Classical Logic , Computationally

The meaning of a proposition in classical logic is ordinarily given denotationally by saying that a proposition stands for either > or ⊥. The logical connectives are defined by truth tables displaying the graph of their associated functions on truth values. The denotational semantics of classical logic emphasizes the “absolute” truth or falsehood of a proposition, independently of how we, as humans, may come to learn it. In contrast, the meaning of a proposition in constructive logic is given operationally by defining what counts as a proof of it. The logical connectives are defined by rules for proving propositions formed with each connective. The operational semantics of constructive logic emphasizes the process of learning and communicating the truth of a proposition in a proof. The close connection between constructive logic and computer programming stems from its operational semantics. As we have seen, a proposition can be read as a specification, or problem statement, and a proof as a program, or solution to that problem. Classical logic, with its emphasis on absolute truth, ignores the operational aspects of proof, and consequently neglects the connection with programming. Despite the difference in semantics, the social processes of classical and constructive mathematics are largely the same. In particular, the conduct of classical mathematics centers on the communication of knowledge through proof, just as in constructive mathematics. This suggests that there may be an alternative semantics for classical logic that exposes its operational content, thereby establishing a connection to computer programming. The purpose of this note is to explore this connection.