A Higher-Order Solution to the Problem of the Concept Horse

Abstract

This paper uses the resources of higher-order logic to articulate a Fregean conception of predicate reference, and of word-world relations more generally, that is immune to the concept horse problem. The paper then addresses a prominent style of expressibility problem for views of broadly this kind, versions of which are due to Linnebo, Hale, and Wright.

 
Central to an account of the relationship between language and reality is an account of that between predicates and reality. Frege’s problem of the concept horse (Section 1) threatens one prominent style of account, on which predicates refer to extra-linguistic entities fundamentally different in kind from those to which singular terms refer. Here, I explore the interaction between this classic problem in the analytic tradition’s history and a contemporary view about the interpretation of higher-order logic, developing a higher-order Fregean semantics and metaphysics that is immune to the problem. At the heart of my approach is a seemingly anti-Fregean semantic framework that appears to withhold reference from predicates. I argue (Section 2) that this appearance is misleading: from a higher-order perspective, this seemingly anti-Fregean semantics can be seen to treat predicates referentially. After articulating the conception of higher-order logic on which this argument depends (Section 3), I show how to combine this approach with other central principles of Fregean semantics and metaphysics without giving rise to the concept horse problem (Section 4). I then (Section 5) address a prominent style of objection to views of broadly this kind, versions of which are due to Øystein Linnebo, Bob Hale, and Crispin Wright. The lesson is that the concept horse problem for Fregean conceptions of predicate reference—and of the language-reality relationship more generally—and their apparent conflict with non-referential treatments of predicates, both stem from a failure to take seriously the existential import of higher-order quantification alongside unwarranted insistence on formulating Fregean semantics in first-order terms.

To be clear, my goal is not scholarly reconstruction of Frege’s view. Rather, my goal is to articulate and explore a recognisably Fregean position that captures the core of his conception of predicate reference. Although I do think my proposal captures much of what Frege intended, I won’t argue for that here. I leave it to the reader to decide the extent to which my proposal deserves the label ‘Fregean’.

1. The Problem of the Concept Horse

This section outlines a Fregean conception of predicate reference, and its associated problem of the concept horse.

2. From Non-Referential Semantics to Referential Semantics

This section argues that an apparently non-referential semantic analysis of predicates is really a referential semantics compatible with rejection of (1) (and all other principles of its form).

A semantic analysis of a language’s singular terms and predicates is adequate only if it combines with the language’s compositional rules to assign truth-conditions to all atomic predications in the language. Focussing on the monadic case for simplicity, the natural compositional rule for atomic predication is:[16]

Given Atom, the semantic clauses for terms and predicates must jointly encode two kinds of information. Firstly, they must encode what the terms refer to. The simplest way of doing so is for the semantic clauses for terms to be principles like:

I’ll assume henceforth that the semantic clauses for singular terms take this form. Note that two singular terms flank ‘refers to’.

Secondly, the clauses must encode what the predicates apply to. On the referential Fregean semantics driving the concept horse problem, the semantic clauses for predicates are principles like:

As with (a) and (b), two singular terms flank ‘refers to’. Premiss (1) of the concept horse argument is of this form. However, such clauses alone do not settle what ‘F’ and ‘G’ apply to. Further machinery is needed to connect predicate reference with applicability. The natural candidate is:

The concept horse argument shows that this semantic analysis of predicates is inadequate, given Objecthood, Concepthood, Exclusivity, (2), and (3), for those principles—or suitable analogues in the case of (2) and (3)—are inconsistent with (1), (Fi), and (Gi).

Semantic theories needn’t follow this two-part specification of a referent for each predicate plus an independent theory of application. The semantic clauses for predicates may instead specify their application conditions directly, as in:

Combined with (a) and (b), (Fii) and (Gii) imply, e.g.:

So compositional rule Atom assigns truth-conditions to ‘F(a)’ and ‘G(b)’ thus:

Given matching clauses for all terms and predicates of the language, the result is a systematic assignment of truth-conditions to all its atomic predications.

Semantic theories of this kind appear to treat predicates non-referentially:[17] (Fii) and (Gii) do not specify referents for ‘F’ and ‘G’, and there is no need for a separate theory of the relationship between predicate reference and applicability. Defenders of this approach can thus consistently reject (1), (Fi), (Gi), and all similar principles that use singular terms to specify the referents of predicates. Such principles play no role in semantic theories of this kind, rendering them immune to the concept horse problem. Yet this comes at a cost to the Fregean: it appears to conflict with Reference, and hence with the Fregean theory of predication. I’ll now argue that this appearance is misleading. Semantic theories employing principles like (Fii) and (Gii) really treat predicates referentially, even when coupled with rejection of (1), (Fi), and (Gi).

