Ontological Pluralism and Notational Variance Bruno Whittle July 18, 2017 Ontological pluralism is the view that there are dierent ways to exist. It is a position with deep roots in the history of philosophy. For example, Aristotle seemed to endorse it when he said that 'there are many senses in which a thing may be said to 'be''.1 Although the view fell out of favour, there has recently been a resurgence of interest, sparked by defences from Kris McDaniel [2009, 2010a, 2010b] and Jason Turner [2010, 2012]. Indeed, while the position may still have relatively few adherents in quite these terms, the inuential Fregean approach to higher-order quantication-according to which this is over 'concepts' rather than objects- would seem to be an instance of it.2 In contemporary presentations, the view is stated in terms of fundamental languages.3 at is, languages whose expressions 'carve nature at the joints', or whose meanings are natural in the sense of Lewis [1983, 1986].us stated, it is the claim that such languages have more than one type of quantication, ranging over dierent domains. For example, ∃a ranging over abstract objects, and ∃c ranging over concrete ones.4 1Metaphysics IV.2. For more on the history of ontological pluralism, see McDaniel [2009] and Caplan [2011]. 2See Frege [1891, 1892]. A similar point could be made about many other approaches to highorder quantication, e.g. those of Russell [1908], Prior [1971] and Williamson [2003]. On the relation between ontological pluralism and higher-order quantication, see Turner [2010: 12–13] and Caplan [2011]. 3See McDaniel [2009] and Turner [2010, 2012]. On fundamental languages more generally, see Sider [2011]. 4In fact, there is some debate about how exactly the position should be stated: see McDaniel [2009], Turner [2010], Caplan [2011] and Spencer [2012]. For example, it has been suggested that Lewisian naturalness might be replaced by Schaerian fundamentality (see Caplan [2011: 91–93]), or that, in the spirit of Wittgenstein's Tractatus, it be stated without reference to quantication at all, but rather 'sorts' of terms and argument places (see Turner [2010: 10–11]). Most of these con1 ere is, however, a worry that one might have about ontological pluralism. is is not a worry that it is false, but rather that it is not substantively dierent from its supposed rival, (ontological) monism. In particular, one might worry that the languages proposed by the pluralist are mere notational variants of those proposed by the monist. For example, one might fear that the pluralist language LP with quantiers ∃a and ∃c is a mere such variant of the monist language LM , with the single quantier ∃ but predicates Abstract and Concrete. By way of analogy, consider the languages L◻ and L◇, identical except that the rst contains ◻, but not ◇, while the second is the opposite. If one came across two people arguing over which of these is a fundamental language, one would presumably not think that this was an important philosophical question that had been too long neglected. Rather, one would think that it is not a substantive question at all. e worry is that LP stands in the same sort of relation to LM that L◻ stands in to L◇, and thus that the debate between the pluralist and the monist is similarly insubstantive. However, Turner [2012] has recently given an ingenious response to this concern, employing a principle that he calls 'logical realism'.5 According to this, if two fundamental languages are notational variants, under a given translation, then this translationmust preserve logic.at is, a formulamust be a consequence of a given set of formulas i this is also true of their translations. But, Turner argues, unlike in the case ofL◻ andL◇, the translation that threatens to show thatLP andLM are notational variants fails to preserve logic. e purpose of this paper is to oer a counter-response on behalf of the 'notationalist'. I argue that, properly applied, the principle of logical realism is no threat to the claim thatLP andLM are notational variants. Indeed, there seems to be every reason to think that they are. e structure of the paper is as follows. §1 contains preliminaries. §2 gives the worry, and §3 Turner's response. §4 argues against this response. An appendix extends the argument to a variety of pluralism not considered in themain text: that according to which the fundamental language is 'sorted'. siderations are orthogonal to the concerns of this paper.e exception is the 'Tractarian' question: the arguments to follow are in terms of quantication, and it would take work to reframe them in other terms. However, almost all recent discussion of ontological pluralism has similarly assumed that is stated in quanticational terms. 