Hypersequent Calculi for Lewis’ Conditional Logics with Uniformity and Reflexivity

Abstract

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.

Nested Sequents for the Logic of Conditional Belief

“…These axioms represent an "internalised" version of the AGM belief revision postulates in a multi-agent setting, e.g., axioms 5 and 6 encode the Minimal Change Principle 5 An alternative axiomatization of CDL taking i as primitive essentially amounts to a multi-agent version of Lewis' counterfactual logic system VTA [7] and contains in addition to classical propositional logic the following:…”

“…For VTA this yields an equivalence relation, thus characterizing modal logic S5. The relation can be used to evaluate formulas KA, i.e., formulas ⊥ ¬A 7 . For A ∈ F S5i , define T (A) ∈ F CDL to be the formula obtained by replacing every occurrence of K i A with ⊥ i ¬A.…”

“…Similarly to the calculi for Lewis' conditional logics in [7], our calculus N CDL takes as primitive the comparative plausibility operator, albeit in its multi-agent version A i B. In order to obtain an internal calculus for CDL, the simple hypersequent structure used to capture Lewis' logics in [7], is no longer adequate.…”

“…Similarly to the calculi for Lewis' conditional logics in [7], our calculus N CDL takes as primitive the comparative plausibility operator, albeit in its multi-agent version A i B. In order to obtain an internal calculus for CDL, the simple hypersequent structure used to capture Lewis' logics in [7], is no longer adequate. To keep track of the "locality" of information for each agent, and to account for beliefs of an agent occurring within the beliefs of another, we use a nested structure, which is not necessary in the single-agent case.…”

VINTE: An Implementation of Internal Calculi for Lewis’ Logics of Counterfactual Reasoning

“…α ∀ ¬B or α ∃ A. 7 It is worth noticing that absoluteness can be equally stated as local absoluteness:…”