Abstract
Many important ideas discussed in analytic philosophy since the mid-20th century have been phrased in terms of possible worlds understood as mutually incompatible but intrinsically possible alternative scenarios . Such worlds involve a number of objects that enjoy various properties and are interrelated in different ways. Further, they provide circumstances of evaluation of suitable declarative sentences, allowing one to determine such sentences as true or false.
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Notes
- 1.
The notion of possible world is abstract, in the sense of admitting many interpretations. This is not to say that possible worlds are abstract. Of the various things that qualify as worlds, instants are, for example, more likely to count as abstract than the actual physical universe. For not understanding worlds as entire possible world histories as Lewis [76] does, see, e.g., Hintikka [47], [49, pp. 22–3].
- 2.
Individuals do not normally count as worlds in the sense discussed: declarative sentences are not normally taken to be true or false with respect to individuals. (In Prior’s egocentric logic, however, individuals are precisely viewed as contexts of evaluation of certain types of sentences; cf. Sect. 6.5.) I follow this standard view in this book and do not consider individuals as worlds.
- 3.
Different metaphysical interpretations of the notion of local object are possible. If we are interested in temporally extended worlds, then local objects could be metaphysically interpreted as world-bound temporal parts or time-slices of individuals. A sense-datum theorist of perceptual experience could view sense data as local objects that in suitable circumstances serve as signs of spatiotemporally extended material objects; in this vein, Russell [106, pp. 115–6] defines physical things as those series of aspects (sense-data, appearances) that obey the laws of physics. A proponent of neutral monism could interpret local objects as portions of neutral stuff out of which physical and mental entities are constructed.
- 4.
As noted in the Preface, this is essentially the distinction to which Carnap [11] referred by speaking of logical identity and genidentity.
- 5.
Various philosophers have found it perplexing to consider identity as a binary relation; for a discussion, see [125]. For my part, I assume the standard mathematical understanding of relations (relations-in-extension). Thus, whenever A is a set, the set \(\{\langle a,a \rangle : a \in A\}\) is a binary relation—namely, the identity relation on A. In Sect. 2.3, I point out that in the modal language I introduce, the semantic value of a formula of the form \(x=y\) can be taken to be a ternary relation whose terms are a pair of world lines and a world. Yet, this fact does not make the relation of extensional identity disappear: precisely when the mentioned ternary relation prevails among world lines \(\mathbf {I}\) and \(\mathbf {J}\) and a world w, the realization of \(\mathbf {I}\) in w and the realization of \(\mathbf {J}\) in w are extensionally identical local objects of the world w.
- 6.
The semantic notions of extensional identity and local object are understood with reference to relatively simple worlds. It will depend on the type of language that is considered which sorts of contexts count as relatively simple worlds: their ‘simplicity’ need not be absolute in any sense. For example, having started with a language \(L_1\) used for talking about temporally unanalyzed worlds, we might turn our attention to the instantaneous phases of those scenarios and formulate a language \(L_2\) suitable for making claims about temporally structured worlds. Entities that are considered as local objects (not as individuals) in connection with \(L_1\) are seen as temporally extended individuals in connection with \(L_2\), linking together local objects, each of which is bound to a specific temporal phase. In \(L_2\), the only unproblematic notion of identity is identity relative to temporal phases. By contrast, the world-relative notion of local identity that was unproblematic in \(L_1\) becomes a notion of cross-temporal identity in \(L_2\). It may be noted that the notion of ‘individual’ in the semantics of first-order logic is formally analogous to the notion of local object in the following sense. Formulas of first-order logic are evaluated relative to a domain whose elements are referred to as ‘individuals’. We can consider entities of any internal complexity as values of first-order variables (as individuals), provided that we are merely interested in talking about those entities themselves and not of their potential constitution. Among such individuals, there could be, for example, sets of natural numbers (that is, elements of the power set of \(\mathbb {N}\)). Should we wish to talk not only about such sets but also about their elements, we would need to leave aside first-order logic evaluated relative to the power set of \(\mathbb {N}\) and turn attention to second-order logic evaluated relative to the set \(\mathbb {N}\) itself, so that first-order variables would take natural numbers as values, while second-order variables would take as values sets of natural numbers.
- 7.
