In a series of recent articles, Benj Hellie, Adam Murray, and Jessica Wilson (2012; forthcoming) have developed a view of metaphysical necessity according to which it exhibits a type of relativity.Footnote 9 According to their view, what is metaphysically necessary is relative to which world is considered as ‘indicatively actual’. This view of relativised metaphysical modality, hereafter RMM, promises interesting solutions to a variety of modal puzzles and paradoxes such as Chisholm’s Paradox, Fine’s (2005) arguments against Modal Monism, and arguments for the necessity of existence. One of the most attractive features of these solutions is their advertised capacity to maintain an S5 logic for metaphysical necessity. Moreover, as Murray and Wilson (2012, pp. 190–198) note, an advantage of RMM is that it is ‘formally analogous’ (2012, pp. 190–198) to the epistemic interpretation of the two-dimensional semantic framework, which provides a natural class of models that are ‘heuristically useful’ (2012, pp. 190–191; forthcoming, p. 2) for studying RMM. Nevertheless, despite the promise of RMM, the purpose of this section is to argue that the moral of equivalence creates significant awkwardness for RMM’s solution to Chisholm’s Paradox. To this end, this section will present the details of RMM in the context of a two-dimensional modal logic. The strategy will first be to outline the epistemic interpretation of the two-dimensional semantic framework and then clarify the formal analogy between it and RMM.
According to Chalmers’ version of two-dimensionalism, with every expression of a language—not just each predicate letter—is associated a two-dimensional intension. One can think of a two-dimensional intension as complex semantic value from which two subsidiary intensions can be recovered, a primary intension and a secondary intension. In the simple case of monadic predicates, primary intensions are functions from metaphysically possible worlds considered as actual to sets of individuals, and secondary intensions are functions from metaphysically possible worlds considered as counterfactual to sets of individuals.Footnote 10 Consider some world \(w_{1}\) where the clear, transparent, drinkable liquid which runs from taps, makes up roughly 55% of the human body, etc., has the chemical composition XYZ. The primary intension of ‘water’ takes \(w_{1}\) to XYZ. But the secondary intension we actually assign ‘water’ takes \(w_{1}\) to H\(_{2}\)O. Roughly speaking, the thought is that when we consider a world as actual, we consider what the expression ‘water’ would pick out at that world had its usage been the same as it actually is. When we consider a world as counterfactual, we consider what water (i.e. H\(_{2}\)O) would have been like at that world.
Clearly, two-dimensionalism and the twin-earth thought experiments used to motivate semantic externalism are closely connected. The lesson of semantic externalism is that there are aspects of an expression’s meaning which in some sense depend on how the non-linguistic world turns out. If the clear, transparent, drinkable liquid which runs from taps, makes up roughly 55% of the human body, etc., has the chemical composition XYZ, then ‘water’ refers to XYZ; if the clear, transparent, drinkable liquid which runs from taps, makes up roughly 55% of the human body, etc., has the chemical composition H\(_{2}\)O, as it actually does, then ‘water’ refers to H\(_{2}\)O. Two-dimensionalism incorporates the fact that the semantic values of expressions like ‘water’ exhibit non-trivial dependence on the non-linguistic world into its semantic theory by assigning them a primary intension whose extension varies with which world is considered as actual.
An upshot to the two-dimensional picture is that truth-value assignments to sentences governed by metaphysical modal operators will be sensitive to which world is considered as actual. For instance, considering the actual world as actual it is metaphysically necessary that water is H\(_{2}\)O. But considering \(w_{1}\) as actual, it is metaphysically impossible that water is H\(_{2}\)O, for under that supposition necessarily it is XYZ. It can help to see a tabular representation of the two-dimensional intension the epistemic two-dimensionalist assigns to a given expression. Take the sentence ‘water is H\(_{2}\)O’. In the following table, the labels of rows correspond to which world is considered as actual, and the labels of the columns correspond to which world is considered as counterfactual:
‘Water is H\(_{2}\)O’
|
H\(_{2}\)O world
|
XYZ world
|
H\(_{2}\)O world
|
T
|
T
|
XYZ world
|
F
|
F
|
The diagonal of Ts and Fs reaching from the top left T to the bottom right F corresponds to the sentence’s primary intension—hence its aptronym ‘the diagonal intension’. The rows correspond to the secondary intension that the sentence is assigned when the H\(_{2}\)O-world and XYZ-world are respectively considered as actual, which clearly represent two different functions. Thus an expression’s secondary intension along with its extension varies with which world is considered as actual, hence why metaphysical necessities are sensitive to which world we consider as actual.
