Abstract
In this chapter, I discuss, first, the nature of the proposal, according to which it is a ‘transcendental precondition’ of the way in which we speak and think about individuals in modal settings that they are categorized as world lines (Sect. 2.2).
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Notes
- 1.
This said, certainly nothing prevents X and Y from being modal statements—being, for example, of the form possibly Z or necessarily Z. I merely wish to stress that this is not a part of what saying ‘Y is a necessary condition for the possibility of X’ means.
- 2.
All references of the form An/Bm or An or Bm are to Kant [60].
- 3.
The unspecific task would be to argue for \(P_1\) or for \(P_2\). However, it is beyond the scope of this book to undertake a global defense of realism or transcendental idealism.
- 4.
I take it that neither physical nor intentional objects should be considered as local objects, but both types of objects must be viewed as world lines. In the case of intentional objects, this is so even if we limit attention to objects that are not thought of as being temporally extended or having modal properties. This is because there is normally a great variety of scenarios compatible with an agent’s intentional state; intentional objects are construed as world lines considered in relation to the totality of all such scenarios.
- 5.
The syntax is conveniently specified in Backus–Naur form. It should be understood as follows. If Q is a predicate and the \(x_i\) are variables, both \(Q(x_1,\ldots ,x_n)\) and \(x_1=x_2\) are formulas. The result of prefixing a formula by \(\lnot \), \(\square \), or \(\exists x\) is likewise a formula. Finally, the result of combining a formula with a formula using \(\wedge \) is a further formula. The symbol \(\phi \) represents schematically any expression generated by the grammar, and distinct occurrences of \(\phi \) need not represent the same expression.
- 6.
There is a sense in which the question ‘Is the set \( dom (w) \cap dom (w')\) non-empty?’ is ill formed: in any single world w, we only encounter its local objects, and remaining within w, we can never effect the relevant comparisons allowing us to meaningfully affirm that every \(a \in dom (w)\) is distinct from every \(a' \in dom (w')\). From a metatheoretic perspective, we can answer this question. Worlds partition the class of local objects into cells so that local objects a and \(a'\) can be compared in terms of numerical identity iff they belong to the same cell. The set \( dom (w) \cap dom (w')\) is empty if w and \(w'\) correspond to distinct cells of this partition; otherwise, w equals \(w'\). Cf. footnote 15 in Sect. 4.5.
- 7.
The distinction between availability and realization is considered in detail in Sect. 3.3. For a recent discussion of quantifying-in in general attitudinal contexts, see Jespersen [56].
- 8.
Cf. the discussion in footnote 5 in Sect. 1.1.
- 9.
Hintikka [40] showed that in first-order logic, the semantics of quantifiers can be interpreted ‘inclusively’ or ‘exclusively’ and that in the latter case, the expressive power of first-order logic is not diminished by disallowing the use of the identity symbol in the syntax (supposing we restrict attention to vocabularies in which neither constant nor function symbols occur). Unlike in the standard ‘inclusive’ interpretation, according to the ‘exclusive’ interpretation, a formula \(\exists x\, \phi (x,x_1,\ldots ,x_n)\) is satisfied in a model
under an assignment \(\varGamma \) iff there is in the domain of 
an individual b other than any of the individuals \(\varGamma (x_1),\ldots ,\varGamma (x_n)\) such that \(\varGamma [x/b]\) satisfies the formula \(\phi (x,x_1,\ldots ,x_n)\) in 
: the range of x excludes the values of all variables \(x_1,\ldots ,x_n\) that are free in the scope of the quantifier \(\exists x\). Wehmeier [125] attempts to argue that we can dispense with the binary relation of identity, and as a partial motivation, he refers to Hintikka’s result. However, as Hintikka [40, p. 228] himself stressed and contrary to what Wehmeier [125, Sect. 2] suggests, Hintikka’s result merely shows that we do not need the identity symbol in the syntax of first-order logic; the result by no means suggests that we can dispense with the notion of identity. The exclusive interpretation of quantifiers merely provides an alternative way of dealing with the notion of extensional identity in first-order logic. - 10.
Given the syntax of \(L_0\), the only possible quantifier meeting this criterion could be \(\exists x\). The obvious syntactic notion of subformula gives rise to the notion of scope in the usual way.
- 11.
As in first-order logic, also in \(L_0\), the satisfaction of a formula \(\phi \) under an assignment g evidently depends only on the values of g on those variables that are free in \(\phi \).
