singular terms revisited

Synthese manuscript No.(will be inserted by the editor)

Singular Terms Revisited

Robert Schwartzkopff

Received: date / Accepted: date

Abstract Neo-Fregeans take their argument for arithmetical realism to de-pend on the availability of certain, so-called broadly syntactic tests for whethera given expression functions as a singular term. The broadly syntactic testsproposed in the neo-Fregean tradition are the so-called inferential test andthe Aristotelian test. If these tests are to subserve the neo-Fregean argument,they must be at least adequate, in the sense of correctly classifying paradigmcases of singular terms and non-singular terms. In this paper, I pursue twomain goals. On the one hand, I show that the tests’ current state-of-the-artformulations are inadequate and, hence, cannot subserve the neo-Fregean ar-gument. On the other hand, I propose revisions that are adequate and, hence,can subserve this argument.

Keywords Singular Terms ¨ Neo-Fregeanism ¨ Philosophy of Mathematics

1 Introduction

According to neo-Fregeans, the fact that an expression α functions as a sin-gular term in true sentences of a certain kind guarantees that α dischargesits semantic function—i.e. that of denoting some object a—and consequently

For the financial support during the early and middle stages of this paper’s development, Iwish to express my gratitude to the Arts and Humanities Research Council and the RoyalInstitute of Philosophy.

Robert SchwartzkopffResearch Group Nominalizations: Philosophical and Linguistical Perspectives, Departmentof Philosophy, University of Hamburg, Von-Melle-Park 6, 20146 Hamburg, GermanyTel.: +49 (40) 428 38 - 9255Fax: +49 (40) 428 38 - 3983E-mail: [email protected]

2 Short form of author list

guarantees that a exists.1 The neo-Fregean argument for arithmetical realismthen is based on three assumptions.2 First, that the language of pure arith-metic provides for sentences that contain expressions such as, for instance,“simple numerals and various complex numerical expressions”3 that functionas ‘arithmetical singular terms’, i.e. as expressions with the semantic functionto denote certain mathematical objects, viz. natural numbers. Second, that atleast some of these sentences are of the prerequisite kind, i.e. that their truthguarantees that the arithmetical singular terms they contain discharge theirsemantic function. Third, that at least some sentences of the latter kind are,in fact, true. If so, the arithmetical singular terms contained in the pertinentsentences are guaranteed to denote. Hence, natural numbers, i.e. the objectsthey aim to denote, are guaranteed to exist.

In the context of this argument, the broadly syntactic neo-Fregean testsfor singular-termhood are usually called upon to lend support to the first ofthe three assumptions above.4 Moreover, it is clear why these tests have to bebroadly syntactic. Since the conclusion of the neo-Fregean argument assertsthat arithmetical singular terms denote (and denote objects), the argumentwould be question-begging if the semantic function of an expression as an(arithmetical) singular term had to be determined by, as it were, first findingout whether it denotes (and whether its denotation is an object).

In order to be able to subserve the neo-Fregean argument, any proposedbroadly syntactic tests for singular-termhood need to be adequate in the sensethat they

. . . admit as singular terms all (uses of) expressions whose status assuch may in the present context be taken as (relatively) unproblematic,and which . . . exclude (uses of) expressions which clearly do not playthat role. . . . [I]n other words, we are approaching the task, not emptyhanded, but equipped with a range of clear positive and negative caseswhich any satisfactory general tests . . . should fit . . ..5

This is because, as Hale puts it, only

. . . general tests which meet this adequacy condition may then be ap-plied to give a ruling in less clear, or otherwise problematic cases.6

The broadly syntactic tests proposed in the neo-Fregean tradition—i.e.the so-called inferential test and the Aristotelian test—trace back to MichaelDummett’s Frege: Philosophy of Language7 and have since been adopted, de-

1 See, e.g., Hale (1984, 225f.) and Hale and Wright (2001, 8). The main contributions tothe neo-Fregean program are collected in Hale and Wright (2001). See also Wright (1983);Hale (1987). For an overview see MacBride (2003).

2 See, e.g., Hale (1996, 32), Hale (1994, 48), and Hale and Wright (2001, 8).3 Hale (1996, 31).4 However, see §5 pp. 17.5 Hale (1996, 32).6 Hale (1996, 32).7 See Dummett (1973, 54-80). A precursor to the Aristotelian test can be found in Straw-

son (1971, §5). Building on Strawson, Kunne (1983, 24-27) defends a modally strengthenedversion of the Aristotelian test.

Singular Terms Revisited 3

fended, and refined by, most notably, Bob Hale and Crispin Wright.8 The tests’current state-of-the-art formulations are given in Hale’s recent book NecessaryBeings.9

Hale’s formulations of the two tests are the only extant worked-out candi-dates for broadly syntactic tests that could subserve the neo-Fregean argumentfor arithmetical realism. Thus, the argument would suffer a rather devastatingblow if these tests proved to be inadequate. The aim of this paper is two-fold.First, I show that the tests’ current state-of-the-art formulations are inade-quate. Second, I soften this blow by showing how the tests can be revised soas to restore their adequacy.

The plan is as follows. In §2 and §3, I present Hale’s latest formulations of,respectively, the inferential test and the Aristotelian test. In §4, I demonstratethat Hale’s Aristotelian test is inadequate and show how a modest revision canrestore its adequacy. In §5, I establish the inadequacy of Hale’s Inferential Testand formulate an adequate version that constitutes a more radical departurefrom Hale’s formulation. In §6, I conclude and look ahead.

2 Hale’s Inferential Test

The inferential test is motivated by the observation that paradigm singularterms display a different inferential behaviour than (some) paradigm non-singular terms (most notably, quantifiers): the former validate certain inferenceforms that are invalidated by the latter. Following Dummett, this differencein inferential behaviour can be illustrated as follows.

First, the following inference, which involves the singular term ‘Plato’, isvalid:10

1 Plato is wise.6 Something is such that it is wise.

In contrast, the corresponding inference with the non-existential quantifier ‘noman’ in place of ‘Plato’ is invalid:

2 No man is wise.6 Something is such that it is wise.

Second, the following ‘Plato’-involving inference is valid:

3 Plato is wise and Plato is snub-nosed.6 Something is such that (it is wise and it is snub-nosed).

8 See, e.g., Wright (1983, 53-64), Hale (1987, 15-44), Hale (1994, 1996), and Hale andWright (2003, 2009).

9 See Hale (2013, pp. 40-46).10 Note that it is generally agreed upon that, in the context of the inferential test, the

notion of validity can be understood neither model-theoretically nor in terms of uniformsubstitution or derivability. Rather, it has to be understood in terms of strict implication:the inference from ϕ to ψ is valid iff it is not possible that (ϕ is true and it is not the casethat ψ is true); see, e.g., Rumfitt (2003, 203).


In contrast, the corresponding inference with the existential quantifier ‘someman’ in place of ‘Plato’ is invalid:

4 Some man is wise and some man is snub-nosed.6 Something is such that (it is wise and it is snub-nosed).

Finally, the following ‘Plato’-involving inference is valid:

5 Plato is such that (he is wise or he is snub-nosed).6 Plato is wise or Plato is snub-nosed.

In contrast, the corresponding inference with the universal quantifier ‘everyman’ in place of ‘Plato’ is invalid:

6 Every man is such that (he is wise or he is snub-nosed).6 Every man is wise or every man is snub-nosed.

The inferential test is based on the assumption that the above inferen-tial differences can be exploited to state a general test for singular-termhood.However, as witnessed by the vivid and ongoing debate, formulating this testin an adequate way is no trivial task.11 The intricacies of this task are bestillustrated against the background of Hale’s most recent formulation of theinferential test.

Before I present Hale’s test, four preliminary remarks. First, Hale’s formu-lation of the inferential test is restricted to what he calls ‘substantival expres-sions’. Paradigm cases of substantival expressions are expressions headed byan overt determiner such as ‘no/some/every/the/this man’ and expressionslike proper names that can stand in their place. Second, Hale formulates thetest as a test for whether a sub-sentential English expression functions as asingular term in an English sentence.12 However, note that Hale intends thetest to be further relativised to uses of the relevant sentences.13 Third, in mypresentation of Hale’s formulation, I will use the following notation (i) If S is asentence containing an expression α, S will be referred to using ‘ăα, βą’.14 In‘ăα, βą’, β refers to the result of deleting (one occurrence of) α from S. Forinstance, if S is ‘Plato is wise’, α refers to ‘Plato’ and β refers to ‘is wise’, thenS is referred to by ‘ă‘Plato’,‘is wise’ą’. (ii) The result of replacing α in S withsome appropriate pro-noun will be referred to using ‘ă it, βą’. For instance,‘He is wise’—i.e. the result of replacing ‘Plato’ in ‘Plato is wise’ with the pro-noun ‘he’—will be referred to by ‘ă it,‘is wise’ą’. (iii) I use ‘^’ and ‘_’ toabbreviate the English sentential connectives ‘and’ and ‘or’. (iv) I use ‘x’ and‘y’ as devices of quasi-quotation.15 Thus, if ‘ăα, β1ą’ refers to ‘Plato is wise’

11 For relevant discussion see, e.g., Wetzel (1990) vs. Hale (1994), Rumfitt (2003) vs. Haleand Wright (2003), MacFarlane (2009) vs. Hale and Wright (2009), and most recently Hale(2013).12 The language-relativisation will henceforth be left tacit.13 This use-relativisation is made explicit in Hale (1987, 18-22) but has been left tacit

since.14 I use ‘α’ and ‘β’ as metalinguistic variables ranging over sub-sentential expressions (of

English).15 See, e.g., Quine (1981, §6).


and ‘ă α, β2 ą’ refers to ‘Plato is snub-nosed’, then xă α, β1 ą ^ ă α, β2 ąy

refers to ‘Plato is wise and Plato is snub-nosed’.With these preliminaries out of the way, Hale’s formulation of the inferen-

tial test can be captured as follows:16

Hale’s Inferential Test A substantival expression α functions as a singular term in a sen-tence S1 = ăα, β1ą iff

H.1 for S1, the following inference is valid:

from ăα, β1ą to xSomething is such that ă it, β1ąy,

H.2 for S1 and some sentence S2 = ăα, β2ą, the following inference is valid:

from xăα, β1ą ^ ăα, β2ąy to xSomething is such that ă it, β1ą ^ ă it, β2ąy,

H.3 for S1 and some sentence S2 = ă α, β2ą, the following inference is valid:

from xα is such that ă it, β1ą _ ă it, β2ąy to x ăα, β1ą or ăα, β2ąy,

where:

Constraint 1.1 S2 neither strictly implies nor is strictly implied by S1,

Constraint 1.2 the displayed occurrences of β1 and β2 in (H.2) and (H.3) are essential,

Constraint 1.3 the displayed occurrence of α in S2 in (H.2) itself meets condition(H.3),

Constraint 1.4 neither the conclusions of the inferences displayed in (H.1) nor theconclusions of the inferences in (H.2) are such that a point may be reached wherea well-formed question for further specification may be rejected as not requiring ananswer,17

Constraint 1.5 if valid, the validity of the inferences in (H.1) - (H.3) can be immedi-ately recognised by any suitably competent speaker.

