31 - Classical Logic

Summary

In constructive logic a proposition is true exactly when it has a proof, a derivation of it from axioms and assumptions, and is false exactly when it has a refutation, a derivation of a contradiction from the assumption that it is true. Constructive logic is a logic of positive evidence. To affirm or deny a proposition requires a proof, either of the proposition itself or of a contradiction, under the assumption that it has a proof. We are not always in a position to affirm or deny a proposition. An open problem is one for which we have neither a proof nor a refutation-so that, constructively speaking, it is neither true nor false.

In contrast, classical logic (the one we learned in school) is a logic of perfect information in which every proposition is either true or false. We may say that classical logic corresponds to “god's view” of the world-there are no open problems; rather, all propositions are either true or false. Put another way, to assert that every proposition is either true or false is to weaken the notion of truth to encompass all that is not false, dually to the constructively (and classically) valid interpretation of falsity as all that is not true. The symmetry between truth and falsity is appealing, but there is a price to pay for this: The meanings of the logical connectives are weaker in the classical case than in the constructive.