Many-valued propositional intuitionism

The semantics for intuitionistic logic is based on “proof” rather than on “truth”. Since we frequently classify proofs in various ways, we are naturally motivated to consider many-valued theories of intuitionism. In this paper we provide a general syntactic and semantic theory for many-valued propositional intuitionism, where the values intuitively refer to types of proofs. Soundness and completeness results are presented, and it is shown that the many-valued theory is a conservative extension of the usual intuitionistic propositional calculus.