Abstract
This paper is an exposition of some recent results concerning various notions of strength and weakness of the concept of truth, both published or not. We try to systematically present these notions and their relationship to the current research on truth. We discuss the concept of the Tarski boundary between weak and strong theories of truth and we give an overview of non-conservativity results for the extensions of the basic compositional truth theory. Additionally, we present a natural strong theory of truth which admits a number of apparently unrelated axiomatisations. Finally, we discuss other possible explications of the notion of ‘strength’ of axiomatic theories of truth.
References
Cantini, A. (1990), A Theory of Formal Truth Arithmetically Equivalent to ID1, Journal of Symbolic Logic” 55(1), s. 244–259.
Cieśliński, C. (2017, w przygotowaniu), The Epistemic Lightness of Truth. Deflationism and Its Logic, Cambridge: Cambridge University Press.
Cieśliński, C. (2010), Deflationary Truth and Pathologies, „Journal of Philosophical Logic” 39(3), s. 325–337.
Cieśliński, C. (2010), Truth, Conservativeness, and Provability, „Mind” 119(474), s. 409–422.
Enayat, A., Visser, A. (2015), New Constructions of Satisfaction Classes, w: T. Achourioti, H. Galinon, J.M. Fernández (eds.), Unifying the Philosophy of Truth, s. 321–335, Dordrecht: Springer Netherlands.
Franzen, T. (2003), Inexhaustibility. A Non-Exhaustive Treatment, Association for Symbolic Logic.
Fujimoto, K. (2010), Relative Truth Definability of Axiomatic Truth Theories, „Bulletin of Symbolic Logic” 16(3), s. 305–344.
Halbach, V. (2011), Axiomatic Theories of Truth, Cambridge: Cambridge University Press.
Kaye, R. (1991), Models of Peano Arithmetic, New York: Clarendon Press.
Kaufmann, M., Schmerl, J. (1987), Remarks on Weak Notions of Saturation in Models of Peano Arithmetic, „Journal of Symbolic Logic” 52(1), s. 129–148.
Ketland, J. (1999), Deflationism and Tarski’s Paradise, „Mind” 108(429), s. 69–94.
Kotlarski, H. (1986), Bounded Induction and Satisfaction Classes, „Zeitschrift für mathematische Logik und Grundlagen der Mathematik” 32(31–34), s. 531–544.
Kotlarski, H., Krajewski, S., Lachlan, A. (1981), Construction of Satisfaction Classes for Nonstandard Models, „Canadian Mathematical Bulletin” 24, s. 283–293.
Leigh, G. (2015), Conservativity for Theories of Compositional Truth via Cut Elimination, „Journal of Symbolic Logic” 80(3), s. 845–865.
Łełyk, M., Wcisło, B. (2017), Notes on Bounded Induction for the Compositional Truth Predicate, „The Review of Symbolic Logic”.
Łełyk, M., Wcisło, B. (2017), Models of Weak Theories of Truth, „Archive for Mathematical Logic”.
Shapiro, S. (1998), Proof and Truth: Through Thick and Thin, „Journal of Philosophy” 95(10), s. 493–521.