- 12.
No atomic formula P(x) can be satisfied by a value \(x:=\mathbf {I}\) in a world w unless the world line \(\mathbf {I}\) is realized in w. Thus, if ‘haired’ and ‘bald’ are construed as atomic predicates and \(\mathbf {I}\) is not realized in w, the assignment \(x:=\mathbf {I}\) satisfies neither haired(x) nor bald(x) in w. As these predicates are used in English, qualifying anything as haired or bald in a context w would perhaps be taken to presuppose that the thing is present in w, rather than its presence in w being viewed as part of what is affirmed by such a qualification. In any event, natural-language semantics agrees that ‘haired’ and ‘bald’ cannot be satisfied in w by anything not realized in w. The same holds for such predicates as ‘unfair’. If the value \(\mathbf {I}\) of x is realized in w, we can safely say that \(x:=\mathbf {I}\) satisfies unfair(x) in w iff it satisfies \(\lnot \)fair(x) in w, but generally, we could have \(w,x:=\mathbf {I}\models \lnot \)fair(x) because \(\mathbf {I}\) is not realized in w. In that case, we would not have \(w,x:=\mathbf {I}\models \) unfair(x), since this would require that \(\mathbf {I}\) be realized in w. That is, unfair(x) is not simply the negation of fair(x). This said, unfair(x) can be defined in terms of fair(x): the formula unfair(x) is equivalent to \(x=x\wedge \lnot \)fair(x), since this latter is satisfied in w by \(x:=\mathbf {I}\) iff \(\mathbf {I}\) is realized in w and \(\mathbf {I}(w)\) fails to be fair. One could define a strong notion of negation (\(\sim \)) in L, by stipulating that \(\sim \!\!P(x)\) means \(x=x \wedge \lnot P(x)\). Then, unfair(x) could indeed be defined as a negation of fair(x) in a certain sense, because unfair(x) is equivalent to \(\sim \)fair(x).
- 13.
We say that y is free for x in \(\phi \) iff x does not occur free in the scope of the quantifier \(\exists y\) in \(\phi \). If y is not free for x in \(\phi \), substituting y for a certain free occurrence of x in \(\phi \) results in a formula in which that occurrence of y is bound.
- 14.
For sortals, see, e.g., Grandy [35], Lowe [81], Wiggins [126].
- 15.
For variants of four-dimensionalism, see, e.g., Lewis [76], Hawley [38, 39], Sider [112].
- 16.
- 17.
Let X be a set, \(\kappa \) a cardinal number, and
a collection of non-empty, not necessarily pairwise disjoint subsets of X. If \(X = \bigcup _{i < \kappa } C_i\) and \(X \subseteq Y\), then 
is a cover of X and a subcover of Y. If 
is a cover of X and the elements of the collection 
are pairwise disjoint, then 
is a partition of X and a subpartition of Y. - 18.
If S is a set, I write |S| for its cardinality.
- 19.
We could opt for a symmetric concept of \(\mathbf{b}\) and \(\mathbf{c}\) being ‘co-realized’. However, I prefer to view B as providing the contexts in which elements of C may or may not be realized.
- 20.
We could go much further in generalizing the notion of system of modal unities, but the type of language to be used for talking about such systems imposes limits to what are reasonable generalizations. Modal operators do not carry syntactic variables, and they are evaluated in terms of binary relations among worlds, which is why we cannot end up evaluating a formula relative to n worlds for \(n \ge 2\). Neither is the set of worlds accessible at a given world dependent on values of first-order variables. Thus, it would be pointless to replace B by a collection of subcovers of A indexed by elements of C or to replace the relation R (or, more generally, a collection of binary relations) by a collection of relations on B with arbitrary arities. When predicate symbols are interpreted extensionally, it suffices to define the function Int as above. Otherwise, its values should be defined as sets of tuples of elements of the set \(\bigcup _{\mathbf{b} \in \mathbf{B}}{} \mathbf{C}_{\mathbf{b}}\).
- 21.
