Suppose that we are willing to deviate from the path of Lewis and abandon his, Quinean in spirit, regimentation of modal claims in formulas of first order logic enriched with appropriate basic predicates, adapting instead the model theoretic apparatus used for interpretation of quantified modal logic to counterpart- theoretic view on modality, letting a Lewisian structure be F_{?} = {W, R, C, D, D_{O}), where D_{?} = {D_{n}D_{/}), D_{n} and D working as above, D_{?} assigning to each point w a set D_{r} (w) with the requirement that for any of distinct w., w,, D_{n}(w.) П D_{n} (w,) = 0 (which takes care of Lewis’s P2, another axiomatic assumption about counterparts that may be questioned and abandoned, see e. g. Belardinelli (2007), and allows numbers or sets to belong to domains of distinct worlds), an inner domain of objects which together may be a domain of quantification in its demanding sense; and D_{O} assigning to each point w an outer domain of objects—a distinction familiar from various developments of the idea of building a logic free of existential commitments (see Lambert (1959, 1963, 1965, 1967, 1972, 1974a,b, 1985, 1987, 1991, 2001, 2002); Lambert and Bencivenga (1986); Meyer and Lambert (1968), Antonelli (2000); Bencivenga (1986, 2002)) and applicable also in providing semantics for modal logics (following the advice of Scott (1970); see e. g. Kracht and Kutz (2002), Belardinelli (2007), Schwarz (2012)). C assigns to each {w.,w,) a set {{d,d) : d. e D_{r} (w) U D_{O} (w.) and d, e D_{?} (w) U D_{O} (w,)}—a

counterpart relation, in short. Let I be a function assigning to each point w and each predicate letter P. a set of objects {d : d. e D_{r} (w,) U D_{O} (w.)}. For an instance of ^Зх.П(х)^{п} to be true in a structure M_{?} = {F_{?}, Г) at a point w with respect to an assignment of objects to variables g it is required that it be true under some assignment g[_{d} Д _{(}w) ]. Let C_{?} be like C except that both objects related by the counterpart relation belong to inner domains of respective points of evaluation, the relation being therefore both left- and right-restricted and each {w,w,) being in effect assigned a set {{d., d,) : d. e D_{?} (w.) and d. e D_{?} (w,)}. If an instance of a quantified statement ^Зх.0П(х)^{п} is to be true in M_{?} at a point w with respect to an assignment of objects to variables g, it is required that ^П(х)^{п} be true at some point w' such that {w, w') e R(w) with respect to some assignment g [ _{d:dE}{_{d ;}< _{g}(_{x}),d >e_{Ci} (_{w>w}')} ]. It may be assumed that counterpart relations which are established in models which are relevant for the syntax-semantics mapping belong to CE-relations in the sense of Kracht and Kutz (2005); although stipulative in nature, it seems a reasonable point for the C-I component to begin with the procedure of interpreting syntactic objects without getting lost in models with objects suddenly lacking counterparts (see Forbes (1982) for another stance on this point).

Another amendment to the original formulation of the counterpart theory affects the axiom P5, one which forbids the counterpart relation to link objects in their own worlds, and together with P6 ensures that an object is its own unique counterpart in a world it inhabits. Outside this specific situation the counterpart theory admits of modeling fission and fusion of objects via counterpart relations; strictly speaking, an object may have multiple representatives in another world, and several objects may have just one. These possibilities are excluded on the standard formulation of the counterpart theory within a world, where the only admissible counterpart relation by P5 and P6 links objects of the domain to themselves. Lewis (1986), however, contemplates ‘the thought that I might have been someone else’:

Here am I, there goes poor Fred; there but for the grace of God go I; how lucky I am to be me, not him. Where there is luck there must be contingency. I am contemplating the possibility of my being poor Fred, and rejoicing that it is unrealised. (...) Like any other possible person, he is a possible way for a person to be. And in a sense he is even a possible way for me to be. He is my counterpart under an extraordinarily generous counterpart relation, one which demands nothing more of counterparts than that they be things of the same kind. (Lewis 1986: 231-232)

Figure 3.2: Fission of d and fusion of d and d in a counterpart model

A footnote adds: ‘...I took it as axiomatic that nothing can have any counterpart besides itself in its own world. I would now consider that requirement appropriate under some but not all resolutions of the vagueness of the counterpart relation (Lewis (1986: 232 n. 22); see also Lewis (1983: 42-43)). In fact, the restriction concerning counterparthood in an object’s own world— ‘a restriction that demonstrably does no work in the model theory, and has semantic consequences only on certain theories of possible worlds’ (Hazen (1979: 331 n. 17); see also Hazen (1977: 112-113))—was what made the semantic apparatus behave outside modal contexts in a Kripkean way (incidentally, it is also the requirement that looks dubious as a restriction on a fundamentally qualitative similarity relation, see Fara (2009: 291 n. 10))—P5 and reflexivity of the counterpart relation guaranteed by P6 work together to turn the intra-world counterparthood into the identity relation. In the absence of modal operators, de re presentation amounts to presentation of the object itself and nothing else. If one avails oneself of a language with an abstraction operator and makes use thereof in formalizing statements de re, there is no difference whatsoever with regard to truth conditions between expressions like ^(Хх.П(х)) (a)^{n} and ^П(а)^{п} (their instances, to take the risk of being pedantic); the very same object enters into an explication of truth conditions in both cases, since no modal operators intervene. This happens both on the Kripkean approach and on the Lewisian one—both accessibility relation and counterpart relation are in effect idle. Although there may be sought other differences between the two expressions—as e. g. it may be argued that they express distinct propositions despite enjoying truth-conditional equivalence (a view defended by Salmon (2010))—the difference in question turns extremely subtle, and one may wonder with Kripke (2005):

