Øystein Linnebo
University of Bristol
Sets, Properties, and Semantics
In standard set theoretic semantics, both the domain and the semantic values of predicates
are taken to be sets. But as has long been known, this gives rise to difficulties when we
seek to develop a semantics for the language of set theory itself. For instance, if the
domain of set theory is taken to be a set, then Russell’s paradox re-emerges. I examine
this push towards paradox and find it to depend on two assumptions: firstly, that the
domain and the semantic values of predicates are objects; and secondly, that these objects
have an extensional nature.
A currently fashionable response to this threat of paradox denies the first assumption but
holds on to the second (Boolos, Lewis, Rayo). Others deny both assumptions
(Williamson). I argue that the correct response is to hold on to the first assumption but
deny the second. That is, I argue that both the domain and the semantic values of
predicates should be taken to be objects with an intensional nature—or properties, as I
will call them. I end with some remarks about the individuation and existence of
properties.