In any event, this is the most natural way of understanding what Kripke says. Admittedly, if the referents of rigid designators were world-bound objects, there would by hypothesis be no issue of cross-world identity concerning them. Since Kripke rejects the domain constraint, he could speak of ascribing predicates to a world-bound object of world w relative to a distinct world \(w'\). However, in fact, Kripke does not assume that referents of rigid designators are world-bound but allows them to be objects that exist in several worlds. Independently of this interpretive issue, we may note, as Kaplan [61, pp. 492–3] does, that Kripke characterizes his notion of rigid designator in mutually incoherent ways. At times, he says that a rigid designator refers to the same object in all worlds [71, p. 48]. At other times, he takes a rigid designator to refer to the same object in all those worlds in which the object exists ([68, p. 146], [71, p. 49]). The two formulations are equally problematic from the viewpoint of cross-world identity adopted in this book. Kripke’s notion of rigid designator was anticipated in the work of Marcus [82]. She speaks of proper names as ‘identifying tags’ whose descriptive meaning is lost or ignored.
- 22.
As was explained above, already the language-independent idea of considering how this object behaves in counterfactual circumstances presupposes the notion of cross-world identity. A fortiori, then, this same presupposition is involved in the idea of taking the actual referent of a linguistic expression as one’s starting point and considering how this referent behaves in counterfactual circumstances. The linguistic detour cannot remove the heart of the problem, though it can serve to hide it. In particular, the identity of a proper name does not translate into the identity of its actual referent: it may be unproblematic to say that two occurrences of ‘Hesperus’ are two occurrences of the same name, but this linguistic fact has no bearing on the issue of whether it makes sense to say that ‘Hesperus’ refers to one and the same non-linguistic entity on the two occasions.
- 23.
For a critique of the substitutional interpretation applied in modal logic, see [49, p. 28], [51].
- 24.
- 25.
Hawley [39] calls attention to the fact that in Lewis’s analysis, the analog between time and modality is not complete (ordinary objects are world-bound but not time-bound) and discusses the possibility of achieving uniformity by explicating not only sameness across worlds but also sameness over time in counterpart-theoretic terms. In Sider’s stage theory [111], statements about identity over time are indeed accounted for in terms of temporal counterpart relations.
- 26.
For contingently satisfied predicates, see Sect. 4.2. The fact that Kripke’s Humphrey-objection is not applicable against my analysis does not depend on how we choose to deal with proper names in our semantics. In this book, I opt for construing proper names as standing for local objects relative to a world: for every world w in the modal margin of \(\mathbf {J}\), the interpretation of ‘Humphrey’ in w equals \(\mathbf {J}(w)\). Another option would be to construe proper names intensionally, letting ‘Humphrey’ to stand for the world line \(\mathbf {J}\). For a discussion, see Sect. 3.4 and footnote 29 in Sect. 6.6.
- 27.
See Bricker [7, p. 287]. Lewis considered different versions of the thesis. Arguably, his final formulation [78] does not even count as a supervenience thesis (cf. Weatherson [124]).
- 28.
Lewis defines haecceitism as the doctrine that there are at least some cases of ‘haecceitistic difference’ between worlds: there are worlds that do not differ qualitatively in any way but differ in what they ‘represent de re’ concerning some individual. According to anti-haecceitism, there are no cases of haecceitistic difference. Haecceitism is, for instance, compatible with there being the worlds \(w_0\), \(w_1\), and \(w_2\) satisfying the following conditions: (1) each world \(w_i\) has exactly two inhabitants (\(a_i\) and \(b_i\)); (2) individual \(a_1\) is P but \(b_1\) is not P, while \(a_2\) is not P though \(b_2\) indeed is P; and (3) both individuals \(a_1\) and \(a_2\) are counterparts of \(a_0\). The worlds \(w_1\) and \(w_2\) are qualitatively exactly alike, but \(w_1\) represents de re concerning \(a_0\) that it is P, while \(w_2\) represents de re concerning \(a_0\) that it is not P. Lewis [76, p. 225] maintains that a haecceitist need not accept that an individual is distinguished from all other individuals by a haecceity—an unanalyzable non-qualitative property that this individual has and all other individuals lack. In the example, haecceitism without haecceities would mean that not only is the fact that \(a_1\) and \(a_2\) are counterparts of \(a_0\) not triggered by qualitative properties of the three worlds \(w_0\), \(w_1\), and \(w_2\), but this is a primitive fact not determined by any property at all that would belong to the individuals \(a_0\), \(a_1\), and \(a_2\). It must be noted that Lewis’s characterization of haecceitism is not neutral but depends on the idea of de re representation conceptualized in terms of counterparts of world-bound individuals.
- 29.