5Strictly speaking, Turner uses 'logical realism' for the view that motivates the principle, and (LR) for the principle itself. But for simplicity I use 'logical realism' for the principle. 2 1 Preliminaries roughout the paper I focus on the example from the introduction: the instance of pluralism onwhich the fundamental language has exactly the quantiers ∃a and ∃c. However, everything that I say carries straightforwardly over to other instances.6 More specically, in the main text I assume that the pluralist proposes a language, LP, that is just like a standard rst-order language, except that it has ∃a and ∃c in place of ∃. at is, LP is 'unsorted': with one sort of term and argument place, rather than distinct sorts corresponding to the two quantiers. However, as I show in the appendix, a version of the argument to follow can also be given in the case where the a sorted language is proposed. e instance of monism under consideration, then, is that on which the fundamental language, LM , is the standard rst-order language that is just like LP, except that it contains ∃ and unary predicate symbols Abstract and Concrete, in place of ∃a and ∃c. Finally, I assume that LP and LM are languages with equality. To discuss the question of notational variance, we need the concept of a translation. us, if L and L′ are languages, a translation between L and L′ is a pair ⟨t, t′⟩ such that t is a function from formulas of L to those of L′, and t′ is one in the opposite direction. However, I oen abuse notation and use the same symbol for both functions, allowing context to disambiguate. A paradigm example of notational variants are thus a pair of languages along the lines of L◻ and L◇, under the translation that sends ◻φ to ¬◇¬t(φ), and◇ψ to ¬◻¬t(ψ). Similarly, L¬∧ and L¬∨, or L∃ and L∀. For simplicity, I build into the denition of notational variance the requirement that the languages involved are fundamental, although one could also, if desired, give a more inclusive denition. us, I say that L and L′ are notational variants under translation t if: (i) L and L′ are fundamental languages; and (ii) for any formula φ of L, t(φ) 'says the same thing' as φ, and similarly in the other direction.is characterization is of course less than completely precise, but it will suce for our purposes. I should though say something by way of explanation. Concerning (i): we need some clause along these lines because we want notational variants to bemetaphysically on a par. But (ii) on its ownwould not establish this: for example, one might hold that there is a translation between LP and LM that satises (ii), but that nevertheless only one of these languages is fundamental; 6is includes those in which the fundamental language has innitely many quantiers, as long as one is willing to countenance formulas of innite length. 3 in which case one would be metaphysically superior, and the debate between the pluralist and the monist would be substantive. Concerning (ii): which notion of content is at issue? It cannot be an extremely ne-grained notion, requiring a perfect isomorphism between a formula and what it expresses: since that would rule out paradigm examples such as L◻ and L◇. On the other hand, nor can it be a very coarse-grained notion, such as that which identies contents with sets of possible worlds: for we would then get the result that any two mathematical languages that satisfy (i) are notational variants, which is surely unacceptable. What is needed is a middle way: perhaps the sort of notion Frege had in mind when he famously said that 'the direction of a is the same as the direction of b' has the same content as 'a is parallel to b'.7 Such a notion is notoriously dicult to make precise. Fortunately, however, we can to a great extent side-step this issue here: since the heart of the matter is whether the translation between LP and LM (to be given in §2) satises the principle of logical realism (i.e. preserves logic); and this question can be addressed quite independently of that of how to explicate such a 'medium-grained' notion of content.8 Quite generally, if L and L′ are notational variants, then any debate between their proponents would seem to be insubstantive. 