Philosophical Studies

, Volume 156, Issue 3, pp 389–416

Predicate abstraction, the limits of quantification, and the modality of existence

Authors

    • Department of PhilosophyUniversity of Nottingham
Article

DOI: 10.1007/s11098-010-9609-x

Cite this article as:
Percival, P. Philos Stud (2011) 156: 389. doi:10.1007/s11098-010-9609-x
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Abstract

For various reasons several authors have enriched classical first order syntax by adding a predicate abstraction operator. “Conservatives” have done so without disturbing the syntax of the formal quantifiers but “revisionists” have argued that predicate abstraction motivates the universal quantifier’s re-classification from an expression that combines with a variable to yield a sentence from a sentence, to an expression that combines with a one-place predicate to yield a sentence. My main aim is to advance the cause of predicate abstraction while cautioning against revisionism. In so doing, however, I shall pursue a secondary aim by conveying mixed blessings to those who hold the view that in the logical sense of “existence” some existing object is such as to exist contingently. Advocates of this view must concede Williamson’s recent contention that the domain of unrestricted objectual quantification could not have been narrower than it is actually, but predicate abstraction affords them some hope of accommodating this concession.

Keywords

Predicate abstractionQuantificationModalityExistenceSingular propositionLogical form

1 I

1.1 I.i

This paper focuses on syntactic, semantic and metaphysical issues that arise from predicate abstraction. It must be emphasised at the outset, however, that predicate abstraction is distinct from the more familiar phenomenon of property abstraction. A device of property abstraction is a variable-binding device for forming a singular term from an open sentence, the intended referent being the property “expressed” by the sentence. One rationale for property abstraction proceeds from the thought that a sentence ‘Barak is tall and Barak is handsome’ does not attribute a single complex property to Barak: at best it attributes to him one (relatively) simple property, being tall, and then attributes to him another (relatively) simple property, being handsome. Property abstraction is a formal device by means of which to construct a parallel sentence that does attribute a single complex property to him. It is commonly symbolised by λ. Applied to an open sentence ‘x is tall and x is handsome,’ λ yields a singular term ‘(λx)(x is tall and x is handsome)’ the intended reference of which is the property of being both tall and handsome. To employ this term to form a sentence that says that Barak has the single property of being both tall and handsome, one must link it, by means of a relational term—the copula, symbolised say by ‘H’—to a singular term that refers to Barak. The relational form of the resulting sentence, ‘H(Barak, (λx)(x is tall and x is handsome),’ contrasts with the conjunctive form of the original sentence ‘Barak is tall and Barak is handsome.’

Predicate abstraction is akin to property abstraction. In one respect, however, it is fundamentally different. Whereas property abstraction yields a singular term from an open sentence, a device of predicate abstraction is a variable binding device for forming predicates out of sentences. Somewhat confusingly, it too is commonly symbolised by λ. When applied to an open sentence ‘x is tall and x is handsome’ it results in a complex predicate ‘(λx)(x is tall and x is handsome)’ the intended interpretation of which is (something like) ‘is such as to be both tall and handsome.’ Like any other (one-place) predicate it may by conjoined with a singular term to yield a subject-predicate sentence. In particular, it may be combined with ‘Barak’ to yield a sentence ‘(λx)(x is tall and x is handsome) (Barak)’ the subject-predicate form of which is the same as that of the simple ‘Barak is tall.’

In effect, the presupposition of property abstraction is that properties ‘expressed’ by open sentences, atomic or complex, are themselves objects i.e. that fall within the range of first order variables. Property abstraction is a formal device by means of which singular terms referring to such objects can be constructed. In contrast, predicate abstraction carries no such presupposition. One might well wonder, then, as to its point. If one has an ontology of intensional objects—complex properties—then of course one has need of a formal device by means of which to refer to them systematically. In the absence of such an ontology, however, what is the point of a device by means of which to construct sentences such as ‘(λx)(x is tall and x is handsome) (Barak)’ that are of subject-predicate form from complex open sentences such as ‘x is tall and x is handsome’ that are not of subject-predicate form?

One reason for adding a predicate abstraction operator to classical formal syntax is that so doing allows one to take at face value the appearance that in natural languages singular terms exhibit certain scope distinctions that are semantically significant.1 An apparent scope distinction involving modality has attracted most attention amongst philosophers,2 but it is convenient to illustrate the point by focusing upon a distinction involving negation.

The distinction I have in mind is most striking with respect to improper definite descriptions such as “The president of the UK,” and empty proper names such as “Vulcan.” Consider two sentence pairs:
  1. (1)

    a) It is not the case that the President of the UK is a woman

    b) The President of the UK is not a woman

     
and
  1. (2)

    a) It is not the case that Vulcan is hot

    b) Vulcan is not hot.

     

In each pair, the first sentence strikes one as true whereas the second sentence does not.3 A plausible explanation of this asymmetry postulates a semantically significant distinction of scope: (na) is the result of performing an operation—negation—on a sentence, whereas (nb) is not constructed in this manner and the scope of negation within it is more narrow. Since my purpose in this section is merely to explain one motivation for introducing a predicate abstraction operator, I shall assume that this explanation is correct.4

Strikingly, the classical syntax of first order logic does not permit scope distinctions between singular terms and negation, however. It symbolises negation as an S/S sentence operator (i.e., an expression that combines with a sentence to yield a sentence)5 and it adopts two principles that together preclude scope distinctions between singular terms and sentence operators:
  1. (C1)

    n singular terms combine with an n-place predicate to form a sentence.

     
  2. (C2)

    The only predicates are the primitive predicates.

     
Given (C1) and (C2), if ζ is a sentence operator and τ a singular term, τ may fall within ζ‘s scope, but not vice versa. If ϕ is a predicate, ζ may be applied to the sentence ϕτ to form a sentence, but there is no expression containing ζ with which τ may be combined to form a sentence. By (C1), τ may only be combined with predicates; by (C2) no predicate contains ζ.6

It follows that if definite descriptions and proper names are formally represented as singular terms, the scope distinctions exhibited in the pairs (1a,b) and (2a,b) have no classical formal representation. So we are left with two options: (i) represent formally (improper) definite descriptions and (empty) proper names other than by singular terms, or (ii) revise classical formal syntax so as to permit complex predicates in which negation is a constituent.

A Russellian development of option (i) proceeds in two stages. Firstly, it appeals to a description theory of proper names (or, at least, of those proper names that are (epistemically) possibly empty), i.e. according to which all proper names (of that kind) are shorthand for definite descriptions; it thereby reduces the problem of explaining the distinction exhibited by the pair (2a,b) to the problem of explaining the distinction exhibited by the pair (1a,b). Secondly, it eliminates definite descriptions in favour of quantifiers, and thereby facilitates an explanation to the effect that the distinction exhibited by the pair (1a,b) arises from the fact that with respect to negation, the particular quantifier held to be implicit in the definite description has narrow scope in (1a) but wide scope in (1b).7 From this Russellian viewpoint, then, the formal representations of (na) and (nb), respectively, might then take the forms:
  1. (R)

    a) ¬∃x(∀y (βy ↔ x = y) ∧ ϕx)

    b) ∃x(∀y (βy ↔ x = y) ∧ ¬ϕx).

     
Well-known objections to Russellian theories of definite descriptions and proper names have encouraged several philosophers to pursue option (ii), however.8 Adding a variable-binding predicate abstraction operator to the primitive syntax of classical formal languages is one of several proposals to this end.9
Standardly, the predicate abstraction operator is symbolised as λ (written to the left of an additional occurrence of the variable it binds) or, less commonly, as ^ (written above an additional occurrence of the variable it binds).10 I shall employ the notation λ. Restricting predicate abstraction to the formation of one-place predicates,11 the formation rule for λ is:
  1. (λ)

    If σ is a sentence and ν a variable, (λν)(σ) is a one-place predicate.

     
λ binds ν, so any free occurrences of ν in σ are bound in (λν)(σ). The semantic value of (λν)(σ) under an interpretation (of predicate letters and names), relative to an assignment of objects to variables, is the class comprising all and only the objects x in the domain such that, when what the assignment assigns to variables other than ν is kept fixed, assigning x to ν results in σ being true.12
For each one-place sentence operator ξ, predicate ϕ, and singular term τ, (λ) induces a scope distinction between the non-predicational sentence ξϕτ, which is obtained by combining ξ with the predicational sentence ϕτ, and the predicational sentence (λv)(ξϕν)τ, which is obtained by combining τ with the complex predicate (λν)(ξϕν) that results from combining λ (and ν) with the open sentence ξϕν. Accordingly, putting ¬ for ξ, the scope distinctions exhibited within the pairs (1a,b) and (2a,b) may now be represented formally even if definite descriptions and proper names are treated as singular terms. Sentences (1a) and (2a) are represented by formulae of the form
  1. (3a)

    ¬ϕτ

     
whereas sentences (1b) and (2b) are represented by formulae of the form
  1. (3b)

    v)(¬ϕν)τ

     

To complete this account of the differences between (1a) and (1b), and between (2a) and (2b), semantic clauses must be provided to the effect that when a term τ is empty, (closed) sentences of the form (3a) are true whereas those of the form (3b) are not. This is straightforward, although the details depend on whether (3b)’s untruth when τ is empty is equated with falsity, or with neither truth nor falsity. The base clause for atomic sentences will allow for singular terms to be empty while ensuring that if ϕ is a predicate, a sentence ϕτ is only true if τ has a bearer. In that case, neither sentences of the form (3b), nor the constituent sentences in sentences of the form (3a), are true if τ is empty. Hence, provided the clause for negation ensures that negation takes untruth of any kind into truth, sentences of the form (3a) come out true when τ is empty.

