Higher Order Logic

For this session of PhiMath, we read Bob Hale’s paper “Properties and the Interpretation of Second-Order Logic”. He defends a deflationary conception of properties, which are the things we quantify over in second order logic. According to Hale, something is a property iff there could be a predicate that stands for it.

He looks into Quine’s claim that second-order logic is “set theory in sheep’s clothing”, which he understands as a broader charge motivated by the idea that second-order quantification carries heavy existential commitments. According to him, his deflationary interpretation of second-order logic can resist Quine’s critique. During the meeting, we consider Hale’s proposal and its implications for second-order logic.

We are continuing our discussion about Higher-Order Logic (HOL), this time focusing specifically on arguments against adopting Higher-Order logics for various purposes. These arguments range from (HOL) being set theory in disguise, the (lack of) applicability of (HOL) for its intended purposes, and the serious metalogical issues the logic faces.

The reading group in the philosophy of mathematics is back and we hold the first meeting on the 12th of October. The topic will be Higher-Order Logic. We will base the discussion around Stewart Shapiro’s chapter “Higher-order logic” in The Oxford Handbook of Philosophy of Mathematics and Logic.

Tibo will give a short presentation supporting the (in)famous Quine thesis that SOL is in reality ST, followed by a debate. The rules will be uploaded in due time.

Slides used during the presentation are available here (link now unfortunately broken).

Suggested Bibliography (They are freely accessible from your UvA student account)

• Quine W.V.O., Philosophy of Logic (2nd Edition), Chapter 5: ‘The Scope of Logic’, Harvard University Press, 1986. [PDF accessible from the UvA Library]