Abstract
Neo-Fregeans take their argument for arithmetical realism to depend on the availability of certain, so-called broadly syntactic tests for whether a given expression functions as a singular term. The broadly syntactic tests proposed in the neo-Fregean tradition are the so-called inferential test and the Aristotelian test. If these tests are to subserve the neo-Fregean argument, they must be at least adequate, in the sense of correctly classifying paradigm cases of singular terms and non-singular terms. In this paper, I pursue two main goals. On the one hand, I show that the tests’ current state-of-the-art formulations are inadequate and, hence, cannot subserve the neo-Fregean argument. On the other hand, I propose revisions that are adequate and, hence, can subserve this argument.
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Notes
Hale (1996, p. 31).
However, see Sect. 5, p. 16.
Hale (1996, p. 32).
Hale (1996, p. 32).
See Hale (2013, pp. 40–46).
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 uniform substitution or derivability. Rather, it has to be understood in terms of strict implication: the inference from \(\varphi \) to \(\psi \) is valid iff it is not possible that (\(\varphi \) is true and it is not the case that \(\psi \) is true); see, e.g., Rumfitt (2003, p. 203).
The language-relativisation will henceforth be left tacit.
This use-relativisation is made explicit in Hale (1987, pp. 18–22) but has been left tacit since.
I use ‘\(\alpha \)’ and ‘\(\beta \)’ as metalinguistic variables ranging over sub-sentential expressions (of English).
See, e.g., Quine (1981, Sect. 6).
Apart from the different notation, the below differs from Hale’s exact wording in four negligible respects. First, Hale formulates clause (H.2) in terms of the inference from \(S_1\) and \(S_2\) to the pertinent conclusion (rather than from their conjunction). Second, I formulate the premiss of the inference in clause (H.3) as \(^\ulcorner \alpha \) is such that \((<it , \beta _1> \vee <it , \beta _2>)^\urcorner \), whereas Hale formulates it as \(^\ulcorner \)It is true of \(\alpha \) that \((<it , \beta _1> \vee <it , \beta _2>)^\urcorner \). Third, for presentational 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).
Some foreshadowing: (Constraint 1.4) is meant to distinguish first-order uses of ‘something’ as in ‘Something is wise’ from higher-order uses as in ‘Plato is something’ when validly inferred from, say, ‘Plato is wise’. For details, see p. 7f.
See Quine (1960, Sect. 30).
Other potentially problematic cases are ones in which the test expression is embedded in a modal or epistemic context.
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 only arguments with truth-evaluable premisses and conclusions can be valid or invalid.
See Hale (1994, p. 66). I say ‘roughly’ because Hale’s official characterisation is more complex, a complexity we can safely ignore in the present context.
This idea traces back to Dummett (1973, p. 61, 67ff.). For further discussion, see e.g. Wright (1983, p. 58, 61ff.), Hale (1987, p. 16f.), Hale (1994, p. 37ff.) and Hale (1994, pp. 53–58). To get 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 is wise?, and you answer Plato. In the case of (11), you assert ‘Something is such that Plato is it’, someone enquires What is such that Plato is it?, and you answer a philosopher. In both cases, 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 not require 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 rights to reject her request because you have already answered it: I just told you, the author of ‘Euthyphro’.
See Hale (1994, p. 70).
Hale (2013, p. 44) credits Rumfitt (2003, p. 203ff.) as the originator of this objection and one Paul McCallion with the discovery that Rumfitt’s objection can be generalised to cover considerably more cases than the one discussed by Rumfitt himself. This attribution is somewhat generous. For one, Rumfitt (2003, p. 204n21) is aware that his objection generalises. And for another, Hale (1994, p. 70f.) had already observed that individual quantifiers pose a problem for the formulation of the inferential test he advocated at the time.
This example is in the style that Hale ascribes to McCallion. Rumfitt’s example of an individual quantifier is ‘some even prime number’.
Hale (2013, p. 45).
Hale (2013, p. 45n83).
Hale (1994, p. 52).
Hale (1994, p. 52).
The formulation of sub-clause (ii) slightly deviates from how Hale would have put it, i.e. from ‘\(\beta _2\) in \(S_2\) \(=\) \(<\!\alpha , \beta _2\!>\) does not fail Hale’s Inferential Test’. My formulation is an improvement over Hale’s because Hale’s Aristotelian Test is supposed to also render verdicts for adjectives like ‘wise’ in ‘Some man is wise’. However, deleting ‘wise’ from this sentence 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, ‘Some man is’ contains the substantival ‘some man’ that fails this test.