Following Russell, Prior claims that when τ is a logically proper name the scope distinction exhibited by (3a) and (3b) is of no semantic consequence.13 More recently, however, others have claimed that the distinction remains semantically significant even when τ is not empty,14 and their grounds for so doing extend to logically proper names. Their claim is of great metaphysical interest. I address it in Sect. IV.ii below.

1.2 I.ii

The rationale just given for introducing a predicate abstraction operator makes no mention of the treatment of quantification. It should come as little surprise, therefore, that many authors adopt what I call the “conservative” view according to which predicate abstraction has no impact on the classical syntax of quantification.15 In their systems, the following classical quantifier rule is left unchanged:
  1. (CQ)

    ∀ combines with variables to form sentences out of sentences, i.e. so that ∀νσ is well-formed iff ν is a variable and σ a sentence.

     
On the conservative view, therefore, the introduction of λ merely extends (CQ)’s application to novel sentences involving λ. It leaves the familiar quantified sentences of classical syntax quite undisturbed.
In contrast, according to what I shall call the “revisionist” view, predicate abstraction warrants ∀‘s re-classification to an S/P1 expression that combines with a one-place predicate P1 to yield a sentence S.16 This view holds that once a predicate abstraction operator is added to formal syntax, (CQ) should be replaced by:
  1. (SQ)

    ∀ combines with one-place predicates ϕ to yield sentences ∀ϕ.

     
Classically, (CQ) is accompanied by a semantic treatment whereby quantifiers bind variables. If ν is free in a sentence σ, so that the semantic value of σ depends (potentially) on which object is assigned to ν, the semantic value of ∀νσ is independent of this assignment; any free occurrence of ν in σ is bound in ∀νσ. In contrast, (SQ) is accompanied by a semantic clause whereby ∀ does not bind variables. Under an interpretation of primitive names and primitive predicates, and relative to an assignment of objects to variables, the semantic value of a sentence ∀ϕ is 1 (true) iff everything in the domain belongs to the class that is ϕ‘s semantic value.17 Hence, for no variable ν or predicate ϕ is ν free in ϕ but bound in ∀ϕ: for each variable ν, if, under an interpretation, the semantic value of ϕ depends on the assignment to ν, then so too does the semantic value of ∀ϕ. Accordingly, on the revisionist view ∀ as classically conceived is usurped by λ: its variable-binding role is transferred to λ.18

In order to have a concrete basis for subsequent discussion, it is convenient to give a concrete illustration of the syntactic consequences of replacing (CQ) by (SQ) when λ is introduced. Let L be a first order language with classical syntax: its descriptive primitive symbols are the terms (specifically, individual variables) x, y, z …, and for each n, the denumerably many n-place predicates, including the one place predicates F, G, H …; its logical symbols are the symbols ¬, ∧, = , ∀, (, and). Let Lλ be the language that results from L when a predicate abstraction operator λ is added to the primitive symbols and the rule (λ) for this operator is added to the formation rules. Syntactically, Lλ extends L. Viewed syntactically, the quantified sentences of L form a set properly included in the set of quantified sentences of Lλ: Lλ includes all the quantified sentences of L and in addition quantified sentences such as ∀x(λy)(Fy)x and ∀x(λy)(¬(Fy∧Gy))x that are not sentences of L.

On the other hand, let S be the language obtained from L when λ is added to the primitive logical symbols, (λ) is added to the formation rules, and the classical quantifier rule (CQ) is replaced by the revisionist quantifier rule (SQ). Syntactically, S is not an extension of L; nor does it contain any of the quantified sentences by which Lλ extends L. Indeed, viewed syntactically S has no quantified sentences in common with L or Lλ. In S, all expressions ∀νσ are ill-formed. In particular, the sentences ∀y¬(Fy∧Gy) of L, and ∀x(λy)(¬(Fy∧Gy))x of Lλ, are ill-formed. In lieu of such sentences, S contains a single category of unfamiliar looking quantified sentences, such as ∀(λy)(¬(Fy∧Gy)), that purport to do duty both for the quantified sentences common to L and Lλ, and for the novel quantified sentences by which Lλ (syntactically) extends L.

The question arises as to whether these syntactic differences between the revisionist language S on the one hand and the classical languages L and Lλ on the other are superficial. In particular, the question arises as to whether S is a notational variant of the language Lλ. In what follows, this question is my primary concern. My fundamental thesis is that in the following sense the answer to it is negative: whether S has the same expressive power as Lλ turns on fundamental metaphysical issues that a revisionist adoption of (SQ) would obscure.

1.3 I.iii

To help get a feel for this question let us reflect not on the languages L, Lλ, and S themselves, but on extended languages Lc, Lλc, and Sc that are obtained, respectively, by adding names a, b, c, … to the primitive terms. As before, Lλc extends Lλ but Sc has no quantified sentences in common with either Lc or Lλc. It is noticeable, however, that whereas the novel quantified sentences of Sc are supposed to do duty for the familiar quantified sentences of Lc and Lλc, from a classical viewpoint the novel quantified sentences of Sc are unable to exercise their duties in full. They fail to mirror the syntactic role of the quantified sentences of Lc itself, and they mirror the syntactic role of only some of the novel quantified sentences by which Lλc extends Lc.19

To see this, notice that in classical first order syntax, and in particular in the language Lc, the operation of generalisation is “ubiquitous” in the following sense. Syntactically, every sentence … τ … admits of both particular and universal generalisation with respect to τ: to generalise the sentence with respect to τ, one-first substitutes a variable ν for τ that does not occur in … τ … so as to obtain a sentence … ν … in which ν is free, and one then binds ν by means of the appropriate quantifier. Moreover, when the classical language Lc is extended to Lλc and (CQ) is retained the operation of generalisation remains ubiquitous. For example, sentences of the form (3a) admit universal generalisation, as do sentences of the form (3b). In particular, the sentences in the pair
  1. (4)

    a) ∀x¬Fx

    b) ∀x(λy)(¬Fy)x

     
are obtained by performing the operation of universal generalisation with respect to b, respectively, on the sentences in the pair
  1. (5)

    a) ¬Fb

    b) (λy)(¬Fy)b.

     
Generalisation would cease to be ubiquitous when a predicate abstraction operator is added, however, were (CQ) replaced by the revisionist rule (SQ). In Sc the operation of universal generalisation with respect to τ is restricted to those sentences … τ … that are “predicational” with respect to τ i.e. in that they are of the form ϕτ, where ϕ is a predicate. In Sc implementing the operation of universal generalisation (with respect to τ) upon such sentences consists in substituting ∀ for τ and then re-ordering the sentence. For example, in Sc the universal generalisation of (5b) is not (4b), but
  1. (6)

    ∀(λy)(¬Fy)20

     
Because the operation of generalisation assumes this narrow form, there are sentences of Sc on which it cannot be performed. (5a) is among them. In particular, in Sc neither (4a) nor
  1. (7)

    ∀¬F

     
result from performing the operation of universal generalisation with respect to b on (5a). In Sc both (4a) and (7) are ill formed. From a syntactic viewpoint, therefore, in Sc sentences such as (5b) may be straightforwardly generalised but there is no straightforward way to generalise sentences such as (5a).
It would be natural to seek in Sc an indirect means by which to achieve the effect of universally generalising (5a). To this end, notice that although (7) is ill-formed in Sc, ∀F is well formed. Hence, so too is
  1. (8)

    ¬∀F

     
Since ¬∀ is classically equivalent to ∃¬, one might therefore propose that (8) is at any rate logically equivalent to the particular generalisation of (5a), and, correlatively, that irrespective of how things are syntactically, from a semantic viewpoint an equivalent to the universal generalisation of (5a) may be obtained in Sc by substituting the particular for the universal quantifier in (8) so as to obtain
  1. (9)

    ¬∃F

     
The first thing to note about this proposal, however, is that as it stands it does not respect the impact of the revisionist’s rule (SQ) on the classical relation between ∃ and ∀. In Sc, as in L and its classical extensions, ∃ is a defined term. But its definition cannot be the definition “¬∀¬” it receives in classical languages, since by (SQ) no expression “…¬∀¬…” is well-formed.21 Rather, if ∃ is to be defined in a way that legitimises (9) its definition must take some such form as “∃ϕ =df ¬∀(λν)(¬ϕν),” where ν does not occur in ϕ. In that case (9) becomes the double negation of (6), i.e.
  1. (10)

    ¬¬∀(λy)(¬Fy)

     
Syntactically, the sentence that is doubly negated in (10) is not the universal generalisation of (5a): it is obtained by applying the operation of universal generalisation to (5b). Consequently, the proposal under discussion effectively reduces to the suggestion that while the language Sc does not permit universal generalisation to be applied directly to the sentence (5a), it does include a sentence—namely, (6)—that is equivalent to the universal generalisation of (5a).

