Tableau calculus for preference-based conditional logics

Logics in Artificial Intelligence

Pattinson

2012

Abstract. We present unlabelled cut-free sequent calculi for Lewis' conditional logic V and extensions, in both the languages with the entrenchment connective and the strong conditional. The calculi give rise to Pspace-decision procedures, also in the language with the weak conditional. Furthermore, they are used to prove the Craig interpolation property for all the logics under consideration, and yield a Pspace-decision procedure for a recently considered hybrid version of V.

Section: Resultsmentioning

confidence: 99%

Sequent Systems for Lewis’ Conditional Logics

Logics in Artificial Intelligence

Pattinson

2012

“…9 The proof uses in an essential way the fact that a backwards application of jump reduces the modal degree of a sequent. Although rule A i plays a similar role as jump, it does not reduce the modal degree when applied backwards.…”

Section: Definition 17 a Sequent Is Saturated If It Has The Formmentioning

confidence: 99%

“…Calculi for some weaker conditional logics are given, e.g., in [1,18] and more recently in [19,15]. Regarding Lewis' counterfactual logics, external labelled calculi have been proposed in [9] and in [16], both based on a relational reformulation of the sphere semantics. We are interested in internal sequent calculi, where a sequent denotes a formula of the language.…”

Section: Introductionmentioning

confidence: 99%

Standard Sequent Calculi for Lewis’ Logics of Counterfactuals

Girlando

Logics in Artificial Intelligence

Olivetti

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.

“…In previous work [7], we defined calculi with many of these properties for weaker logics of the Lewis' family. For the logics with uniformity to the best of our knowledge no internal calculi are known; the only known external calculi for these adopt a hybrid language and a relational semantics [6]. We also consider logics with absoluteness, a property stronger than uniformity stating that all worlds have the same system of spheres.…”

Section: Introductionmentioning

confidence: 99%

Hypersequent Calculi for Lewis’ Conditional Logics with Uniformity and Reflexivity

Girlando

Lecture Notes in Computer Science

Olivetti

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.