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, pp. 43–46). On the one hand, replacing, say, ‘some man’ in ‘some man is wise’ with ‘no man’ has the desired effect:
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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 ‘no man’ may fail to have the desired effect:
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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’ are scope-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 \(\beta _2\)-constituent of ‘Some man is wise’—‘Every woman loves’—i.e. the \(\beta _2\)-constituent of ‘Every woman loves some man’—violates this constraint. For like ‘some man’ in ‘Some man is wise’, ‘every woman’ in ‘Every woman loves some man’ fails Hale’s Inferential Test. Thanks to an anonymous referee for this journal for pointing out that this point needs to be made explicit.
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Hale (2013, p. 44) is aware of this requirement. However, as we will see presently, contrary to his intentions, Hale’s Aristotelian Test is formulated in a way that renders singular terms incapable of passing it.
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 \(\alpha '\), it needs to be ensured that they are expressions—i.e. singular terms—that saturate ‘is wise’ rather than expressions that ‘is wise’ saturates. However, the requirement imposed by (Constraint 2.1) that permissible \(\alpha '\)s be syntactically congruous with \(\alpha \) is unable to effect this restriction.
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) quantifiers are not syntactically congruent because, syntactically speaking, singular terms can do things quantifiers cannot. For instance, given Frege’s (1893, Sects. 29–32) syntactic formation rules, 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 that the quantifier ‘some girl’ combines with ‘loves’ to form the unary predicate ‘loves some girl’. This, so the referee’s worry, shows that singular terms and quantifiers cannot, strictly speaking, be syntactically congruent. In response, let me make two remarks (and express my 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 perfectly grammatical English sentence. Thus, the phrase-structure rules of English allow for the construction of ‘Romeo loves some girl’ from the name ‘Romeo’, the verb ‘loves’, and the quantifier ‘some girl’ in the same way as they allow for the construction of ‘Romeo loves Juliet’ from ‘Romeo’, ‘loves’, and the name ‘Juliet’; for more on phrase-structure rules, see e.g. Poole (2002, ch. 2). This shows that the phrase-structure rules of English are more permissive than Frege’s formation rules and that, consequently, there is a good sense in which the English expression ‘some girl’ is syntactically congruous with ‘Juliet’ in the English sentence ‘Romeo loves Juliet’. And since we are dealing with English sentences, rather than sentences of some formal language such as Frege’s, all the point I made in the main text seems to 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 formation rules are permissive in this way engenders a problem once we turn to the task of semantic interpretation. Roughly, the problem is that, unlike ‘Romeo loves Juliet’, ‘Romeo loves some girl’ is not straightforwardly interpretable because ‘loves’ denotes a relation between objects such as Romeo and Juliet, whereas the quantifier ‘some girl’ denotes, in Fregean terminology, a (second-level) concept. This interpretability problem is a well-known staple of contemporary linguistics and linguists usually resolve it in one of two ways. Either by positing a type-shifting operation that can alter the denotation of quantifiers so as to become interpretable 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 Form of ‘Romeo loves some girl’ is relevant for its semantic interpretation and (ii) on the level of its Logical Form, ‘some girl’ occupies a syntactic position that allows it to be interpreted in its usual concept-denoting type, see e.g. Heim and Kratzer (1998, ch. 7). As far as I can see, there is nothing that prevents proponents of the neo-Fregean tests from adopting the linguists’ solution. Thus, to the extent that such interpretability problems appear to spell spell trouble for the neo-Fregean tests, it seems that appearances are deceptive.
Hale (2013, p. 44) appears somewhat aware of this fact but fails to draw the consequence advocated below.
For instance, the inference in clause (H.1) from ‘Nothing that is Plato is wise’ to ‘Something is such that it is wise’ is invalid.
See, e.g., Hofweber (2000).
See, e.g., Hofweber (2005, pp. 217–20).
This important observation is due to Heck (2002, p. 3ff.).
See, e.g., Sainsbury (2005, pp. 46, 65).
Heck (2002, p. 5) makes a similar point using a negative existential such as the above ‘Sherlock Holmes does not exist’.
See, e.g. Hale and Wright (2009).
See, e.g., Hale and Wright (2009, p. 464n14).
Of course, such a revision would also require that the test’s left-hand side and clause (H.1) be accordingly use-relativised.
The reason clause (H.2+3) is formulated to require that the inferences be valid for some sentence in only some rather than all of its uses is that the latter would “demand, unrealistically, that [\(\alpha \)] be a term having only one referential use—a requirement clearly not met in the case of most ordinary proper names”; Hale (1987, 19). Uses of ‘Aristotle’ in which it respectively denotes Aristotle of Stagira and Aristotle Onassis are a case in point.
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’ indicates that it is only a interim formulation, which in the light of the problems to be discussed presently will be replaced by a ‘?’-less modification. By a similar token, Hale’s Improved Aristotelian Test should also be relativised to uses and restricted to expressions in denotation-demanding position. I will explicitly do so in Sect. 5.3.
