Atom-canonicity in algebraic logic in connection to omitting types in modal fragments of L_{\omega, \omega}

TL;DR

This paper investigates the failure of the omitting types theorem and Vaught's theorem in finite variable fragments of first-order logic and related modal logics, using algebraic constructions to demonstrate non-atom canonicity and non-axiomatizability.

Contribution

It introduces new algebraic constructions showing the failure of classical theorems in finite variable fragments and modal expansions, and explores their implications for logic axiomatizability and canonicity.

Findings

Omitting types theorem fails for L_n with clique guarded semantics and S5^n.

Vaught's theorem fails almost everywhere for L_n, with explicit algebraic examples.

Certain modal logics between K^n and S5^n cannot be axiomatized by canonical equations.

Abstract

Fix 2<n<\omega. Let L_n denote first order logic restricted to the first n variables. CA_n denotes the class of cylindric algebras of dimension n and for m>n, Nr_n\CA_m(\subseteq CA_n) denotes the class of n-neat reducts of CA_m's. The existence of certain finite relation algebras and finite CA_n's lacking relativized complete representations is shown to imply that the omitting types theorem (OTT) fails for L_n with respect to clique guarded semantics (which is an equivalent formalism of its packed fragments), and for the multi-dimensional modal logic S5^n. Several such relation and cylindric algebras are explicitly exhibited using rainbow constructions and Monk-like algebras. Certain CA_n constructed to show non-atom canonicity of the variety S\Nr_n\CA_{n+3} are used to show that Vaught's theorem (VT) for L_{\omega, \omega}, looked upon as a special case of OTT for L_{\omega, \omega}, fails almost everywhere (a notion to be defined below) when restricted to L_n. That VT fails everywhere for L_n, which is stronger than failing almost everywhere as the name suggests, is reduced to the existence, for each n<m<\omega, of a finite relation algebra R_m having a so-called m-1 strong blur, but R_m has no m-dimensional relational basis. VT for other modal fragments and expansions of L_n, like its guarded fragments, n-products of uni-modal logics like K^n, and first order definable expansions, is approached. It is shown that any multi-modal canonical logic L, such that $K^n\subseteq L\subseteq S5^n$, L cannot be axiomatized by canonical equations. In particular, L is not Sahlqvist. Elementary generation and di-completeness for L_n and its clique guarded fragments are proved. Positive omitting types theorems are proved for L_n with respect to standard semantics.