Another enrichment of the original formulation of the counterpart theory concerns domains assigned to distinct points/worlds. Pairwise disjoint, they were intended to contain individuals tout court, parts of spatiotemporally disjoint worlds understood in a realist way; such domains supply objects over which it is legitimate to quantify in the language of the counterpart theory. Existence understood in this way, as intimately tied with existential quantification over objects belonging to such sets, was not enough for all distinctions required by analysis of modal claims; and, while retaining P2, Lewis introduces additional varieties of existence:

I took as primitive the notion of an individual being in a possible world. (...) Any possible individual is part of a world, and in that sense it is in a world. (...) However, the possible individuals are not all the individuals. I wish to impose no restrictions on mereological summation of individuals, hence I must grant that there are individuals consisting of parts from several worlds. But such a cross-world sum is not a possible individual. (...) it is partly in each of many worlds, overlapping different worlds in virtue of different ones of its parts. Finally, there are the non-individuals: that is, the sets. (...) no set is in any world in the sense of being a part of it. Numbers, properties, propositions, events—all these are sets, and not in any world. Numbers et al. are no more located in logical space than they are in ordinary time and space. Even a sequence of possible individuals all from the same world is not, strictly speaking, itself in that world. (...) If we evaluate a quantification at a world, we will normally omit many things not in that world, for instance the possible individuals that inhabit other worlds. But we will not omit the numbers, or some of the other sets. Let us say that an individual exists from the standpoint of a world iff it belongs to the least restricted domain that is normally— modal metaphysics being deemed abnormal—appropriate in evaluating the truth at that world of quantifications. (Lewis 1983: 39-40)

There are thus, according to Lewis (1983), three ways to ‘be in a world’; and the distinction reappears in a modified form in Lewis (1986), wherein actuality is at stake, and the distinction is ‘between being actual, being partly actual, and being actual by courtesy’ (Lewis 1986: 96 n. 61), with differences between the two views resulting also from changes in Lewis’s views on sets and their nature. Details are closely intertwined, as usual in the canonical formulations of the counterpart theory, with metaphysical views of its founder; from a general point of view, however, separating them from the interpretive apparatus, such modifications open the way to various modelings of the domain of a model. Thus, taking the basic case to consist of varying domains, there being a function which assigns to every point of evaluation a set of objects—the domain of quantification in the most restricted sense—one may extend such sets so as to include objects which are ‘extra-worldly’. The precise nature of this ‘extra-worldliness’ depends in the case of the standard counterpart theory upon metaphysical choices with regard to objects which may enter interpretations of quantified statements, but the most general sense is purely technical in nature (see Iacona (2007) for a discussion with regard to the simplest quantified modal logic of Linsky and Zalta (1994)): on a standard treatment of varying domain semantics, the domain of a Kripkean structure F_{k} = {W R, D), where D is a function assigning to each index w a set D(w), is defined as U {D(w) : w e W}—a stipulation which may well be useful for many purposes and may make it easy to switch back and forth between varying and constant domain semantics, but a stipulation nonetheless. Take a monadic predicate P and a structure M_{k} = {F_{k}, I), with I an interpretation function such that for every point w, an object d e D(w) iff d satisfies P(x) at w. Interpretation of the predicate P assigns then to every point exactly the set of objects assigned to it by D, but without further stipulation it is possible that there are objects which do not satisfy P(x) at any point at all. If the interpretation of P is invariant across models built on a structure F_{k}, the domain of a structure is not the smallest set satisfying the definition above—there are objects which are not values of D for any w. A predicate constant with an interpretation invariant in this way is hardly unheard of; depending upon the intended interpretation of P, the structure will partition the whole domain of objects in different ways, according to their being actual, concrete, existing etc. at various points or entirely non-actual, abstract, non-existing at all (given the tie between the domain of a world/point and actualist quantification, the predicate in question will be then evaluated at every point as nothing more and nothing less than the familiar x3y.y = x). A Kripkean may then want to enrich his structures and make them rather F_{k}+ = {W, R, D^ D), with D_{n} performing the task of D in a structure F_{k}, and D assigning to each point a subset (possibly improper) of the set of objects which do not belong to D (w) for any w e W. An instance of a simple quantified statement’s ^Зх.П(х)^{п} being true in a model M_{k}+ = {F_{k}+, I) at a point w with respect to an assignment of objects to variables g demands that ^П(х)^{п} be true under an assignment g_{x} which agrees with g except possibly with regard to the variable x and is such that it assigns to x an object which belongs to D_{n}(w) U D (w). For a(n instance of a) quantified statement ^3x.0n(x)^{n} in a model M with respect to an assignment of objects to variables g to be true requires then that there be a variant g_{x }of g such that it agrees with g except possibly with regard to the variable x and such that ^П(х)^{п} be true at some point W such that wRw’ with respect to g_{x}; which, in turn, requires that an object satisfying such a formula belong to D_{n}(w') U D (w^{r}) for some w' such that wRw'.

Returning to counterpart theory, it is to be immediately noted that adding D to the frame in such teoretical context is not equivalent to, but, taken generally, rather a special case of, enriching a frame with a function assigning to each point w a set consisting of objects from the set of entities {d: d g D_{n}(w)}. It differs from the standard use of the latter form of enrichment of a frame since, first, the domain of the frame is enriched so as to contain objects not being members of any D_{n}; second, since, it is explicitly restricted to such objects; third, since elements of the set selected for each w in this way are in the range of existential quantification—all in contrast with outer domains as they are exploited in vari ous incarnations of systems of free logic—logic free of existential assumptions, denying that inference from ^П(я)^{п} to ^Зх.П(х)^{п} is valid. In this way, objects selected by Dare available as values of variables bound by the 3-quantifier and in their case the counterpart relation is identity—quite a sensible behaviour for mathematical objects, for example (although one may raise doubts as to whether this kind of ‘being in a world’ is still not subject to a modified version of a criticism that ‘it makes five and twelve much too much like photons,’ as (Putnam 2013: 202) summarized his opinion on the Quinean stance on the existence of sets): one does not need to deny that they exist, one may easily quantify over them, one does not make them possibilia, and still, while sticking to the tenets of the counterpart theory, one is not immediately committed to an admittedly strange view that the number two may have a counterpart which has, in some other world, properties only similar to the number two in the actual world, not to mention a stranger yet situation in which the empty set would have counterparts, not identical, but qualitatively similar to it. In the case of the standard counterpart theory, it is a strict interpretation of ‘being in a world’ as expressed in P1, P3 and P4 that makes them work together to ensure that there are no extraworldly entities (P1) which might be connected by a counterpart relation, which has both its domain and the range restricted to objects belonging to domains of relevant worlds (P3 and P4); these axioms need not be rejected, but their interpretation might be relaxed to embrace ‘existence from the standpoint of a world’ as the third way of ‘being in world’ is characterized in the quote from Lewis (1983) above.