Why think of singular terms as referential under a semantics that employs clauses (a) and (b)? One might answer: because those clauses employs the notion of reference; they say what ‘a’ and ‘b’ refer to. Yet what does employing the notion of reference amount to? Although the word ‘refers’ appears in both (a) and (b), that’s neither here nor there. That word could have been used to mean all sorts of different things. And we could, if we wanted, reformulate Atom, (Fii), and (Gii) so that ‘refers’ appears in them too:

That is a cheap way of making predicates referential. It renders debate about predicate reference merely terminological. A non-terminological debate requires an account of reference that isn’t tied to any particular mode of linguistic expression.

We obtain a better notion of referentiality by focussing on word-world relations. The idea is that an expression is referential when its semantic role involves a relation with some particular aspect of extra-linguistic reality, different aspects for expressions with different semantic roles. What it is for the expression to function semantically as it does is for it to bear such-and-such relation to such-and-such aspect of reality. The hallmark of such a relationship is the possibility of existential generalisation into the expression’s semantic clause (in an adequate semantic theory for the language). That semantic clause captures the expression’s semantic role; it says how the expression semantically behaves, which governs its interaction with other expressions, via the compositional rules, to determine the semantic properties of complex expressions in which it occurs. When existential generalisation into a semantic clause is possible, the relevant semantic role involves a relation with something, namely whatever was generalised out of the original clause. Whatever else that something may be, it’s certainly an aspect, or constituent, of reality. Hence:

The non-triviality restriction rules out two cases. (i) The term for e can always be existentially generalised out of e’s semantic clause, e.g.: ‘ ‘a’ refers to Al’ entails ‘∃x(x refers to Al)’. (ii) One can existentially generalise a sentence without binding any variable within it, e.g.: ‘ ‘a’ refers to Al’ entails ‘∃x(‘a’ refers to Al)’. Neither case suffices for an interesting word-world relation between ‘a’ and anything else.

By way of illustration, consider the case of singular terms. Semantic clauses like (a) and (b) are open to existential generalisation since they entail:

So by the Criterion: singular terms governed by such clauses are referential. The account correctly classifies singular terms as referential.

What about predicates? It depends on their semantic clauses. Clauses like (1), (Fi), and (Gi) are open to existential generalisation since they entail:

So by the Criterion: predicates governed by such clauses are referential.

What about a semantics based instead around clauses like (Fii) and (Gii), or (Fii*) and (Gii*)? At a first pass, such a semantics appears non-referential. The only expressions in those clauses open to existential generalisation are ‘F’ and ‘G’ themselves. So by the Criterion: predicates governed by such clauses are not referential. Those clauses entail the existence of the predicates they concern, but not anything else to which they’re related. The Criterion thus appears to vindicate the assessment as non-referential of semantic analyses of predicates based around clauses like (Fii) and (Gii). As I’ll now argue, however, that appearance is misleading.

First-order quantifiers bind variables occupying the positions of singular terms. The only expressions in (Fii) and (Gii) open to first-order existential generalisation are ‘F’ and ‘G’. Second-order quantifiers bind variables occupying the positions of predicates.[18] In a second-order setting, ‘F’ and ‘G’ are not the only expressions in (Fii) and (Gii) open to existential generalisation. The predicates ‘eats cookies’ and ‘is generous’ are open to second-order existential generalisation, and (Fii) and (Gii) entail:[19]

So by the Criterion of Referentiality: predicates governed by semantic clauses like (Fii) and (Gii) are referential. Although their semantic role doesn’t involve a relation to any first-order aspects of reality, it does involve a relation to second-order aspects of reality.

We can rewrite some of these principles slightly, to clarify what’s going on. We replace quantifiers over objects with familiar first-order quantifiers. We also employ the following defined, mixed-order predicate:

Note that ‘2-refers’ takes a term or term-variable in its first argument position, and a predicate or predicate-variable in its second. In this respect it’s unlike the reference predicate in (1), (a), (b), (Fi), and (Gi), which takes a term or term-variable in each of its argument positions. To differentiate them, I’ll call the original notion 1-reference, and this newly defined notion 2-reference. My key theses are that: (i) 2-reference is the appropriate notion of reference for predicates; (ii) the Criterion of Referentiality classifies 2-reference as a genuine notion of reference because it expresses a relation between expressions and second-order reality, as revealed by the possibility of existential generalisation into the second argument position of ‘2-refers’; and (iii) a referential semantics for predicates formulated using 2-reference is immune to the concept horse problem because principles like (1) play no role in such a semantics, that is, because such a semantics employs 2-reference in place of 1-reference in its clauses for predicates.

