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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.

##### MSC:
 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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