The following two aspects of Hale’s Inferential Test demand explanation.First, why are clauses (H.2) and (H.3) formulated in the way they are? Inparticular, why do they require that the relevant inferences be valid for S1

and only some rather than every α-containing sentence S2? Second, what arethe rationales that underlie the test’s five constraints? As we will see presently,each of these constraints relates to the task of formulating the inferential testin an adequate way, i.e. in a way that correctly classifies paradigm cases ofsingular terms and non-singular terms.

As for the ‘existential’ formulation of clause (H.2) and clause (H.3), Haleargues that its adoption is mandated by the fact that the alternative ‘univer-

16 Apart from the different notation, the below differs from Hale’s exact wording in fournegligible respects. First, Hale formulates clause (H.2) in terms of the inference from S1 andS2 to the pertinent conclusion (rather than from their conjunction). Second, I formulatethe premiss of the inference in clause (H.3) as xα is such that pă it, β1 ą _ ă it, β2 ąqy,whereas Hale formulates it as xIt is true of α that pă it, β1 ą _ ă it, β2 ąqy. Third, forpresentational purposes, I have inverted Hale’s order of the first four constraints. Fourth,rather than stating (Constraint 1.5) separately, Hale incorporates it directly into clauses(H.1) - (H.3).17 Some foreshadowing: (Constraint 1.4) is meant to distinguish first-order uses of ‘some-

thing’ as in ‘Something is wise’ from higher-order uses as in ‘Plato is something’ when validlyinferred from, say, ‘Plato is wise’. For details, see p. 7f.


sal’ formulation would render the test too strong, in the sense of excludingexpressions that it should not exclude.18 This problem arises from expressionsthat, within a given sentence, occupy what Quine called ‘not purely referentialpositions’.19 Thus, consider ‘Plato’ in the sentences S1 = ‘Plato is wise’ andS2 = “Plato’ has five letters’.20 The following inference is not valid:

7 Plato is wise and ‘Plato’ has five letters.6 Something is such that (it is wise and ‘it’ has five letters).21

Thus, a universal formulation of clause (H.2) would have incorrectly ruledthat ‘Plato’ in S1 does not function as a singular term. This is because forS1 and some ‘Plato’-containing sentence S2—viz. “Plato’ has five letters’—theinference in clause (H.2) is not valid. Similarly for clause (H.3). Consequently,formulating the test in terms of the universal alternatives of clause (H.2) andclause (H.3) would have rendered the test too strong and, hence, inadequate.Now, the existentially formulated clauses (H.2) and (H.3) may not render thetest too strong. However, if not for (Constraint 1.1) and (Constraint 1.2),they would have rendered the test too weak—in the sense of not excludingexpressions it should exclude—and, hence, once more inadequate.22

As for (Constraint 1.1), consider ‘some man’ in the sentences S1 = ‘Someman is wise’ and S2 = ‘Some man is wise and snub-nosed’. The followinginference is valid:

8 Some man is wise and some man is wise and snub-nosed.6 Something is such that (it is wise and it is wise and snub-nosed).

The validity of (8) suffices to show that, without (Constraint 1.1), Hale’sInferential Test would have been too weak. As an existential quantifier ‘someman’ in S1 should have failed clause (H.2). But without (Constraint 1.1) itwould not have because, for S1 and some ‘some man’-containing sentence S2—viz. ‘Some man is wise and snub-nosed’—the inference in clause (H.2) is valid.As Hale observes, the notable difference between the above invalid inference(4) and the valid inference (8) is the following. In (4), the sentence S2 thatacts as the ‘side-premiss’ does not strictly imply the main premiss S1, whereasin (8) it does. Thus, Hale proposes to forestall potential counterexamples inthe style of (8) by means of (Constraint 1.1).

As for (Constraint 1.2), consider ‘everything’ in S1 = ‘Everything is wise’and S2 = ‘Everything is such that snow is white’.23 The following inference isvalid:

18 See Hale (1987, 18ff.), Hale (1994, 58f, 62).19 See Quine (1960, §30).20 Other potentially problematic cases are ones in which the test expression is embedded

in a modal or epistemic context.21 Note, though, that arguably (7) is not invalid either, at least not if its conclusion is

meaningless. Plausibly, lack of meaning entails lack of truth-evaluable content and onlyarguments with truth-evaluable premisses and conclusions can be valid or invalid.22 This point has been made by Wetzel (1990) against Hale’s (1987, 15-21) unconstrained

existential formulation of the test. In response, Hale (1994) added (Constraint 1.1) and(Constraint 1.2). Similarly for (Constraint 1.3).23 This example is somewhat artificial. However, more natural, if also more complex ex-

amples are possible; see Wetzel (1990, 246). For discussion, see Hale (1994, 63-67).


9 Everything is such that (it is wise or it is such that snow is white).6 Everything is wise or everything is such that snow is white.

The validity of (9) suffices to show that, without (Constraint 1.2), Hale’sInferential Test would have been too weak. As a universal quantifier ‘everyman’ in S1 should have failed clause (H.3). But without (Constraint 1.2) itwould not have because, for S1 and some ‘everything’-containing sentence S2—viz. ‘Everything is such that snow is white’—the inference in (H.3) is valid.As Hale observes, the notable difference between the above invalid inference(6) and the valid inference (9) is the following. In (6), the occurrence of theβ-constituent—viz. ‘is snub-nosed’—of the side-premiss is, whereas in (9) theoccurrence of the β-constituent—viz. ‘is such that snow is white’—in the side-premiss is not essential in roughly the following sense: the occurrence of anexpression β in a sentence S is essential only if there is no sentence S* that islogically equivalent to S but does not contain β.24 In this sense, the occurrenceof ‘is such that snow is white’ in S2 is inessential because it is logically equiva-lent with ‘Snow is white’, a sentence in which ‘is such that snow is white’ doesnot occur. Consequently, Hale proposes to forestall potential counterexamplesin the style of (9) by means of (Constraint 1.2).

To illustrate the need for (Constraint 1.3), consider ‘a sheep’ in S1 = ‘Johnowns a sheep’. As an existential quantifier ‘a sheep’ should fail clause (H.2).However, on the face of it, it does not. This is because there is a sentenceS2—viz. ‘A sheep is a ruminant’—such that the pertinent inference is valid:

10 John owns a sheep and a sheep is a ruminant.6 A sheep is such that (John owns it and it is a ruminant).

The problem is that S2 can be read in two ways— existentially: Some sheep isa ruminant, and universally: Every sheep is a ruminant—and that the latterreading does, whereas the former does not validate (10). Hale thus proposes(Constraint 1.3) to forestall such cases by requiring that ‘a sheep’ in S2 alsosatisfy clause (H.3), the clause typically not satisfied by universal quantifiers,thus disbarring S2 from acting as the side-premiss in the pertinent applicationof the test.

The need for (Constraint 1.4) arises from the fact that Hale formulates theinferential test as giving individually necessary and jointly sufficient condi-tions for the singular-termhood of substantival expressions. The problem thisconstraint forestalls can be illustrated as follows. Consider ‘a philosopher’ inS1 = ‘Plato is a philosopher’. The following inference is no less valid that theabove (1):

11 Plato is a philosopher.6 Something is such that Plato is it.

Moreover, the same holds, mutatis mutandis, for the corresponding inferencesfrom and to, respectively, the conjunction and disjunction of ‘Plato is a philoso-pher’ and, for instance, S2 = ‘Socrates is a philosopher’. Thus, it seems that

24 See Hale (1994, 66). I say ‘roughly’ because Hale’s official characterisation is more com-plex, a complexity we can safely ignore in the present context.


although ‘a philosopher’ in S1 does not function as a singular term, it is ableto sustain all relevant inferences. Commonly, this problem is diagnosed as aris-ing from the fact that ‘something’ can be used to express different kinds ofgenerality: in (1) it expresses first-order generality, whilst in (11) it expressessecond-order generality. (Constraint 1.4) is meant ensure that, in an appli-cation of the test, ‘something’ is used to express first-order generality. Foras Hale argues, uses of ‘something’ in which it respectively subserves first-and second-order existential generalisation into the position of a substantivalexpression, contrast in the way enshrined in (Constraint 1.4).25 Thus, Haleproposes to preempt potential counterexamples in the style of (11) by meansof (Constraint 1.4).

Finally, (Constraint 1.5) is meant to ward off what we may call the ‘Gen-eralised Rumfitt Objection’. This objection arises from the fact that there aresubstantival non-singular term—which following Hale we might call ‘individ-ual quantifiers’26—that validate all relevant inferences but cannot be cordonedoff by the previous constraints.27 Individual quantifiers are exemplified by ex-pressions of the form ‘Some F ’ and ‘Something that is F ’ in which ‘F ’ is apredicate that is necessarily satisfied by a unique object if it is satisfied at all.To illustrate this problem, consider ‘Something that is (identical with) Plato’in S1 = ‘Something that is (identical with) Plato is wise’ and S2 = ‘Somethingthat is (identical with) Plato is snub-nosed’.28 Notably, S1 satisfies clause (H.1)of Hale’s Inferential Test. And, together with S2, S1 also satisfies clauses (H.2)and (H.3). For instance, the following inference is as valid as the above (3):

12 Something that is Plato is wise and something that is Plato is snub-nosed.6 Something is such that it is wise and it is snub-nosed.