Using this notation, the supposedly non-referential clauses (Fii) and (Gii) become:

Those entail:

The referential similarity between (a), (b), (Fi), and (Gi) on the one hand, and (Fii) and (Gii) on the other is now clear. Just as the former say that ‘a’, ‘b’, ‘F’ and ‘G’ 1-refer to certain first-order aspects of reality, the latter say that ‘F’ and ‘G’ 2-refer to certain second-order aspects of reality. The existence of such aspects is expressed by (a∃), (b∃), (Fii∃), and (Gii∃), which follow by existential generalisation from (a), (b), (Fii), and (Gii). The exact word-world relation differs between the two cases, as does the order of quantification needed to satisfy the Criterion of Referentiality. But some word-world relation is present in each case, and the Criterion classifies predicates governed by clauses like (Fii) and (Gii) as referential. The defined higher-order reference predicate ‘2-refers’ allows us to combine this referential similarity with linguistic similarity, thereby making vivid (Fii)’s and (Gii)’s referential standing. Those semantic clauses specify the 2-referents of predicates explicitly, and without using singular terms to do so. The concept horse problem does not arise because the appropriate reference clause for ‘est un cheval’ is not (1) but:

Note that since ‘is a horse’ is a predicate, this is well-formed.

I’ve argued that a seemingly non-referential semantics for predicates is revealed as a referential semantics, when viewed from a second-order perspective. The argument rests on the following assumption, which hasn’t yet been made explicit: just as first-order existentially quantified sentences express existence claims, so too do second-order existentially quantified sentences; it’s just that they express second-order existence, rather than first-order. Without that assumption, (Fii∃) and (Gii∃) would not capture the existence of (second-order) aspects of reality to which ‘F’ and ‘G’ 2-refer. The Criterion of Referentiality would then be misapplied in the case of second-order quantification, and predicates would not count as referential. What justifies this assumption? And what does second-order quantification mean? The next section addresses these questions. Section 4 then formulates a Fregean theory of predication in higher-order terms.[20]

3. Second-Order Quantification and Existence

What does second-order quantification mean? What is the relationship between second-order quantification and existence? I cannot hope to settle these matters of major controversy here. In answer to the first question, I simply assume without argument a popular and attractive conception of second-order quantification and its relationship to the first-order. My primary concern is with second-order quantification’s role in a Fregean theory of predication given this conception of it, not with a defence of that conception. In answer to the second question, I offer some preliminary justification for my preferred answer, though I won’t pretend that my discussion is utterly decisive.

What do second-order quantifiers mean, and how do they relate to first-order quantifiers? My preferred answer is as follows.[21]

Second-order quantification is a perfectly legitimate and intelligible form of genuine (non-substitutional) quantification whose truth-conditions cannot be explicated in first-order terms. It is not a disguised form of first-order quantification, or quantification over a special kind of object in the range of unrestricted first-order quantification or to which singular terms may in principle refer. Second-order quantification is a sui generis form of (non-substitutional) quantification that must be understood in its own terms or not at all. There is thus a fundamental semantical distinction between these two orders of quantification: first- and second-order quantified sentences express fundamentally different kinds of quantificational truth-conditions. The familiar classification of quantifiers into the substitutional and the objectual is therefore not exhaustive, for second-order quantification is neither substitutional nor objectual (since the objects constitute the range of unrestricted first-order quantification). The semantic role of second-order quantifiers is both genuinely (non-substitutionally) quantificational and yet irreducibly second-order.

This fundamental semantic distinction between the first- and second-order is essential to my proposal’s success. To see why, suppose the truth-conditions of second-order quantified sentences can be explicated in first-order terms, using first-order quantification over some privileged range of objects. Then the second-order existential generalisations—i.e., (Fii∃) and (Gii∃)—of the reference-clauses for predicates—i.e., (Fii) and (Gii)—are semantically equivalent to first-order quantified sentences of (1∃)’s form: ∃xR(‘est un cheval’, x). This ‘R’-predicate is then the appropriate reference predicate for predicates, the result of translating ‘2-refers’ into the first-order language used to explicate the truth-conditions of second-order quantification. So a new singular term t could be introduced for the quantifier’s witness, and the concept horse argument run just as before, substituting t for ‘the concept horse’ throughout. If second-order quantification is to save the Fregean theory of predication, a fundamental semantic distinction between the first- and second-order is required.

The best arguments for this semantic distinction employ variants of Russell’s Paradox and Cantor’s Theorem.[22] The very argument Frege thought had undermined his philosophy of mathematics may thereby help to save his theory of predication from the concept horse problem by motivating the conception of higher-order logic needed for a consistent Fregean theory of predication. The concept horse argument turns on blurring the semantic distinctions between expressions of different semantic types in a manner akin to Basic Law V.[23] Both Basic Law V and the problematic version of the Fregean theory of predication—i.e., the version employing semantic clauses like (1), (Fi), and (Gi)—require an object in the range of first-order quantification for each (possible) predicate, to serve as its referent or extension, different objects for non-coextensive predicates. Despite this common source, however, the nature of the problems is quite different. Whereas elementary logic reveals Basic Law V’s inconsistency, the concept horse problem also requires Objecthood, Concepthood, Exclusivity, (2), and (3).