Thus reduced, the proposal factors into a presupposition and a claim: (i) pace the limitation of the language Sc, in principle each sentence ¬ϕτ having the same form as (5a) may be subject to universal generalisation; and (ii) the universal generalisation of any sentence ¬ϕτ of Sc is equivalent to a sentence ∀(λy)(¬ϕy) of Sc. In the next section I shall argue that the presupposition (i) is true, but by no means trivially so. In Sect. IV.ii I shall explain why a proponent of the commonplace doctrine that some existing objects are such as to exist only contingently should reject the claim (ii).

2 II

2.1 II.i

The limitation the revisionist quantifier rule (SQ) imposes on the operation of universal generalisation may be summed up as follows. Predicate abstraction forces us to distinguish between two kinds of non-atomic sentence: the ‘predicational’ sentences that are obtained by conjoining a singular term with a predicate, and the ‘non-predicational’ sentences that are obtained by other means. Although the classical quantifier rule (CQ) is indifferent to this distinction, (SQ) exploits it so as to restrict the operation of generalisation to predicational sentences. But in no predicational sentence ϕτ does the subject term τ fall within the scope of a sentential operator ξ: the scope of any such operator must be confined within the predicate ϕ. Consequently, since (SQ) precludes generalisation of a non-predicational sentence ξϕτ in which the term τ does fall within the scope of the sentence operator ξ, from a purely syntactic viewpoint (SQ) precludes quantification into positions that fall within the scope of even truth-functional sentential operators.

To those steeped in classical syntax, (SQ’s) ban on performing the syntactic operation of quantification into the scope of any sentential operator must seem strange. Part of (SQ)’s interest, however, is that if this ban were justifiable on conceptual grounds to the effect that quantification into non-predicational contexts is incoherent, i.e. so that the presupposition (i) at the end of the last section is false, the effect would be to dispel a major obstacle to the consensus that some object is such as to exist contingently.

This obstacle is couched in terms of two key notions. The first is the notion of a proposition that is singular with respect to an object. Sometimes this notion is explained in terms that presuppose that a proposition is a fine-grained structured object. From this viewpoint, a proposition is singular with respect to an object x iff it has x as a constituent. Let us call this the ‘Russellian’ notion of singularity. In part because I wish the argument that follows to engage the views of those who reject fine-grained propositions of this ilk, however, I would like to apply the notion ‘singular with respect to x’ to coarse-grained propositions, i.e. sets of possible worlds. So applying it would be trivial, of course, were the Russellian notion legitimate: a coarse-grained proposition P1—a set of possible worlds—could be said to be singular with respect to x iff there is a fine-grained proposition P2 such that (a) P2 is true in exactly the worlds that belong to P1, and (b) P2 is singular with respect to x in the Russellian sense. Prima facie, however, the notion of a proposition that is singular with respect to an object is neutral with respect to the issue as to whether propositions are fine-grained or coarse-grained. After all, one rationale for the notion of singular proposition begins with a thought that might be entertained and found credible prior to any consideration of what propositions are and whether they are fine-grained or coarse-grained. The thought is this: there is an object such that e.g. the proposition Barak Obama is tall is directly about it in a way that neither the proposition that some man is uniquely President of the U.S. is directly about it (since this proposition is not about any object) nor the proposition that the President of the U.S. in 2010 is tall is directly about it (since although this proposition is about Obama, it is only indirectly so). We label the distinction between the three types of proposition by saying that only the first proposition is singular with respect to Obama. There is no reason why this thought should not survive the subsequent conviction that a proposition is a set of possible worlds. Moreover, there is a more general way to articulate the thought in terms to which a proponent of the coarse-grained view need not object. In the case of the proposition that Barak Obama is tall, but not in the other two cases, there is an object x (namely Barak Obama) which is such that, for every possible world w, the proposition’s truth value at w turns on how things are with x according to w. Here then is a definition of singularity (with respect to objects x, y, z …) that is compatible with the coarse-grained view: a proposition is singular with respect to each and only the objects x, y, z … iff, for every world w, the truth value of the proposition at w turns exactly on how things are, according to w, with x, y, z ….

The second key notion is that of a proposition that is predicational with respect to an object. Hitherto I have distinguished sentences that are predicational i.e. in that they are of the form ϕτ, where ϕ is a predicate and τ a singular term. I now extend the notion ‘predicational’ to propositions in the obvious way: a proposition is predicational with respect to an object x iff the proposition is expressible by a predicational sentence of the form ϕτ in which τ directly refers to x.22

Having explained the two key notions we are now in a position to appreciate the obstacle to the consensus that some object is such as to exist contingently. It comprises two theses I shall defend in Sect. IV.ii, namely:
  1. (M)

    For every object x, there is a proposition that is singular with respect to x and metaphysically necessarily true.

     
  2. (N)

    For every proposition p that is singular with respect to object x, if q is an unrestricted particular generalisation which claims exactly of (unrestricted) something what p claims of x, then it is metaphysically necessary that if p is true then q is true.

     
From the classical perspective (CQ) affords, (M) and (N) together entail that for every object x, it is metaphysically necessary that x belongs to the domain of unrestricted quantification. In turn, this entails, given the Quinean view that to exist is to fall within the domain of unrestricted quantification, that no existing object is such as to exist contingently.
Were it possible to ground (SQ) in the thesis that quantification must be restricted to predicational contexts, however, this obstacle to the standard view that some existing object is such as to exist contingently could be overcome in an especially pleasing way. In particular, the first entailment would fail: if, in accordance with (SQ), for each object x only propositions that are predicational with respect to x have particular generalisations, then (M) does not suffice to yield the conclusion that it is metaphysically necessary that x belongs to the domain of unrestricted quantification. Rather, what would be required to reach this conclusion would be the stronger thesis:
  1. (M*)

    For every object x, there is a proposition that is singular with respect to x and predicationalwith respect to x that is metaphysically necessarily true.

     
In Sect. IV.ii we will see that several philosophers reject (M*) while accepting the weaker thesis (M).23

Clearly, were it possible to provide (SQ) with a strong conceptual grounding to the effect that it is only if a singular proposition is predicational that it has a particular generalisation, an extremely novel variant of the doctrine that some existing object is such as to exist contingently would become available. This variant would be Quinean (because accepting that necessarily, the existing objects are the objects in the domain of the unrestricted quantifiers) but neither Priorean (because accepting (M)) nor Orthodox (because accepting (N)).24

Unfortunately, however, no such conceptual justification of (SQ) can be given. In effect, a justification of this kind would extend to even truth-functional sentential operators a stance that some have taken with respect to epistemic operators like “it is believed that.” But the ground that has been adduced for believing quantification into epistemic contexts incoherent is quite different from any that might be proposed for banning quantification into the scopes of truth-functional operators. According to this ground quantification into epistemic contexts is conceptually incoherent because it fails to conform to the requirement that in quantifiable position, co-referring terms are inter-substitutable salva veritate. Positions that merely fall within the scopes of truth functional operators clearly do satisfy this requirement, however. If the candidate term for generalisation is empty, then a fortiori there is no counterexample. So instead of considering (1a), or (2a), let us consider
  1. (11)

    It is not the case that Obama is bald.

     
With respect to this context any co-referring singular term may be substituted for “Obama” salva veritate. In particular, given
  1. (12)

    Obama is the President of the U.S.

     
it follows from (11) that
  1. (13)

    It is not the case that the President of the U.S. is bald.

     
Mutatis mutandis, the point holds good for conjunction (and hence for all truth-functional sentential operators).25

More generally, it is equally clear that quantification into positions that are not predicated upon is coherent. This is attested to by the classical semantic clause for ∀, i.e. according to which, under an interpretation of names and primitive predicates, a sentence ∀νσ is true with respect to an assignment of objects to variables iff σ is true with respect to every assignment that differs at most in what is assigned to ν. Even in non-vacuous cases, in which ν is free in σ, this clause is no less coherent when σ is non-predicational with respect to ν than it is when σ is predicational with respect to ν.

