Sequent Systems for Lewis’ Conditional Logics

Lecture Notes in Computer Science

et al. 2017

Self Cite

We present the first internal calculi for Lewis' conditional logics characterized by uniformity and reflexivity, including non-standard internal hypersequent calculi for a number of extensions of the logic VTU. These calculi allow for syntactic proofs of cut elimination and known connections to S5. We then introduce standard internal hypersequent calculi for all these logics, in which sequents are enriched by additional structures to encode plausibility formulas as well as diamond formulas. These calculi provide both a decision procedure for the respective logics and constructive countermodel extraction from a failed proof search attempt. This paper is a preliminary version accepted for the conference TABLEAUX 2017.

“…where is the outer modality defined by A ≡ (⊥ ¬A). The rules of the calculi H L extend the calculi from [12] to the hypersequent setting and are given in Fig. 1.…”

Section: Definitionmentioning

confidence: 99%

Hypersequent Calculi for Lewis’ Conditional Logics with Uniformity and Reflexivity

Girlando

Lellmann

Lecture Notes in Computer Science

et al. 2017

Self Cite

“…We can give quick alternative completeness proofs for the proposed calculi by simulating derivations in the corresponding sequent calculi from [13,12], shown in Tab. 4.…”

Section: Completeness Via Translationmentioning

confidence: 99%

“…We are interested in internal sequent calculi, where a sequent denotes a formula of the language. Calculi of this kind have been proposed by Gent [7] and de Swart [20], and more recently in [12,13]. They are analytical and provide a decision procedure for the respective logics; on the other hand, they comprise an infinite set of rules with a variable number of premises.…”

Section: Introductionmentioning

confidence: 99%

Standard Sequent Calculi for Lewis’ Logics of Counterfactuals

Girlando

Lellmann

Logics in Artificial Intelligence

et al. 2016

Self Cite

Abstract. We present new sequent calculi for Lewis' logics of counterfactuals. The calculi are based on Lewis' connective of comparative plausibility and modularly capture almost all logics of Lewis' family. Our calculi are standard, in the sense that each connective is handled by a finite number of rules with a fixed and finite number of premises; internal, meaning that a sequent denotes a formula in the language, and analytical. We present two equivalent versions of the calculi: in the first one, the calculi comprise simple rules; we show that for the basic case of logic V, the calculus allows for syntactic cut-elimination, a fundamental proof-theoretical property. In the second version, the calculi comprise invertible rules, they allow for terminating proof search and semantical completeness. We finally show that our calculi can simulate the only internal (non-standard) sequent calculi previously known for these logics.

“…A first step in this direction is the calculus for the flat version of CK+ID+CSO, corresponding to KLM logic of cumulativity C and characterized by cumulative models, we conjecture that it can be extended to whole CK+CSO+ID, although the preferential semantics for the whole system is unknown. Since for preferential logics the semantics is given in terms of ordered models, it might turn out that the basic connective to consider is no longer the conditional operator, but a kind of entrenchment operator A B, whose meaning is "A is more plausible than B" (similarly to what is done in [24]) which reflects the ordered semantics more naturally. The conditional operator would then be obtained as a derived operator:…”

Section: Preferential Extensions Of Ckmentioning

confidence: 99%

“…including cumulativity (CM) and the or-axiom (CA), although the resulting systems are fairly complicated. Finally in [24] the authors provide internal calculi for Lewis' conditional logic V and some extensions. Their calculi are formulated for a language comprising the entrenchment connective, the strong and the weak conditional operator, both conditional operators can be defined in terms of the entrenchment connective.…”

Section: Introductionmentioning

confidence: 99%

Nested sequent calculi for normal conditional logics

Alenda

G.L.

2013

J Logic Computation

Nested sequent calculi are a useful generalization of ordinary sequent calculi, where sequents are allowed to occur within sequents. Nested sequent calculi have been profitably employed in the area of (multi)-modal logic to obtain analytic and modular proof systems for these logics. In this work, we extend the realm of nested sequents by providing nested sequent calculi for the basic conditional logic CK and some of its significant extensions. We provide also a calculus for Kraus Lehman Magidor cumulative logic C. The calculi are internal (a sequent can be directly translated into a formula), cut-free and analytic. Moreover, they can be used to design (sometimes optimal) decision procedures for the respective logics, and to obtain complexity upper bounds. Our calculi are an argument in favour of nested sequent calculi for modal logics and alike, showing their versatility and power.