However, although inferences (3) and (12) are inferentially on a par, Haleargues that they differ, as it were, epistemically. That is, the validity of (12)

25 This idea traces back to Dummett (1973, 61, 67ff.). For further discussion, see e.g.Wright (1983, 58, 61ff.), Hale (1987, 16f.), Hale (1994, 37ff.) and Hale (1994, 53-58). Toget an impression of the intended contrast, compare the following scenarios. In the case of(1), you assert ‘Something is such that it is wise’, someone enquires What is such that it iswise?, and you answer Plato. In the case of (11), you assert ‘Something is such that Plato isit’, someone enquires What is such that Plato is it?, and you answer a philosopher. In bothcases, your interlocutor might press on and enquire Which Plato? and Which philosopher?.In the second case, you would be within your rights to reject her request because it does notrequire an answer. In the first case, though, you might answer the author of ‘Euthyphro’.But if pressed further—Which author of ‘Euthyphro’?—you would be within your rightsto reject her request because you have already answered it: I just told you, the author of‘Euthyphro’.26 See Hale (1994, 70).27 Hale (2013, 44) credits Rumfitt (2003, 203ff.) as the originator of this objection and

one Paul McCallion with the discovery that Rumfitt’s objection can be generalised to coverconsiderably more cases than the one discussed by Rumfitt himself. This attribution issomewhat generous. For one, Rumfitt (2003, 204n21) is aware that his objection generalises.And for another, Hale (1994, 70f.) had already observed that individual quantifiers pose aproblem for the formulation of the inferential test he advocated at the time.28 This example is in the style that Hale ascribes to McCallion. Rumfitt’s example of an

individual quantifier is ‘some even prime number’.


cannot, whereas that of (3) can be “immediately recognised as valid by anycompetent speaker—that is, recognized as valid without the need of interme-diate reasoning”. 29 For according to Hale, to validly draw inference (12)

. . . one must reason somewhat as follows: Suppose [Something thatis Plato is wise and something that is Plato is snub-nosed]. Let s beanything [that is Plato]. Then [s is wise]. Likewise [s is snub-nosed].But then [Something is such that it is wise and it is snub-nosed].30

Trading on this epistemic difference between genuine singular terms and indi-vidual quantifiers, Hale proposes to preempt the Generalised Rumfitt Objec-tion by means of (Constraint 1.5).

This concludes my presentation of Hale’s Inferential Test, the current state-of-the-art formulation of the inferential test. As noted above, in order to sub-serve the neo-Fregean argument this test has to be at least adequate. However,as I will argue in §5, this formulation is inadequate. First, though, let us takea look at Hale’s formulation of the Aristotelian test.

3 Hale’s Aristotelian Test

Assuming for the moment that Hale’s Inferential Test is adequate, it wouldbe capable of correctly classifying the paradigm cases of singular and non-singular terms within the class of substantival expressions. However, since it isrestricted to substantival expressions, it does not apply to and, hence, cannotexclude non-substantival expressions such as, for instance, ‘is wise’ in ‘Platois wise’. The exclusion of such expressions is the main task of the Aristoteliantest.

The Aristotelian test is commonly introduced as a linguistic counterpartof Aristotle’s dictum that “whereas any quality has a contrary, a (primary)substance has not”.31 Thus, Hale introduces this test as appealing to the ideathat

. . . whereas for any given predicate there is always a contradictory pred-icate, applying to a given object just in case the original predicate failsto apply . . ., we do not have for any given singular term a ‘contradic-tory’ singular term such that the statement incorporating the one istrue if and only if the corresponding statement incorporating the otheris not true.32

Building on this idea and using the notation introduced above, Hale’s formu-lation of the Aristotelian test can be captured as follows:33

29 Hale (2013, 45).30 Hale (2013, 45n83).31 Hale (1994, 52).32 Hale (1994, 52).33 See, e.g., Hale (1994, 47), Hale (1996, 52), Hale (2013, 42). Like Hale, I use ‘ ’ and ‘Ø’

to abbreviate the natural language connectives ‘it is not the case that’ and ‘if and only if’,respectively.


Hale’s Aristotelian Test An expression α functions as a singular term in a sentence S1 =ăα, β1ą only if

H.4 there is no expression α1 such that for all expressions β2:

xăα1, β2ąØ ăα, β2ąy is true,

where:

Constraint 2.1 α1 is syntactically congruous with α in S1,

Constraint 2.2 (i) β2 is syntactically congruous with β1 in S1, (ii) β2 in S2 = ăα, β2ąneither is nor contains an expression that fails Hale’s Inferential Test.34

The rationales that underlie (Constraint 2.1) and the first part of (Con-straint 2.2) are straightforward. Together they ensure that the range of thevariables ‘α1’ and ‘β2’ is restricted in such a way that both α and all valuesof α1 can combine with any value of β2 to yield well-formed sentences S2 andS12. Consequently, Hale’s formulation effectively boils down to the following

requirement: an expression α only functions as a singular term in a sentenceS1 if there is no expression α1 that can act as what we may call α’s complementexpression with respect to all pertinent sentences S2, i.e. no expression α1 thatcan replace α in every relevant sentence S2 such that, in each instance, theresult of this substitution is a sentence S1

2 that is true just in case the negationof the original sentence S2 is.

The rationale that underlies the second part of (Constraint 2.2) can beillustrated as follows. Consider ‘is wise’ in S1 = ‘Plato is wise’ and supposethe test had been formulated without this part. Now, take every sentenceS2 obtained from S1 by replacing ‘Plato’ with some syntactically congruousexpression β2. That is, amongst other sentences, take sentences like ‘Plato iswise’ itself as well as sentences like ‘Some man is wise’. Under the presentsupposition, the test would classify ‘is wise’ as a non-singular term only ifthere was an expression α1, that could replace ‘is wise’ in all of the abovesentences with the desired effect. As witnessed by the truth of the following,there is indeed such an expression—viz. ‘is not wise’—for the case of ‘Plato iswise’:

13 Plato is not wise iff it is not the case that Plato is wise.

However, the falsity of the following shows that replacing ‘is wise’ in ‘Someman is wise’ with ‘is not wise’ does not have the desired effect:

14 Some man is not wise iff it is not the case that some man is wise.

Thus, without the second part of (Constraint 2.2), Hale’s Aristotelian Testwould have been unable to classify ‘is wise’ in S1 as a non-singular term. The

34 The formulation of sub-clause (ii) slightly deviates from how Hale would have put it,i.e. from ‘β2 in S2 = ă α, β2 ą does not fail Hale’s Inferential Test ’. My formulation isan improvement over Hale’s because Hale’s Aristotelian Test is supposed to also renderverdicts for adjectives like ‘wise’ in ‘Some man is wise’. However, deleting ‘wise’ from thissentence yields ‘Some man is’. Since this is not a substantival expression, it can and, hence,does not fail Hale’s Inferential Test because the test does not apply. Nevertheless, ‘Someman is’ contains the substantival ‘some man’ that fails this test.


constraint’s second part preempts this problem because, although ‘some man’is syntactically congruous with ‘Plato’ in S1 , ‘Some man is wise’ is a sentencewith respect to which ‘some man’ fails Hale’s Inferential Test.35

This concludes my presentation of Hale’s Aristotelian Test, the currentstate-of-the-art formulation of the Aristotelian test. As with Hale’s InferentialTest, I will argue below that it is inadequate. However, before I do so, twofurther remarks.

The first remark concerns the relation between Hale’s Aristotelian Testand individual quantifiers, which Hale’s Inferential Test excludes by meansof (Constrain 1.5).36 On the one hand, note that the former test would nothave been compromised even if the latter test had been formulated withoutthis constraint—i.e. even if it would not exclude individual quantifiers. Forinstance, the following is as true as the above (13):

15 Something that is Plato is not wise iff it is not the case that something that isPlato is wise.

On the other hand, as witnessed by the truth of the following, it appearsthat individual quantifiers need not be excluded via constraint (Constrain1.5) of Hale’s Inferential Test but can also be excluded by means of Hale’sAristotelian Test :

16 Nothing that is Plato is wise iff it is not the case that something that is Platois wise.

In Hale and Wright (2003), these two facts provided the basis of their dismissalof the Rumfitt Objection discussed in §2. The reason for the change of heartin Hale (2013) will be discussed in §4.

As for the second remark, it bears highlighting that, unlike Hale’s Infer-ential Test, the application of Hale’s Aristotelian Test is not restricted toexpressions of a certain syntactic type, substantival or otherwise. Thus, thetest also applies to genuine singular terms such as ‘Plato’ in ‘Plato is wise’.

35 Note that a similar problem arises not only with respect to expressions such as ‘is wise’but also with respect to quantifiers; for discussion, see Hale (1996, 43-46). On the one hand,replacing, say, ‘some man’ in ‘some man is wise’ with ‘no man’ has the desired effect:

i No man is wise iff it is not the case that some man is wise.

But on the other hand, replacing ‘some man’ in ‘every woman loves some man’ with ‘noman’ may fail to have the desired effect:

ii Every woman loves no man iff it is not the case that every woman loves some man.

The problem is that ‘Every woman loves some man’ and ‘Every woman loves no man’ arescope-ambiguous and that (ii) will be true if ‘some man’ and ‘no man’ take wide scope over‘every woman’ but false if ‘every woman’ takes wide scope over one or both of ‘some man’and ‘no man’, respectively.(Constraint 2.2) takes care of this problem as well. For unlike ‘is wise’—i.e. the β2-constituentof ‘Some man is wise’—‘Every woman loves’—i.e. the β2-constituent of ‘Every woman lovessome man’—violates this constraint. For like ‘some man’ in ‘Some man is wise’, ‘everywoman’ in ‘Every woman loves some man’ fails Hale’s Inferential Test. Thanks to an anony-mous referee for this journal for pointing out that this point needs to be made explicit.36 See Hale (1994, 71), Hale and Wright (2003, 257), Rumfitt (2003, 204).