2.2 II.ii

The fact that it is semantically coherent to incorporate a predicate abstraction operator without disturbing the classical syntax of quantification does not settle the dispute in favour of the conservative view, however. Even when a syntactic classification proposed for a symbol is coherent, two kinds of argument might be given against it. If the classification is a hypothesis regarding syntactic facts that pertain to a language that enjoys an independent existence, a theoretical argument might be given: such an argument would present evidence for the claim that the classification fails to meet an independent standard of correctness (or, even, truth). In contrast, if the classification is a stipulation that serves in the construction of a language, then a pragmatic argument might be given: such an argument would turn on the purposes for which the language is being constructed.

I shall evaluate these two kinds of argument for the revisionist’s rule (SQ) in turn. My evaluation of the second argument will reveal that those who maintain that some existing object is such as to exist contingently must accommodate both (M) and (N). I will then explain how they are best advised to try to do so.

3 III

Let us first consider the possibility of a theoretical argument for the revisionist quantifier rule (SQ). It might be objected that because ∀ is a formal symbol, the idea of such an argument is absurd: what makes a linguistic symbol “formal” is that its syntax is stipulated when a formal language to which it belongs is constructed; hence, the syntax of a formal symbol is not a matter fit for theoretical speculation.

Were it tenable, this objection would gel with my overall case for a conservative view regarding the consequences of predicate abstraction for ∀‘s syntax. It is not tenable, however. Admittedly, some formal symbols have no syntactic “life,” so to speak, beyond syntactic rules that have been stipulated for them. For example, all of the uses that have been made of the symbol ∀ of the language L (above) are subject to the explicit stipulation that this symbol is governed by (CQ). Consequently, no question can arise as to whether, with respect to this symbol, (CQ) is correct.

The dispute between conservatives and revisionists regarding the syntactic classification of ∀ need not be construed according to this model, however. The focus of the dispute may be taken to be a symbol of a language that is given independently of any stipulative syntactic classification. The language I have in mind is the language of classical first order logic.26

This language is a lingua franca across several disciplines. Many students acquire a rudimentary understanding of it without ever knowing its syntax explicitly. In many academic contexts—this paper included—its sentences may be used, or mentioned, unreflectively. For example, although I gave no explicit explanation of how (Ra) above, i.e. ¬∃x(∀y (φy ↔ x = y) ∧ ϕx), is to be understood, I would not have been permitted to publish the suggestion that (1a) be formalised as ¬x∀(∃y(ϕy → x=y)∧φx. With respect to this language, Quine’s observation about the limited power of semantical rules applies equally to rules of syntax: in use a symbol (individuated typographically) is soon cast free, conceptually, from any stipulative anchorings by which it was first introduced. Even if the language of classical first order logic originated in a syntax that was stipulative, what was then stipulated might now be incorrect. The syntax (and semantics) of the universal quantifier ∀ of the language of classical first order logic is no more susceptible to stipulation than is the syntax (and semantics) of the English expression “all.” Syntactic and semantic proposals regarding it must answer to our current linguistic practice.27

A conservative might concede this point only to object that the classical account of the syntax of the language of classical first order logic is nevertheless clearly correct: it is weakly adequate, in that it generates all and only the sentences of this language, and this much is all that is required. This objection has a false presupposition, however, since a syntax can be weakly adequate and yet still be incorrect. For a syntax to be correct, and therefore strongly adequate, its parsings must match those effected by processing mechanisms in speakers of the language.

A conservative might also object that reflection on predicate abstraction cannot possibly show that the classical account of the syntax of the language of first order logic is incorrect, since this language does not include a predicate abstraction operator. The syntax of a language cannot be illuminated by an extraneous symbol! This objection is shortsighted, however. What underlies the conceptual revisionist thought that the symbol ∀ of the language of classical first order logic is an S/P1 expression is the phenomenon of predicate abstraction, not the predicate abstraction operator itself. In principle, the language of classical first order logic could permit predicate abstraction even though it does not contain any such symbol as the predicate abstraction operator λ.

To see this, notice that in a sense the role played by λ in the characterisation of predicate abstraction is redundant. The purpose to which predicate abstraction is put might be served by eschewing the use of any such symbol as λ and replacing the rule (λ) by:
  1. (ν)

    If σ is a sentence and ν a variable, νσ is a one-place (complex) predicate.

     
Let Sν be the language that results from S by deleting λ from the primitive symbols and replacing the rule (λ) by (ν). In an obvious sense, Sν is a notational variant of S. Yet the effect of the change in notation is dramatic. Although the syntax of ∀ in Sν is given by the revisionist rule (SQ), typographically, the quantified sentences of the language L are restored! It is just that in Sν these sentences are parsed differently. For example, in L the sentences ∀y¬(Fy∧Gy) and ∀yFy are obtained, respectively, by conjoining ∀ (and y) with the open sentences ¬(Fy∧Gy) and Fy, whereas in Sν they are obtained by conjoining ∀ with the closed one-place complex predicates y¬(Fy∧Gy) and yFy.
The fact that a change of notation can reconcile the quantified sentences of L with the non-classical quantifier-rule (SQ) invites an argument to the effect that the S/P1 classification of ∀ is a corollary of the thesis that the correct parsing of the language of classical first order logic does recognise complex predicates—namely, predicates such as y¬(Fy∧Gy) and yFy. This argument runs as follows:
  1. (i)

    Typographically, the set s comprising all and only the quantified sentences of the language of classical first order logic with identity but without names is included in the set comprising all and only the quantified sentences of L.

     
  2. (ii)

    Some of the quantified sentences in the set s have complex predicates as constituents.

     
  3. (iii)

    If (ii) is true, the sentence ∀y¬(Fy∧Gy) of the language of classical first order logic contains the complex predicate y¬(Fy∧Gy).

     
  4. (iv)

    If the sentence ∀y¬(Fy∧Gy) of the language of classical first order logic contains the complex predicate y¬(Fy∧Gy), then in this language ∀ is an S/P1 expression that combines with a one-place predicate to form a sentence.

     
So
  1. (v)

    The symbol ∀ of the language of classical first order logic combines with a one-place predicate to yield a sentence.

     
This argument is hinted at when, speaking of L and a language ^L (“El-cap”) that is a notational variant of Sν,28 Stalnaker (1977) writes:

[El-cap] is a more natural formulation than L. Frege’s variable binding device, used in both ^L and L is essentially a device for forming predicates out of sentences by the omission of a singular term. ^L shows this device for what it is: a means of constructing complex predicates by omission of a singular term. In [L, however,] the device is tied to quantification; complex predicates can be formed only in the process of representing universal and existential quantifications. It may be granted that the original motivation for the device for forming complex predicates was that it was needed to represent quantification, but the device itself is separable from its motivation: it need have nothing to do with generalized statements.29

Notice that even though the syntax stipulated for L does not countenance the formation of complex predicates, in the parts of the passage I have underlined Stalnaker says that a device of L is “essentially a device for forming predicates out of sentences,” one that leads “complex predicates [to be] … formed … in the process of representing universal and existential quantifications.” Since the device is present in all and only the quantified sentences of L, he appears to imply that the explicit syntax of L notwithstanding, a sentence such as ∀y¬(Fy∧Gy) does contain a complex predicate—namely, a complex predicate formed from the open sentence ¬(Fy∧Gy). Taken at face value, the implication is that the syntax explicitly given for L parses e.g. ∀y¬(Fy∧Gy) incorrectly, since this sentence contains the complex predicate y¬(Fy∧Gy) as a constituent. The S/P1 view of ∀ is then a corollary: if the sentence ∀y¬(Fy∧Gy) has a complex predicate y¬(Fy∧Gy) as a constituent, ∀ combines with a predicate to form a sentence.30Mutatis mutandis, a parallel argument can be framed with respect to the language Lc and the language of classical first order logic with identity and individual names.

There is an obvious difficulty with revisionist arguments of this kind, however. The language of classical first order logic includes neither sentences of the form ∀ϕ, where ϕ is a primitive predicate, nor sentences of the form ϕτ, where ϕ is of the form νσ and τ is an individual term. Since combining the rule (ν) with (SQ) yields the former, while combining it with the classical rule (C1) for forming sentences from terms and predicates yields the latter,31 a weakly adequate syntax for the language of classical first order logic that includes the rule (ν) cannot include (SQ) or (C1). Rather, it would have to restrict these principles as follows:
  1. (SQ*)

    ∀ combines with a complex one-place predicate to yield a sentence.

     
  2. (C1*)

    n singular terms combine with an n-place simple predicate to form a sentence.

