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Construction of a canonical model for a first-order non-Fregean logic with a connective for reference and a total truth predicate. (English) Zbl 1283.03058
Summary: 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
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