Consequently, genuine singular terms are required to pass Hale’s AristotelianTest, lest its results contradict those of Hale’s Inferential Test.37

In the last two sections, I have presented Hale’s Inferential Test and Hale’sAristotelian Test, i.e. Hale’s current state-of-the-art formulations of the in-ferential test and the Aristotelian test, the two broadly syntactic tests forsingular-termhood proposed in the neo-Fregean tradition. According to neo-Fregeans, these tests play a crucial role in their argument for arithmeticalrealism. Neo-Fregeans also admit that the tests will only be able to play thisrole if they are adequate, in the sense of correctly classifying paradigm casesof singular terms and non-singular terms. Over the course of the next two sec-tions, I will show that Hale’s formulations of these tests are both inadequate.Consequently, Hale’s formulations cannot subserve the neo-Fregean argument.However, I will also show that both tests can be formulated in a way thatavoids the inadequacies of Hale’s formulations. Reverting the presentationalorder, I begin with discussing Hale’s Aristotelian Test.

4 Revising the Aristotelian Test

In this section, I show that Hale’s Aristotelian Test is inadequate because itclassifies paradigm cases of singular terms as non-singular terms. Moreover, Ipropose a modest revision that removes this inadequacy.

4.1 The Stirton Objection

That Hale’s Aristotelian Test is inadequate is shown by what we may callthe ‘Stirton Objection’, which can be articulated as follows.38 In §3, we sawthat Hale’s Aristotelian Test excludes individual quantifiers such as ‘some-thing that is Plato’ because for each such quantifier there is an anti-individualquantifier—‘nothing that is Plato’ in the case at hand—that can act as itscomplement expression. That is, where β2 is restricted to expressions thatobey the test’s (Constraint 2.2) we have:

i @β2 xă ‘nothing that is Plato’, β2ąØ ă ‘something that is Plato’, β2ąy is true.

From (i), we obtain the following result, which is crucial for the exclusion of‘something that is Plato’ by Hale’s Aristotelian Test :

ii Dα1@β2 xăα1, β2ąØ ă ‘something that is Plato’, β2ąy is true,

37 Hale (2013, 44) is aware of this requirement. However, as we will see presently, contraryto his intentions, Hale’s Aristotelian Test is formulated in a way that renders singular termsincapable of passing it.38 The following argument is in the style of Hale (2013, 44), who ascribes it to McCallion.

However, the gist of the following objection is already present in Stirton (2000, 206), whomHale does not credit.


where α1 is restricted to expressions that obey the test’s (Constraint 2.1).However, it is also the case that:

iii @β2 xă ‘something that is Plato’, β2ąØă ‘Plato’, β2ąy is true.

Finally, from (ii) and (iii) it follows that:

iv Dα1@β2 xăα1, β2ąØ ă ‘Plato’, β2ąy is true.

And (iv) entails that, according to Hale’s Aristotelian Test, ‘Plato’ does notfunction as a singular term in any sentence ă ‘Plato’, β2ą such as, for instance,‘Plato is wise’. But since ‘Plato’ does so function, this shows that Hale’s testis inadequate.

Now, Hale is aware of the Stirton Objection. In fact, in Hale (2013) he statesit himself. However, since Hale misdiagnoses its source, he fails to counter it.According to Hale, this objection arises from the (generalised) Rumfitt Objec-tion concerning the inability of his earlier formulation of the inferential test toexclude individual quantifiers. Thus, he contends that the Stirton Objectionwould be preempted by a formulation of the inferential test that excludes in-dividual quantifiers. It is because of this contention that Hale choses to amendhis 1994 formulation by imposing the additional (Constraint 1.5). However,since the two objections are largely independent, Hale’s amendment misses itsmark.

The Stirton Objection arises due to a combination of two factors. First,because any anti-individual quantifier can act as the complement expression forboth the corresponding individual quantifier and the corresponding singularterm. Second, because the formulation of Hale’s Aristotelian Test allows thatanti-individual quantifiers are in the range of α1 if α is a singular term evenif individual quantifiers are excluded by Hale’s Inferential Test. After all, theonly restriction on the values of α1 is imposed by (Constraint 2.1), whichrequires that they be syntactically congruous with α as it occurs in S1.39

But this constraint is evidently satisfied.40 Thus, Hale’s attempt to counter

39 In Fregean terminology, the source of the Stirton Objection can be described as follows.In order to prevent ‘anti-individual quantifiers’ from being permissible values of α1, it needsto be ensured that they are expressions—i.e. singular terms—that saturate ‘is wise’ ratherthan expressions that ‘is wise’ saturates. However, the requirement imposed by (Constraint2.1) that permissible α1s be syntactically congruous with α is unable to effect this restriction.40 An anonymous referee for this journal has expressed the worry that this is not strictly

speaking true. The worry was that, in a certain sense, singular terms and (individual) quan-tifiers are not syntactically congruent because, syntactically speaking, singular terms cando things quantifiers cannot. For instance, given Frege’s (1893, §§29-32) syntactic formationrules, the singular term ‘Juliet’ can combine with the binary (first-level) predicate ‘loves’to form the unary predicate ‘loves Juliet’. In contrast, Frege’s formation rules forbid thatthe quantifier ‘some girl’ combines with ‘loves’ to form the unary predicate ‘loves somegirl’. This, so the referee’s worry, shows that singular terms and quantifiers cannot, strictlyspeaking, be syntactically congruent. In response, let me make two remarks (and expressmy gratitude to the referee for pressing me on this).First, it can hardly be disputed—not that the referee did—that ‘Romeo loves some girl’—i.e. the result of substituting ‘some girl’ for ‘Juliet’ in ‘Romeo loves Juliet’ is a perfectlygrammatical English sentence. Thus, the phrase-structure rules of English allow for the


the Stirton Objection by preempting the Rumfitt Objection is bound to beinsufficient. Of course, the Rumfitt Objection and the fact that individualquantifiers could be excluded by Hale’s Aristotelian Test with the help of anti-individual quantifiers made the latter’s existence particularly vivid. But thisdoes not change the fact that the exclusion of the former by Hale’s InferentialTest does not suffice as a response to the Stirton Objection.

4.2 Minor Revsions

Now, the fact that Hale’s Inferential Test excludes individual quantifiers maynot suffice to counter the Stirton Objection, but as we will see presently,it can support an adequate solution. As we have seen, the objection arisesbecause, if the test expression α is a singular term, anti-individual quantifiersare permissible values of α1.41 Thus, an adequate solution of the problemmust consist in imposing some restriction on the values of α1 that rendersanti-individual quantifiers impermisssable and is also congenial to the test’sbroadly syntactic nature. Fortunately, such a suitable restriction is close tohand. For note that a genuine singular term like ‘Plato’ in ‘Plato is wise’ andthe corresponding anti-individual quantifier ‘Nothing that is Plato’ in ‘Nothingthat is Plato is wise’ behave differently vis-a-vis Hale’s Inferential Test. Unlikethe former, the latter fails this test.42

construction of ‘Romeo loves some girl’ from the name ‘Romeo’, the verb ‘loves’, and thequantifier ‘some girl’ in the same way as they allow for the construction of ‘Romeo lovesJuliet’ from ‘Romeo’, ‘loves’, and the name ‘Juliet’; for more on phrase-structure rules, seee.g. Poole (2002, ch. 2). This shows that the phrase-structure rules of English are morepermissive than Frege’s formation rules and that, consequently, there is a good sense inwhich the English expression ‘some girl’ is syntactically congruous with ‘Juliet’ in the Englishsentence ‘Romeo loves Juliet’. And since we are dealing with English sentences, rather thansentences of some formal language such as Frege’s, all the point I made in the main text seemsto require is that, with respect to the relevant sentences, singular terms and (individual)quantifiers are syntactically congruous in this sense.Second, as the referee correctly pointed out, the fact that English’s syntactic formationrules are permissive in this way engenders a problem once we turn to the task of semanticinterpretation. Roughly, the problem is that, unlike ‘Romeo loves Juliet’, ‘Romeo lovessome girl’ is not straightforwardly interpretable because ‘loves’ denotes a relation betweenobjects such as Romeo and Juliet, whereas the quantifier ‘some girl’ denotes, in Fregeanterminology, a (second-level) concept. This interpretability problem is a well-known staple ofcontemporary linguistics and linguists usually resolve it in one of two ways. Either by positinga type-shifting operation that can alter the denotation of a quantifiers so as to becomeinterpretable in situ. Or by positing different levels of a sentence’s syntactic representation—called ‘Surface Structure’ and ‘Logical Form’, respectively—such that (i) the Logical Formof ‘Romeo loves some girl’ is relevant for its semantic interpretation and (ii) on the level ofits Logical Form, ‘some girl’ occupies a syntactic position that allows it to be interpretedin its usual concept-denoting type, see e.g. Heim and Kratzer (1998, ch. 7). As far as I cansee, there is nothing that prevents proponents of the neo-Fregean tests from adopting thelinguists’ solution. Thus, to the extent that such interpretability problems appear to spellspell trouble for the neo-Fregean tests, it seems that appearances are deceptive.41 Hale (2013, 44) appears somewhat aware of this fact but fails to draw the consequence

advocated below.42 For instance, the inference in clause (H.1) from ‘Nothing that is Plato is wise’ to ‘Some-

thing is such that it is wise’ is invalid.


I thus propose that Hale’s Aristotelian Test be revised by replacing theconstraint that restricts the permissible values of α1, i.e. (Constraint 2.1), withthe following constraint to the effect that the permissible values of α1 behave,vis-a-vis Hale’s Inferential Test, in the same way as the test expression α itself:

Constraint 2.1* (i) α1 is syntactically congruous with α in S1, (ii) α1 in S12 = ă α1, β2ą

passes/fails/neither passes nor fails Hale’s Inferential Test just in case α1 in S2 does.