     
Prima facie, however, these restrictions would be entirely arbitrary. This gives reason to suspect that with respect to the language of classical first order logic, although the syntax to which (SQ*), (C1*) and (ν) belong is weakly adequate, it is not strongly adequate.32

Similar reasoning is appropriate with respect to first order languages the primitive syntax of which includes a predicate abstraction operator. Perhaps some of these languages are sufficiently prominent to have gained a “life” that is independent of any syntactic stipulations by which they were first introduced. Among them are languages the sentences of which coincide typographically with the sentences of Lλ and Lλc respectively. In such cases, ∀ appears to have a classical syntax captured by (CQ): combined with a variable, it combines with a sentence to form a sentence. But, as before, this appearance might be countered on behalf of (SQ) by supposing that in a sentence such as ∀x(λy)¬(Fy∧Gy)x, x(λy)¬(Fy∧Gy)x is a complex predicate formed by prefacing x to an open sentence (λy)¬(Fy∧Gy)x. In effect, on this conception such languages include two means by which to form complex predicates out of sentences: one is given by (λ), the other by (ν).

The reply is the same. If (ν) is correct for such languages, (SQ) and (C1) are not. For if (SQ) were also correct, such languages would include e.g. ∀F, while if (C1) were also correct, they would include e.g. x(λy)¬(Fy∧Gy)xτ. So (ν) could only be accommodated if restrictions were imposed on these principles. That such restrictions would be entirely arbitrary suggests that a weakly adequate syntax that employs them would not be strongly adequate.

4 IV

Let us turn to pragmatic arguments that might be given in favour of (SQ). These arguments concern the best way of stipulating syntactic rules for the universal quantifier when constructing a first order language that contains a predicate abstraction operator. Among them, different arguments claim different pragmatic virtues for (SQ). I shall consider two. The first claims that (SQ) more faithfully represents quantification in natural language. The second claims that (SQ) brings greater simplicity and elegance without loss of expressive power. Again, at the very least Stalnaker (1977, p. 330) hints at both arguments:

[O]nce one introduces a means of representing complex predicates explicitly, it can be seen that it is redundant to treat quantifiers as variable binding operators. One can treat quantifiers as something more like the quantifiers of natural language without losing any of the expressive power which the variable binding device brought to quantification theory.33

I shall consider these two arguments in turn.

4.1 IV.i

If natural language does not permit quantification into the scope of truth-functional sentential operators such as negation then (SQ) represents quantification in natural language more faithfully than does (CQ). Any suggestion that natural language does not permit quantification of this kind might seem ridiculous, however. After all, it contains sentences like
  1. (14)

    Someone is not bald.

     
But the matter is not so easily resolved. While (14) is the particular generalisation of
  1. (15)

    Obama is not bald

     
we presumed at the outset that in sentences such as (15) negation does not have the wide scope it enjoys in sentences such as
  1. (11)

    It is not the case that Obama is bald

     
It follows that (14) is not a counterexample to the claim that natural language does not permit quantification into the scope of truth-functional sentential operators.
A counterexample to this claim would be a sentence that stands to (11) as (14) stands to (15). Obviously, merely substituting “someone” for “Obama” in (11) does not yield such a sentence. A more likely candidate is
  1. (16)

    Someone is such that it is not the case that he is bald.

     
Nevertheless, (16) is not the particular generalisation of (11). It could hardly fail to be the particular generalisation of
  1. (17)

    Obama is such that it is not the case that he is bald.

     
But in (17) the position occupied by the term “Obama” is predicated upon. Indeed, such sentences are paradigmatically predicational. To suppose otherwise would be to suppose that the sentence
  1. (18)

    The President of the UK is such that it is not the case that he is a woman

     
might strike one as true (just as (1a) does). It does not so strike one, however (anymore than (1b) does). On the contrary, once predicate abstraction is recognised, the contrast between (15) and (17) appears to consist in the fact that predicate abstraction is tacit in the former but explicit in the latter: (17) contains a predicate “is such that it is not the case that he is bald” that is obtained by prefacing a natural language predicate abstraction operator “is such that” to the open sentence “it is not the case that he is bald.” Accordingly, the formalisation of (17) should have the same form as that of (1b), namely
  1. (3b)

    v)(¬ϕv

     
Since (16) is the particular generalisation of (17), the relative scopes of its negation sign and bound variable should mirror the relative scopes of the negation sign and the term ‘Obama’ in (17). Negation does not govern the term ‘Obama’ in (17), however: its scope is confined to a constituent open sentence. Likewise, then, the position upon which quantification occurs in (16) does not fall within the scope of the negation sign. Since, by hypothesis, in (11) the term ‘Obama’ does fall within the scope of the negation sign it follows that (16) is not the particular generalisation of (11).

At this point one might suspect that any attempt to provide an English sentence σ that stands to (11) as (14) stands to (15) must encounter an insurmountable difficulty. σ is to contain a particular quantifier having broad scope with respect to a negation sign. But if σ is of the form “someone is … not …,” it generalises a sentence σ* of the form “τ is … not …” in which, likewise, the name “Obama” falls outside the scope of the negation sign. In that case, σ cannot stand to (11) as (14) stands to (15): in (11) “Obama” falls within the scope of the negation sign.

All such reasoning reveals, however, is that in natural language quantification with respect to non-predicational position cannot take the form “someone is … not ….” In natural language, quantification can take other forms, and the fact is that some of these forms do implement quantification into positions that are not predicated upon. For example, two such forms are:
  1. (19)

    For someone x, it is not the case that x is a Republican

     
and
  1. (20)

    There is a person regarding whom it is not the case that he is a Republican.

     
It might be objected that (19) is not natural language because it employs variables, and that (20) does not employ a true quantifier. But this objection should be overruled on both counts: the construction “there is/are” has long been taken to be a form of quantification, and if (19) is not natural language, what is it? In any case, the objection is sidestepped by further forms in which elements of (19) and (20) are combined, such as
  1. (21)

    Regarding someone, it is not the case that he is a Republican.

     
(21) is a natural language quantification into a position that is not predicated upon—specifically, into a position that falls within the scope of sentential negation.

Accordingly, provided the revisionist claim that re-classifying ∀ as an S/P1 expression makes for a more faithful representation of quantification in natural language is construed as the claim that natural language forbids quantification into positions that are not predicated upon, it fails. Moreoever, if it were construed as the view that certain natural language expressions—specifically, “someone,” “everything” etc.—combine with one-place predicates to form sentences, it would still fail. Expressions like “some,” “any,” “most” etc., have much stronger claims to the title “quantifier,” and their syntax is quite different. More importantly, as (19) and (21) illustrate, even though “someone” and “everything” etc. often occur conjoined with a predicate, on other occasions they do not.

4.2 IV.ii

The second pragmatic argument for rejecting the classical quantifier rule (CQ) in favour of the revisionist rule (SQ) is to the effect that in combination with the introduction of a predicate abstraction operator, (SQ)’s re-classification of ∀ achieves greater elegance without loss of expressive power. That (SQ) is more elegant than (CQ) may be granted. So the question to be addressed is whether (SQ) incurs loss of expressive power.

“Loss of expressive power” may be understood in different ways. I shall focus on a strong construal of it according to which a move from one language to another involves a loss of expressive power iff the consequence of the move is simply that under natural interpretations of the primitive predicates some coarse-grained propositions can no longer be articulated (a “coarse-grained” proposition being a set of metaphysically possible worlds). I shall argue that those who hold that some existing object is such as to exist contingently should hold that (SQ)’s re-classification of ∀ does lead to a loss of expressive power. In particular, they should hold that the languages S and Sc are less expressive, respectively, than the languages Lλ and Lλc: under a natural interpretation of the primitive predicates, the proposition expressed by the sentence ∀x¬Fx of Lλ and Lλc cannot be expressed in the language S (or Sc). It should be noted, however, that my main point—that revisionist advocacy of (SQ) threatens to blind us to certain metaphysical issues that should be kept in focus—requires no more than that my argument be worthy of serious consideration.

Let us assume that some existing object is such as to exist contingently. Since the vast majority of those who believe this assumption take ordinary physical objects, such as Mars, to fall into this category, let us assume further that Mars exists contingently.34 Let w be a world with respect to which the proposition that it is not the case that Mars exists is true. In particular, let w be a world that comprises a single gold sphere. Let the predicate F be interpreted as “is green,” and let the other primitive predicates be interpreted by similarly (relatively) simple properties. The first (and main) step of my argument is to show that under this interpretation the proposition expressed by the sentence ∀x¬Fx is not modally equivalent to the proposition expressed by the sentence ∀(λy)¬Fy. Of course, the fact that some sentence of a language L2 does not express the proposition expressed by some sentence of a language L1 does not entail that no sentence of L2 expresses that proposition: another sentence might do so. I suggest, however, that under the natural interpretation of the primitive predicates provided, if, on the assumption that the existence of Mars is contingent, the sentence ∀(λy)(¬Fy) of S (and Sc) and does not express the proposition expressed by the sentence ∀x¬Fx of Lλ (and Lλc), then in fact no sentence of Sc does so.

