Abstract

A logic/consequence relation |– is Carnap Categorical (CC) if the only interpretation of the logical constants consistent with |– is the standard interpretation. Categoricity results have been obtained for first-order logic, modal logics, intuitionistic propositional logic, and many others. In all of these cases, the proof that the conjunction symbol must have its standard interpretation is almost trivial; it is the other logical constants that require some work. This triviality looks suspicious. In fact, it essentially comes from  the definition of consistency. For example, in case of propositional logic with classical 2-valued semantics, the definition goes like this: an interpretation I is consistent with |–, if  A1,…,An |– B  implies that (for each assignment v of truth values to the propositional variables) if the value of A1 and … and A2 under I (and v) is 1, then the value of B under I (and v) is also 1. With an &-introduction rule of the form  A,B |– A&B,  it is no surprise that & has to mean ‘and’. In other words, the possible interpretations of & are biased by the fact that the commas to the left of |– are already taken to mean ‘and’. (Similarly, if we allow multiple conclusions; then commas to the right are taken to mean ‘or’.)

How can we avoid this bias? In this talk I avoid it by disallowing commas altogether, considering only so-called binary logics: consequence relations with exactly one premise and one conclusion. Note that the &-introduction rule above is not in this format; it can be replaced by a conditional rule in the binary format: if  A |– B  and  A|– C,  then  A |– B&C. I investigate what happens to the categoricity results when ‘and-bias’ is removed in this way. Categoricity of & (and other connectives) is no longer trivial: it still holds for (binary) classical propositional logic, but fails for some weaker logics when we impose the binary format. Results and counter-examples are presented in the setting of algebraic semantics, and of Kripke semantics for intuitionistic logic; the technical machinery that I use will be explained.