Do second-order quantified sentences express existence claims? The argument from the Criterion of Referentiality to the referentiality of predicates governed by (Fii) and (Gii) presupposes that second-order formulae like

do indeed express existence claims. So one might respond by denying that they do. Perhaps only first-order quantified sentences expresses existence claims,[24] and so second-order quantification doesn’t really bring ‘ontological commitment’, in Quine’s (in)famous phrase. On this view, the Criterion of Referentiality is misapplied in the second-order case, and the apparently non-referential semantics really is non-referential.

Should we regard only first-order quantified sentences as expressing existence claims? I know of no compelling reason to do so.

The best reason I can find is that the English word ‘exists’ does not permit second-order usage. We can say that particular objects exist, e.g.: ‘Al exists’. We can also say that objects of particular kinds exist, e.g.: ‘cookies exist’. Both uses can be captured with first-order quantification, e.g: ‘∃x(x = Al)’, ‘∃x(cookie(x))’. But we cannot, in English, attribute existence to the predicational aspects of reality generalised by second-order sentences like ‘∃X(X(Al))’. Strings like ‘is a cookie exists’ are not grammatical English. The point is not that second-order quantification isn’t expressible in English. The point is that even if second-order quantification is expressible in English, it doesn’t use the word ‘exists’. So we speakers of English should not regard the contents of second-order quantified sentences as concerning existence.

The problem with this argument is its linguistic focus. It shows, at most, that the English word ‘exists’ can’t be used to express second-order existential generalisation. It remains open that ‘exists’ expresses only the first-order restriction of the background fundamental notion: existential generalisation.[25] If so, it would be more perspicuous to identify existence with the more general and fundamental notion of which ‘exists’ expresses a restriction. On this view, the fully general notion of existence goes with existential quantification regardless of order, and regardless of expressibility with the English ‘exists’.

The issue is liable to appear merely verbal. First- and second-order quantifiers both express kinds of generalisation. The question is whether they both express existence too. What turns on this issue? Why not regard both views as equally acceptable precisifications of our ordinary notion of existence? I see no reason not to do so. One precisification stays closer to ordinary usage, restricting ‘existence’ to the first-order. The other precisification expands ordinary usage to track the more general and fundamental underlying notion.[26] Both precisifications are equally legitimate. Dispute about which is correct is no more substantive than dispute about whether uses of ‘mass’ prior to the discovery of special relativity express relativistic mass or proper mass.[27] Prior usage didn’t distinguish between two interpretations which we now see come apart. As a result, a semantic decision must now be made, with neither choice ruled out by earlier usage as incorrect. There is thus a perfectly legitimate sense in which all orders of existential quantification express existence.[28] On this precisification, the Criterion of Referentiality classifies predicates governed by clauses like (Fii) and (Gii) as genuinely referential because those clauses entail the existence of second-order aspects of reality to which the predicates 2-refer; that’s what’s expressed by (Fii∃) and (Gii∃). Combining that with the view about second-order quantification from the beginning of this section, we obtain two fundamentally different but intimately connected notions of existence, first- and second-order existence.

4. A Higher-Order Fregean Theory of Predication

I’ve argued that a seemingly non-referential semantics for predicates is really a referential semantics, one that sees the semantic roles of singular terms and predicates as involving parallel structures of word-world relations. That argument turned on views about the relationship between first- and second-order quantification, and between second-order existential quantification and existence, that I articulated in the preceding section. It is now time to return to the Fregean theory of predication. I’ll use higher-order resources to formulate a version of that theory that survives refutation by the concept horse argument.[29]

5. Expressibility Problems

My higher-order Fregean proposal is now complete. To close the paper, I consider a prominent style of objection to views of broadly this kind. Section 5.1 outlines some versions of the problem that occur in the literature, to which Section 5.2 responds. Section 5.3 then turns to a more difficult version of the problem and some speculative remarks about how to resolve it.

Acknowledgements

Thanks to Bill Brewer, Craig French, Anil Gomes, David Liggins, Ian Phillips, Scott Sturgeon, Lee Walters, Caspar Wilson, and Al Wilson for comments and discussion. Thanks also to an audience at KCL for their feedback. An Area Editor and anonymous referee for Ergo offered some very helpful suggestions.