2 e Worry Consider the translation t between LM and LP, dened by induction as follows. Going rst from LP to LM : if φ is a formula of the form ∃axψ, then t(φ) is ∃x(Abstract(x) ∧ t(ψ)); and similarly if φ is ∃cxψ, but with Concrete in place of Ab7See [1884: 74–75]. For discussion of this line of thought in Frege, see Hale and Wright [2001]. 8Turner [2012: 423] gives a denition of notational variance that promises to avoid this issue. is is in terms of theories rather than languages, and (simplifying inessentially) is as follows.eories T and T ′ are notational variants under t if: (I) the languages of T and T ′ are fundamental; and (II) φ is a theorem of T i t(φ) is one of T ′, and similarly in the other direction.e problem is that this denition is inadequate by Turner's own lights, for it is incompatible with logical realism.is can be seen by letting T = T ′ be some theory in a fundamental language such that for some atomic sentence α = Fs1 . . . sn , ¬α is a theorem of T , but α is logically contingent (i.e. neither logically true nor false). For consider the translation that sends atomic formulas φ of the form Fr1 . . . rn to φ∧¬α; that is the identity on other atomic formulas; and that handles non-atomic formulas in the obvious way.is satises (II), but does not preserve logic: because the logically contingent α is sent to the contradiction α∧¬α.ere does not seem to be any way, therefore, of ultimately avoiding the dicult questions about content raised in the text. I should note, however, that this issue with Turner's denition in no way undermines his argument, since he could just as well use the characterization that I have given. 4 stract; formulas of other forms are handled in the obvious way.9 Going in the opposite direction: if φ is ∃xψ, then t(φ) is ∃axt(ψ) ∨ ∃cxt(ψ); t(Abstract(s)) is ∃axx = s; and t(Concrete(s)) is ∃cxx = s. Formulas of other forms are handled as before.us, for example, ∃axSet(x) is translated as ∃x(Abstract(x)∧ Set(x)), while ∃xx = harry is translated as ∃axx = harry ∨ ∃cxx = harry. e problem is that we seem to be dealing here with translations that say the same thing as the formulas translated-and with equal metaphysical transparency. Just as in the case of L◻ and L◇, for example. But if that is right, then LP and LM are notational variants, and the debate between the pluralist and the monist would seem to be insubstantive. 3 Logical Realism Turner has a response, however, which uses the following principle (see [2012]). (LR) If L and L′ are notational variants under t, then t preserves logic. at is, for any set Γ ∪ {φ} of formulas of L, Γ ⊧ φ i t(Γ) ⊧ t(φ); and similarly for any set Γ ∪ {φ} of formulas of L′.10 is principle is supported by the following, apparently very plausible, line of thought (see Turner [2012: 426–27]). If Γ ⊧ φ, then this corresponds to an important metaphysical relation between the contents of these formulas-it is not a mere accident of notation. But then this relation should be captured by any metaphysically perspicuous expression of these contents. However, if L and L′ really are notational variants under t, then the members of t(Γ) ∪ {t(φ)} oer just such an expression, and so we have t(Γ) ⊧ t(φ). at is, if Γ ⊧ φ, then t(Γ) ⊧ t(φ). Similarly for the converse: giving (LR). Armed with this principle, however, Turner has an answer to the worry of §2. He points out that the translation t between LP and LM does not seem to preserve logic. For consider the following sentence αM of LM : ∀x(Abstract(x) ∨ Concrete(x)).11 is does not seem to be a logical truth. However its translation does seem to be one. For this (simplifying slightly) is αP ∶ ∀ax(∃a yy = x ∨ ∃c yy = x) ∧ ∀cx(∃a yy = x ∨ ∃c yy = x), 9I.e. t sends atomic formulas to themselves; t(¬ψ) = ¬t(ψ); and t(ψ ∧ χ) = t(ψ) ∧ t(χ). 10Here Γ ⊧ φ means that φ is a logical consequence of Γ, and t(Γ) = {t(ψ) ∶ ψ ∈ Γ}. 