2016-12-07: Jamie Gabbay (Heriot-Watt)

Tue, 14 Oct 2025 16:51:42 +0000

Jamie Gabbay (Heriot-Watt)

Nominal filters and semantics of predicate logic

Abstract

A filter P is a consistent deductively closed set of predicates. A filter is prime when

(φ ∨ ψ) ∈ P ⇒ (φ ∈ P ∨ ψ ∈ P)
In words: if phi-or-psi is in P then phi is in P or psi is in P. Primeness gives soundness for disjunction.

Using this it is not hard to construct a semantics to propositional logic in which a predicate φ "means" the set of prime filters containing it. This is a standard "trick" for building semantics and is an extremely useful proof-method.

I have developed a semantics for predicate logic and also for the lambda-calculus based on similar notions of filter, but in a nominal context — meaning that filters are developed using Fraenkel-Mostowski (FM) set theory instead of Zermelo-Fraenkel (ZF) set theory. What matters here is that FM sets have additional name structure over ZF sets, and this additional structure can be exploited to give semantics to the extra structure that predicates have over propositions, and in particular the additional name structure lets us write down primeness conditions for soundness for universal quantification.

The resulting semantics is rich and interesting. In a sentence: nominal techniques help us to extend the notion of Stone representation and duality from propositional logic to full first-order logic (also with equality, if we wish, and also to other logics and calculi with variables and quantifiers).

I will give a detailed description of the filter-style conditions involved, and discuss some of what I think they tell us about predicates and quantification in logic and computation. More information can also be found in two papers here:
http://www.gabbay.org.uk/papers.html#semooc
http://www.gabbay.org.uk/papers.html#repdul

2016-12-07 11:00:00 UTC
LT1310