Completeness and Correspondence in Chellas–Segerberg Semantics

Abstract

We investigate a lattice of conditional logics described by a Kripke type semantics, which was suggested by Chellas and Segerberg -Chellas-Segerberg (CS) semantics -plus 30 further principles. We (i) present a non-trivial frame-based completeness result, (ii) a translation procedure which gives one corresponding trivial frame conditions for arbitrary formula schemata, and (iii) nontrivial frame conditions in CS semantics which correspond to the 30 principles.

Connexive Conditional Logic. Part I

“…David Nelson's paraconsistent constructive logic [Almukdad and Nelson, 1984] C paraconsistent constructive connexive logic [Wansing, 2005] CK basic conditional logic [Chellas, 1975;Segerberg, 1989;Unterhuber, 2013;Unterhuber and Schurz, 2014] CKR conditional logic CK extended by A A (reflexivity) ( [Unterhuber, 2013, Ch. 7.1]; see also [Chellas, 1975;Segerberg, 1989;Unterhuber and Schurz, 2014]) This paper CK FDE basic conditional logic CK based on FDE 6 CKR FDE conditional logic CKR based on FDE cCL basic weakly connexive conditional logic CCL basic connexive conditional logic, cCL extended by A A cCCL basic weakly connexive constructive conditional logic CCCL basic connexive constructive conditional logic, cCCL extended by A A Table 1. Overview of the systems investigated in the paper and relevant systems in the literature tableau proof systems for cCCL and CCCL and show these calculi to be sound and complete.…”

“…We shall sometimes omit outermost brackets of formulas. 7 As in [Unterhuber, 2013;Unterhuber and Schurz, 2014], in addition to L we shall use a language L FC for talking about (general) frame conditions. The language L FC is a two-sorted, set-theoretic language which contains (i) variables w, w ′ , w ′′ , .…”

“…Moreover, it can be shown that any conditional logic whatsoever from the mentioned lattice of systems is complete with respect to some class of Chellas models. To avoid such a triviality result for Segerberg frames, trivial and non-trivial frame conditions are distinguished, where only the former make use of use of inessential expressions as defined in [Unterhuber and Schurz, 2014] and can be characterized in terms of a standardized translation from logical principles into frame conditions. For example, the principle (A B) ⊃ B corresponds to the trivial frame condition…”

“…, where the trivial frame condition uses inessential expressions, such as Y and w ′ , and results from a standard translation of (A B) ⊃ B (cf. [Unterhuber and Schurz, 2014]). 9 Note that none of the frame conditions employ the parameter P , although the parameter P plays an integral role in the completeness proof.…”

“…Later Krister Segerberg [1989] considered general Chellas frames  which we call Segerberg frames [see also Unterhuber, 2013, Ch. 4.3;Unterhuber and Schurz, 2014]. 3 Since our base logic is the paraconsistent and paracomplete logic FDE, the set W is now to be understood as a set of states that may fail to support the truth or the falsity of a formula or support both the truth and the falsity of a formula.…”