(Call the result of this revision ‘Hale’s Improved Aristotelian Test’.)Unlike Hale’s own, this revised formulation avoids the inadequacy revealed

by the Stirton Objection. Moreover, Hale’s Improved Aristotelian Test makesit possible to appreciate the importance of Hale’s Inferential Test, i.e. the im-portance of formulating the inferential test in a way that excludes individualquantifiers. This is because unlike Hale’s Aristotelian Test, Hale’s ImprovedAristotelian Test would be unable to exclude individual quantifiers if theywere allowed to slip past the inferential test. For on the one hand, their exclu-sion would require that anti-individual quantifiers be among the permissiblevalues of α1 if α is an individual quantifier. But on the other, (Constraint2.1*) would render anti-individual quantifiers impermissible because, unlikeindividual quantifiers, they would be excluded by the inferential test.

In this section, I have shown that Hale’s Aristotelian Test, the test’s currentstate-of-the-art formulation, is inadequate. Moreover, I have shown that itsadequacy can be restored by a rather modest revision, i.e. Hale’s ImprovedAristotelian Test. In the next section, I will show that the current state-of-the-art formulation of the inferential test, i.e. Hale’s Inferential Test, is inade-quate as well but that providing for an adequate revision calls for more drasticmeasures.

5 Revising the Inferential Test

Hale’s Inferential Test is deficient in several respects. For one, its formula-tion is dialectically disadvantageous. More importantly, though, it is outrightinadequate. The test’s problems arise from two directions, viz. its appeal toexistential generalisation using ‘something’ and the various constraints dis-cussed in §2. It is the aim of this section to discuss these problems and torevise Hale’s formulation in a way that is adequate and also cuts down on thenumber of required constraints.

5.1 Problems of Hale’s Test

The first problem concerning the appeal to existential generalisation using‘something’ in the test’s clauses (H.1) and (H.2) is dialectical. In order toregard these clauses as giving necessary conditions, ‘something’ has to be af-forded what might be called a ‘domain condition reading’ in the sense that


the truth of existential generalisations turns on how things stand with someentities that comprise the quantifier’s domain. However, although ‘something’clearly has this interpretation, it may well not be its only one. For instance,Thomas Hofweber has recently argued that ‘something’ also has a different,quasi-substitutional interpretation, an interpretation that, according to Hofwe-ber, is on display when ‘Something does not exist’ is validly inferred from,say, ‘Sherlock Holmes does not exist’.43 I will not venture into the details ofHofweber’s view. Nor do I wish to defend it. Rather, I merely wish to pointout that, according to Hofweber, it is this use of ‘something’ that is on displaywhen ‘something’ is used to validly quantify into the position of numerals inarithmetical statements.44 Hofweber may be wrong, of course. Nevertheless, itappears desirable to formulate the inferential test in a way that is independentof the outcome of this particular debate.

Fortunately, the appeal to existential generalisation using ‘something’ isdispensable. There are other simple forms of inference that do not involve‘something’ but which can be appealed to in order to disqualify those kinds ofquantifiers that Hale’s Inferential Test rules out by appeal to the ‘something’-involving clauses (H.1) and (H.2).45 For instance, the singular term ‘Plato’ in‘Plato is wise’ validates not only inferences such as the above (3) but, togetherwith ‘Plato is snub-nosed’, also validates inferences like:

17 Plato is wise and Plato is snub-nosed6 Plato is such that (he is wise and he is snub-nosed).

In contrast, the existential quantifier ‘some man’ in ‘some man is wise’ invali-dates not only inferences such as the above (4) but, together with ‘Some manis snub-nosed’, also invalidates inferences like:

18 Some man is wise and some man is snub-nosed6 Some man is such that (he is wise and he is snub-nosed).

By a similar token, the singular term ‘Plato’ in ‘Plato is wise’ validates notonly inferences such as the above (1) but, together with ‘Plato is snub-nosed’,also validates inferences like:

19 Plato is wise or Plato is snub-nosed6 Plato is such that (he is wise or he is snub-nosed).

In contrast, the non-existential quantifier ‘no man’ in ‘no man is wise’ invali-dates not only inferences such as the above (2) but, together with ‘No man issnub-nosed’, also invalidates inferences like:

20 No man is wise or no man is snub-nosed6 No man is such that (he is wise or he is snub-nosed).

Thus, it seems possible to reformulate clauses (H.1) and (H.2) by avertingto the inference forms on display in (17) – (20), thereby dispensing with thedialectically problematic appeal to existential generalisation using ‘something’.

43 See, e.g., Hofweber (2000).44 See, e.g., Hofweber (2005, pp. 217-20).45 This important observation is due to Heck (2002, 3ff.).


In addition to the above dialectical problem, the appeal to existential gen-eralisation using ‘something’ also threatens to render Hale’s Inferential Testinadequate even if it was possible to restrict the test’s application in a waythat would enforce the required domain condition reading of ‘something’. Thiscan be seen as follows. Evidently, ‘Plato’ functions as a singular term in both‘Plato is wise’ and its sentential negation ‘It is not the case that Plato is wise’.Now, suppose that ‘Plato is wise’ has what are sometimes called Ockhamisttruth-conditions—i.e. that it is true iff ‘Plato’ denotes a unique individual thatsatisfies ‘is wise’—and that falsity is failure of truth.46 On these assumptions,‘Plato is wise’ is false and, hence, its sentential negation true, if ‘Plato’ isempty. But then the inference from ‘It is not the case that Plato is wise’ to‘Something is such that it is not the case that it is wise’ is invalid if ‘some-thing’ is taken in its domain condition reading.47 Of course, these assumptionare debatable. However, they form part and parcel of the kind of negative freelogic Hale himself advocates.48 Thus, Hale’s Inferential Test is inadequate byhis own lights.

What the revealed inadequacy shows is that the inferential test shouldnot be formulated as a test for whether (substantival) expressions function assingular terms in just any sentence. Rather, it should be formulated as a testfor whether they do so function in sentences in which their positions are what,following Hale and Wright, we may call ‘denotation-demanding’.49 Roughly,the position of any expression α in a given sentence S is denotation-demandingif S’s truth requires that α denote. Thus, the position of ‘Plato’ in ‘Plato iswise’ is denotation-demanding, whereas its position in ‘It is not the case thatPlato is wise’ is not.

I take it that the lack of generality of a thus-restricted formulation is un-problematic, especially in the context of the neo-Fregean argument for arith-metical realism. For one, this argument also depends on the assumption thatat least some sentences that contain arithmetical singular terms are such thattheir truth requires that these singular terms denote. And for another, sen-tences whose truth requires that some ingredient singular term denote arethose in which the term’s position is denotation-demanding. Thus, a test forwhether expressions function as singular terms in denotation-demanding po-sitions would discharge, in one fell swoop, two of the assumptions of the neo-Fregean argument.

During its presentation in §2, we saw that Hale’s Inferential Test had to besubjected to a variety of diverse constraints lest it falls prey to a correspond-ing variety of equally diverse counterexamples. However, the imposition of theproposed constraints has at least two drawbacks. First, this method of safe-guarding against counterexamples is dialectically disadvantageous and should,if possible, be avoided. Given the constraints’ highly specific nature, there is

46 See, e.g., Sainsbury (2005, 46,65).47 Heck (2002, 5) makes a similar point using a negative existential such as the above

‘Sherlock Holmes does not exist’.48 See, e.g. Hale and Wright (2009).49 See, e.g., Hale and Wright (2009, 464n14).


no antecedent guarantee that there will not be counterexamples of differentkinds that these constraints are ill-equipped to preempt. This is not to saythat the lack of such a warranty should be taken to undermine any faith in thetest’s adequacy. In fact, I am somewhat sympathetic to the idea that Hale’sInferential Test may be presumed adequate until proven otherwise. However,this is to say that we should dispense with as many of these constraints aspossible if this could be done by formulating the inferential test in a way thatwould preempt present and potential future counterexamples in a more prin-cipled manner. Second and even more pressingly, the appeal to (Constraint1.1) – (Constraint 1.3), renders Hale’s Inferential Test inadequate.

According to (Constraint 1.3), the displayed occurrence of the test expres-sion α in the sentences S2 that are allowed to act as the side-premisses inapplications of clause (H.2) itself must satisfy clause (H.3). However, as for-mulated, (Constraint 1.3) fails to do what it is supposed to do. Recall, thisconstraint was introduced to get around the difficulty that, without it, thevalidity of the below inference would ensure that ‘a sheep’ in S1 = ‘John ownsa sheep’ passes Hale’s Inferential Test :

10 John owns a sheep and a sheep is a ruminant.

6 A sheep is such that (John owns it and it is a ruminant).

The idea was that since the validity of (10) depends on taking S2 = ‘Asheep is a ruminant’ in its universal rather than its existential reading, theproblem could be forestalled by requiring that S2 satisfy clause (H.3), i.e. theone typically not satisfied by universal quantifiers. However, although the ideais sound, its execution is flawed. For as witnessed by the valid inference below,there is a sentence—e.g. the above S1—that together with S2 sustains thepertinent inference:

21 A sheep is such that (John owns it or it is a ruminant)6 John owns a sheep or a sheep is a ruminant.

Of course, (21) is only valid on the existential reading of S2, i.e. the readingon which (10) is invalid. But since nothing in the formulation of (Constraint1.3) prevents this change in readings, this formulation is insufficient for thetask at hand. Now, it may be possible to reformulate (Constraint 1.3) in a waythat requires that the reading of S2 that validates (10) should also be the onethat validates (21). However, there is a different and more elegant solution,which renders (Constraint 1.3) obsolete. Moreover, this solution is congenialto Hale’s aforementioned intention of relativising the inferential test to usesof the relevant sentences.

Plausibly, the difference between the existential and universal reading of S2

is a matter of how it is used. Thus, counterexamples in the style of (10) could beforestalled by a revision of Hale’s Inferential Test that replaces clauses (H.2)and (H.3) with the following, explicitly use-relativised clause that, moreover,combines the forces of clause (H.2) and clause (H.3):50

50 Of course, such a revision would also require that the test’s left-hand side and clause(H.1) be accordingly use-relativised.


H.2+3 for S1 in use u1 and some sentence S2 = ăα, β2 ą in some use u2, the following

inference are valid:51

from x ă α, β1 ą and ă α, β2 ąy to xSomething is such that ă it, β1 ą and ă it, β2 ąy,

and

fromxα is such that ă it, β1ą or ă it, β2ąy to x ăα, β1ą or ăα, β2ąy.