Murray and Wilson (2012) offer a ‘metaphysical’ interpretation of the Chalmersian framework, whilst retaining the “formal analogy” (p. 198) to it. To be specific, Murray & Wilson depart from epistemic two-dimensionalism in their interpretation of secondary intensions. The traditional line is to treat all secondary intensions, except the one assigned when the actual world is considered as actual, as mere epistemic necessities for all we know a priori. However, Murray and Wilson (2012, p. 211) regard them as genuine ‘relativised’ metaphysical necessities. Hellie, Murray and Wilson (forthcoming, p. 15; their emphasis) later describe this view as follows.
Our intended picture is moderate modal naturalism: facts about natural law (and, in turn, about metaphysical possibility) have a hybrid explanation: some such facts are underdetermined either by any abstract metaphysical principles alone, or by any actual contingent ‘categorical’ facts alone, but instead only become determined through the concretization of certain true such principles in actual categorical facts.
Nevertheless, whatever departure Murray and Wilson make from epistemic two-dimensionalism, they still endorse (2012, p. 203) the claim that not all truths vary with which worlds are considered as actual, since there must be some ‘basic’ ways of individuating or specifying worlds which are invariant as such. Indeed, like Chalmers, they (2012, p. 202) hold that “the basic individuation of worlds [...] might proceed by way of, e.g., ‘semantically stable’ (Bealer 2000) or ‘canonical’ (Chalmers 2006) descriptions.” Effectively, this guarantees that Murray and Wilson’s ‘metaphysical’ interpretation of the Chalmersian framework does not depart foundationally from Chalmers’ preferred epistemic construal. At points in the following discussion, this will become pertinent.Footnote 11\(^,\)Footnote 12
A central moral that Murray and Wilson want to draw from Chisholm’s Paradox is that truth-value assignments to essentialist claims about Woody’s material makeup are highly sensitive to which world is considered as actual. As one way of predicting this sensitivity, they (2012, p. 206) entertain a view on which proper names like ‘Woody’ function similarly to natural kind terms. On this implementation of RMM, just as the denotation of ‘water’ varies with which world is considered as actual, so does the denotation of ‘Woody’. As a consequence, the various denotations of ‘Woody’ differ in their modal profiles, specifically regarding the matter from which they can originate. Advocates of RMM suggest that acknowledging this sensitivity dissolves Chisholm’s Paradox.
To provide more detail, regarding Chisholm’s Paradox Murray and Wilson would claim that when we consider world 6 as actual it is necessary that amongst all six of Woody’s original parts are, say, at least three of those which it has at 6. However, when we consider world 4 as actual it is not necessary that amongst all six of Woody’s original parts are at least three of those which it has at 6, for when world 4 is considered as actual ‘Woody’ changes its denotation. Thus, when world 4 is considered as actual, it is necessary that amongst all six of Woody’s parts are at least three of those it has at 4 (and so one of those which it has at 6).
