Beyond Accessibility

Abstract

Possible worlds semantics for Modal Logic evolved in the fifties, through the work of Kanger, Hintikka and Kripke. Its main ideas are the use of possible worlds (which may stand for worlds in some grand sense, but also for points in time, situations, information stages or computer states), structured by a pattern of accessibility—with individual objects living in domains per world and having properties there, which may change in passing from one world to another. In propositional modal logic, where only worlds and accessibility matter (plus a ‘valuation’ for interpreting atomic propositions over the whole pattern), this picture has always seemed perfectly obvious. But in modal predicate logic, there has been recurrent debate concerning appropriate choices to be made in the semantics, starting from early doubts in Quine [25] about the very coherence of ascribing necessary properties to objects, and continuing into the sixties and seventies with various accounts of ‘trans-world identity’ for individuals across worlds. Noticeable are the ‘counterpart theory’ of Lewis [21], denying that objects can sensibly be identical across different worlds, or the ‘rigid designation’ theory in Kripke [20], affirming that only such objects make sense. More elaborate accounts of various possible approaches are given in Fine [9] and Carson [13]. Thus, the philosophical literature shows a variety of possible options in the semantics of modal predicate logic, concerning both the nature of individuals, and the interpretation of necessary propositions concerning them. Still, a widespread standard view exists (cf. Hughes and Cresswell [17] with domains growing along accessibility patterns (i.e., whenever xRy, then EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa % aaleaacaWG4baabeaakiabgAOinlaadseadaWgaaWcbaGaamyEaaqa % baaaaa!3BE4!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${D_x} \subseteq {D_y} $$), and calling a statement □φ(x _l,..., x _n ) true of objects d ₁,..., d _n in a world iff φ holds of those same objects in all accessible worlds.