To show that the proposition expressed by ∀x¬Fx is not modally equivalent to the proposition expressed by ∀(λy)(¬Fy) it suffices to show that only the former is true with respect to the world w comprising a single gold sphere. Clearly, the proposition expressed by ∀x¬Fx is true with respect to w: since w comprises a single gold sphere it is a world with respect to which nothing is green.35 Given two further theses, which I call (T) and (N), however, the proposition expressed by ∀(λy)(¬Fy) is false with respect to w. I elaborate and defend these theses in turn (my defence of (T) being conditional on the assumption that the proposition “it is not the case that Mars exists” is true with respect to w).

4.2.1 (Conditional) defence of (T)

Let “b” designate Mars. The first thesis I use to derive the conclusion that the proposition expressed by ∀(λy)(¬Fy) is false with respect to w (assuming Mars does not exist with respect to w) is
  1. (T)

    (i) The proposition expressed by ¬Fb is true with respect to w.

    (ii) The proposition expressed by (λy)(¬Fy)b is false with respect to w.

     
Part (i) of (T) is relatively uncontroversial, although Prior rejects it on the grounds that each contingently existing object is such that had it not existed there would have been no propositions that are singular with respect to it, and, hence, that no actual proposition concerning it would have been true. His rationale is very hard to stomach, however, and rejection of T(i) should be viewed as a measure of the very last resort. For one thing, Prior is obliged to hold that the only sense in which the proposition that Mars does not exist might have been true is a sense in which the proposition that (Mars is green and it is not the case that Mars is green) might also have been true.36 For another, Prior’s rationale is self-defeating (or even self-contradictory): whereas it appeals to certain (possible) circumstances as being (possible) circumstances in virtue of which the proposition that Mars does not exist might have been true, these circumstances are such that they cannot be identified without saying, albeit entirely negatively, how, with respect to them, things are with Mars.37

Part (ii) of (T)—namely, that the proposition expressed by (λy)(¬Fy)b is false with respect to w—is much more controversial. Nevertheless, it has been defended by several philosophers who maintain that some existing objects are such as to exist merely contingently, including philosophers who maintain part (i) of (T).38 In effect, by adopting (T) in its entirety these philosophers embrace one of the additional rationales for the introduction of a predicate abstraction operator to which I alluded in Sect. I.i: pace Russell and Prior they hold that even proper names having a bearer exhibit a semantically significant distinction of scope with respect to negation. In contrast with the case of empty names and improper definite descriptions, they hold this distinction to be a matter not of actual truth value, but of truth value with respect to some merely possible circumstance.

Advocates of the view that some existing object is such as to exist contingently should follow this lead. They too should embrace the thesis that the proposition expressed by (λy)(¬Fy)b is false with respect to w. This is because the thesis follows, given the assumption that some object is such as to exist contingently, from a doctrine everyone has reason to endorse—namely, “predicate actualism” i.e. the doctrine that it is metaphysically necessary that the objects that satisfy predicates are exactly the existing objects.39 Predicate actualism in this sense is a natural corollary of a logical constraint on the existence predicate E. Where ϕ is any predicate, the constraint is:
  1. (W)

    ϕτ |- Eτ

     
Since any two existence predicates governed by this constraint are logically equivalent, the concern that is commonly and justifiably felt that parties to many ontological disputes talk past one another would be dispelled were all parties to agree to adopt (W).40,41

4.2.2 Defence of (N)

In order to derive from (T) the conclusion that on our interpretation of F and the other primitive predicates the proposition expressed by ∀(λy)¬Fy is not modally equivalent to the proposition expressed by ∀x¬Fx, one requires a further doctrine, namely:
  1. (N)

    For every proposition p that is singular with respect to object x, if q is an (unrestricted) particular generalisation which claims exactly of (unrestricted) something what p claims of x, then it is metaphysically necessary that if p is true then q is true.

     
By (T), the proposition actually expressed by “it is not the case that Mars is green” is true with respect to w. By (N), therefore, this proposition’s particular generalisation is true with respect to w. Therefore, with respect to w, Mars falls within the domain of the unrestricted quantifier in the proposition expressed by ∃x¬Fx and, hence, within the domain of the unrestricted quantifier in the propositions respectively expressed by ∀x¬Fx and ∀(λy)¬Fy. Since by (T), the proposition expressed by “Mars is such that it is not green” is false with respect to w, it follows that its universal generalisation—namely, the proposition expressed by ∀(λy)¬Fy—is false with respect to w. By construction, however, the universal generalisation of the proposition actually expressed by “it is not the case that Mars is green”—namely, the proposition expressed by ∀x¬Fx—is true with respect to w.

In my view the truth of (N) is no less compelling than is that of (T). Since (N) is a consequence of the doctrine that the domain of unrestricted quantification could not have been narrower than it actually is, (N) is supported by the arguments recently marshalled in defence of this doctrine by Williamson (1999, 2000, 2002). The variant of one of Williamson’s arguments I think especially compelling is this: to identify an object—say Mars—as being a counterexample would be to hold, absurdly, that the following sentence expresses a proposition that might have been true: “Mars is possibly identical to Mars and (unrestrictedly) absolutely nothing is possibly identical to Mars.” But reflection on the nature of particular generalisation yields a demonstration of (T) that is somewhat different from Williamson’s. Every proposition p that is singular with respect to x is such that its particular generalisation q (on x) is obtained by excising information—namely, x’s identity—from p. But to excise information from a proposition is to weaken it, and every proposition strictly implies its weakenings.42

Notwithstanding such powerful considerations, (N) is highly controversial. Indeed, it is rejected by the orthodox variant of the doctrine that some existing object is such as to exist contingently. In my view, however, orthodoxy is rooted in considerations that are entirely negative: it denies (N) only as a desperate measure of last resort. It does so in order to overcome the obstacle to the thesis that some object is such as to exist contingently which in Sect. II we saw to be constituted by (N) in combination with:
  1. (M)

    For every object x, there is a proposition that is singular with respect to x and metaphysically necessarily true.

     
The fact is that the considerations just adduced are so powerful that orthodoxy’s rejection of (N) is no less desperate than is Prior’s rejection of (M).43 Of course, advocates of orthodoxy will reject my account of the relation singular propositions bear to their particular generalisations. Citing Quine’s treatment of existence they will object that, on the contrary, the particular generalisation q of a proposition p that is singular with respect to x differs from p in part because it contains additional information—namely, that what p claims regarding x is true of an existing object. This objection is misguided, however. If “exists” is taken to mean “falls within the domain of the unrestricted quantifiers” the question is begged. But if “exists” has some other meaning the objection’s order of priorities is distorted.
To appreciate this distortion of priorities, consider two further theses:
  1. (Q)

    It is metaphysically necessary that all and only the objects within the domain of the (unrestricted) particular quantifier are the existing objects.

     
  2. (C)

    Some existing object is such as to exist contingently.

     
Given the classical perspective on quantification I defended in Sect. II, whereby for every proposition that is singular with respect to x there is a particular generalisation of that proposition (on x), the set {(M), (N), Q), (C)} is inconsistent. Given this perspective, (M) and (N) combine to entail a thesis—
  1. (X)

    The domain of the (unrestricted) particular quantifier could not have been narrower than it is actually.

     
—that is inconsistent with the conjunction of (Q) with (C).

It would be a travesty to reject (N) on the grounds that (M), (Q) and (C) together entail its falsity, however. Admittedly, pace Prior, (M) is secure. In effect, I defended it when arguing for (T): for if any proposition that is singular with respect to Mars is true with respect to a world with respect to which, ex hypothesi, Mars does not exist, then there is no reason to reject the compelling appearance that e.g. “either Mars is self-identical or it is not the case that Mars is self-identical” actually expresses a proposition that is singular with respect to Mars and true with respect to every metaphysically possible world (the choice of Mars as the object of discussion being arbitrary). But (N) is no less fundamental than (M). It has a conceptual grounding that is far superior to any that might be given for the conjunction of (Q) with (C). Consequently, the conjunction of (Q) with (C) cannot bear the weight orthodoxy places upon it. Instead of arguing from the conjunction of (Q) with (C) to the rejection of (N), one should employ modus tollens. In combination with (M), (N) is a ground on which to maintain the negation of (Q) or the negation of (C).

4.3 IV.iii

In effect, I have concluded that the consensus view (C) that some existing object exists contingently must (i) accept Williamson’s thesis (X) that the domain of unrestricted quantification could not have been narrower than it is actually and (ii) reject Quine’s view (Q) that to exist is to belong to the domain of unrestricted quantification. To many advocates of (C), there are two reasons why this conclusion will appear not so much bad as catastrophic. Firstly, acceptance of (X) would seem to make inevitable acceptance of Williamson’s thesis that each object that exists in the “logical” sense is such that its existence is metaphysically necessary: for what Williamson has in mind when he speaks of the “logical” sense of existence is membership of the domain of the unrestricted quantifiers.44 Secondly, many advocates of (C) associate rejection of (Q) with an undisciplined Meingonianism that embodies a (primitive) notion of existence that is incomprehensible.

