Stone duality for first-order logic: a nominal approach to logic and topology

Abstract

What are variables, and what is universal quantification over a variable? Nominal sets are a notion of 'sets with names', and using equational axioms in nominal algebra these names can be given substitution and quantification actions. So we can axiomatise first-order logic as a nominal logical theory.We can then seek a nominal sets representation theorem in which predicates are interpreted as sets; logical conjunction is interpreted as sets intersection; negation as complement. Now what about substitution; what is it for substitution to act on a predicate-interpreted-as-a-set, in which case universal quantification becomes an infinite sets intersection?Given answers to these questions, we can seek notions of topology. What is the general notion of topological space of which our sets representation of predicates makes predicates into 'open sets'; and what specific class of topological spaces corresponds to the image of nominal algebras for first-order logic?The classic Stone duality answers these questions for Boolean algebras, representing them as Stone spaces. Nominal algebra lets us extend Boolean algebras to 'FOL-algebras', and nominal sets let us correspondingly extend Stone spaces to '∀-Stone spaces'. These extensions reveal a wealth of structure, and we obtain an attractive and self-contained account of logic and topology in which variables directly populate the denotation, and open predicates are interpreted as sets rather than functions from valuations to sets.

Quantifiers in logic and proof-search using permissive-nominal terms and sets

“…An investigation of models followed, in several styles. This includes translations of theories in permissive-nominal syntax to higher-order terms [GM09b,DG12b]-see also [LV12], which translates a simpler syntax and does not consider theories, but follows the same general idea-and it includes the topological model of first-order logic [Gab11b] mentioned above (essentially, a topological treatment of [GM08b] and of a first-order version of Figures 7 and 8 from this paper).…”

“…The similarity of syntax and semantics can make it look like nothing is happening: Definition 5.16 could be mistaken for a Hilbert axiomatisation; Definition 5.27 might seem to just translate syntax trivially to semantics. But this is deceptive: Subsection 7.2 constructed a functional model, and if that model is least somewhat familiar, in [GM11] and even more so in [Gab11b] we build (topological) models that are not derived from either syntax or functions, in which substitution and abstraction are interpreted completely independently of the apparatus of syntax and functions-and yet they still satisfy the axioms and have a natural construction.…”

Equivariant ZFA with Choice: a position paper

“…∈ A of atoms; elements that can be compared for equality but which have few if any other properties. This deceptively simple foundational assumption has many applications-nominal abstract syntax (syntax-with-binding) [9,11]; as implemented in Isabelle [12]; an open consistency problem [6]; duality results [5,8]; generalised finiteness for automata and regular languages [10,1]; rewriting with binding [3]; and more.…”

“…2. We sometimes specifically want non-supported elements; for example two recent papers [5,8] are concerned with sets that have a notion of nominal support, but whose elements do not. 3.…”

“…In the Isabelle implementation of FM in my PhD thesis[4] this was literally so: I used Quine atoms such that a = {a}. This removes the condition a, b ∈ A in (Ext), at a cost of some extremely mild nonwellfoundedness 5. To see this made precise see Subsection 2.6 and Lemma 4.17 of[2].…”