It might be objected that Hale’s Revised Inferential Test is inadequate because it misclassifies singular terms in positions that are not denotation-demanding as terms whose positions are denotation-demanding. For is not the inference from, say, ‘It is not the case that 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? I contend that it is not since I consider the position of ‘Plato’ in the premiss as not being denotation-demanding whilst I consider its position in the conclusion denotation-demanding.
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 that can do without them would be dialectically desirable.
Apart from the explicit relativisation to uses and the restriction concerning denotation-demanding positions, the below differs from Heck’s (2002, pp. 8–15) holistic inferential test in the following respects. First, Heck’s test is one for whether an expressions functions as a singular term with respect to classes of sentences that contain it (rather that a single sentence that does). Second, Heck’s test neither provides a sufficient condition nor is it restricted to substantival expressions. Third, Heck’s test appeals to a fourth but redundant inference form, viz. the converse of (A). I chose the formulation below in order to remain as close as possible to Hale’s original one.
Although originally formulated for Hale’s test involving ‘something’, Heck (2002, p. 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.
See the atypically valid inferences (8) and (9) in Sect. 2, p. 6f.
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.
Heck (2002, p. 8) characterises non-triviality differently. However, as Heck (p.c.) acknowledges, his characterisation is inferior to mine.
The reason Non-Triviality is formulated in terms of ‘all classes A \(_1\) and A \(_2\)’ rather than the simpler ‘all classes A \(_1\)’ is this. Consider A \(=\) {‘Everything is self-identical’ , ‘Everything is such that snow is white’}, a class that satisfies the Inferential Condition. It seems to me that there is no single class A \(_1\), which satisfies this condition, such that \(\mathbf A \cup \mathbf A _1\) fails it. However, there are plenty of pairs of relevant classes A \(_1\) and A \(_2\) such that \(\mathbf A \cup \mathbf A _1 \cup \mathbf A _2\) fails the Inferential Condition. The classes A \(_1\) \(=\) {‘Everything is such that snow is white’,‘Everything is wise’} and A \(_2\) \(=\) {‘Everything is such that snow is white’,‘Everything is snub-nosed’} are a case in point.
The ‘C’ below stands for ‘Complement expression’.
Thanks to an anonymous referee for this journal for pressing me on the following point.
Strictly speaking, I think that the best and strongest possible atomistic version of the inferential would be obtained from Hale’s Revised Inferential Test by a method similar to the one I am about to sketch. However, for the sake of simplicity, I ignore the complications that 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 Revised Inferential Test.
Cp. Sect. 2, p. 6. How the necessary restriction can be effected may be a matter of debate. However, it appears that restricting the values of \(S_2\) to sentences that, with respect to the test expression \(\alpha \), themselves satisfy clause (H.1) of Hale’s Inferential Test would do the trick. At the very least, such a restriction would preempt the problem of excessive strength revealed 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 Revised Inferential Test. Since this revision has dispensed with Hale’s clause (H.1), the needed restriction cannot be effected in its terms. However, it seems to me that it would suffice to restrict ‘for every sentence \(S_2\)’ to sentences \(S_2\) \(=\) \(<\alpha ,\beta _2>\) that strictly imply \(\ulcorner \alpha \) is such that \(<it,\beta _2>\urcorner \). For the inference from “Plato’ has five letters’ to ‘Plato is such that ‘it’ has five letters’ is no more valid than the inference from the same premiss to ‘Something is such that (‘it’ has five letters)’.
Similarly for Hale’s Revised Inferential Test.
Of course, there will be some expressions \(\alpha \) in some sentences \(S_1\)—viz. those that invalidate the inference in clause (H.1)—that Hale’s test can rule out conclusively. But that does not change the fact that in many cases Hale’s Inferential Test is bound to be as ‘inconclusive’ as the Holistic Inferential Test.
Of course, any class that fails the Inferential Condition is vacuously non-trivial.
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Acknowledgments
Since reading his manuscript ‘What is a Singular Term?’ got me thinking on the matters discussed above, I am grateful to Richard G. Heck. Moreover, I am grateful for many helpful discussions with and comments from Christian Folde, Stephan Krämer, Stefan Roski, Benjamin Schnieder, Nathan Wildman, and Richard Woodward as well as the participants of the Oberseminar Sprache und Welt at the University of Hamburg in the winter term 2013, the audience of the workshop Talking of Something or Talking of Nothing? in Gothenburg in January 2014, and two anonymous referees for this journal. For the financial support during the early and middle stages of this paper’s development, I wish to express my gratitude to the Arts and Humanities Research Council and the Royal Institute of Philosophy.
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Schwartzkopff, R. Singular terms revisited. Synthese 193, 909–936 (2016). https://doi.org/10.1007/s11229-015-0777-2
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DOI: https://doi.org/10.1007/s11229-015-0777-2