If Salmon is right, there are distinct propositions 0(a), Ххфх(а), Xykx0x(y)(a), and so on ad infinitum, all closely related but distinct. If n-place relations are involved, the situation comes to involve complicated infinite trees. Is all this really plausible? (Kripke 2005: 1025 n. 45)

Figure 3.3: Standard counterpart relation within a world in a counterpart model

Now, an expression like ^{l_}(Ax.n(x)) (a)^{n} (its instance, again) offers more possibilities when interpreted with an apparatus of counterparts freed from the assumptions that every object is its own unique counterpart in its own world: retaining the re- flexivity of the counterpart relation ensures that an object belongs to the set of its counterparts without necessarily making this set a singleton. ^(Лх.П(х)) (a)^{n} will be true in a model iff all counterparts of whatever object is denoted by Th"^{1} belong to the set determined by ^(Лх.П(х)^{п}. The difference comes as very real (and so does the difference in the case of binary relations between ^(Лх.П(х, x)) (a)^{n} and ^П(а, a)^{n}), despite there being no modal operators involved; only if the coun- terparthood relation is reduced to its extreme case—identity—may doubts as to such distinctions arise. There are certainly applications of the counterpart theory in which such restrictions are welcome and understandable, and this is reflected in Kripke’s remark:

The very term ‘propositional function’ clearly suggests that Russell did not intend any distinction between Ххфх(а) and ф(а). Nor does a mathematician analogously intend any distinction between Ax(x!)(3) and the number 6. Nor did Church, inventor of the lambda notation, intend any such distinction. (Kripke 2005: 1025 n. 45)

We have already observed above that consideration related to the behaviour of mathematical objects led Lewis to postulate a distinct way of ‘being in a world,’ according to which the counterpart relation between such objects turns to be the identity relation, in contrast to mundane entities which do not appear in domains of distinct worlds and there is therefore no possibility for them to be related to their counterparts by the identity relation. Answering the question about the behaviour of the C-I component elicited by the presence of chains requires that the realm of mathematical objects be considered not that central for the choice of interpretive properties of structures interpreted as/translated as involving А-bound variables and arguments of such А-terms. The C-I component, facing structures with several occurrences of objects which have undergone internal merge, has to be fixed upon an interpretive procedure of sufficient generality of use—applicable whenever chains appear in a structure—and as free of aprioristic commitments as possible. Restricting the counterpart relation to identity means assuming throughout that objects taken as values of variables may be identified and cross-identified without further ado, lest the interpretive process hang in the air, providing merely truth conditions without prospects of making actual use thereof (this does not imply that the domain of the model must be understood in a realist way, though); differentiating the interpretive procedure according to the absence or presence of modal operators in the body of the evaluated expression means giving up local computation of meaning. All this does not exclude possible use of interpretations so restricted—it only does not seem like the strategy that the C-I component should be pursuing in the general case.

Figure 3.4: A noncanonical counterpart relation within a world in a counterpart model

Different variants of the theory of senses and their denotations developed gradually in Church (1946, 1951, 1973, 1974, 1993) (for further modifications see e. g. Jespersen (2010), Anderson (2001))—not designed to handle specifically the behaviour of expressions of natural language, but, with appropriate adjustments, suitable as the starting point for an analysis of natural language expressions for analogous purposes (as Church (1974: 149) himself notes, ‘the stronger are the conditions required in order that two names shall express the same sense, or that concepts shall be identical, the more closely will the abstract theory of concepts resemble the more concrete theory of the names themselves—with the relations symbolized by ^_{oala} serving as analogues of the relations of denoting in the semantical theory.)—witness to the legitimacy of introducing a distinction between Xxfx(a) and f(a) for justified purposes. In the case of modeling the workings of the C-I component, the possibility of making counterpart relations taking care of such distinctions both across and within modal contexts suggests that it may be the option chosen for interpretive purposes as the default one, restrictions being possibly applied for more specific purposes—one of cases in which the C-I component may avail itself of a range of interpretive options, but one is destined to serve the purpose of interpreting objects delivered from narrow syntax as they enter the realm of interpretation. Once the counterpart relation is so relaxed, the truth of the simplest expressions '"(Ах.Щх)) (a)^{n} in a structure M_{?} = (F_{?}, Г) at a point w with respect to an assignment of objects to variables g involves already the requirement that ^{1_}П(а:)^{_|} be true in this structure with respect to all

assignments g ^_{dde|d (g(x}f_{>EC(wwи} J. All this happens with assignment functions

providing the link to objects in domains assigned to each point of evaluation— elements of domains, to avoid misunderstanings about their relationship to familiar inhabitants of the external world.