References

• Dummett, Michael (1981). Frege: Philosophy of Language (2nd ed.). Duckworth.
• Field, Hartry (1973). Theory Change and the Indeterminacy of Reference. The Journal of Philosophy, 70(14), 462–481.
• Frege, Gottlob (1892). On Concept and Object. Vietreljahrsschrift fur Wissenschaftliche Philosophie, 16, 192–205. (Page references are to the version in Frege 1997: 181–193).
• Frege, Gottlob (1997). The Frege Reader. Michael Beaney (Ed.). Blackwell.
• Gallin, Daniel (1975). Intensional and Higher-Order Modal Logic: With Applications to Montague Semantics. North-Holland.
• Hale, Bob (2013). Necessary Beings: An Essay on Ontology, Modality, and the Relations Between Them. Oxford University Press. http://dx.doi.org/10.1093/acprof:oso/9780199669578.001.0001
• Hale, Bob and Crispin Wright (2001). The Reason’s Proper Study. Oxford University Press. http://dx.doi.org/10.1093/0198236395.001.0001
• Hale, Bob and Crispin Wright (2012). Horse Sense. The Journal of Philosophy, 109(1/2), 85–131. http://dx.doi.org/10.5840/jphil20121091/24
• Krämer, Stephan (2013). Semantic Values in Higher-Order Semantics. Philosophical Studies, 168(3), 709–724. http://dx.doi.org/10.1007/s11098-013-0157-z
• Liebesman, David (2015). Predication as Ascription. Mind, 124(494), 517–569.
• Linnebo, Øystein (2006). Sets, Properties, and Unrestricted Quantification. In Agustin Rayo and Gabriel Uzquiano (Eds.), Absolute Generality (149–178). Oxford University Press. http://dx.doi.org/10.1093/mind/fzu182
• Linnebo, Øystein and Agustin Rayo (2012). Hierarchies Ontological and Ideological. Mind, 121(482), 269–308. http://dx.doi.org/10.1093/mind/fzs050
• MacBride, Fraser (2008). Predicate Reference. In Barry C. Smith and Ernest LePore (Eds.), The Oxford Handbook of Philosophy of Language (422–475). Oxford University Press. http://dx.doi.org/10.1093/oxfordhb/9780199552238.003.0019
• MacBride, Fraser (2011). Impure Reference: A Way Around the Concept Horse Paradox. Philosophical Perspectives, 25 (Metaphysics)(1), 297–312.
• Magidor, Ofra (2009). The Last Dogma of Type Confusions. Proceedings of the Aristotelian Society, 109(1), 1–29. http://dx.doi.org/10.1111/j.1467-9264.2009.00256.x
• Oliver, Alex (2005). The Reference Principle. Analysis, 67, 177–187.
• Prior, A. N. (1971). Objects of Thought. Oxford University Press. http://dx.doi.org/10.1093/analys/65.3.177
• Rayo, Agustin and Timothy Williamson (2003). A Completeness Theorem for Unrestricted First-Order Languages. In J. C. Beall (Ed.), Liars and Heaps: New Essays on Paradox (331–356). Oxford University Press. http://dx.doi.org/10.1111/0029-4624.00288
• Rayo, Agustin and Stephen Yablo (2001). Nominalism Through De-Nominalization. Nous, 35(1), 74–92.
• Rumfitt, Ian (2014). Truth and Meaning. Proceedings of the Aristotelian Society: Supplementary Volume, 88(1), 21–55. http://dx.doi.org/10.1111/j.1467-8349.2014.00231.x
• Schiffer, Stephen (2003). The Things We Mean. Oxford University Press. http://dx.doi.org/10.1093/0199257760.001.0001
• Shapiro, Stewart (1991). Foundations Without Foundationalism: A Case for Second-Order Logic. Oxford University Press.
• Trueman, Robert (2014). The Concept Horse with No Name. Philosophical Studies, 172(7), 1889–1906. http://dx.doi.org/10.1111/j.1520-8583.2003.00017.x
• Trueman, Robert (2016a). Realism and the Truth in Nominalism. Manuscript in preparation.
• Trueman, Robert (2016b). Substitution in a Sense. Manuscript in preparation.
• Williamson, Timothy (2003). Everything. Philosophical Perspectives, 17(1), 415–65.
• Williamson, Timothy (2013). Modal Logic as Metaphysics. Oxford University Press.
• Wright, Crispin (1998). Why Frege did not Deserve his Granum Salis: A Note on the Paradox of ‘the Concept Horse’ and the Ascription of Bedeutungen to Predicates. Grazer Philosophische Studien, 55, 239–263. (Reprinted in Hale & Wright 2001: Chapter 3).
• Wright, Crispin (2007). On Quantifying into Predicate Position. In Mary Leng, Alexander Paseau, and Michael Potter (Eds.), Mathematical Knowledge (150– 174). Oxford University Press. http://dx.doi.org/10.5840/gps19985522

Notes:

1. I leave the restriction to semantically non-defective expressions tacit henceforth, and likewise for the “typically” qualification needed to accommodate talk about language. I also ignore the possibility of semantically non-defective yet non-referring expressions, putative examples of which include ‘Santa’, ‘Zeus’, and ‘Mary Poppins’.