11Universal quantiers are dened in terms of existential ones in the usual way. 5 which appears to be a logical truth in virtue of the fact that ∀ax∃a yy = x and ∀cx∃c yy = x are.12 Perhaps, then, we are not dealing with notational variants aer all?13 4 Notational Variance Even So One way of trying to counter this response would be to insist that either Abstract or Concrete be dened in terms of the other: for example, that Abstract be dened as ¬Concrete. If such a denition was permitted, then αM would be a logical truth (a quantied instance of the law of excludedmiddle), and we would no longer have a violation of (LR). But there seem to be a number of drawbacks with this idea. For one thing, to give such a denition is to treat one of Abstract and Concrete as more fundamental than the other. But it is at least plausible that the properties of being abstract and concrete are equally fundamental. For another, this idea (at least on the most obvious implementations) requires that nothing can be within the range of more than one of the pluralist's quantiers (see Turner [2012: 428–29]). Perhaps in the case of ∃a and ∃c this is uncontentious. But there are other versions of pluralism where it is very far from so: e.g. that with a quantier ∃p over the physical, and another ∃m over the mental. us, this way of trying to counter Turner's response seems unsatisfactory. e strategy that I pursue is quite dierent.e basic idea is this. We only get Turner's result-that αP is a logical truth, while αM is not-if we assume that ∃a and ∃c are logical constants, but Abstract and Concrete are not. (For it is only the thought that these are standard, non-logical, predicate symbols that justies the claim that αM is not a logical truth.) However, the work that ∃a and ∃c do in LP is precisely that which is done by Abstract and Concrete (together with ∃) inLM . But then, if we are treating the former as logical (as Turner quite naturally is), we should surely also so treat the latter. Indeed, this dierential treatment seems particularly 12ere is no comparable example in the opposite direction: i.e. a set Γ ∪ {φ} of formulas of LP such that we similarly seem to have Γ ⊧ φ ⇎ t(Γ) ⊧ t(φ). But no matter: one counterexample is enough. (Note that we cannot tweak our translation exploiting the fact just mentioned, because the relevant function from LP to LM is not surjective.) 13I should note that Turner [2012] in fact gives a whole range of arguments aimed at establishing that LP and LM are not notational variants. Specically, aimed at establishing that there is no other translation under which they are such. However, these general arguments make the same assumptions about the logic ofLP andLM that the particular one does-and that I argue against in §4.us, if the argument of that section succeeds, then it refutes these general arguments as well. 6 problematic if one is trying to refute the suggestion that LP and LM are notational variants. For the whole thrust of that suggestion is that the relevant expressions are essentially interchangeable. But then, if we are treating one set as logical, we must also treat the other as logical. To argue as Turner does would seem akin to arguing against the claim that L◻ and L◇ are notational variants (under the standard translation), on the basis that if one counts ◻ as logical, but does not so count ◇, then this translation fails to preserve logic (since one would have ◻F(a) ⊧ F(a) but ¬ ◇ ¬F(a) ⊭ F(a), for example). But that would of course be a very unpersuasive way to argue against this claim of notational variance. e contention that ∃a and ∃c, on the one hand, and Abstract and Concrete, on the other, are on a par as far as logicality is concerned can be further supported by considering standard accounts of logical constancy.ere is, of course, no universally accepted such account.14 But two of the most widely invoked are those in terms of (a) topic neutrality, or (b) permutation invariance.15 Consider (a) rst: the claim is that an expression is a logical constant i it can be used to talk about any subject matter. But, of course, the expressions in question are exactly tied in that regard: Abstract can be used to talk about abstract objects completely generally (but not others); exactly what is true of ∃a; and similarly for Concrete and ∃c. As for (b), the usual statement of such an account is that an expression e is logical i for any interpretation I with domain D, and any permutation π of D,16 the semantic value of e in I, eI , is unchanged under π. at is, π(eI) = eI .17 For example, = passes this test, because =I= {⟨d , d⟩ ∶ d ∈ D} = π(=I) (since π is a surjection). Similarly, the test is passed by ∃, given that ∃I = {X ⊆ D ∶ X ≠ ∅} = {π(X) ∶ X ⊆ D ∧ X ≠ ∅} = π(∃I).18 On the other hand, a standard non-logical unary predicate symbol F will not pass the test: just choose I with F I ∉ {∅,D}, and π such that for some d ∈ F I , π(d) ∉ F I . We then have π(F I) ≠ F I . 