In both respects this is an overreaction, however. The inference from (X) to Williamson’s further thesis is valid only if Williamson’s two presuppositions—namely (i) there is only one “logical” sense of existence, and (ii) existence in this sense is membership of the domain of the unrestricted quantifiers—are true. As we have seen, however, there are good reasons not to grant either presupposition: the existence predicate should be held to be defined by the inference rule (W), and a notion of existence that is so defined is clearly “logical.” Correlatively, rejection of (Q) need not descend into unintelligibility: the notion of existence defined by (W) is perfectly intelligible.

We have seen (in Sect. II) that the revisionist quantifier rule (SQ) is powerless to aid (C) against the obstacle (M) constitutes in combination with (N). Nevertheless, advocates of (C) are well-advised to invoke the predicate abstraction operator. So doing introduces a distinction between predicational and non-predicational singular propositions, and thereby facilitates a reconciliation of the inference rule (W)—and therefore predicate actualism—with the conjunction of (M) and (N). Predicate abstraction is central to the best variant of the doctrine that some existing object is such as to exist contingently.45

5 V

Enriching classical formal syntax by adding a predicate abstraction operator sharpens the question as to whether the domain of unrestricted quantification must coincide with the domain of predication. If the classical syntax of formal quantification, as given by (CQ), is abandoned in favour of the revisionist quantifier rule (SQ), however, one way in which the two domains might fail to coincide is closed off, and only the possibility that the domain of predication is wider than the domain of unrestricted quantification remains.

Some authors have found this latter possibility attractive. Indeed, some think it not only metaphysically possible, but actually realised: they hold that certain objects—Socrates, for example, or Brama (the person who would have resulted had ovum x from Michelle Obama been fertilised by sperm y from Gordon Brown under such-and-such conditions)—about which certain propositions are true are such that it is actually the case that these objects remain outside the domain of even unrestricted quantification.46 In arguing that for every proposition that is singular with respect to an object, there is a particular generalisation of the proposition with respect to that object that the proposition strictly implies, I have argued that their view is not even metaphysically possible. By appealing to predicate abstraction, however, I have done so in such a way as to lend credibility to the thesis that the domains of quantification and predication might fail to coincide from the opposite direction: assuming that some existing object is such as to exist contingently, it is metaphysically possible that the domain of quantification is wider than the domain of predication.

Of course, if the assumption that some existing object is such that it exists contingently were false, the case I have made for the possibility that some objects satisfy no predicates would collapse. The syntax of formal quantification could then be revised upon the introduction of a predicate abstraction operator, and the classical quantifier rule (CQ) replaced by (SQ), without loss of expressive power. Much of my case against revisionist advocacy of (SQ) would survive this eventuality, however. Consequently, I have not attempted to defend the doctrine that some existing objects are such as to exist contingently. On the contrary, I am sceptical of it. In effect, I have argued that this doctrine turns on whether it is indeed metaphysically possible for the truth value for a non-predicational singular proposition to come apart from that of the associated predicational proposition. I am inclined to think that it cannot.47

My main point is that irrespective of how the facts pertaining to the modalities of existence, quantification, and predication should turn out, all options should be kept firmly in view, and explored. In developing this point I do not deny Williamson’s observation that it is permissible, indeed, inevitable, for logic to embody metaphysical theses.48 I am not concerned to banish metaphysics from logic. Rather, my concern is to banish metaphysics from syntax. To keep ourselves alert to the possibility—epistemic if not metaphysical—that the domain of quantification is broader than the domain of predication, we need to investigate formal languages the syntax of which facilitates expression of this possibility. In such languages, the syntax of predicate abstraction is combined with the classical formal syntax of quantification. For some purposes, formal first order languages whose quantificational syntax is given by (SQ) no doubt suffice. At best, however, these languages should be employed sparingly and with caution. Otherwise, we might end up blind to metaphysical issues we should keep in focus.

Footnotes
1

Additional reasons include the following. Thomason and Stalnaker (1973) argue that predicate abstraction facilitates the formal representation of scope distinctions between adverbs that are predicate modifiers, and adverbs that are sentential operators. Lambert and Bencivenga (1986) claim both that “Vulcan is self-identical” is ambiguous—there is a reading on which it is truth-valueless, and a reading on which it is false—and that appeal to a predicate abstraction operator affords an explanation of the ambiguity.

 
2

For example, consider the distinction between “The Prime Minister of the UK might not have been the Prime Minister of the UK” which is true on the most immediate interpretation, and “It might have been the case that the Prime Minister of the UK is not the Prime Minister of the UK” which is not. To my knowledge, Thomason and Stalnaker (1968) were the first to conceive this distinction as a scope distinction best represented formally by means of predicate abstraction.

 
3

Following Russell, Kripke (2005, pp. 1014–1016) and many other commentators report that sentences relevantly like (1b) can be read in different ways, and that on one reading they seem true. In my idiolect, however, (1b) does not have a reading on which it seems true. More rarely, some commentators—such as Atlas (1977)—report that sentences relevantly like (1a) can be read different ways, and that on one reading they seem not to be true. In my idiolect, however, (1a) is not ambiguous in this way: it does not have a reading on which it does not seem true. Similarly, Grice (1969, p. 120) distinguishes readings on which sentences like (2b) are true, from readings on which they are not true. Perhaps there is agreement about the appearances to this extent: on readings that are most immediate, (1a) and (2a) strike one as true, whereas (1b) and (2b) do not.

 
4

For very different explanations, see Atlas (1977) and Horn (1985). The issue as to whether scope distinctions between singular terms and negation have an explanatory role to play with respect to such sentences cuts across the issue raised in Footnote 3, because if e.g. (1b) has different readings on which it has different truth values, predicate abstraction would be motivated by the thought that a plausible explanation of the ambiguity is that (1b) permits both wide-scope and narrow-scope readings of the definite description with respect to negation. Again, however, although this explanation is typically adopted by those who do report an ability to read (1b) in different ways, some who report this ability—such as Atlas (1977), Horn (1985), and Glanzberg (2008)—reject this explanation.

 
5

In my usage, the sentences include open sentences as well as closed sentences.

 
6

Sometimes complex open sentences—formulae such as ¬Fy etc.—are spoken of as “predicates.” But such formulae are not predicates in the sense of the classical base clause for forming sentences out of predicates. They do not combine with terms to form sentences. I follow Neale (2005, p. 822) in defining the scope of an expression as the smallest sentence in which the expression occurs.

 
7

In my terminology, the paradigmatic “particular” quantifier is “some.” I refrain from calling it “existential” so as not to beg the question as to whether its domain includes objects that do not exist.

 
8

For a survey and discussion of these objections, see Ludlow (2007). Since my aim in this section is merely to motivate the modification of classical syntax by means of a predicate abstraction operator, I ignore non-Russellian developments of option (i). It should be noted that if the considerations adduced by those who claim that even proper names that have a bearer exhibit a semantically significant scope distinction with respect to negation are correct (see Sect. IV.ii below), their claim would extend to even logically proper names—in which case the Russellian development of option (i) would collapse.

 
9

Another proposal is to allow negation to operate not only on sentences, but on predicates. This effect might be achieved in two ways. Firstly, the classical (“external”) negation sign of category S/S might be complemented by a second (“internal”) negation sign ¬p of category P/P that combines with an n-place predicate ϕ to yield an n-place predicate ¬pϕ. Secondly, as in Frege, the syntactical rule for the classical negation sign might be relaxed so as to assign it a dual function captured in the rule: if ϕ is a sentence, ¬ϕ is a sentence; if ϕ is an n-place predicate ¬ϕ is an n-place predicate. The introduction of a predicate abstraction operator is more systematic than either alternative, however. For example, it alone automatically accommodates what would appear to be scope distinctions between singular terms and modalities that are analogous to the scope distinctions exhibited by the pairs (1a,b) and 2(a,b) (cf. Thomason and Stalnaker (1973, p. 198) and Footnote 2 above).

 
10

The notation λ for predicate abstraction has its origins in Church’s (1932) notation for functional abstraction. In contrast, the notation ^ for predication abstraction, which, beginning with his collaborative papers with Thomason, Stalnaker (1977, 1995, 2003) has consistently employed, has its origins in Whitehead and Russell (1910–13)’s notation for propositionalfunction abstraction. Nevertheless, it turns out that the two notations have a common root. Barendregt (1997, p. 182) relates that although Church had intended to employ Whitehead and Russell’s notation, whereby ^ is written above the variable to be bound, one of his typesetters, being unable to set ^ above the variable, had set it to the left, only for a second typesetter to change ^ positioned thus to λ. For more on the notation ^ for predicate abstraction, and its rationale, see Sect. III below.

 
11

This is for convenience. See Stalnaker (1977, Footnote 3) and Lambert and Bencivenga (1986, p. 249) for rules that permit the abstraction of n-adic complex predicates, for any n.