2. I use ‘may in principle’ to abstract away from the merely practical and cognitive linguistic limitations of finite beings like ourselves. Read ‘a singular term may in principle refer to y’ as meaning roughly: singular terms are not precluded from referring to y by their semantic role alone. A singular term may in principle refer to y despite it being impossible for any such term to refer to y, for the impossibility may flow not from the semantic role of singular terms alone, but also from the impossibility of, say, cognitive contact between language users and y. (Hale 2013: Chapter 1, 193–195) employs a similar but more restrictive modality in his Fregean explication of objecthood.

3. Of course, Objecthood and Concepthood are not the only possible semantic analyses of ontological categories. See Hale (2013: 31–34) for another.

4. Frege (1892: 193) uses the unity of the proposition to motivate Exclusivity. Since that problem is highly obscure, I won’t consider it here.

5. MacBride (2011) resolves the problem by, in effect, rejecting Determination.

6. Determination is a cousin of Crispin Wright’s (1998: 73) Reference Principle (RP): if expressions e, e∗ are co-referential, then substitution of e for e∗ preserves grammaticality in all contexts, and preserves truth-value in all extensional contexts. Since the exact relationship between RP and Determination is somewhat complicated, I won’t examine it properly here. Moreover, the theoretical basis for RP is unclear. It entails that expressions with the same referent belong to the same grammatical category. Why should reference constrain syntax in that way? I see no obvious reason why it should, and Schiffer (2003: 93) and Oliver (2005) argue that it doesn’t, though see Trueman (2016b) for a defence of RP based on a more sophisticated notion of substitution. My own view is that Determination captures whatever theoretical support RP may enjoy by replacing RP’s route from reference to syntax with one from reference to semantics. Note also that Determination is neutral about how rich the referents of expressions must be. For example, an extensionally individuated conception of predicate referents is inadequate for modal languages, where grammaticality-preserving substitution of coextensive predicates can affect truth-value. That shows that Frege’s extensional conception of predicate referents was inadequate for modal languages. It does not show that Determination is flawed, since an intensionally individuated conception of predicate referents is adequate for modal languages.

7. Dummett (1981: 181, 190–191, 210 and elsewhere) discusses the conception of reference as semantic role.

8. Suppose expression e refers to object o. By Objecthood: e is co-referential with some potential term t. By Determination: e functions semantically just like t. So if e refers to an object, e functions semantically as a term. Likewise mutatis mutandis for concepts and predicates.

9. Qualification: only ambiguous expressions can function semantically as both singular term and predicate, with those different semantic functions corresponding to different disambiguations. Alternatively, no semantically individuated expression can function semantically as both singular term and predicate.

10. Frege (1892: 183–184) considers and rebuts the best apparent examples.

11. I say that Linguistic Exclusivity is the best explanation, not that it is the only explanation. One might instead appeal to incompatibility between the syntactic functions of singular terms and predicates. But surely syntax is up to us in a way that semantics is not. If I choose to use e so that it can grammatically occur in the positions of both singular terms and predicates, then surely it thereby does come to have the syntactic function of both singular terms and predicates in my idiolect. Whether both kinds of sentence are meaningful and express truth-conditions (under a single assignment of semantic properties to e) is a different matter entirely. Whereas we can choose what strings count as grammatical in our own idiolects, we cannot choose what truth-conditions there are to express, or how they can be compositionally generated from the semantic properties of other expressions.

12. See Wright (1998) and MacBride (2008: Section 4.2) for critical discussion of key attempts to specify the referents of predicates without using singular terms. My proposal is another attempt to do so.

13. Wright (1998: 79) suggests a description-quantifier binding variables in predicate position. But then ‘the concept horse’ cannot grammatically occupy the second argument position of ‘refers to’, which requires a singular term, and (1) doesn’t instantiate (1∃).

14. Assumption: description-quantifiers have existential import, so ^⌜(the x : A)B^⌝ entails ^⌜(∃x : A)B^⌝.

15. What about the conception of reference as encoding semantic role that I used to motivate Exclusivity, via Determination? The difficulty is that there may be multiple ways to fix semantic properties, other than via assignments of extra-linguistic referents. The conception ensures that if an expression refers, its referent determines its semantic properties. But the conception alone doesn’t ensure that all expressions refer.

16. World-relativised analogues of Atom and the other semantic principles that follow are needed for languages containing modal vocabulary. I ignore this complication in the main text.

17. Wright (1998: 76) regards clauses like (Fii) and (Gii) as non-referential, or at best only implicitly referential, and rejects them on those grounds. I’ll argue that Wright’s wrong to do so.

18. An excellent overview of second-order logic is Shapiro (1991).

19. Assumption: the following rule of second-order existential generalisation is valid: from A, one may infer ^⌜∃VA^V/Φ⌝, where A^V/Φ results from A by substituting zero or more occurrences of the monadic predicate Φ with the monadic second-order variable V, so that all and only those newly substituted in variables are bound by the initial quantifier in ^⌜∃VA^V/Φ⌝. This is standard amongst advocates of second-order logic, e.g., Shapiro (1991: 66), Hale (2013:181).

