Indicative Conditionals and Dynamic Epistemic Logic

Abstract

Recent ideas about epistemic modals and indicative conditionals in formal semantics have significant overlap with ideas in modal logic and dynamic epistemic logic. The purpose of this paper is to show how greater interaction between formal semantics and dynamic epistemic logic in this area can be of mutual benefit. In one direction, we show how concepts and tools from modal logic and dynamic epistemic logic can be used to give a simple, complete axiomatization of Yalcin's [16] semantic consequence relation for a language with epistemic modals and indicative conditionals. In the other direction, the formal semantics for indicative conditionals due to Kolodny and MacFarlane [9] gives rise to a new dynamic operator that is very natural from the point of view of dynamic epistemic logic, allowing succinct expression of dependence (as in dependence logic) or supervenience statements. We prove decidability for the logic with epistemic modals and Kolodny and MacFarlane's indicative conditional via a full and faithful computable translation from their logic to the modal logic K45.Logic and the formal semantics of natural language are related by blood and yet somewhat estranged. Today it is rare that formal semanticists consider questions of axiomatizability or decidability of the consequence relations defined by model-theoretic accounts of natural language fragments. Meanwhile logicians focus more on logics motivated by mathematical or philosophical concerns than on logics arising from semantic theories in linguistics.The cost of estrangement is that insights from one field that would be useful for the other may go unnoticed or efforts may be unnecessarily duplicated. The aim of this paper is to help encourage a family reunion between logic and formal semantics of natural language, by way of concrete examples. The topic of modals and conditionals is a prime example of overlap between formal semantics and logic. In this paper, we consider the case of epistemic modals and indicative conditionals.In §1, we show how concepts and tools from modal logic and dynamic epistemic logic can be used to give a simple, complete axiomatization of Yalcin's [16] semantic consequence relation for a language with epistemic modals and indicative conditionals. Then in §2, we show that the formal semantics for indicative conditionals due to Kolodny and MacFarlane [9] gives rise to a new dynamic operator that is very natural from the point of view of dynamic epistemic logic, allowing succinct expression of dependence (as in dependence logic) or supervenience statements. We prove decidability for the logic with epistemic modals and Kolodny and MacFarlane's indicative conditional via a full and faithful computable translation from their logic to the modal logic K45.There are other examples of clear overlap between formal semantics and dynamic epistemic logic, such as the connection between the dynamic logical consequence of [14] and the dynamic consequence of [1], between the notions of epistemic contradictions in [15] and of Moorean sentences i...

Path Semantics for Indicative Conditionals

“…1 if for all p * p, A p = 1 0 if for some p * p, A p = 0 # otherwise 32 For a similar algorithm, see Kolodny &MacFarlane 2010 andHolliday &Icard 2017. Conditionals quantify over maximal updates of paths. For shorthand, let 'p A ' be a variable ranging over maximal updates of p with respect to A.…”

Indicative Conditionals in Objective Contexts

“…In this fragment it is not possible to formalize adequately most of the data discussed in Section 3. In Holliday and Icard (2018) a deductive calculus is introduced and completeness proved for what the authors call Yalcin's semantics. This semantics is almost identical with Bledin's semantics with one important difference.…”

Epistemic Modals in Hypothetical Reasoning

“…showed that the non-implicational fragment ofYalcin's (2007) informational consequence is the logic of S5-validity, i.e. where A and B are non-implicational sentences, A is an informational consequence of B iff there is a proof of B from A in the modal logic S5 Holliday and Icard (2017). extend this to full informational consequence by providing axioms for Yalcin's informational conditional in S5.…”