Revising Hale’s Inferential Test in terms of clause (H.2+3), would avoidthe above difficulty because there is no single way to use S2 = ‘A sheep is aruminant’ that validates both (10) and (21). If it is used universally, (10) isvalid but (21) is not. And if it is used existentially, (21) is valid but (10) is not.Moreover, this revision would have the additional advantage that it reducesthe number of required constraints.

At this point, let me pause briefly to assemble the proposed solutions to theproblems just discussed into a revised version of Hale’s Inferential Test. Thisrevision differs from Hale’s own formulation in three crucial respects. First, itdispenses with the problematic appeal to ‘something’ and instead makes useof the inference forms on display in the above (17) – (20). Second, it will bea test for whether an expression functions as a singular term that occupies adenotation-demanding position. Third, it will contain a single explicitly use-relativised clause in the combined style of clause (H.2+3). Thus, the revisiontakes the following form:52

Hale’s Revised Inferential Test A substantival expression α functions as a singular termin a sentence S1 = ăα, β1ą in which α occupies a denotation-demanding iff

Inferential Condition? the following inferences are valid, for S1 in use u and somesentence S2 = ăα, β2ą in some use u2:

A from xăα, β1ą ^ ăα, β2ąy to xα is such that ă it, β1ą ^ ă it, β2ąy,

O.1 from xăα, β1ą _ ăα, β2ąy to xα is such that ă it, β1ą _ ă it, β2ąy ,

O.2 from xα is such that ă it, β1ą _ ă it, β2ąy to xăα, β1ą _ ăα, β2ąy.

(Modulo, of course, the imposition of suitably use-relativised versions of allconstraints except the now obsolete (Constraint 1.3).)

As one can easily check, this revision has all of the virtues of Hale’s orig-inal, but solves the latter’s adequacy problems discussed above.53 Moreover,

51 The reason clause (H.2+3) is formulated to require that the inferences be valid forsome sentence in only some rather than all of its uses is that the latter would “demand,unrealistically, that [α] be a term having only one referential use—a requirement clearly notmet in the case of most ordinary proper names”; Hale (1987, 19). Uses of ‘Aristotle’ in whichit respectively denotes Aristotle of Stagira and Aristotle Onassis are a case in point.52 In the names of the inference forms below, ‘A’ and ‘O’ stand for ‘And’ and ‘Or’, the

sentential connectives that feature in these forms. The ‘?’ on ‘Inferential Condition’ in-dicates that it is only a interim formulation, which in the light of the problems to bediscussed presently will be replaced by a ‘?’-less modification. By a similar token, Hale’sImproved Aristotelian Test should also be relativised to uses and restricted to expressionsin denotation-demanding position. I will explicitly do so in §5.3.53 It might be objected that Hale’s Revised Inferential Test is inadequate because it mis-

classifies singular terms in positions that are not denotation-demanding as terms whose


it has the dialectical advantage of being subject to a smaller number of con-straints. Hale’s Revised Inferential Test thus constitutes a not insignificantimprovement over Hale’s Inferential Test. However, Hale’s Inferential Test isbeset by a couple of further problems that even Hale’s Revised Inferential Testis unable to solve.

These problems are engendered by (Constraint 1.1) and (Constraint 1.2).Let us address them in turn. (Constraint 1.1) required that there be no strictimplications between the sentence S1 that contains the test expression α andthe choices of the sentences S2 that are allowed to act as the side-premissesin the application of Hale’s (Revised) Inferential Test. This makes for thefollowing problem. Consider ‘Plato’ in S1 = ‘Plato is self-identical’. Plausibly,every sentence S2 that together with S1 validates the pertinent inferences willstrictly imply S1. Thus, the test rules that ‘Plato’ does not function as asingular term in S1. But since ‘Plato’ in S1 does so function, Hale’s (Revised)Inferential Test is inadequate.

A similar point can be made against (Constraint 1.2), which concernedthe essential occurrence of expressions. The sentence ‘Plato is such that snowis white’ seems logically equivalent with ‘Plato exists and snow is white’. Ifso, the occurrence of ‘is such that snow is white’ in ‘Plato is such that snowis white’ would be inessential and Hale’s (Revised) Inferential Test wouldincorrectly rule that ‘Plato’ does not function as a singular term. Thus, apartfrom the dialectical advantage of doing so, the adequacy of the inferential testrequires that it be formulated in a way that dispenses with (Constraint 1.1)and (Constraint 1.2).

5.2 Major Revisions

In light of the last two problems, it should be clear that even Hale’s (Revised)Inferential Test is in need of revisions.54 For its current formulation is man-ifestly inadequate and, hence, unable to subserve the neo-Fregean argumentfor arithmetical realism. I contend that these problems can only be solved bya more radical departure from Hale’s Inferential Test. This departure is in-spired by the holistic approach developed by Richard Heck in his unpublished2002 manuscript ‘What is a Singular Term?’. However, although I take Heck’sproposal as a starting point, mine differs from Heck’s in several respects.

In a nutshell, my Heck-inspired proposal is the following. The questionof whether an expression α functions as singular term in (a use of) a givensentence S that contains it should be approached by testing whether S can

positions are denotation-demanding. For is not the inference from, say, ‘It is not the casethat Plato is wise and it is not the case that Plato is snub-nosed’ to ‘Plato is such that(it is not the case that he is wise and it is not the case that he is snub-nosed)’ valid? Icontend that it is not since I consider the position of ‘Plato’ in the premiss as not beingdenotation-demanding whilst I consider its position in the conclusion denotation-demanding.54 Note, too, that even if one considers the above problems concerning (Constraint 1.1)

and (Constraint 1.2) less pressing than I do, one should still agree that a formulation thatcan do without them would be dialectically desirable.


be an element of a whole class of sentences such that any two elements ofthis class sustain the inferences in (A), (O.1), and (O.2) of Hale’s RevisedInferential Test. Since my proposal is best illustrated against the backgroundof a definitive formulation, I begin by providing this formulation and thencontinue to explain its rationale:55

Holistic Inferential Test A substantival expression α functions as a singular term in someuse u of a sentence S = ăα, βą in which α occupies a denotation-demanding positioniff S is an element of at least one class A of sentences, each containing α, such that:

Inferential Condition the following inferences are valid, for all of A’s elements S1 =ăα, β1ą and S2 = ăα, β2ą in, respectively, some use u1 and u2:

A from xăα, β1ą ^ ăα, β2ąy to xα is such that ă it, β1ą ^ ă it, β2ąy,

O.1 from xăα, β1ą _ ăα, β2ąy to xα is such that ă it, β1ą _ ă it, β2ąy ,

O.2 from xα is such that ă it, β1ą _ ă it, β2ąy to xăα, β1ą _ ăα, β2ąy.

provided that:

Constraint I.1 the sentence class A is non-trivial (in the sense to be explainedbelow),

Constraint I.2 in use u1 and u2 of, respectively, S1 and S2, neither the conclusionsof the inferences in (A) and (O.1) nor the premisses of the inference in (O.2)are such that a point may be reached where a well-formed question for furtherspecification may be rejected as not requiring an answer,56

Constraint I.3 if in use u1 and u2 of, respectively, S1 and S2 the inferences in(A), (O.1), and (O.2) are valid, their validity can be immediately recognisedby every suitably competent speaker.

(Constraint I.2) and (Constraint I.3) are merely relabelled and explicitlyuse-relativised versions of the already familiar (Constraint 1.4) and (Constraint1.5). Thus, they do not need to be discussed further. However, two other fea-tures of the Holistic Inferential Test demand special attention. First, how doesthe holistic appeal help to solve the problems posed by (Constraint 1.1) and(Constraint 1.2) of Hale’s (Revised) Inferential Test? Second, what purposeserves (Constraint I.1) and how is the relevant notion of non-triviality to beunderstood? As we will see presently, the answers to these questions go handin hand.

Recall, (Constraint 1.1) and (Constraint 1.2) were needed because, with-out them, Hale’s (Revised) Inferential Test would incorrectly rule that, forinstance, ‘Some man’ and ‘Everything’ in, respectively, ‘Some man is wise’

55 Apart from the explicit relativisation to uses and the restriction concerning denotation-demanding positions, the below differs from Heck’s (2002, 8-15) holistic inferential test inthe following respects. First, Heck’s test is one for whether an expressions functions as asingular term with respect to classes of sentences that contain it (rather that a single sentencethat does). Second, Heck’s test neither provides a sufficient condition nor is it restricted tosubstantival expressions. Third, Heck’s test appeals to a fourth but redundant inferenceform, viz. the converse of (A). I chose the formulation below in order to remain as close aspossible to Hale’s original one.56 Although originally formulated for Hale’s test involving ‘something’, Heck (2002, 11)

shows that (Constraint I.2), i.e. the old (Constraint 1.4), can also be applied to the‘something’-free inferences forms appealed to above.


and ‘Everything is wise’ function as singular terms.57 Conversely, though, theimposition of these constraints had the intolerable consequence that the testincorrectly rules that ‘Plato’ in, respectively, ‘Plato is self-identical’ and ‘Platois such that snow is white’ does not function as a singular term.

Now, even without (Constraint I.1), the Holistic Inferential Test would lackthe latter of the above two consequences. After all, there are many sentenceclasses that contain ‘Plato is self-identical’ such as, for instance A = {‘Platois self-identical’, ‘Plato is wise’} that satisfy the test’s Inferential Condition.Thus, the test does not misclassify ‘Plato’ in ‘Plato is self-identical’.

However, without (Constraint I.1), the Holistic Inferential Test would havemisclassified ‘some man’ and ‘everything’ in, respectively, ‘Some man is wise’and ‘Everything is wise’. After all, there are many sentence classes that contain‘Some man is wise’ or ‘Everything is wise’ but that also satisfy the InferentialCondition.58 The classes B = {‘Some man is wise’, ‘Some man is wise andsnub-nosed’} and C = {‘Everything is wise’, ‘Everything is such that snow iswhite’} are a case in point.

