Adding a Conditional to Kripke’s Theory of Truth

Abstract

Kripke's theory of truth [29] has been very successful but shows wellknown expressive difficulties; recently, Field has proposed to overcome them by adding a new conditional connective to it. In Field's theories, desirable conditional and truth-theoretic principles are validated that Kripke's theory does not yield. Some authors, however, are dissatisfied with certain aspects of Field's theories, in particular the high complexity. I analyze Field's models and pin down some reasons for discontent with them, focusing on the meaning of the new conditional and on the status of the principles so successfully recovered. Subsequently, I develop a semantics that improves on Kripke's theory following Field's program of adding a conditional to it, using some inductive constructions that include Kripke's one and feature a strong evaluation for conditionals. The new theory overcomes several problems of Kripke's one and, although weaker than Field's proposals, it avoids the difficulties that affect them; at the same time, the new theory turns out to be quite simple. Moreover, the new construction can be used to model various conceptions of what a conditional connective is, in ways that are precluded to both Kripke's and Field's theories.

A Unified Theory of Truth and Paradox

“…45 For strong Kleene logic, see Urquhart (2001). For extensions of strong Kleene logic with conditionals obeying stronger introduction rules, see Rossi (2016).…”

“…For the use and relevance of the Łukasiewicz conditional in theories of truth, seeRossi (2016).32 Con L Ł in effect contains the elements of V Ł to be used as constants in solving equation systems definable in L Ł . I do not put Con L Ł = V Ł to avoid confusion in the definition of L Ł , although the two sets are identified in practice.…”

Fixed-point models for paradoxical predicates

“…Combining ingredients of a Kripkean semantics with some revision theoretic techniques[GB93], his main goal consisted in defining a "nice" conditional satisfying all instances of the T-Schema, while avoiding revenge phenomena (see in particular[Fie04,Fie08,Fie14]). For a critical evaluation see, e.g.,[RW07,Wel11,Wel14], and for similar investigations see e.g [Bac13,Ros16]…”

“…I should mention that the idea of adding a third set for paradoxical sentences is not new. As an anonymous referee correctly pointed out to me, in the context of semantic theories of truth it was suggested by Rossi[Ros16], although for different purposes and with a different formal development.29 Similarly, the value u (undefined) only formally represents the lack of truth-values. Compare also [Kri75, footnote 18].Australasian Journal of Logic (18:7) 2021, Article no.…”

Model-theoretic semantics and revenge paradoxes

“…11See Field (2003, §4) and Field (2008, Chapter17.4) for a partial list of the principles validated in Field’s theory. See Rossi (2016) for a self-contained presentation of Field’s theory, and a discussion of its conditional.…”