14See MacFarlane [2015] for a useful survey. 15For (a), see Ryle [1954: 111–29] and Peacocke [1976: 229], for (b), Tarski [1986] and McGee [1996]. 16I.e. any bijection of D into itself. 17In the rst instance, π is dened for members of D, but it can be extended to ordered tuples of members of D, subsets of D etc. in the obvious way: e.g. if a, b ∈ D, then π(⟨a, b⟩) = ⟨π(a), π(b)⟩; if X ⊆ D, then π(X) = {π(d) ∶ d ∈ D}; and so on. 18As usual, I take the semantic value of a quantier Q to be a set of subsets of the domain. e idea is that Qxφ is true i the extension of φ is in this set. 7 What happens when we apply this test to the expressions of LP and LM? Consider rst ∃a and ∃c. In any LP-interpretation I, each of these will be assigned its own domain, Da and Dc.19 If π is a permutation of the total domain, i.e. Da ∪ Dc, then: π(∃Ia) = ∃Ia and π(∃Ic) = ∃Ic i for any d ∈ Da ∪ Dc, d ∈ Da ⇔ π(d) ∈ Da, and d ∈ Dc ⇔ π(d) ∈ Dc. at is, the semantic values of ∃a and ∃c are invariant under precisely those permutations that respect the 'abstract'/'concrete' divide: i.e. that send 'abstract' objects of the interpretation to other such objects, and 'concrete' objects to other such ones. But the situation with Abstract and Concrete is exactly similar. If I is an LMinterpretation, then it will have a single domain D. And a permutation of D will leave AbstractI and ConcreteI unchanged i it similarly respects the 'abstract'/'concrete' divide: i.e. sends 'abstract' members of the domain, which are in this case the members of AbstractI , to other such objects, and similarly for 'concrete' ones. Again, then, the two sets of expressions would seem to be precisely tied as far as logicality is concerned. us, given that we are taking ∃a and ∃c to be logical, we should also so take Abstract andConcrete.20Wewill see, however, that oncewe do this, our translation t satises (LR) aer all. 4.1 Logic for LP and LM I want to show, then, that once we take Abstract and Concrete to be logical, the translation t of §2 does indeed preserve logic. As is standard, I assume that logical consequence for a language L is dened in terms of interpretations of L. ConsiderLP rst.e natural notion of an interpretation ofLP is as follows (see Turner [2012: 432]). An LP-interpretation I is a triple ⟨Da ,Dc , i⟩ such that Da and Dc are non-empty21 sets (I use D∪ for Da ∪Dc); and i is a function that sends every individual constant of LP to a member of D∪; every n-ary (n ≥ 1) function symbol to an n-ary function from D∪ to D∪; and every n-ary (n ≥ 1) predicate symbol to a subset of Dn∪. One might also insist that Da and Dc are disjoint. And everything that I say would straightforwardly carry over, but for deniteness I assume that Da and Dc are allowed to overlap. An LP-valuation is an LP-interpretation together with an assignment of values to variables. Satisfaction is then dened in the obvious way: with ∃a taken to range 19As with ∃, ∃Ia = {X ⊆ Da ∶ X ≠ ∅}, and similarly for ∃c . 20What about the possibility of taking none of these expressions to be logical? I consider that towards the end of this section. 21We could also allow one or both to be empty, but for simplicity I do not do this. 8 over Da, and ∃c over Dc. Finally, for a set Γ ∪ {φ} of formulas of LP, we say that Γ ⊧ φ if, whenever an LP-valuation satises every member of Γ, it also satises φ. What, now, about interpretations of LM? at is, what is the cash value of the claim that Abstract and Concrete are logical? I suggest that the natural denition of an interpretation of LM is in fact exactly the same as in the LP case. It is just that, now, Da is not the range of ∃a, but the extension of Abstract (and similarly for Dc). Further, we do not need to supply an additional domain for ∃, because this is simply taken to range over D∪. is would seem to be the