Notre Dame Journal of Formal Logic

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

Steffen Lewitzka

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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.

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Notre Dame J. Formal Logic Volume 50, Number 3 (2009), 275-301.

First available: 10 November 2009

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Zentralblatt MATH identifier

Mathematical Reviews number (MathSciNet)

Primary: 03B20: Subsystems of classical logic (including intuitionistic logic) 03B60: Other nonclassical logic
Secondary: 03B65: Logic of natural languages [See also 68T50, 91F20]

truth theory non-Fregean logics self-reference intuitionistic logic semantic paradoxes intensional semantics extension intension denotation


Lewitzka, Steffen. ∈ I : An Intuitionistic Logic without Fregean Axiom and with Predicates for Truth and Falsity. Notre Dame Journal of Formal Logic 50 (2009), no. 3, 275--301. doi:10.1215/00294527-2009-012.

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