Truth, Partial Logic and Infinitary Proof Systems

Abstract

In this paper we apply proof theoretic methods used for classical systems in order to obtain upper bounds for systems in partial logic. We focus on a truth predicate interpreted in a Kripke style way via strong Kleene; whereas the aim is to connect harmoniously the partial version of KripkeFeferman with its intended semantics. The method we apply is based on infinitary proof systems containing an ω-rule. MotivationThis paper is part of a systematic investigation of syntactical predicates based on proof theoretic methods. In this paper we apply proof theoretic methods used for classical systems in order to obtain upper bounds for systems in partial logic. We focus on a truth predicate, interpreted in a Kripke style way via strong Kleene; whereas the aim is to connect harmoniously the partial version of Kripke-Feferman with its intended semantics. The method we apply is a variation of Schütte's based on infinitary proof systems.The conception of truth we investigate is a version of Kripke's theory. Kripke made use of inductive definitions in order to characterize fixed-points as intended interpretations of the truth predicate. Of special interest for Kripke is the minimal fixed-point. We will use an infinitary proof system for which the set of theorems coincides with the minimal fixed-point. One part of our investigation designs an infinitary proof system, SK ∞ . The system SK ∞ is a sequent calculus with an ω-rule and truth introduction rules. With SK ∞ we can characterize the minimal strong Kleene fixed-point, I sk . In a second part we consider an internal axiomatization of the semantical theory; the resulting thoery is called PKF. This theory is a good candidate and does not contain an infinitary rule, however it is not based on classical logic but on partial logic. The system PKF was introduced by Halbach & Horsten [11] for achieving a faithful axiomatization of Kripke's fixed-point construction of Strong Kleene. The theory PKF has several interesting features for our purposes. On the one hand PKF is close to the semantic construction as it directly incorporates the closure conditions of the fixed-points. Moreover PKF is true in and only in the fixed-point models as an adequacy result shows. On the other hand PKF is close to the infinitary system SK ∞ as it can be embedded into it in a straight way. A closer look at 1

, , and Reinhardt’s Program

“…6 See Section 2 for details on notation. 7 An anonymous referee suggests that since Reinhardt identifies the significant part of KF with the set of significant sentences and not with the set of admissible inferences, our proposal is not a different version of Reinhardt's program but a different project altogether. This really depends on what one takes Reinhardt's program to be, as Reinhardt himself remains rather vague and unspecific in this respect.…”

“…Rather it also seems crucial to ask which inferences are admissible within the significant part of KF. 7 Of course, in classical logic this difference 5 To be precise, Reinhardt [20] asked whether there was a KF-proof ϕ 1 , ... , ϕn, ϕ rather than a KF-proof ϕ 1 , ... , ϕn, Tr ϕ . But this presupposes that all KF-provably true sentences are also theorems of KF.…”

“…The theory Partial Kripke-Feferman (PKF) may be seen as the non-classical counterpart to KF. It was developed by Halbach & Horsten [10], and variants of PKF have been introduced and studied by, e.g., Halbach & Nicolai [11], Nicolai [14], and Fischer & Gratzl [7]. 21 We begin by introducing different rules of induction employed in the formulation of the theories.…”

Cut elimination for systems of transparent truth with restricted initial sequents

“…We relate the infinitary system introduced above and a fixed-point model for L Tr . The results of this section are intended to compare to [FG18] and, less directly, to [Can90]. We show that the form of invertibility allowed by LGT ∞ enables one to preserve fully disquotational truth rules with context while capturing Kripkean grounded truth.…”

“…Of course, the price to pay is the adoption of properly weaker truth rules. This is for instance the strategy considered in [Can90,FG18] in the context of Strong Kleene logic and supervaluational logic respectively. Alternatively, one can restrict the contraction rule, keep the node-based measure of applications of semantic rules, and still obtain a full cut elimination proof [Gri82,Can03].…”