That sentence classes like these are potentially problematic is hardly sur-prising. For they satisfy the Inferential Condition for the same reasons thatcaused Hale to subject his test to (Constraint 1.1) and (Constraint 1.2). How-ever, unlike Hale’s own formulation of the inferential test, the above holisticformulation puts us in the position to resolve this difficulty in a more elegantand dialectically more satisfying manner. (Constraint 1.1) and (Constraint1.2) only remedy the symptoms on display in the specific counterexamplesthey help preempt. Thus, they do not preempt these problems by analysingand then trying to forestall the common cause of these symptoms. Conse-quently, there is no guarantee that this common cause will not manifest itselfin different symptoms, thereby creating problems that (Constraint 1.1) and(Constraint 1.2) cannot preempt.

Of course, this assumes that the problems these constraints address havea common cause. However, it seems quite clear that they do. In each case,the fact that certain expressions pass the inferential test is, as it were, a falsepositive which arises from the fact that some feature of the embedding con-texts prevents the expressions in question from betraying their true inferentialcolours. For instance, in the case of the above sentence class B, this featurewas the modal connection between ‘is wise’ and ‘is wise and snub-nosed’. Sinceas a matter of necessity everything that is wise and snub-nosed is also wise,‘Some man is wise’ was strictly implied by ‘Some man is wise and snub-nosed’.And it is due to this fact that these two sentences sustain each of the threepertinent inferences. Similarly, in the case of the above C, where this ‘maskingfeature’ was the non-essential occurrence of ‘is such that snow is white’. Thus,what is needed is some general way to ensure that the holistic version of theinferential test only be applied to sentence-classes in which every expression

57 See the atypically valid inferences (8) and (9) in §2 p. 6f.58 Of course, there are also many such sentence classes such as, for instance, {‘Some man

is wise’, ‘Some man is snub-nosed’} and {‘Everything is wise’, ‘Everything is snub-nosed’}that do not satisfy the Inferential Condition.


whose inferential behaviour has the potential to differ from that of singularterms gets the chance to realise this potential. In other words, we need someway to ensure that, if an expression sustains all inferences with respect to agiven sentence-class, this is not due to some special features of the particularsentences that comprise this class. As we will see, the restriction of the appli-cation of the Holistic Inferential Test to non-trivial sentence-classes does justthat.

As for the relevant notion of non-triviality, note that there is a crucialdifference between, on the one hand, the above class A and, on the other, theclasses B and C. For consider the classes A1 = {‘Plato is self-identical’, ‘Platois snub-nosed’}, B1 = {‘Some man sits’, ‘Some man sits and drinks’}, and C1

= {‘Everything is snub-nosed’, ‘Everything is such that grass is green’}, whichall satisfy the Inferential Condition. The difference is that, whereas the unionof A and A1 is again a class that satisfies this condition, neither the union of Band B1 nor the union of C and C1 satisfies it. This observation suggests thata sentence class’s non-triviality can be characterised along the following lines.Unless the fact that a given sentence class A satisfies the Inferential Conditiondepends on some special features of the sentences that comprise it, it shouldalways be possible to enlarge A by conjoining it with other classes that satisfythe Inferential Condition to obtain a class that also satisfies this condition.This idea can be precisified as follows. Let S be a sentence that contains thetest expression α and let A be a class of sentences, each containing α, that inparticular contains S. Then A is non-trivial iff:59

Non-Triviality If A satisfies the Inferential Condition, then all classes A1 and A2 of

α-containing sentences, which contain S and satisfy the Inferential Condition, are such

that AYA1 YA2 also satisfies the Inferential Condition.60

As one can easily check, the above above classes B and C are trivial in thissense and are thus excluded by (Constraint I.1). Consequently, the HolisticInferential Test steers free of the problems that Hale’s (Revised) InferentialTest had to forestall bey means of (Constraint 1.1) and (Constrain 1.2). How-ever, unlike Hale’s (Revised) Inferential Test, the Holistic Inferential Test canavoid these problems without thereby being rendered inadequate. Thus, itconstitutes a significant improvement over Hale’s current state-of-the-art for-mulation of the inferential test. In particular, since, unlike Hale’s formulation,the Holistic Inferential Test is adequate, only the former can subserve theneo-Fregean argument for arithmetical realism.

59 Heck (2002, 8) characterises non-triviality differently. However, as Heck (p.c.) acknowl-edges, his characterisation is inferior to mine.60 The reason Non-Triviality is formulated in terms of ‘all classes A1 and A2’ rather than

the simpler ‘all classes A1’ is this. Consider A = {‘Everything is self-identical’ , ‘Everythingis such that snow is white’}, a class that satisfies the Inferential Condition. It seems tome that there is no single class A1, which satisfies this condition, such that A YA1 failsit. However, there are plenty of pairs of relevant classes A1 and A2 such that A Y A1 Y

A2 fails the Inferential Condition. The classes A1 = {‘Everything is such that snow iswhite’,‘Everything is wise’} and A2 = {‘Everything is such that snow is white’,‘Everythingis snub-nosed’} are a case in point.


5.3 Parting Thoughts

Before I bring this section to a close, I would like to make three further remarks.First, it should be clear that adopting the Holistic Inferential Test at the

expense of Hale’s Inferential Test requires some further if modest modifica-tion of Hale’s Improved Aristotelian Test. For one, it should also be relativisedto uses of the relevant sentences and restricted to expressions in denotation-demanding positions. Doing so ensures that, modulo of course the former’srestriction to substantival expressions, the two tests are tests for the samething, viz. for whether an expression α functions as a singular term in a useof some α-containing sentence in which α occupies a denotation-demandingposition. Moreover, the two constraints of Hale’s Improved Aristotelian Testrefer back to the results of Hale’s Inferential Test. Once Hale’s test has beensupplanted by the Holistic Inferential Test, these back-references have to beamended accordingly. Implementing these revisions is straightforward and re-sults in:61

Hale’s Revised Aristotelian Test An expression α functions as a singular term in a useu1 of a sentence S1 = ăα, β1ą in which α occupies a denotation-demanding positiononly if

Aristotelian Condition there is no expression α1 such that, for all expressions β2,the following holds for the resulting sentences S2 = ăα, β2ą and S1

2 = ăα1, β2ąin, respectively, some use u2 and u1

2:

C xăα1, β2ąØ ăα, β2ąy is true,

where:

Constraint A.1 (i) α1 is syntactically congruous with α in S1, (ii) α1 in S12 passes/

fails/neither passes nor fails the Holistic Inferential Test just in case α1 in S2

does,

Constraint A.2 (i) β2 is syntactically congruous with β1 in S1, (ii) β2 in S2

neither is nor contains an expression that fails the Holistic Inferential Test.

The second remark concerns my earlier contention that the problems thatbeset Hale’s Inferential Test can only be solved by means of the holistic revi-sions proposed above.62 So far I have shown that the Holistic Inferential Testdoes, in fact, solve these problems. What I have not shown, though, is that‘going holistic’ is the only way to do so. Nevertheless, I am confident thatno revision along Hale’s non-holistic or atomistic lines can be expected to befully adequate. If this confidence was warranted, this would go a long way todemonstrate that adopting the Holistic Inferential Test is mandatory ratherthat optional.

Whence, then, my confidence? In a nutshell, the reason is this. It seems tome that even the best and strongest possible atomistic reformulation of theinferential test, which is not too strong and makes do without the inadequacy-inducing (Constraint 1.1) and (Constraint 1.2), would still be too weak and,

61 The ‘C’ below stands for ‘Complement expression’.62 Thanks to an anonymous referee for this journal for pressing me on the following point.


hence, inadequate. Let me explain. Plausibly, the best and strongest possibleatomistic version of the inferential would be obtained from Hale’s InferentialTest by the following three-step procedure, the result of which we may call‘Hale’s Universalised Inferential Test ’.63 First, Hale’s Inferential Test is refor-mulated in terms of the following, universally formulated variants of clauses(H.2) and clause (H.3):H.2@ for S1 and every sentence S2 = ăα, β2ą, the following inference is valid:

from xăα, β1ą ^ ăα, β2ąy to xSomething is such that ă it, β1ą ^ ă it, β2ąy.

H.3@ for S1 and every sentence S2 = ă α, β2ą, the following inference is valid:

from xα is such that ă it, β1ą _ ă it, β2ąy to x ăα, β1ą or ăα, β2ąy.

Second, some constraint on the values of the universal quantifier ‘for everysentence S2’ is imposed to ensure that the resulting test does not fall preyto the problem of excessive strength, which motivated Hale to formulate histest in terms of the weak, existentially formulated clauses (H.2) and (H.3).64

Third, the problematic (Constraint 1.1) and (Constraint 1.2) are dropped.Now, when developing the notion of non-triviality in §5.2, I observed the

following. If the fact that two α-containing sentences S1 and S2 validate certaininference forms is to be taken as indicative of α’s singular-termhood, it mustbe ensured that the inferences’ validity is not a mere artefact of certain specialfeatures of S1 and S2 , either individually or jointly. At first glance, it mayseem as if Hale’s Universalised Inferential Test would manage to do just that.Thus, compare the following pairs of inferences, repeated from §2:

4 Some man is wise and some man is snub-nosed.

6 Something is such that (it is wise and it is snub-nosed).

8 Some man is wise and some man is wise and snub-nosed.

6 Something is such that (it is wise and it is wise and snub-nosed).

5 Every man is such that (he is wise or he is snub-nosed).

6 Every man is wise or every man is snub-nosed.

63 Strictly speaking, I think that the best and strongest possible atomistic version of theinferential would be obtained from Hale’s Revised Inferential Test by a method similar tothe one I am about to sketch. However, for the sake of simplicity, I ignore the complicationsthat doing so would engender. With one possible exception to be addressed in fn. 64 below,everything I say in terms of Hale’s Inferential Test can be transposed to Hale’s RevisedInferential Test.64 Cp. §2 p. 6. How the necessary restriction can be effected may be a matter of debate.

