Logics for Propositional Determinacy and Independence

Abstract

This paper investigates formal logics for reasoning about determinacy and independence. Propositional Dependence Logic D and Propositional Independence Logic I are recently developed logical systems, based on team semantics, that provide a framework for such reasoning tasks. We introduce two new logics L D and L I , based on Kripke semantics, and propose them as alternatives for D and I, respectively. We analyse the relative expressive powers of these four logics and discuss the way these systems relate to natural language. We argue that L D and L I naturally resolve a range of interpretational problems that arise in D and I. We also obtain sound and complete axiomatizations for L D and L I . * Visiting professorship 1 Also available as a reprint in [SvE88].

Supervenience, Dependence, Disjunction

“…With such a broad terrain we promise something for everyone (with interests in this area) understood in the usual ∀∃ sense rather than the ∃∀ sense, so the remainder of this paragraph offers advice as to which sections will treat the topics of interest to which readers, for whom the cross-references supplied can be followed for any pre-requisites from elsewhere in the paper. A useful focus for such a comparative discussion is provided by Goranko and Kuusisto [2018], in which the authors propose a logic of dependence, or determinacy, as they put it, which they urge is better than the modal dependence logic of Väänänen [2008], in particular because of the treatment of disjunction in the latter logic. We get to that in sections 3-5, and move on to the related area of inquisitive semantics in Section 6 and to another issue (the "or" in whether-constructions) it raises, in Section 7, before returning in Section 8 to some aspects of property supervenience  partly philosophical and partly formal  alluded to in the preceding sections, and a postscript tying up one conceptual loose end (Section 9).…”

“…We treat L D in the present section, returning to D in the next. The latter goes back to Väänänen [2008], as a modal variation on the dependence and independence logics of Väänänen [2007a], themselves inspired by the game-theoretic seman-1 Two other logics, focussed on a notion of independence rather than dependence, I and L I , in something of the same spirit as D and L D , respectively, are also treated in [Goranko and Kuusisto, 2018] but will not be discussed here.Supervenience, dependence, disjunction 5 tics for Henkin (or 'branching') quantifiers of Hintikka and co-authors, later re-packaged and marketed as independence-friendly (or 'IF') logic, and by the provision for the latter of a compositional model-theoretic semantics in [Hodges, 1997]. …”

“…(See further Remark 1.1 below. )There is a standard possible worlds semantics, in the style of Kripke or more accurately, of Carnap, given for the propositional logic of determinacy in [Goranko and Kuusisto, 2018], using models M = W, V  at least as I shall expound matters here, moving from a Carnapian to a Kripkean approach  with W a non-empty set, subsets of which are assigned to propositional variables p i (i = 1, . .…”

“…None of these historical roots will concern us here; for more information on them, see [Goranko and Kuusisto, 2018] and the references there supplied, as well as [Tulenheimo, 2016]. (In a final Postscript section we return to quantificational dependence logic for contrastive purposes.)…”

“…[Goranko and Kuusisto, 2018] simply relativizes truth to the pair consisting of W and some element w thereof, taking W to be a set, specifically, of truth-value assignments to the propositional variables, and they 2 See also [Hodges, 2007]. An earlier gesture in the direction of a compositional treatment, though this time in the 'homophonic truth-theory' tradition, can be found in [Humberstone, 1987]; Patton [1989] (perhaps justifiably) provides an unfavourable assessment of its prospects.…”