In “Tonk, Plonk and Plink” Belnap  defended the claim that natural deduction rules define the meaning of a connective, provided at least that those rules form a conservative extension of the structural rules of deduction. In this paper we will investigate a stronger criterion of success for defining the meaning of a connective. Each connective comes with an intended interpretation, (for example, the intended interpretation of & is recorded by the truth table for &). For a set of rules to define the meaning of a connective, we would expect it to be categorical, i.e., we expect (roughly) that it force the intended interpretation of the connective on all its “models”. To put it another way, rules define a connective when they are strong enough to eliminate any non-standard interpretations.1
KeywordsModal Logic Categorical Semantic Ontological Commitment Canonical Model Semantical Condition
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