Presentation

Some central questions of metaphysics revolve around propositions, properties and relations and around how they interact with modality: What does it mean for a necessity to be broader than another? Is there such a thing as a broadest necessity? what does it mean for a property or relation to be fundamental? Are fundamental properties and relations `freely recombinable’ in the sense that they can instantiate any logically consistent pattern? Are propositions structured entities constructed out of fundamental properties and relations?   More generally, to what extent can we make good on the metaphor of reality as a language, and the corresponding metaphor of the fundamental as the primitive constants of that language? 

Unfortunately, debates on such questions are often formulated in ways that are prone to lead to paradoxes. Higher-order logic is a precise framework for regimenting such debates. It is a generalization of first-order logic that allows one to quantify not just into singular position but into any given grammatical position. It allows one to replace quantification over propositions, properties and modalities by quantification into sentence, predicate or operator position. Higher-order logic is often studied in a broadly Fregean setting  which in effect assumes that there are only two propositions. Once this assumption is lifted there are a surprising number of choice points regarding the logic and model theory of higher-order logic, corresponding to a number of different metaphysical pictures of the granularity of properties and propositions. Many of the technical tools in this area have been developed by computer scientists and category theorists but aren’t widely known to philosophers.