natural denition of an LM-interpretation, given that ∃ is intended to range over abstract and concrete things-and nothing else. Of course, there are distinct monist languages in which ∃ is also intended range over other sorts of thing (e.g. mental items). Such languages will still be notational variants of pluralist ones: but those with additional quantiers beyond ∃a and ∃c (e.g. ∃m). Here is another way in which one can justify this denition. One aspect of our treating ∃a and ∃c as logical is to insist that the other expressions of the language have their interpretations drawn from the 'abstract' and 'concrete' objects of this interpretation: i.e. the objects that constitute the interpretations of ∃a and ∃c, which is to say the members of D∪.us, we require that the interpretations of individual constants are members of D∪, the interpretations of n-ary predicate symbols are subsets of Dn∪, etc. In the case of =, for example, this amounts to =I= {⟨d , d⟩ ∶ d ∈ D∪}. e natural way of treating Abstract and Concrete as logical would seem to be by imposing a similar requirement. at is, to insist that the interpretations of other expressions be drawn from the objects that constitute those of Abstract and Concrete. In the case of =, this again amounts to =I= {⟨d , d⟩ ∶ d ∈ D∪}. In that of ∃, it amounts to ∃I = {X ⊆ D∪ ∶ X ≠ ∅}. But then we have precisely the notion of an LM-interpretation that I have proposed, together with a denition of satisfaction on which ∃ ranges over D∪. We're then home and dry, by the following result (which is proved by a straightforward induction on the degrees of φ and ψ).22 Proposition 1. Let σ be an LP-valuation (i.e. LM-valuation), φ a formula of LP and ψ one of LM . en: (i) σ ⊧ φ i σ ⊧ t(φ); 22If σ is an LP-valuation, and φ is a formula of LP , then σ ⊧ φ means that σ satises φ. And similarly for LM . Note that although any LP-valuation is an LM-valuation, and vice versa, 'σ ⊧' means something rather dierent, depending on whether we are treating σ as an LP-valuation, or anLMone.us, slightly more verbosely, (i), for example, might be written: σ ⊧P φ i σ ⊧M t(φ). 9 (ii) σ ⊧ ψ i σ ⊧ t(ψ). It follows immediately that (LR) is satised. For suppose that Γ ∪ {φ} is a set of formulas of LP.en: Γ ⊧ φ i any LP-valuation satisfying Γ, satises φ; which, by (i), holds i any LM-valuation satisfying t(Γ), satises t(φ). us, Γ ⊧ φ i t(Γ) ⊧ t(φ). e case where Γ ∪ {φ} is a set of formulas of LM is exactly similar. As promised, then, (LR) is met aer all. ere is one nal loose end to tie up. We saw above that if we are taking ∃a and ∃c to be logical, then we should also so take Abstract and Concrete. Just as: if we take ◻ to be logical, then we should also so take◇. In the case of L◻ and L◇, however, (LR) is satised bothwhenwe treat◻ and◇ as logical, andwhenwe treat neither as. Might one insist that if LP and LM are genuine notational variants, then (LR) must similarly be satised both when we treat all of the relevant expressions as logical, and when we treat none of them as? In fact, unlike in the case of L◻ and L◇, (LR) is not satised when we treat none of these expressions-i.e. ∃a, ∃c, ∃, Abstract and Concrete-as logical.23 On reection, however, to insist that (LR) is satised even in this case is unreasonable. is can be seen by considering a simple variant of the modal example: L◻◇ and L◻.24 It seems clear that these are notational variants (under the obvious translation s). But (LR) is not satised if we treat neither ◻ nor ◇ as logical: since if δ is ¬ ◻ ¬F(a) and ζ is ◇F(a), then δ ⊭ ζ ; but s(ζ) = δ = s(δ) and so s(δ) ⊧ s(ζ). It seems, then, that notational variance does not require (LR) to hold even in the casewherewe treat the expressions being (non-homophonically) translated as nonlogical. Indeed, what the example of L◻◇ and L◻ seems to show is that we can only expect notational variants to satisfy (LR), if we are treating the expressions being translated as logical. A result that lends further-if slightlymore indirect-support 23e following would be a counterexample to (LR). Let β be ∃x(¬Abstract(x)∧¬Concrete(x)), and γ, ∃xx ≠ x. If we are treating none of the expressions as logical, then β ⊭ γ. However, the 'double' translation of β, t(t(β)), is ∃x(Abstract(x) ∧ ¬Abstract(x) ∧ ¬Concrete(x)) ∨ ∃x(Concrete(x) ∧ ¬Abstract(x) ∧ ¬Concrete(x)), while t(t(γ)) is ∃x(Abstract(x) ∧ x ≠ x) ∨ ∃x(Concrete(x) ∧ x ≠ x). And t(t(β)) ⊧ t(t(γ)) (since in each formula ∃x is applied to contradictions). 