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.
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Notes
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.
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.
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.
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.
In my usage, the sentences include open sentences as well as closed sentences.
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.
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.
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.
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).
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 propositional function 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.
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.
Prior (1971, p. 149).
Strictly speaking the “domain” here is the domain of quantification (see Footnote 12).
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.
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.
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.
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.)
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).
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.)
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.
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.
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.
Stalnaker (1977, p. 330). The underlining is mine.
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).
See Sect. I.i above.
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.
Stalnaker (1977, p. 330).
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.
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.
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 the truths 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.
Cf. Fine (2005, p. 200).
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.
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).
In my usage, p strictly implies q iff it is metaphysically necessary that if p is true then q is true.
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.
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.
See, for example, Salmon (1987).
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.’
Williamson (2000).
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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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Percival, P. Predicate abstraction, the limits of quantification, and the modality of existence. Philos Stud 156, 389–416 (2011). https://doi.org/10.1007/s11098-010-9609-x
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DOI: https://doi.org/10.1007/s11098-010-9609-x