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\(\in_I\): an intuitionistic logic without Fregean axiom and with predicates for truth and falsity. (English) Zbl 1190.03016
Summary: 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.

03B20 Subsystems of classical logic (including intuitionistic logic)
03A05 Philosophical and critical aspects of logic and foundations
03B60 Other nonclassical logic
03B65 Logic of natural languages
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