20. Shortly after this paper had been accepted for publication, I encountered Trueman (2016a), the main ideas of which are in many ways very similar to those just outlined. Trueman and I both regard 2-reference as the appropriate notion of reference for predicates. Unlike me, however, Trueman has no role for the Criterion of Referentiality. He focusses instead on the fact that 1-reference and 2-reference both allow formulation of disquotational principles, using that to justify regarding both as genuine notions of reference. He also uses the impossibility of formulating disquotational principles for predicates in terms of 1-reference to argue against semantic clauses like (1), (Fi), and (Gi). I’d like to thank Trueman for some very interesting correspondence about these issues.

21. For discussion of this kind of approach, see Prior (1971: ch3), Rayo and Yablo (2001), Williamson (2003: esp. Section 9), Rayo and Williamson (2003: Section 1), Wright (2007), MacBride (2008: Section 19.2.2), and Williamson (2013: 254–261).

22. See Williamson (2003: Section 9), Rayo and Williamson (2003: Section 1), and Linnebo (2006). By way of illustration, here’s a quick version of one such argument. Assume that second-order quantification and unrestricted first-order quantification are both legitimate. Were second-order quantifiers in the same kind of semantic business as first-order quantifiers, a first-order account of what second-order quantifiers range over would be possible. But a version of Cantor’s Theorem shows that to be impossible. Such an account would require a first-order representative for every second-order “entity”. In an extensional setting, that minimally requires an f from the first-order to the second-order such that (α) ∀X∃y(X ≡ f(y)), where ‘≡’ expresses coextensiveness. Note that ‘ f ’ forms a predicate from a term, since it’s from the first-order to the second-order. First set Cx = _df: ¬f(x)x. Pick any first-order y. Either Cy or ¬Cy. Consider the first case: Cy. By definition of C: ¬f (y)y. So C ≢ f(y). Now consider the second case: ¬Cy. By definition of C: ¬¬f(y)y. Hence f(y)y. So again C ≢ f(y). Either way: C ≢ f(y). Since y was arbitrary, we’ve got a counterexample to (α): ¬∃y(C ≡ f(y)). So f assigns no first-order representative to C. Since f was arbitrary, we can generalise: for every f assigning first-order representatives to second-order “entities”, some second-order C lacks first-order representative under f , and since the first-order quantifiers in the argument are unrestricted, no f ∗ with more inclusive domain is available. So a first-order account of what second-order quantifiers range over is impossible, and the truth-conditions of second-order quantification cannot be explicated in first-order terms. It is also worth noting that, given the initial assumptions, this argument undermines all responses to the concept horse problem that permit singular terms with which to refer to—and hence objects corresponding to—each concept. Such proposals include Wright (1998: 85-90), Hale and Wright (2012), and Hale (2013: 31–34); though Liebesman (2015: Section 5) disagrees.

23. Think of Basic Law V as the second-order principle: ∀X, Y(ext(X) = ext(Y) ↔ X1 − 1Y). ‘ext’ is the ‘extension of’ functor, which forms a singular term from a predicate or predicate variable. ‘X1 − 1Y’ says that the Xs are in one-one correspondence with the Ys.

24. More carefully: perhaps non-first-order quantified sentences expresses existence claims only if interpreted using first-order quantification. That’s ruled out by the fundamental semantic distinction between first- and second-order quantification outlined at the beginning of this section.

25. Although convenient, this probably isn’t quite the right way to put it. One can analyse English ‘exists’-sentences using first-order existential quantification without being committed to ‘exists’ expressing such quantification. It follows only that certain first-order quantified sentences express what English ‘exists’-sentences express. The point in the text is then that even if no second-order quantified sentence expresses what any English ‘exists’-sentence expresses, it remains open that English ‘exists’-sentences express only the first-order restriction of the fundamental background notion of what’s expressed by existentially generalised sentences regardless of order.

26. Section 5.3 discusses a Fregean problem with this notion, and some potential solutions.

27. See Field (1973) for discussion.

28. More strongly: we should reject the ordinary notion of existence as lacking distinctive theoretical import because: (i) our ordinary word ‘exists’ can be precisified in two equally good ways; and (ii) both precisifications mark distinctions that are readily marked in other, less emotive and loaded, ways.

29. As far as I’m aware, the closest extant proposal is due to Michael Dummett (1981: 211–218), which Dummett attributes to Frege. There isn’t space here to properly compare our approaches. One key difference is that Dummett’s goal is to explicate Frege’s semantics and metaphysics within natural language. This brings commitments in natural language semantics—about the interpretations of English quantification and the copula inter alia—that I avoid. It is also unclear how to extend Dummett’s proposal beyond the ordinary predicates considered so far to the rest of Frege’s type-theoretic hierarchy. Section 5 discusses the corresponding generalisation of my proposal and addresses some recent criticisms of Dummett in Hale and Wright (2012: 96–98) and Hale (2013: 27–28).

