$\in_I$ : An Intuitionistic Logic without Fregean Axiom and with Predicates for Truth and Falsity

Lewitzka, Steffen

Notre Dame J. Formal Logic, Tome 50 (2009) no. 1, p. 275-301 / Harvested from Project Euclid

Résumé

We present $\in_I$ -Logic (Epsilon-I-Logic), a non-Fregean intuitionistic logic with a truth predicate and a falsity predicate as intuitionistic negation. $\in_I$ is an extension and intuitionistic generalization of the classical logic $\in_T$ (without quantifiers) designed by Sträter as a theory of truth with propositional self-reference. The intensional semantics of $\in_T$ offers a new solution to semantic paradoxes. In the present paper we introduce an intuitionistic semantics and study some semantic notions in this broader context. Also we enrich the quantifier-free language by the new connective < that expresses reference between statements and yields a finer characterization of intensional models. Our results in the intuitionistic setting lead to a clear distinction between the notion of denotation of a sentence and the here-proposed notion of extension of a sentence (both concepts are equivalent in the classical context). We generalize the Fregean Axiom to an intuitionistic version not valid in $\in_I$ . A main result of the paper is the development of several model constructions. We construct intensional models and present a method for the construction of standard models which contain specific (self-)referential propositions.

Détail

Publié le : 2009-07-15
Classification:  truth theory,  non-Fregean logics,  self-reference,  intuitionistic logic,  semantic paradoxes,  intensional semantics,  extension,  intension,  denotation,  03B20,  03B60,  03B65