World lines should not be viewed as generated by any sorts of properties—in particular, not by anything qualifiable as ‘haecceities’; cf. the comments on essences in Sect. 2.7.4. My position is certainly closer in spirit to haecceitism without haecceities than to anti-haecceitism—although literally, my position is anti-haecceitist by Lewis’s criteria, since I maintain that there are no cases of haecceitistic difference between worlds, for the simple reason that I maintain that there are no Lewisian de re representations in the first place. World lines are not supervenient on world-internal facts, so there could be a world line \(\mathbf {I}\), worlds \(w_1\) and \(w_2\), and local objects \(a_1 \in dom (w_1)\) and \(a_2 \in dom (w_2)\) such that as to their internal qualitative features, \(w_1\) and \(w_2\) are exactly alike, the local object \(a_1\) is P while \(a_2\) is not P, and yet \(a_1\) and \(a_2\) could both be realizations of the world line \(\mathbf {I}\). (For ‘exact likeness’, cf. the notion of internal indistinguishability discussed in Sect. 4.5.) Even though my view resembles haecceitism, I do not subscribe to haecceitism in Lewis’s sense: the world \(w_2\) does not involve a representation de re concerning the local object \(a_1\) in virtue of the fact that \(a_2=\mathbf {I}(w_2)\) and \(a_1=\mathbf {I}(w_1)\). Local objects or the worlds in which they are located do not in any sense represent local objects to be found in other worlds. What may but need not happen is that local objects of two worlds are realizations of one and the same world line.
- 30.
In Sect. 5.7, I explain how my modal language \(L_0\) (and its extension L, to be introduced in Sect. 3.4) can be translated into first-order logic. This translation makes it particularly easy to observe the formal similarities between world line semantics and Lewis’s counterpart theory—the latter being standardly presented in terms of first-order logic, cf. [75].
- 31.
Lewis resorts to non-transitive counterpart relations in his reply to Chisholm’s identity paradox (see [14], [76, pp. 243–8]). The paradox can be presented as follows. Take two individuals existing in \(w_0\)—say, Adam and Noah. Suppose Adam has properties \(P_1,\ldots ,P_n\) and Noah the properties \(Q_1,\ldots ,Q_n\) in \(w_0\). (For simplicity, let us suppose that the properties \(P_1,\ldots ,P_n,Q_1,\ldots ,Q_n\) are pairwise independent.) Presumably, at least some of Adam’s and Noah’s properties could get exchanged. If so, by repeated exchanges, we arrive at a sequence of worlds \(w_0,\ldots ,w_n\) such that in \(w_i\), Adam has the properties \(Q_1,\ldots ,Q_{i},P_{i+1},\ldots ,P_{n}\) and Noah the properties \(P_1,\ldots ,P_{i},Q_{i+1},\ldots , Q_{n}\) (for all \(1 \le i \le n\)). In \(w_n\), Adam’s properties are those of Noah in \(w_0\), and vice versa. Supposing there are no other individuals in \(w_0\) and \(w_n\), these worlds are qualitatively exactly alike, so individuals appear to have a ‘bare identity’ entirely unrelated to their properties. This suggests that for any property and any individual, there is a world in which the individual has this property. Lewis blocks this reasoning by appealing to the idea of world-bound individuals. Adam and Noah are located in \(w_0\). Even if there were individuals \(a_1\) of \(w_1\) and \(a_2\) of \(w_2\) such that \(a_1\) is a counterpart of Adam and \(a_2\) is a counterpart of \(a_1\)—with \(a_1\) having the properties of Adam except for \(P_1\), and \(a_2\) having the properties of Adam save for \(P_1\) and \(P_2\)—we cannot infer that \(a_2\) is a counterpart of Adam, since the counterpart relation need not be transitive. From my viewpoint, Adam is a world line. He is realized in \(w_0\). He has a fixed modal margin, which may or may not contain the worlds \(w_1,\ldots ,w_n\). World lines are not supervenient on qualitative world-internal features. Whatever (extensional) predicates a local object of a world \(w_i\) satisfies, it need not be the realization of Adam in \(w_i\). This is why Chisholm’s reasoning cannot be carried out when world line semantics is assumed.
- 32.