The difficulty in settling on a specific reading is due to the specification ‘except itself’. The meaning of the binary connective ‘except if’ is given by the equivalence (p except if q) \(\Leftrightarrow \) \((\lnot p \leftrightarrow q)\). Consequently, (x is not identical to y except if y is x itself) has the form \((\lnot \lnot x=y \leftrightarrow x=y)\). While \(\forall x \square \, \forall y (\lnot \lnot x=y \leftrightarrow x=y)\) is a possible rendering of the phrase ‘nothing is ever identical to anything else except itself’, it is weaker than what is presumably intended, since the condition \((\lnot \lnot x=y \leftrightarrow x=y)\) holds vacuously for any values of x and y—also in the special case that no value of x is identical to any value of y.
- 8.
Because at present we are discussing how to interpret formulas of quantified modal logic, we have as yet no precise semantics relative to which we could rigorously speak of entailment.
- 9.
- 10.
Meinongians reason in terms of a stock of objects of which some exist while others are non-existent. Objects in the former class satisfy the existence predicate; those in the latter class do not. For a discussion on Meinongianism, see Sect. 3.8.
- 11.
For existence-entailing predicates, see Priest [95, Sect. 3.3], Crane [21, Sect. 3.3].
- 12.
Hintikka fails to make a clear distinction between local objects and individuals; cf. Sect. 1.5 below.
- 13.
See Sect. 2.7.2 for a comparison between Lewis’s view and my proposal.
- 14.
Saying that a world line (an individual) ‘exists’ in a world means that it is realized therein. Quantificational locutions such as ‘there is’, again, will be understood in terms of world lines available as values of quantified variables. The important distinction between availability and realization is discussed in detail in Sect. 3.3.
- 15.
The described relation between world lines and the corresponding partial functions can be compared to the relation between variable embodiments and principles of variable embodiment in Kit Fine’s metaphysics. (For a discussion, see Sect. 2.7.3.) Observe that the partial functions \(\mathbf {I}\) as described above are not ‘individual concepts’, if individual concepts are taken to be functions whose values are individuals (possible values of quantified variables). Values of partial functions induced by world lines are local objects, not world lines—not individuals, not entities of the sort that function as values of quantified variables. For a discussion of what world lines and their corresponding partial functions are not, cf. Sect. 2.7.4.
- 16.
For this way of understanding cross-world identity, see Tulenheimo [118, p. 384]. Note that denying the meaningfulness of asking whether local objects of distinct worlds are extensionally identical in no way compromises the notion of partial function whose arguments are worlds and whose values are world-bound local objects. A partial function f is well defined as soon as there is a set A and a family \(\{B_i : i \in A\}\) of sets indexed by the elements of A such that set-theoretically speaking, f is a set of pairs \(\langle a,b \rangle \), where \(a \in A\) and \(b \in B_a\). What counts is that for every \(a \in A\), it is clear whether f is defined on a, and if indeed f is defined on a, it must be clear which element of the set \(B_a\) is being associated with a. In order for there to exist such a function, it is absolutely irrelevant whether comparisons in terms of identity and numerical distinctness can be made between elements of sets \(B_i\) and \(B_j\) with \(i \ne j\). In typical mathematical cases, such comparisons will be possible, but this is by no means essential for the definition of the notion of (partial) function.
- 17.
This is a general analysis of individuals, meant to apply however worlds are interpreted from a substantial viewpoint. Thus, there are no grounds for saying categorically that these individuals are transcendent in the sense ‘passing beyond all experience’. If worlds are entire universes, then yes; if they are spatial perspectives or instants within a structured world, then no. But, under all interpretations, they are transcendent in the sense of not residing in a single world, when ‘world’ is understood in the abstract semantic sense explained in Sect. 1.1.
- 18.
The supervenience of counterpart relations on local properties of worlds is a part of Lewis’s thesis of Humean supervenience . Namely, this thesis leads to the adoption of what Lewis calls anti-haecceitism, according to which facts about any given world supervene on the distribution of qualitative properties and relations within worlds. Modal facts about Lewis’s world-bound individuals (representations de re) are articulated in terms of counterpart relations. These counterpart relations must, then, supervene on local features of the worlds. Otherwise modal facts would be independent of the distribution of qualitative properties and relations within worlds, contrary to anti-haecceitism . For a discussion, see Sect. 2.7.2; cf. [76, Sect. 4.4], [124]. It should be noted that one can defend ‘haecceitism’ without postulating ‘haecceities’ and that world lines are not haecceities, cf. footnotes 28 and 29 in Sect. 2.7.2 .
