## Notre Dame Journal of Formal Logic

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

Steffen Lewitzka

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 10 November 2009

http://projecteuclid.org/euclid.ndjfl/1257862039

Digital Object Identifier
doi:10.1215/00294527-2009-012

Mathematical Reviews number (MathSciNet)
MR2572975

Zentralblatt MATH identifier
05657220

#### Citation

Lewitzka, Steffen. ∈ I : An Intuitionistic Logic without Fregean Axiom and with Predicates for Truth and Falsity. Notre Dame J. Formal Logic 50 (2009), no. 3, 275--301. doi:10.1215/00294527-2009-012. http://projecteuclid.org/euclid.ndjfl/1257862039.

#### References

• [1] Bab, S., $\in_\mu$-Logik---Eine Theorie propositionaler Logiken, Ph.D. thesis, Technische Universität Berlin, 2007.
• [2] Bab, S., "$\in_\mu$-logics---Propositional logics with self-reference and modalities", in Autonomous Systems---Self-Organization, Management, and Control, edited by B. Mahr and S. Huanye, Springer Verlag, Berlin, 2008. Proceedings of the 8th International Workshop, Shanghai Jiao Tong University, October, 2008.
• [3] Bab, S., B. Mahr, and T. Wieczorek, Epsilon-style (of) semantics'' in New Approaches to Classes and Concepts, edited by K. Robering, Studies in Logic, vol. 14, College Publications, 2008.
• [4] Barwise, J., and J. Etchemendy, The Liar. An Essay on Truth and Circularity, Oxford University Press, Oxford, 1987.
• [5] Béziau, J.-Y., Was Frege wrong when identifying reference with truth-value?'' Sorites, vol. 11 (1999), pp. 15--22.
• [6] Bloom, S. L., and R. Suszko, "Investigations into the sentential calculus with identity", Notre Dame Journal of Formal Logic, vol. 13 (1972), pp. 289--308.
• [7] Fitting, M., "Notes on the mathematical aspects of Kripke's theory of truth", Notre Dame Journal of Formal Logic, vol. 27 (1986), pp. 75--88.
• [8] Frege, G., "Über Sinn und Bedeutung", Zeitschrift für Philosophie und philosophische Kritik, vol. 100 (1892), pp. 25--50.
• [9] Kripke, S., "Outline of a theory of truth", The Journal of Philosophy, vol. 72 (1975), pp. 690--716.
• [10] Lewitzka, S., $\in_T(\Sigma)$-Logik: Eine Erweiterung der Prädikatenlogik erster Stufe mit Selbstreferenz und totalem Wahrheitsprädikat, Diplomarbeit, Technische Universität Berlin, 1998.
• [11] Lewitzka, S., and A. B. M. Brunner, "Minimally generated abstract logics", Logica Universalis, 2009, DOI 10.1007/s11787-009-0007-0.
• [12] Mahr, B., W. Sträter, and C. Umbach, "Fundamentals of a Theory of Types and Declarations", Technical Report KIT-Report 82, 1990.
• [13] Mahr, B., "Applications of type theory", pp. 343--55 in Proceedings of the International Joint Conference CAAP/FASE on Theory and Practice of Software Development, (Orsay, 1993), vol. 668 of Lecture Notes in Computer Science, Springer, Berlin, 1993.
• [14] Mahr, B., and S. Bab, "$\in\sb T$"-integration of logics", pp. 204--19 in Formal Methods in Software and Systems Modeling, edited by H.-J. Kreowski, U. Montanari, F. Orejas, G. Rozenberg, and G. Taentzer, vol. 3393 of Lecture Notes in Computer Science, Springer, Berlin, 2005.
• [15] Malinowski, G., "Non-Fregean logic and other formalizations of propositional identity", Bulletin of the Section of Logic, vol. 14 (1985), pp. 21--29.
• [16] Omyła, M., "Remarks on non-Fregean logic", Studies in Logic, Grammar and Rhetoric, vol. 10 (2007), pp. 21--31.
• [17] Shramko, Y., and H. Wansing, "The slingshot argument and sentential identity", Studia Logica, vol. 91 (2009), pp. 429--55.
• [18] Sträter, W., "$\in_T$ eine Logik erster Stufe mit Selbstreferenz und totalem Wahrheitsprädikat", Forschungsbericht, KIT-Report 98, 1992.
• [19] Suszko, R., "Non-Fregean logic and theories", Analele Universitatii Bucuresti, Acta Logica, vol. 11 (1968), pp. 105--125.
• [20] Suszko, R., "Ontology in the Tractatus of L. Wittgenstein", Notre Dame Journal of Formal Logic, vol. 9 (1968), pp. 7--33.
• [21] Suszko, R., "Abolition of the Fregean axiom", pp. 169--239 in Logic Colloquium (Boston, 1972--1973), edited by R. Parikh, vol. 453 of Lecture Notes in Mathematics, Springer, Berlin, 1975.
• [22] Suszko, R., "The Fregean axiom and Polish mathematical logic in the 1920s", Studia Logica, vol. 36 (1977/78), pp. 377--80.
• [23] Tarski, A., "Der Wahrheitsbegriff in den formalisierten Sprachen", Studia Philosophica, vol. 1 (1935), pp. 261--405.
• [24] Tarski, A., "The semantic conception of truth and the foundations of semantics", Philosophy and Phenomenological Research, vol. 4 (1944), pp. 341--76.
• [25] Zeitz, P., Parametrisierte $\in\sb T$-Logik. Eine Theorie der Erweiterung abstrakter Logiken um die Konzepte Wahrheit, Referenz und klassische Negation, Logos Verlag Berlin, Berlin, 2000. Dissertation, Technische Universität Berlin, Berlin, 1999.