To clarify this proposal, it helps to provide a simple model of Chisholm’s Paradox within the RMM framework.Footnote 13 To do so, the previously used propositional language \(\mathcal {L}\) is extended to \(\mathcal {L}^\downarrow \) with the modal operator \(\downarrow \), which will be used in the object-language to express which world is considered as actual. An RMM-frame \(\mathfrak {F}=\langle \mathcal {W}, \mathcal {R}\rangle \) is an \(\mathcal {L}\) frame in which \(\mathcal {R}\) is an equivalence relation. An RMM-model \(\mathfrak {M}\) of \(\mathcal {L}^\downarrow \) based on an RMM-frame \(\mathfrak {F}=\langle W, \mathcal {R}\rangle \) is a triple \(\langle W, \mathcal {R}, \mathcal {V}\rangle \) in which \(\mathcal {V}\) is now a total function from triples of the form \(\langle \phi , w, u\rangle \), for some proposition letter \(\phi \) and \(w,u \in W\), to either 0 (falsity) or 1 (truth). The idea will be that truth at \(\langle w, u\rangle \) corresponds to truth at world u when w is considered as indicatively actual. We thus define ‘(\(\mathfrak {M}, w, u) \models \phi \)’ for \(\mathfrak {M}\) = \(\langle \mathcal {W}, \mathcal {R}, \mathcal {V}\rangle \), \(w, u \in \mathcal {W}\), and \(\phi \) a formula of the language, recursively as follows (where ‘A’ is an atomic formula, and \(\phi \) and \(\psi \) formulae of \(\mathcal {L}_{p})\):
-
\((\mathfrak {M}, w, u) \models A\) iff \(\mathcal {V}(A, w, u) = 1\);
-
\((\mathfrak {M}, w, u) \models \lnot \phi \) iff \((\mathfrak {M}, w, u) \not \models \phi \);
-
\((\mathfrak {M}, w, u) \models \phi \rightarrow \psi \) iff either \((\mathfrak {M}, w, u) \not \models \phi \) or \((\mathfrak {M}, w, u) \models \psi \);
-
\((\mathfrak {M}, w, u) \models \Box \phi \) iff \((\mathfrak {M}, w, u') \models \phi \), for all \(u' \in W\) such that \(u\mathcal {R}u'\);
-
\((\mathfrak {M}, w, u) \models \, {\downarrow }\phi \) iff \((\mathfrak {M}, u, u) \models \phi \).
For an RMM-model \(\mathfrak {M}\) = \(\langle \mathcal {W}, \mathcal {R}, \mathcal {V}\rangle \), a formula \(\phi \) of \(\mathcal {L}\) is true at \(\langle w, u\rangle \) for \(w,u \in \mathcal {W}\) iff \((\mathfrak {M}, w, u) \models \) \(\phi \). A formula \(\phi \) of \(\mathcal {L}\) is true in an RMM-model \(\mathfrak {M}\) = \(\langle \mathcal {W}, \mathcal {R}, \mathcal {V}\rangle \) iff it is true at \(\langle w, u\rangle \) for all \(w,u \in \mathcal {W}\).Footnote 14 A formula \(\phi \) of \(\mathcal {L}\) is RMM-valid iff it is true in every \(\mathcal {L}\) model.
At this point, it is worth highlighting that officially Murray and Wilson (2012, pp. 190–191, n. 1) note that their proposal “does not involve any additional [modal] operators” such as \(\downarrow \), which is typically present in two-dimensional logics.Footnote 15 However, Murray and Wilson (2012, p. 197) offer a specification of Salmon’s argument in terms of subscripted possibility operators in order to clarify their diagnosis of it. The \(\downarrow \) operator is used here as a mere alternative to such subscripted operators.
Consider a particular RMM-model \(\mathfrak {M}_2\) = \(\langle W_2, \mathcal {R}_2, \mathcal {V}_2\rangle \) of \(\mathcal {L}^\downarrow \), defined as follows.
\(\mathcal {W}_2 =\{n \in \mathbb {N} : n \le 6\}\)
\(\mathcal {R}_2\) = \(\mathcal {W}_2\times \mathcal {W}_2\)
\(\mathcal {V}_2(p_{n},w,m) = {\left\{ \begin{array}{ll} 1,&{} \text {if}\,\,m=n\,\,\wedge \,\, |w-n|\le 3 \\ 0, &{}\text {otherwise} \end{array}\right. }\)
This model can also be represented in diagram form. As before, ‘worlds’ are represented as labelled circles with the name of the ‘world’ occurring outside the circle, and proposition letters which are ‘true’ at that ‘world’ in the given model occurring inside the circle. Each row corresponds intuitively to ‘metaphysical modal space’ when a certain ‘world’ (the marked node) is considered as ‘indicatively actual’. Since accessibility is universal, the diagram uses no arrows.
