Substitution, Quantifiers and Identity in Modal Logic

Abstract

The categorical point of view in logic tries to understand the meaning of all the connectives and operators in a structural, conceptual way as opposed to the purely descriptive point of view. What is in question is the explanation of the linguistic phenomena that is not reduced to a mere description, say, of the logical laws that hold or do not hold in a given context. For questions related to philosophy the intuitive appeal to the natural language is fundamental, but we also feel that some motivations and methods of other nature might be needed too, for instance in order to produce systems able to originate satisfactory mathematical research, as happens in the propositional case for the main non-classical logics. This is what we are trying in this paper to do for quantified modal logic: we basically review the material of Ghilardi/Meloni (1988), (1991a), (1991b), by stressing the conceptual methods which made the investigations themselves possible and by emphasizing the philosophical applications that arose almost immediately after the main key points about substitution became clear. On the other hand, readers more interested in the original mathematical motivations and results are referred to the above-mentioned papers.