A Sequent Calculus for Preferential Conditional Logic Based on Neighbourhood Semantics

The basic preferential conditional logic PCL, initially proposed by Burgess, finds an interest in the formalisation of both counterfactual and plausible reasoning, since it is at the same time more general than Lewis' systems for counterfactuals and it contains as a fragment the KLM preferential logic P for default reasoning. This logic is characterised by Kripke models equipped with a ternary relational semantics that represents a comparative similarity/normality assessment between worlds, relativised to each world. It is first shown that its semantics can be equivalently specified in terms of neighbourhood models. On the basis of this alternative semantics, a new labelled calculus is given that makes use of both world and neighbourhood labels. It is shown that the calculus enjoys syntactic cut elimination and that, by adding suitable termination conditions, it provides a decision procedure.

[1]  Sarit Kraus,et al.  Nonmonotonic Reasoning, Preferential Models and Cumulative Logics , 1990, Artif. Intell..

[2]  Deepak Garg,et al.  Countermodels from Sequent Calculi in Multi-Modal Logics , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

[3]  Richmond H. Thomason,et al.  A semantic analysis of conditional logic1 , 2008 .

[4]  John L. Pollock.  A refined theory of counterfactuals , 1981, J. Philos. Log..

[5]  Laura Giordano,et al.  Analytic tableaux calculi for KLM logics of nonmonotonic reasoning , 2006, TOCL.

[6]  Nicola Olivetti,et al.  Nested sequent calculi for normal conditional logics , 2016, J. Log. Comput..

[7]  Dirk Pattinson,et al.  Optimal Tableaux for Conditional Logics with Cautious Monotonicity , 2010, ECAI.

[8]  Sara Negri,et al.  Proofs and Countermodels in Non-Classical Logics , 2014, Logica Universalis.

[9]  Roy Dyckhoff,et al.  Proof analysis in intermediate logics , 2012, Arch. Math. Log..

[10]  Laura Giordano,et al.  Tableau calculus for preference-based conditional logics: PCL and its extensions , 2009, TOCL.

[11]  Joseph Y. Halpern,et al.  On the Complexity of Conditional Logics , 1994, KR.

[12]  Sara Negri,et al.  Proof Analysis in Modal Logic , 2005, J. Philos. Log..

[13]  Dirk Pattinson,et al.  Generic Modal Cut Elimination Applied to Conditional Logics , 2009, TABLEAUX.

[14]  Robert Stalnaker.  A Theory of Conditionals , 2019, Knowledge and Conditionals.

[15]  Laura Giordano,et al.  Analytic Tableaux for KLM Preferential and Cumulative Logics , 2005, LPAR.

[16]  Camilla Schwind,et al.  A sequent calculus and a theorem prover for standard conditional logics , 2007, TOCL.

[17]  Dirk Pattinson,et al.  Sequent Systems for Lewis' Conditional Logics , 2012, JELIA.

[18]  Brian F. Chellas.  Basic conditional logic , 1975, J. Philos. Log..

[19]  D. Nute.  Topics in Conditional Logic , 1980 .

[20]  John P. Burgess,et al.  Quick completeness proofs for some logics of conditionals , 1981, Notre Dame J. Formal Log..