However, it appears that restricting the values of S2 to sentences that, with respect to thetest expression α, themselves satisfy clause (H.1) of Hale’s Inferential Test would do thetrick. At the very least, such a restriction would preempt the problem of excessive strengthrevealed by the non-valid inference (7). For although (7) is not valid, its side-premiss “Plato’has five letters’ does not satisfy clause (H.1) with respect to ‘Plato’.Things would have been somewhat different had we tried to universalise Hale’s RevisedInferential Test. Since this revision has dispensed with Hale’s clause (H.1), the neededrestriction cannot be effected in its terms. However, it seems to me that it would suffice torestrict ‘for every sentence S2’ to sentences S2 = ă α, β2 ą that strictly imply xα is suchthat ă it, β2ąy. For the inference from “Plato’ has five letters’ to ‘Plato is such that ‘it’ hasfive letters’ is no more valid than the inference from the same premiss to ‘Something is suchthat (‘it’ has five letters)’.


9 Everything is such that (it is wise or it is such that snow is white).

6 Everything is wise or everything is such that snow is white.

Had Hale’s Inferential Test been formulated without (Constraint 1.1) and(Constraint 1.2), the validity of (8) and (9) would have sufficed to show thatit does not exclude ‘some man’ and ‘everything’ in, respectively, ‘Some manis wise’ and ‘Everything is such that snow is white’. In contrast, Hale’s Uni-versalised Inferential Test would lack this consequence. By its lights, the factthat some inferences like (8) and (9) are atypically valid is irrelevant. For ‘someman’ and ‘everything’ in the relevant sentences are excluded on account of (4)and (6) being invalid.

Thus, it may seem that Hale’s Universalised Inferential Test avoids theweakness of Hale’s Inferential Test without having to impose the problematic(Constraint 1.1) and (Constraint 1.2). And since the former is as atomistic asthe latter, it may seem that it is possible to formulate the inferential test ina way that is both atomistic and adequate. Not so. The problem is this. Un-like my Holistic Inferential Test, Hale’s Universalised Inferential Test returnsinadequate results when certain inferences are atypically valid due to somespecial feature of the inference’s main premiss (rather than, as in (9), due toa special feature of the side-premiss or, as in (8), due to one that arises froma special relation between the two premisses).

This problem can be illustrated as follows. Consider ‘everything’ in S1 =‘Everything is self-identical’. As a universal quantifier it would have to beexcluded by (H.3@). But in order to be thus excluded, there would have to beat least some sentence S2 = ă ‘everything’, β2ą such that the inference fromxEverything is such that (it is self-identical or ă it, β2ą)y to xEverything is self-identical or ă ‘everything’, β2 ąy is invalid. However, since S1 is a (logically)necessary truth, its disjunction with any sentence is strictly implied by everysentence. A similar problem arises if, for instance, ‘Everything is such that iswhite’ is the inference’s main premiss, i.e. the one with respect to which thesemantic function of ‘everything’ is evaluated.

The two problems above demonstrate that clause (H.3@) would be unableto exclude ‘everything’ in, respectively, ‘Everything is self-identical’ and ‘Ev-erything is such that is white’. Consequently, Hale’s Universalised InferentialTest would be too weak and, hence, inadequate. And on the plausible as-sumption that Hale’s Universalised Inferential Test is the best and strongestpossible atomistic formulation of the inferential test, this means that no atom-istic formulation can be adequate. If so, the Holistic Inferential Test is not onlyrecommended by the fact that, unlike Hale’s Inferential Test, it is, in fact, ad-equate. In addition, it would also be recommended by the fact that we shouldexpect an adequate formulation of the inferential test to be holistic becauseno atomistic formulation can hope to be adequate.

My third and final remark concerns a certain prominent feature of theHolistic Inferential Test. For all its virtues it will be very hard, if not impossi-ble, to conclusively establish whether, according to this test, a given substan-tival expression functions as a singular term in a use of any given sentence.


First, consider the case of non-singular terms. According to the HolisticInferential Test, a substantival expression α does not function as a singularterm in a sentence S if there is no non-trivial sentence class A that containsS and satisfies the test’s Inferential Condition. Thus, showing that some suchclass A fails the Inferential Condition does not conclusively establish that αin S does not function as a singular term. To establish this claim conclusively,one would have to show that all relevant non-trivial classes fail the InferentialCondition. And since there will be at least a sizeable number of, if not infinitelymany, such classes, it will be hard, if not impossible, to reach a conclusiveverdict about whether an expression is not a singular term. Thus, vis-a-visthe task of demonstrating that α in S does not function as a singular term,the best we can do is to regard every non-trivial sentence class that fails theInferential Condition as yet another piece of evidence that α does not sofunction.

For what it is worth, I do not consider this consequence problematic. But tothe extent that is problematic, it should be noted that Hale’s Inferential Testhas virtually the same problem.65 For according to Hale’s test, a substantivalexpression α does not function as a singular term in a sentence S1 if there isno permissible sentence S2 that, together with S1 invalidates the inferencesin the test’s clauses (H.2) and (H.3). Thus, showing that S1 and some suchsentence S2 invalidate at least one of these inferences does not conclusivelyestablish that α in S1 does not function as a singular term. To establish thisclaim conclusively, one would have to show that S1 and all relevant permissiblesentences S2 fail to sustain at least one of the pertinent inferences. And since,once more, there will be at least a sizeable number of, if not infinitely many,such sentences, it will be hard, if not impossible, to reach a conclusive verdictabout whether an expression does not functions as a singular term.66

Now, unlike Hale’s Inferential Test, the Holistic Inferential Test faces asimilar predicament when it comes to delivering verdicts on whether expres-sions do function as singular terms. The problem is that it can only be appliedto non-trivial sentence-classes. However, whereas establishing a class’s trivial-ity is easy, establishing its non-triviality is not. To demonstrate the trivialityof a class A that satisfies the test’s Inferential Condition, all it takes is toproduce two other such classes A1 and A2 and show that AYA1 YA2 doesnot satisfy this condition. Thus, establishing triviality is a comparatively easytask. In contrast, to demonstrate the non-triviality of a class A that satisfiesthe test’s Inferential Condition, we would have to check whether any two suchclasses A1 and A2 are such that AYA1YA2 does satisfy this condition. Thus,since there will be a plethora of such classes, establishing the non-triviality ofA if A satisfies the Inferential Condition is a rather Herculean task.67 As a

65 Similarly for Hale’s Revised Inferential Test.66 Of course, there will be some expressions α in some sentences S1—viz. those that inval-

idate the inference in clause (H.1)—that Hale’s test can rule out conclusively. But that doesnot change the fact that in many cases Hale’s Inferential Test is bound to be as ‘inconclusive’as the Holistic Inferential Test.67 Of course, any class that fails the Inferential Condition is vacuously non-trivial.


consequence, the best we can do vis-a-vis the task of demonstrating that aclass A that satisfies the Inferential Condition is non-trivial is to regard everypair of classes A1 and A2, which satisfy the Inferential Condition and thatconjoined with A form another class that satisfies it, as yet another piece of ev-idence that A is non-trivial. Since we thus cannot conclusively establish thata class, which satisfies the Inferential Condition, is non-trivial, neither canthe Holistic Inferential deliver a conclusive verdict on whether a substantivalexpression α functions as a singular term in some sentence S.

As before, I do not think that this consequence is particularly problematic.For one, on both Hale’s Inferential Test and the Holistic Inferential Test, weare already forced to make peace with this inconclusiveness when it comes tonon-singular terms. Thus, it is hard to see why it should be any more both-ersome when it comes to singular terms. And for another, although Hale’sInferential Test delivers conclusive verdicts in the case of singular terms, wewould do well to remember that Hale’s test has the ultimate drawback of be-ing inadequate. Thus, embracing the fact that the Holistic Inferential Test isdoubly inconclusive is the price of possessing an adequate formulation of theinferential test.

In this section, I have identified several weaknesses of Hale’s Inferential Test,the current state-of-the-art formulation of the inferential test. For one, thisformulation is dialectically disadvantageous. More pressingly, though, it is in-adequate and, hence, unable to subserve the neo-Fregean argument for arith-metical realism. In response, I developed a novel formulation of the inferentialtest that, above all, is adequate and, therefore, able to subserve this argument.

6 Conclusion & Outlook

Neo-Fregeans take their argument for arithmetical realism to crucially dependon the availability of certain broadly syntactic tests for whether a given ex-pression functions as a singular term. The current state-of-the-art formulationsof these tests are what I have called Hale’s Inferential Test and Hale’s Aris-totelian Test. If these formulations are to subserve the neo-Fregean argument,they must be at least adequate, in the sense of correctly classifying paradigmcases of singular terms and non-singular terms. In this paper, I have accom-plished two things. First, I have shown that Hale’s tests are inadequate and,thus, are unable to subserve the neo-Fregean argument. Second, I proposedand defended revised formulations of the two tests—viz. what I have calledthe Holistic Inferential Test and Hale’s Revised Aristotelian Test—that areadequate and, therefore, can subserve the argument in question.

Time to look ahead. There are at least two issues that are worth to beinvestigated further. First, is it possible to dispense with the test’s remainingconstraints (Constraint I.2) and (Constraint I.3) in a way that does not com-promise the ability of the Holistic Inferential Test to subserve the neo-Fregeanargument? Second, is it possible to strengthen our faith in the tests’ results


once they are applied to controversial cases, in particular the case of arith-metical singular terms? With respect to the first question, I contend that atleast (Constraint I.2) is, in fact, dispensable. For recall, this constraint onlyserved to exclude individual quantifiers. However, to the best of my knowl-edge a test that cannot distinguish between singular terms and individualquantifiers, would still be good enough to subserve the neo-Fregean argument.And with respect to the second question, I contend that our faith could bestrengthened considerably by systematically evaluating the tests against thebackground of contemporary formal semantics. For should such an evaluationcorroborate the neo-Fregean tests, this would put them on a even more securefooting. Both these tenets deserve further attention. For today, though, I mustleave their evaluation to the proverbial future research.

Acknowledgements Since reading his manuscript ‘What is a Singular Term?’ got methinking on the matters discussed above, I am grateful to Richard G. Heck. Moreover, Iam grateful for many helpful discussions with and comments from Christian Folde, StephanKramer, Stefan Roski, Benjamin Schnieder, Nathan Wildman, and Richard Woodward aswell as the participants of the Oberseminar Sprache und Welt at the University of Hamburgin the winter term 2013, the audience of the workshop Talking of Something or Talking ofNothing? in Gothenburg in January 2014, and two anonymous referees for this journal.

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