A logic of co-valuations
Maciej Malicki
TL;DR
This work introduces co-valuations as a semantic mechanism for interpreting free variables in logic and develops a full model theory around them, including $Łoś$'s theorem, compactness, and omitting types. It leverages the Baťaroš–Bice–Vignati duality between ω-posets and second-countable compact $T_1$ spaces to connect logical structures with compact topologies, even though topology is not part of the initial formulation. The theory encompasses compact ω-structures, their expressible topological properties, and a Polish space ${\rm Mod}(L)$ parameterizing infinite models, culminating in a Fraïssé-style framework that produces canonical atomic ω-structures corresponding to canonical continua such as the arc, pseudo-arc, and Lelek fan. The results provide a bridge between model theory and topology, enabling topological properties to be studied via logical methods and yielding canonical representations of well-known continua through Fraïssé limits in this enriched setting.
Abstract
A co-valuation is, essentially, a minimal finite cover. We introduce a logic based on co-valuations, which play the role of valuations of free variables in classical first-order logic, and show that the fundamental tools of model theory - such as ultraproducts, compactness, and omitting types - can be developed in this framework. Using a recently discovered duality between certain countable posets and second-countable compact $T_1$ spaces, we prove that these spaces play the role analogous to countable universes in first-order logic. Thus, although no topology appears in the initial formulation, the logic of co-valuations turns out to be naturally suited for studying compact topological objects. Standard topological notions, such as connectedness and covering dimension, are easily expressible, and model-theoretic properties, including atomicity, can be effectively analyzed. The framework also interacts well with Fraïssé-type constructions.