Predicates, predicativity, and second-order logic
- Date: Wednesday 12 December 2018, 16:15 – 17:15
- Location: Roger Stevens LT 18 (8.18)
- Type: Logic, Seminars, Pure Mathematics
- Cost: Free
Kentaro Fujimoto, University of Bristol. Joint Mathematics and Philosophy Logic Seminar. Part of the Logic Seminar Series.
What are the entities of second-order in second-order logic? There are different views, and the answer depends on what is formalised by second-order logic. They may be sets of first-order objects, of course. They may be attributes of first-order objects, or pluralities of them, or mereological parts of the totality of the first-order objects, etc.
In this talk, I will focus on a particular context of the foundation of mathematics. The question is, if the foundational framework of mathematics is given with a second-order formalism, then what are second-order entities of the framework? My claim is that taking predicates of the first-order objects as second-order entities is a good option for many cases (particularly for second-order set theory and second-order arithmetic), and I will try to investigate a few consequences of this view.
In particular, I will consider how predicativism would be conceived in terms of this conception of second-order formalism. The term "predicativity" alludes to the term "predicates." Quine, however, wrote in his textbook that they must be firmly dissociated. Nonetheless, I think that the basic idea of predicativism can be well explained in terms of predicates. I will further argue that the resulting understanding of predicativity might be able to explain the continuity of predicativity and metapredicativity (in the original definitions) and discontinuity of them from impredicativity.