Note that Lewis does not preclude the possibility of an individual having several counterparts in a given world. Unless it is required that every \(\mathbf{C}_{\mathbf{b}}\) with \(\mathbf{b} \in \mathbf{B}\) be a partition of \(\mathbf{A}\), we can indeed have \( CP (\mathbf{a_0}, \mathbf{a_1})\) and \( CP (\mathbf{a_0}, \mathbf{a_2})\) with \(\mathbf{a_1} \ne \mathbf{a_2}\), where \(\mathbf{a_1}, \mathbf{a_2}\) belong to the same element \(\mathbf{b'}\) of \(\mathbf{B}\). Namely, there can be \(\mathbf{b} \in \mathbf{B}\) with \(\mathbf{b} \ne \mathbf{b'}\), and distinct \(\mathbf{c_1}\) and \(\mathbf{c_2}\) in \(\mathbf{C}_{\mathbf{b}}\), such that \(\mathbf{a_0} \in \mathbf{c}_1 \cap \mathbf{c}_2 \cap \mathbf{b}\), while \(\mathbf{a_1} \in \mathbf{c}_1 \cap \mathbf{b'}\) and \(\mathbf{a_2} \in \mathbf{c}_2 \cap \mathbf{b'}\).
- 33.
Fine [25, p. 73] allows even variable embodiments as values of F (cf. Koslicki [63, p. 78]).
- 34.
For further remarks in this direction, see Sect. 4.6, esp. footnote 17.
- 35.
Priest [95, pp. 46–7] takes ‘objects’ to be functions from worlds to ‘identities’ (see Sect. 3.8 below) and notes that in his framework, there is a risk of a similar superabundance problem. If d is an object and d(w) is its identity in w, can d(w) be viewed as a part of d at w? Priest discusses this idea but dismisses it because he sees it as giving metaphysical priority to identities, and he takes it that this would lead to an uncontrolled proliferation to objects: any function from worlds to identities would count as one.
- 36.
For a blatant example, the condition expressed by the definite description ‘the president of the US’ picks out, in every world in which it is applicable at all, the realization of a unique individual, but these different realizations do not belong to any one individual: no world line is first manifested as (a realization of) Bill Clinton and later on as (a realization of) George Bush Jr. This definite description defines a certain partial function from worlds to local objects, but there is no reason to assume that this function could be a value of a quantified variable—i.e., that it corresponds to a world line. Cf. footnote 38 in Sect. 6.7.
- 37.
Also, Kracht and Kutz [66] assimilate world lines to what they call individual concepts, but in the sense in which they take world lines to be individual concepts (world lines being extracted from counterpart relations), these individual concepts could not be constant functions.
- 38.
A haecceity can be viewed as a trivial individual essence, as opposed to an informative or non-trivial individual essence—a set of qualitative properties whose possession by the individual would be a necessary and sufficient condition for its being the individual it is. It was remarked in footnote 29 of this chapter that world lines are not haecceities. See also footnote 28 in this chapter.
- 39.
In the special case that \(\phi (x)\) is atomic, this amounts to a condition concerning the realizations of \(\mathbf {I}\). The world line \(\mathbf {I}\) is necessarily P(x), if for all worlds w in the modal margin of \(\mathbf {I}\), the realization \(\mathbf {I}(w)\) of \(\mathbf {I}\) belongs to the interpretation of P in w.
- 40.
This is basically how Plantinga [94, pp. 76–7] defines the notion of essence.
- 41.
Lowe makes much of Frege’s discussion [29, Sects. 62–9] of identity criteria (Kennzeichen) in connection with the mathematical practice of defining abstract entities (like directions) as equivalence classes of somewhat less abstract entities (like lines). This leads Lowe to postulate that various sortal terms \(\varPhi \) have an associated ‘criterial relation’ R so that whenever x and y satisfy \(\varPhi \), we have \(x = y\) iff R(x, y). This is a dubious generalization, since here, identity is applied to entities of the same logical type as those to which the criterial relation is applied, while Frege applies identity to sets of lines (directions) and the criterial relation—parallelism—to lines themselves.
under an assignment 
an individual b other than any of the individuals 
: the range of x excludes the values of all variables 
a collection of non-empty, not necessarily pairwise disjoint subsets of X. If 
is a cover of X and a subcover of Y. If 
is a cover of X and the elements of the collection 
are pairwise disjoint, then 
is a partition of X and a subpartition of Y.Author information
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Tulenheimo, T. (2017). The Nature of Modal Individuals. In: Objects and Modalities. Logic, Epistemology, and the Unity of Science, vol 41. Springer, Cham. https://doi.org/10.1007/978-3-319-53119-9_2
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