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

*(English)*Zbl 1283.03058Summary: 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 \(\forall x.(K_{i}x \to x)\), and in a truth-theoretical context an analogue to Tarski’s T-scheme can be expressed by the single axiom \(\forall x.(x:\mathit{true}\leftrightarrow x)\). 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. \(\varphi < \psi \) expresses that the proposition denoted by formula \(\psi \) says something about (refers to) the proposition denoted by \(\varphi \). 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 \(\exists x.\varphi \) ranges over all propositions, in particular over the proposition denoted by \(\exists x.\varphi \) 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.

##### MSC:

03B60 | Other nonclassical logic |