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Worlds and counterparts

The basic picture

If Kripkean models have not become a friendly environment for general applications of different combinations of various choices of their domains and their possible interactions with modal operators in a setting which would be developed with an account of the syntax-semantics relationship which would fulfill requirements of the generative theory in view, the major competitor of Kripke’s theory has found even less favour in linguistic analysis. The counterpart theory of Lewis (Lewis 1968, 1986) does not at first view offer too much as a general framework for formal semantics, in its original formulation requiring adoption of several formal solutions not suitable for a theory of the syntax-semantics mapping, beginning with the rejection of quantified modal logic and requiring that sentences belonging to the modal idiom be translated in an extensional language enriched to handle specifically modal claims, with additional predicates interpreted as expressing specifically modal properties (to recall, monadic ‘ W is interpreted as ‘_ is a world’, ‘A’ as ‘_ is actual’, and dyadic predicates ‘I’ as ‘_ inhabits _’, ‘C as ‘_ is a counterpart of _’) and the whole machinery, eliminating the problem of transworld identification and introducing the notion of counterparts, geared towards explication of de re modal talk. Axioms governing the behaviour of specifically modality-related predicates of the language of counterpart theory ensure that the language may serve as a ‘vehicle for formalized discourse about modality’ (Lewis 1968: 116), with specific axioms formulated and explicated by Lewis (1968: 114) as follows:

P1: VxVy(Ixy з Wy)

(Nothing is in anything except a world)

P2: VxVyVz(Ixy & Ixz з y = z)

(Nothing is in two worlds)

P3: V x Vy (Cxy з 3 zIxz)

(Whatever is a counterpart is in a world)

P4: VxVy(Cxy з 3zIyz)

(Whatever has a counterpart is in a world)

P5: VxVyVz(Ixy & Izy & Cxz з x = z)

(Nothing is a counterpart of anything else in its world)

P6: VxVy(Ixy з Cxx)

(Anything in a world is a counterpart of itself)

P7: 3x( Wx & Vy(Iyx = Ay))

(Some world contains all and only actual things)

P8: 3xAx

(Something is actual)

There is an intricate web of assumptions about modality behind the postulates of the classical counterpart theory, and it is possibly one of the reasons that it has not found much favour in formal semantics: whereas on the Kripkean approach it is easier to separate purely technical aspects of semantics as originally developed in Kripke (1963a,b, 1965a,b) and manipulate them as required by the needs of linguistic analysis, and even if some specific extralogical views are adopted, they are easier to accept—as when Kripke (1980: 44) states that ‘A possible world isn’t a distant country that we are coming across, or viewing through a telescope. ... ‘Possible worlds’ are stipulated, not discovered by powerful telescopes’, an attitude which fosters a deflationary approach to possible worlds and thereby does not implicate a metaphysical commitment which formal semantics abstains from as far as possible; on the Lewisian theory of counterparts and modality the set of postulates governing the interpretation of predicates specific to the language of the counterpart theory is adapted to convey clearly defined commitments with regard to the status of objects belonging to the domain and the status of modal claims. The stance of genuine modal realism is not obviously a standpoint which formal semantics might wish to adopt, especially if the semantic inquiry is intended as a companion to the minimalist investigation of narrow syntax. Possible worlds which ‘are not of our own making’ (Lewis 1986: 3) and such that ‘the difference between this and the other worlds is not a categorial difference’ (Lewis 1986: 2), which may be characterized as when main theses of modal realism are introduced in Lewis (1986):

There are countless other worlds, other very inclusive things. Our world consists of us and all our surroundings, however remote in time and space; just as it is one big thing having lesser things as parts, so likewise do other worlds have lesser otherworldly things as parts. The worlds are something like remote planets; except that most of them are much bigger than mere planets, and they are not remote. Neither are they nearby. They are not at any spatial distance whatever from here. They are not far in the past or future, nor for that matter near; they are not at any temporal distance whatever from now. They are isolated: there are no spatiotemporal relations at all between things that belong to different worlds. Nor does anything that happens at one world cause anything to happen at another. Nor do they overlap; they have no parts in common, with the exception, perhaps, of immanent universals exercising their characteristic privilege of repeated occurrence. (...) There are so many other worlds, in fact, that absolutely every way that a world could possibly be is a way that some world is. (Lewis 1986: 2)

—such worlds are as far apart from points of evaluation of the Kripkean apparatus as possible, far away from the Carnapian line of providing theory-neutral logical tools, and, to say the least, do not seem promising as a point of departure for a semantic theory carrying as little ontological burden as possible. It comes as no wonder that the classical counterpart theory has not been the most influential of Lewis’s proposals as far as the contemporary semantic theory is concerned (see recently Partee (2015) and Holton (2003) for discussion): in contrast to the general semantic program as expressed in Lewis (1970), the notion of unselective binding by adverbs in Lewis (1975), the theory of counterfactuals of Lewis (1973b,a), which became so important for formal analyses of conditionals, the counterpart theory with its world-bound individuals inhabiting their own worlds and connected with actual individuals by a similarity relation seemed appropriate only in some specific settings as an alternative to transworld identification; Lakoff (1968), an early attempt to apply counterpart theory in linguistic semantics, warns his readers that ‘adopting counterpart theory as a device for semantic representation may force one into adopting a philosophical position that most modern philosophers would shrink away from in horror (Lakoff 1968: 11). Even if one is ready to put such worries aside, even if one is not troubled by similarity relation’s playing a central role (‘similarity’ being best understood as a purely technical term, as Cresswell (2004) advises, a not as incorporating pretheoretical intuitions about similarity among mundane objects; although one may stick to the requirement that it be based on qualitative similarity, this requirement still allows to ‘turn ‘similarity’ into a technical term for whatever relation is needed,’ (Cresswell 2004: 3)), and even if one is prepared to ‘extract model theory from Lewis’s work’ (to paraphrase Hazen (1979)), the main feature of the counterpart theory—creating an intimate tie between modality and individuals, thereby forcing evaluation of expressions not with respect to indices as the standard Kripkean machinery does, but with respect to individuals-at-worlds; in other words, making modality depend crucially upon the web of relations created by the counterpart relation—does not seem perfectly suited to be a property of an encompassing semantic framework in which to analyze expressions generated by narrow syntax: Kripkean semantics affords a way to interpret expressions containing various modal operators with much more flexibility provided by relative accessibility relations without objects of the domain entering the picture and without constraints imposed by properties of the counterpart relation, a feature which seems closer to properties of natural language counterparts of such operators (even if the formal language replaces operators with explicit quantification over variables of appropriately modal sorts, as is commonly done due to the sentiment that it is required by an adequate semantics of natural language and an overinterpretation of the results of Cresswell (1990); see Yanovich (2015) for a discussion of this issue), and is adopted tout court only exceptionally (Heim (2001) as reported in Sauerland (2014); Sauerland (2014) developes the account).

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