Put informally, in this RMM-model when world 6 is considered as indicatively actual, Woody must originate from at least half of the matter it does at 6. Yet considering world 4 as indicatively actual, Woody can originate from just less than half of the matter it does at 6. One can see this formally by noticing the following facts about the model:
-
\(\mathcal {V}_2(p_{n}, n, n) = 1\), for all \(n \in \mathcal {W}\)
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\(\mathcal {V}_2(p_{2}, 6, n) = 0\), for all \(n \in \mathcal {W}\)
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\(\mathcal {V}_2(p_{2}, 4, 2) = 1\)
As a result \((\mathfrak {M}_2, 6, 6) \not \models \Diamond p_{2}\), yet \((\mathfrak {M}_2, 4, 6) \models \Diamond p_{2}\). However, despite this, \(\mathfrak {M}_2\) does not serve as a countermodel to any instance of axiom schema 4. For all instances of \(\Diamond \Diamond \phi \rightarrow \Diamond \phi \) are true in \(\mathfrak {M}_2\) given the definition of \(\mathcal {R}_2\). Indeed, all the axioms of S5 are true in \(\mathfrak {M}_2\), since S5 is sound over the class of RMM-frames. Indeed, Murray and Wilson (2012, p. 204; p. 208) view the retention of S5 as a central advantage of the RMM solution, particularly over Nathan Salmon’s (1986, 1989) anti-S4 proposal.
Of course, it follows that advocates of RMM view Chisholm’s Paradox, i.e. the above formal line of reasoning in \(\mathcal {L}\), as an RMM-valid argument. Thus, advocates of RMM will reject one of its premises. Yet they will not reject premise (1), since it is a fact about Woody’s actual origins. Moreover, they will not reject premise (2) since in their terms it merely amounts to the claim that (considering the actual world as indicatively actual) Woody could have originated from slightly different matter, which their approach is designed to recognise. Thus they must reject at least one of premises (3)–(7). To be more precise, say that advocates of RMM accept (reject) a claim exactly when that claim is true (false) when the actual world is considered as indicatively actual. Since in the above model world 6 models the actual world, advocates of RMM will reject premise (5), the formula \(\Box (p_3 \rightarrow \Diamond p_2)\). Read informally, (5) is the claim that it is metaphysically necessary that if Woody originates from six parts, \(m_1, m_2, m_3, n_4, n_5, n_6\) then it is metaphysically possible that Woody originates from six parts, \(m_1, m_2, n_3, n_4, n_5, n_6\) (recall that \(p_6\) is understood informally as the claim that Woody originates from six parts, \(m_1,\ldots , m_6\)). However, advocates of RMM would be keen to stress that (5) is true when world 4 is considered as indicatively actual. Hence Murray and Wilson (2012, p. 198–199; my emphasis) diagnose the fallacy of Chisholm’s Paradox to be that accepting its premises involves “iterated or in situ shifts in which world is held fixed as indicatively actual” that “illegitimately shift indicatively actual horses in modal mid-stream”.
Murray and Wilson (2012, p. 197) specify a version of Salmon’s argument in terms of subscripted possibility operators in order to clarify how iterated shifts in which world is held fixed as indicatively actual figure in the reasoning. As mentioned before, the \(\downarrow \) operator is used here as a mere alternative to such subscripted operators. In terms of this alternative, Murray and Wilson would accept all of the following claims:
(RMM1) \(\lnot \Diamond p_2\)
(RMM2) \(\Box (p_3 \rightarrow \, {\downarrow }\Diamond p_2)\)
(RMM3) \(\Diamond {\downarrow }\Diamond p_2\)
Indeed, in the above model one can easily verify that (RMM1)–(RMM3) are all true at \(\langle 6, u\rangle \) for any \(u \in \mathcal {W}\). Thus although the advocate of RMM will reject \(\Box (p_3 \rightarrow \, \Diamond p_2)\), they will accept (RMM2) which involves the iterated shift in which world is considered as indicatively actual
Nevertheless, despite this novel diagnosis of Chisholm’s Paradox, the moral of equivalence creates significant awkwardness for the proposed RMM solution. As emphasised in Sect. 1, the moral of equivalence is that Chisholm’s Paradox and the Modal Sorites are logically equivalent in logics which contain K45. Since S5 is one such logic, this generates the consequence that the advocate of RMM must recognise that (3)–(7) are equivalent with (3\('\))–(7\('\)) respectively. But in the context of RMM, this consequence is bizarre. The crucial point is that the Modal Sorites involved no nested modal claims, so it is extremely difficult to see why it would result from ‘iterated shifts’ in which world is considered as indicatively actual. Indeed, as stressed previously, the distinctive feature of the Modal Sorites is that it can be run solely from the perspective of the actual world since its premises are all claimed to be actually true. However, since the advocate of RMM will reject one of (3)–(7) from Chisholm’s Paradox, given S5 they must also reject one of (3\('\))–(7\('\)) from the Modal Sorites. But there is no scope to ameliorate the denial of one of (3\('\))–(7\('\)) by merely prefixing its consequent with an occurrence of \({\downarrow }\), as the advocate of RMM attempted to with (RMM2) in response to their denial of (5). To appreciate this, notice that both of the following two formulas would be rejected by advocates of RMM in the above model:
(5\('\)) \(\Diamond p_3 \rightarrow \, \Diamond p_2\)
(5\('{\downarrow }\)) \(\Diamond p_3 \rightarrow \, {\downarrow }\Diamond p_2\)
Thus one cannot ameliorate the denial of (5\('\)) by blaming its allure on an iterated shift in perspective.
