Construction of a canonical model for a first-order non-Fregean logic with a connective for reference and a total truth predicate

Abstract

Logics with quantifiers that range over a model-theoretic universe of propositions are interesting for several applications. For example, in the context of epistemic logic the knowledge axioms can be expressed by the single sentences ∀x.(Kix → x), and in a truth-theoretical context an analogue to Tarski's T-scheme can be expressed by the single axiom ∀x.(x:true ↔ x). In this article, we consider a first-order non-Fregean logic, originally developed by Sträter, which has a total truth predicate and is able to model propositional self-reference. We extend this logic by a connective ‘<’ for propositional reference and study semantic aspects. φ < ψ expresses that the proposition denoted by formula ψ says something about (refers to) the proposition denoted by φ. This connective is related to a syntactical reference relation on formulas and to a semantical reference relation on the propositional universe of a given model. Our goal is to construct a canonical model, i.e. a model that establishes an order-isomorphism from the set of sentences (modulo alpha-congruence) to the universe of propositions, where syntactical and semantical reference are the respective orderings. The construction is not trivial because of the impredicativity of quantifiers: the bound variable in ∃x.φ ranges over all propositions, in particular over the proposition denoted by ∃x.φ itself. Our construction combines ideas coming from Sträter's dissertation with the algebraic concept of a canonical domain, which is introduced and studied in this article.