relevant proof, (5) is a derived sentence that is type-identical to ‘Socrates is mortal’. In such a proof, one can then show that sentence (1) is not a premise. A more sophisticated, proof-theoretic version of the paradox is also presented.