The concern can be stated more generally by appeal to the so-called reduction theorem of S5. Say that a formula of \(\mathcal {L}^\downarrow \) is nested iff it belongs to \(\mathcal {L}\) and contains an occurrence of \(\Box \) which occurs within the scope of some other occurrence of \(\Box \). Similarly, say that a formula of \(\mathcal {L}^\downarrow \) is non-nested iff it belongs to \(\mathcal {L}\) and contains no occurrence of \(\Box \) within the scope of some other occurrence of \(\Box \). According to these definitions, \(\Box \Box p\) and \(\Box (p \rightarrow \lnot \Box \lnot q)\) are nested formulas, \(\Box p\) and \((\lnot \Box \lnot p \rightarrow \Box p)\) are non-nested formulas, and any formula which contains any occurrences of \(\downarrow \) is neither nested nor non-nested. The S5 reduction theorem may then be stated as follows.Footnote 16
S5 Reduction Theorem
Any nested
\(\mathcal {L}^\downarrow \)
formula is
S5
equivalent to a non-nested
\(\mathcal {L}^\downarrow \)
formula.
According to the S5 Reduction Theorem, nested modal claims are always S5 equivalent to non-nested claims. Thus, S5 is simply not hospitable to blaming modal paradox on ‘iterated shifts’ of perspective induced by nested modal claims. In doing so, the advocate of RMM rejects a nested claim. However, given the S5 Reduction Theorem, that nested claim is S5 equivalent to a non-nested claim, which the advocate of RMM must therefore reject. But they cannot explain their rejection of the non-nested claim by appeal to ‘iterated shifts’ of perspective induced by nested modal claims.
Supplementing this, it does not seem persuasive to ameliorate the denial of (5\('\)) by merely citing its S5 equivalence to (5), even if we grant that (5)’s denial can be ameliorated by blaming its allure on an iterated shift in perspective. First, there is the point that one should not expect hyperintensional features like allurement to be closed under logical implication (or known logical implication). For example, one might be allured to the axioms of ZFC on the grounds of set-theoretic platonism, yet not be so allured to the Banach-Tarski theorem despite knowing the axioms to imply it. Moreover, in addition, the manoeuvre does not appear to constitute a satisfactory amelioration of one’s rejection of (5\('\)) anyway. To offer an analogy in point, consider two different formulations of an ordinary non-modal sorites paradox. As its major premises, the first employs a collection of ‘no sharp boundary’ theses of the form \(\lnot (p_n \wedge \lnot p_{n+1})\). In contrast, as its major premises the second employs collection of ‘inductive’ conditionals of the form \(p_n \rightarrow p_{n+1}\). Now, consider an advocate of a solution to the sorites paradox who refuses to accept one of the ‘no sharp boundary’ theses but explains its allure in terms of a very specific cognitive mechanism pertaining to embeddings of negations in the context of vagueness. Regardless of the explanation’s idioscyncracy, surely even this character should recognise that either they ought to reject classical logic or find some other way to ameliorate their rejection of the inductive conditional which is classically equivalent to the negated conjunction in question.
In summary, the retention of S5 creates significant awkwardness for RMM’s diagnosis of why Chisholm’s Paradox is fallacious. Like Nathan Salmon’s anti-S4 solution, the RMM diagnosis places great emphasis on the occurrence of nested necessity operators. Yet blaming modal paradox on ‘iterated shifts’ of perspective induced by nested necessity operators is difficult to reconcile with the characteristic feature of S5 that possibility is simply a non-contingent matter. This casts doubt on whether RMM offers a satisfactory solution to Chisholm’s Paradox.