Failures of Categoricity and Compositionality for Intuitionistic Disjunction∗

There are at least two reasonable types of logical inferentialism. The first type of inferentialism views the meaning of a connective in proof-theoretic terms, having no truck with ordinary model theory except in its role as an instrument to illuminate various proof-theoretic features of a connective. The second type of inferentialist is more modest, allowing that we have some prior conception of the meaning that the natural deduction rules for a particular connective determine. Such a position would look to the conditions forced by acceptance of a set of rules against a background model theory for an account of connective meaning. This position is moderate, plausible, and problematic. In the case of classical logic, against the presumption of standard extensional semantics, Carnap long ago demonstrated the general failure of proof-rules to induce the intended boolean interpretation of the connectives. There have been a number of attempts to solve this problem by modifying the natural deduction format, but none have achieved anything like a consensus.