Generalized Update Semantics

Abstract

This paper explores the relationship between dynamic and truth conditional semantics for epistemic modals. It provides a generalization of a standard dynamic update semantics for modals. This new semantics derives a Kripke semantics for modals and a standard dynamic semantics for modals as special cases. The semantics allows for new characterizations of a variety of principles in modal logic, including the inconsistency of ‘p and might not p’. Finally, the semantics provides a construction procedure for transforming any truth conditional semantics for modals into a dynamic semantics for modals with similar properties.

A strictly stronger relative must

“…More generally, the present semantics satisfies Goldstein's desideratum that multiple modal connectives not always collapse into a single modal connective (Goldstein, 2019). □□ p is equivalent to □ p , and ◊ ◊ p is equivalent to ◊ p , but neither □◊ p nor ◊ □ p is for every p equivalent to □ p or ◊ p or p .…”

“…Both contend that it can. Goldstein (2019) also treats “ p ∧ ◊¬ p ” but not “◊ p ∧ ¬ p ” as inconsistent, but Beddor and Goldstein (2018, p. 105) acknowledge the oddity of the latter and provide an account of it.…”

“…7 7 These authors include Veltman (1986), Gillies (2004), Willer (2013), Bledin (2014), Bledin and Lando (2018), Aloni (2018), Goldstein (2019), Hawke and Steinhert‐Threlkeld (2020), and Ciardelli (in press). On von Fintel and Gillies's (2010) account of “must”, the argument from p to “Must p ” fails, but only because “Must p ” carries a presupposition of indirectness that p lacks. …”

“…8 8 Other accounts of modal connectives that build some kind of internal structure into the background set of possibilities include Kratzer (1981), Veltman (1986), Gauker (1994), Willer (2013) and Goldstein (2019). Other semantic theories that define positive and negative values separately in an alternating recursion and define them for modal or conditional sentences only relative to contexts and not relative to individual worlds include those in Veltman (1986), Gauker (1994), Gauker (2005), Aloni (2018), Hawke and Steinhert‐Threlkeld (2020) and Ciardelli (in press).…”

Dynamic Non-Classicality

“…5 This comprises the semantics for atomic updates, negation, and conjunction of Heim 1983, together with the entries for disjunction from Groenendijk et al 1996 and epistemic 'might' (abbreviated ♦) from Veltman 1996. 6 Including prominent developments of Groenendijk et al [1996]'s system, for instance Dekker 1993;Aloni 2001;Beaver 2001;Gillies 2001;Yalcin 2015;Gillies 2018;Goldstein 2019. is the result of applying [ϕ] to c). The semantics is given recursively from a set At of atomic sentences p, q, r .…”