 
12

Strictly speaking, in this gloss “domain” refers to the domain of predication. Classically, this qualification is superfluous: it is assumed that the objects that fall within the range of predicates coincide with the objects that fall within the range of the (unrestricted) quantifiers. In Sect. IV.ii below, however, I present an argument to the conclusion that if existence is contingent, so too is any such coincidence.

 
13

Prior (1971, p. 149).

 
14

Stalnaker (1977, 1995), Cartwright (1997), and Hoffman (2003).

 
15

For example Bencivenga and Woodruff (1981), Lambert and Bencivenga (1986), Fitting (1991, 1996), Fitting and Mendleson (1998).

 
16

Stalnaker (1977, 1995, 2003), Graff Fara (2006).

 
17

Strictly speaking the “domain” here is the domain of quantification (see Footnote 12).

 
18

In contrast with predicate abstraction, even though property abstraction is also a device for binding a free variable in an open sentence (so as to form a singular term that refers to the property ‘expressed’ by the sentence) there is no prospect of it threatening to make the variable binding role of the classical quantifiers redundant.

 
19

The language Lλc extends Lc as well as Lλ. Lλc is obtained from Lc by adding λ to the primitive symbols and (λ) to the rules.

 
20

The generalisation of (5b) is (6), rather than the ill formed (λy)(¬Wy)∀ because, in accordance with the rule (SQ), in Sc, as in S, the convention is to write ∀ to the left of a predicate.

 
21

Speaking of a language that is not relevantly different from Sc, Stalnaker (2003, p. 158) writes, without noting this difficulty, that non-primitive logical symbols, including ∃, “… are to be understood as abbreviations in the usual way.” (I discuss a language of the kind Stalnaker has in mind in Sect. III below.)

 
22

While the notion of a proposition expressed by a predicational sentence is relatively straightforward, some might reject the notion of what is expressible by a predicational sentence. Their scepticism might be accommodated by taking the notion ‘expressed by a predicational sentence ϕτ in which τ directly refers to x’ to be sufficient and necessary for a proposition to be predicational with respect to x. In that case (M*) below would be clearly false (given the distinction between predicational and non-predicational sentences, and the fact that some objects are unnamed).

 
23

They include Stalnaker (1977, 1995), Cartwright (1997), and Hoffman (2003).

 
24

Stalnaker (1977) can be read as advocating this variant. See Sect. IV.ii below for further discussion of Prior’s rejection of (M) and Orthodoxy’s rejection of (N).

 
25

One might object that the inference from (11) and (12) to (13) is unsound because e.g. whereas “It is not the case that Clark Kent is strong” and “Clark Kent is Superman” are both true, “It is not the case Superman is strong” is false (cf. Saul 1997). In my view, however, such examples reveal nothing about this inference. They involve names that are governed by conventions quite different from those governing ordinary names. (In contrast, it appears that in epistemic contexts substitution of any co-referring names can fail to be truth-preserving.)

 
26

I do not mean to imply that the language of classical first order logic differs from the language of e.g. intuitionistic first order logic.

 
27

My talk of “the” language of classical first order logic does not tie this language down to one typographical convention, since e.g. ¬ has no more right to be deemed the negation sign of classical first order logic than ~ has. Rather, I think of this language as being flexible in its choice of primitive symbols. I do so for convenience only. Strictly speaking, no doubt, one should speak only of languages of classical first order logic.

 
28

The notational differences between ^L and Sν are as follows. ^L includes the additional symbol ^, and instead of (ν), it has the following rule: If σ is a sentence and ν a variable, \(\hat{v}\sigma\) is a one-place (complex) predicate.

 
29

Stalnaker (1977, p. 330). The underlining is mine.

 
30

In interpreting the passage from Stalnaker I do not contradict my earlier claim that (SQ) cannot be true with respect to the symbol ∀ of L. My claim is only that Argument A is hinted at in this passage. Moreover, to my mind the only interpretation on which the passage is coherent first distinguishes my (stipulated) language L (above) from Stalnaker’s typographically identical language L, and then identifies the latter with the language I call the language of classical first order logic (with identity but without names).

 
31

See Sect. I.i above.

 
32

Although this suspicion is natural it might turn out to be incorrect. Since the workings of the language module are a posteriori, one cannot rule out a priori that its parser embodies syntactic rules that, from a semantic or metaphysical perspective, are entirely arbitrary.

 
33

Stalnaker (1977, p. 330).

 
34

Among those who hold that some existents are such as to exist merely contingently, anyone who denies that Mars falls into this category should choose a representative they deem more suitable.

 
35

In claiming this much I assume that nothing is essentially green, since it might be held that if an object x is essentially ϕ the proposition that x is ϕ is true with respect to w whether or not x exists with respect to w—and, as I am about to argue, those who hold this much should then conclude that the proposition that (unrestricted) something is ϕ is true with respect to w.

 
36

See Menzel (1991). For an earlier and illuminating discussion of the objection that is more sympathetic to Prior’s position, see Fine (2005, pp. 205 ff.).

 
37

In insisting that, on the assumption that Mars exists contingently, there is a world w with respect to which the proposition (actually) expressed by ¬Fb is true, I do not insist that the proposition ‘the proposition expressed by ¬Fb exists’ is likewise true with respect to w. In the terminology of Fine (2005, p. 194) I am concerned with outer truth, not inner truth: only the latter requires the existence at w of the propositions that are true with respect to w. Correlatively, when I say what propositions are true with respect to w I should be understood not as saying what propositions would have been true had w been actual but as saying what thetruths would have been had w been actual: only the former requires that the propositions that are true with respect to w are such that had w been actual they would (still) have existed. If a proposition is true with respect to a possible world in my sense the world represents not the proposition as being true but that truth: for example, in order for the proposition that grass is green to be true with respect to a world it is enough that the world should represent grass as being green.

 
38

Stalnaker (1977, 1995), Cartwright (1997), and Hoffman (2003).

 
39

Cf. Fine (2005, p. 200).

 
40

I owe this point to Williamson (1987, 1989). To those who might try to resist this recommendation on the grounds that an existence predicate governed by (W) does not quite match up to our concept of existence I say (i) Are you sure that we have one such concept? and (ii) In any case, so what? What is so special about our concept of existence that we should risk a lifetime of equivocation by continuing to employ it? To those who might try to resist this recommendation by contending that (W) has the absurd consequence that Pegasus exists, I would say first that (W) only applies when ϕτ is true, and, hence, when the term τ refers, and second that I won’t say anything further until he or she gets back to me with demonstrations that “Pegasus” refers to Pegasus, that e.g. “Pegasus is white” expresses a truth, and that “Pegasus is white” is of subject-predicate form.

 
41

Predicate actualism also follows from the combination of two doctrines that might be found attractive. They are (i) (necessarily) everything is such that, necessarily if it satisfies predicates then it has properties, and (ii) (necessarily) everything is such that necessarily, if it has properties then it exists (i.e. “serious” actualism). For a recent defence of predicate actualism along these lines see Stephanou (2007).

 
42

In my usage, p strictly implies q iff it is metaphysically necessary that if p is true then q is true.

 
43

Prior rejects (M) for the same reason that he rejects (T)(i) above. He maintains that for no contingently existing object is there a proposition that is singular with respect to it and metaphysically necessarily true on the grounds that had such an object not existed no proposition concerning it—not even the proposition that it does not exist—would have been true.

 
44

Williamson (1999, 2000, 2002).

 
45

This predicate actualist variant of the doctrine that some existing object is such as to exist contingently is not only non-orthodox. Somewhat confusingly, it is also non-actualist (i.e. in that it holds that it is metaphysically possible for there to be objects that do not exist)! It differs in both respects from the predicate actualist version of the doctrine that is advocated by Stephanou (2007). Of course, I take Stephanou’s variant to be refuted by its failure to accommodate (N). The possibility of combining the position standardly known as “predicate actualism” with a rejection of the position standardly known as “actualism” reflects the inadequacy of standard terminology in this area.

 
46

See, for example, Salmon (1987).

 
47

See Fine (2005, pp. 195–199) for a flavour of the rationale for scepticism about this matter. It should be noted that following Plantinga, Fine’s discussion concerns property abstracts (i.e. which are singular terms), not the predicate abstracts that have been my focus. Mutatis mutandis, however, similar considerations to those Fine raises in favour of an unrestricted principle of property abstraction, namely ‘Necessarily, for any x it is necessary that x has (λy)(¬Ay) iff Ax’ apply in favour of an unrestricted principle of predicate abstraction, namely ‘Necessarily, for any x it is necessary that (λy)(¬Ay)x iff Ax.’

 
48

Williamson (2000).

 

Acknowledgments

I am extremely grateful to Kit Fine, Stefano Predelli, Luis Robledo, Tim Williamson, and an anonymous referee for very helpful and insightful comments on an earlier draft of this paper.

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