- 19.
Lewis admits that at best, the principle of Humean supervenience holds contingently; see [77, p. x], [78, pp. 474–5]. The supposed contingent truth of Humean supervenience in ‘worlds like ours’ would not render the mentioned coincidence much less remarkable. It would mean that whenever our actual world is among the worlds considered, these worlds determine a corresponding set of world lines.
- 20.
When discussing objects of experience, Carnap [11, Sect. 128] takes visual things to be conceptually prior with respect to states of these things but notes that ‘it might be more appropriate to construct first the states-of-things’ and only afterwards the things as classes of genidentical states-of-things. In my exposition, I opt for this latter type of procedure.
- 21.
- 22.
He speaks of ‘perception’ instead of ‘perceptual experience’ but says explicitly that he uses the word without presupposing factiveness [44, p. 153].
- 23.
Here and henceforth, when speaking of the actual world—denoted by ‘\(w_0\)’—I mean the scenario that happens to be the one in which the epistemic agent or language-user considered is situated. I do not wish to suggest that among all worlds, one specific world has, once and for all, been chosen as a distinguished world. In my usage, ‘the actual world’ is the current circumstance of evaluation or the situation in which an agent presently finds herself. It is simply convenient to agree that the symbol ‘\(w_0\)’ and the expression ‘the actual world’ stand for the relevant contextually determined scenario. Consequently, \(w_0\) may but need not be the possible world or the spatiotemporally specific location in which the reader of these lines is situated. In linguistic settings, ‘the actual world’ is in my usage synonymous with ‘the (initial) circumstance of evaluation’, a phrase whose denotation evidently varies from case to case.
- 24.
Assimilating worlds to first-order models does not mean ignoring the relevant distinction between worlds and models that Williamson [127, pp. 81, 83] hails as Kripke’s decisive innovation [67]. Carnap [12] employed in his semantics state descriptions, which played simultaneously the role of worlds and first-order models. The totality of state descriptions represented all combinatorially possible ways of interpreting non-logical predicates of the language and constituted the one and only model of modal logic (modal structure). If elements of domains of modal structures are generically called ‘contexts’, all that matters is that we use modal structures in which each context is associated with a set of accessible contexts and that no specific way of selecting the associated sets is given a privileged status. We are free to think of contexts as first-order models if we so wish. This said, possible worlds must certainly not be strictly speaking identified with first-order models. As Hintikka [45] stresses, worlds must be seen as being structured by properties and relations, differently instantiated in different worlds, and interpretations of non-logical predicates must match these instantiations instead of being chosen arbitrarily; cf. footnote 5 in Sect. 3.2.
- 25.
For the notation \(\mathbf {l}(\mu _i)\), cf. [44, p. 92].
- 26.
This view is easily obscured by mathematical models employed in semantic theorizing. Typically, the domains of these models are sets of mathematical objects, such as numbers. If A and B are sets of numbers, we are normally justified in forming the intersection \(A \cap B\) and asking whether \(A \cap B\) is empty. Likewise, if \(a \in A\) and \(b \in B\), we are normally justified in asking whether a equals b. That is, our means of formal representation are in this case, in this respect, misleading.
- 27.
That is, it is possible to formulate a well-founded analogy between Hintikka and Kant—although Kant’s view has a somewhat more explicit epistemic stress than Hintikka’s view has, according to its transcendental interpretation. For Kant, intuitions are awarenesses of individuals. For Hintikka, world lines are individuals. For Kant, intuitions occur in experience, which is always structured according to the forms of sensibility (spatiality, temporality). For Hintikka, world lines occur in many-world settings, which are always structured according to a system of cross-identification. The analogy can be further deepened: I show in Sect. 4.7 that intentional states (including perceptual experience) can be analyzed as structures of worlds and intentionally individuated world lines. What is more, the latter can in suitable circumstances be ‘awarenesses’ (representations) of physically individuated world lines; see Sect. 4.8 .
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Tulenheimo, T. (2017). Individuals and Cross-World Identity. 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_1
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