Completeness via Correspondence for Extensions of the Logic of Paradox

Abstract

Taking our inspiration from modal correspondence theory, we present the idea of correspondence analysis for many-valued logics. As a benchmark case, we study truth-functional extensions of the Logic of Paradox (LP). First, we characterize each of the possible truth table entries for unary and binary operators that could be added to LP by an inference scheme. Second, we define a class of natural deduction systems on the basis of these characterizing inference schemes and a natural deduction system for LP. Third, we show that each of the resulting natural deduction systems is sound and complete with respect to its particular semantics.

Automated Proof-searching for Strong Kleene Logic and its Binary Extensions via Correspondence Analysis

“…We will use the following adaptation of Kooi and Tamminga's definitions 2.1 and 1 from [26] and [55], respectively. Definition 2 (Single Entry Correspondence [26,55] Tamminga [55,Theorem 1] found inference schemes which characterize an f • 's all possible entries (see Theorem 1 in Section 2 below).…”

“…Taking their inspiration from modal correspondence theory [44,59,60], Kooi and Tamminga [26] presented correspondence analysis for the unary and binary extensions of the three-valued logic of paradox LP [2,42,41]. Correspondence analysis allows one to immediately find inference rules for the unary and binary operators added to LP from their truth tables' entries.…”

Natural Deduction for Four-Valued both Regular and Monotonic Logics

“…Note that K 3 (1938) is a fragment of Łukasiewicz's logic Ł 3 (1920) [18]. Natural deduction systems for K 3 and LP, respectively, are presented in [22,24,17]. In K w 3 negation is the same as for K 3 ; conjunction and disjunction, as was shown in Finn's paper [8], are expressed via K 3 's connectives by equations (1) and (2) (see p. 55), respectively.…”

Correspondence analysis for strong three-valued logic

“…To construct a proof system for K 3 (∼) m (•) n , I follow [3]. I first characterize each possible single entry in the truth-table of a unary or a binary operator by a basic inference scheme.…”

“…In this paper, I present a general method for finding natural deduction systems for truth-functional extensions of K 3 . To do so, I use the correspondence analysis for many-valued logics that was presented recently by [3]. In their study of the logic of paradox (LP ) [4], they characterize every possible single entry in the truthtable of a unary or a binary truth-functional operator by a basic inference scheme.…”