24Of course, these are identical except that while L◻◇ contains ◻ and◇, L◻ contains only ◻. 10 to the contention above that when we are considering whether LP and LM are notational variants, we should treat all of the expressions being translated as logical. ere seems, then, to be every reason to think that LP and LM are notational variants, and thus that the debate between the pluralist and the monist is not a substantive one.25 Appendix: Sorted Pluralism In this appendix, I consider the possibility that the pluralist might propose a sorted language, KP.at is, every term and argument place in KP has a sort, a or c, and a string is a formula only if its terms and argument places match. Again, I assume that the quantiers ofKP are ∃a and ∃c. In this case, too, there is a monist language that seems to be a notational variant of the pluralist one. As in the unsorted case, the main choice point in the logic of KP is whether to allow the domains of ∃a and ∃c to overlap. Here it seems most natural not to allow this: the idea of a sorted language goes naturally with that to the eect that we are dealingwith two fundamentally dierent, and thusmutually exclusive, types of things. us, that is the approach that I consider. But one could easily enough derive similar conclusions on the overlap approach. AKP-interpretation is just like an LPone, except that sorts are respected.at is, if I = ⟨Da ,Dc , i⟩ is a KP-interpretation, and ba is an individual constant of sort a, then i(ba) ∈ Da; and if Rac is a binary predicate symbol of sort ⟨a, c⟩ (the rst argument place is of sort a, the second of sort c), then i(Rac) ⊆ Da ×Dc; and so on. Let KM be just like KP, except that it contains Abstracta, Concretec and ∃ in place of ∃a and ∃c, and variables are no longer sorted. us, e.g., Rac(x , x) is a formula of KM . In KM individual constants and argument places are sorted: this is required if even the quantierand variable-free fragment of the language is to be translatable into KP. However, the machinery of quantication-the quantier and variables-is unsorted, and so it is in this sense a genuinely monist language. It will follow from that fact that KM and KP are notational variants that, given this quantierand variable-free base, the choice of whether to quantify in a monist or a pluralist fashion is not a substantive one. As in the unsorted case, a KM-interpretation is simply a KP-interpretation. 25[Acknowledgements.] 11 How can we translate between these languages? To keep things simple, I give a translation only for sentences: this satises (LR) in the sense that the logic of sentences is preserved. If one thinks that true notational variance requires a translation that works for formulas more generally, then one could extend that which I give, but for reasons of space I do not do this here. e translation s is dened as follows. FromKP toKM this works just as before (i.e. like t of §2). e opposite direction is slightly more involved, given the fact that variables are sorted in KP but not in KM : if we tried to translate as before, we would take some formulas of KM to illformed strings. Instead, I dene a function s0 from formulas of KM to strings of KP as follows. If φ is Abstracta(r), then s0(φ) is ∃axaxa =aa r; if φ is Concretec(r), then s0(φ) is ∃cxcxc =cc r; if φ is atomic of some other form, then s0(φ) = φ; if φ is ¬ψ, then s0(φ) = ¬s0(ψ); if φ is ψ ∧ χ, then s0(φ) = s0(ψ) ∧ s0(χ); and if φ is ∃xψ, then s0(φ) = ∃axa[s0(ψ)]a/x ∨∃cxc[s0(ψ)]c/x , where χa/x is the result of replacing every occurrence of an atomic formula with x in an argument place of sort c with , and then replacing every remaining occurrence of x with xa (and similarly for χc/x). It is then easy to see that if β is a sentence ofKM , s0(β) is one ofKP. We thus set s(β) = s0(β). Although for reasons of space I will not make the case in any detail, it is at least very plausible that s preserves both content (in the sense of §1) and metaphysical perspicuity, and thus that KP and KM are notational variants under s. Further, we have the following, which ensures that (LR) is indeed satised for sentences. Proposition 2. 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