30. Might any natural language expression do the work of ‘concept’? Following Dummett (1981: 215), we might try: ‘for anything whatsoever, either it . . . or it’s not the case that it . . . ’, where the dots mark a single argument position reserved for a predicate. Wright (1998: 78–79) observes that Dummett’s proposal’s adequacy depends on the Law of Excluded Middle (LEM). So he suggests a replacement: ‘either nothing . . . or possibly something . . . ’. Just as Dummett’s proposal involves a non-trivial logical commitment, however, Wright’s involves a non-trivial metaphysical commitment: it precludes concepts that necessarily universally violate LEM, that is, concepts X such that, necessarily, for every object y, X(y) ∨ ¬X(y) fails (in whatever sense LEM was initially thought to fail and which motivated Wright’s proposal). It’s only a small step from countenancing concepts that violate LEM to such necessary violators of LEM. An alternative that permits such concepts is ‘for any object whatsoever, it . . . iff it . . . ’, reading ‘iff’ as the material biconditional. Yet even that is adequate only given the Law of Non-Contradiction, for each of its instances is equivalent to something of the form ∀x¬(F(x) ∧ ¬F(x)). More generally, it is doubtful that any adequate analysis of ‘concept’ can or should entirely avoid non-trivial commitments of some kind.

31. Since the English ‘whatever’ is arguably first-order, this is only a rough gloss on Concepthood*.

32. Rumfitt (2014: Section 5) applies a similar notion of rejection to a Liar-like paradox.

33. Talk about object-concepts used to explicate the view will have to be rejected as contentless too. So what kind of contents are we denying the existence of? Section 5 considers some related expressibility problems for higher-order Fregeans.

34. Alongside rejection of Exclusivity*/** as contentless, this approach also requires rejection as contentless of any other forms of words that could be used to talk about object-concepts. Notably, that includes a cross-order taking singular terms in one argument position and predicates in the other. In an interesting recent paper, Trueman (2014: §6) uses, in effect, restrictivism about semantic evaluability to argue against cross-order identity. His argument turns on the impossibility of formulating reflexivity for cross-order identity. It’s impossible because it would require a single order of variable in each argument position of the cross-order identity predicate. Restrictivism then implies that there is no cross-order identity relation. I don’t think that’s decisive. Reflexivity involves a single quantifier binding two occurrences of the same variable, e.g.: ∀x(x = x); ∀X(X = X). Advocates of cross-order identity might reasonably take that to show that reflexivity is essentially intra-order. If so, then inability to formulate cross-order reflexivity is unproblematic for cross-order identity. More problematic, it seems to me, is the impossibility of formulating indistinguishability for cross-order identity, given restrictivism about semantic evaluability; for then ‘∀Φ(Φ(X) ↔ Φ(y))’ is not semantically evaluable, though a predicate expresses identity only if it guarantees complete indistinguishability.

35. This will need complicating to accommodate cross-level expressions like ‘2-refers’ and the cross-level relations to which they apply. See Williamson (2013: 221–222) for a suitable approach, drawn from Gallin (1975: 67–78).

36. (A)–(C) are from Hale and Wright (2012: 97, 98, 104 respectively). (A) and (B) are from Hale (2013: 27–28). (D)–(F) are from Linnebo (2006: §6.4). I’ve changed terminology and wording to fit present usage.

37. Here’s a concrete example. To aid readability, I mix formal and informal vocabulary, and take some grammatical liberties. Compositional principle for second-level atomic predications ^⌜P²(P¹)^⌝: ^⌜P²(P¹ )^⌝ is true iff P² 2-applies to the 2-referent of P¹ . ‘2-applies’ is the higher-level (and cross-level) analogue of the familiar ‘applies to’; its second argument position is for first-level predicates. Semantic clause for monadic ‘F¹ ’: ‘F¹ ’ 2-refers to is a horse. Semantic clause for monadic ‘G²’: ∀Z¹(‘G²’ 2-applies to Z¹ ↔ H²(Z¹); that is, ‘G²’ 2-applies to every 1-concept that H² s (where ‘H²’ is an interpreted second-level predicate of the metalanguage). These semantic clauses and compositional principle together assign the following truth-condition to ‘G²(F¹)’ without the need for any additional semantic principles such as an analysis of 2-application: ‘G²(F¹)’ is true iff ‘G²’ 2-applies to the 1-concept that ‘F¹’ 2-refers to; iff H² (is a horse). The reference predicate ‘3-refers(x, Y²)’ for second-level predicates is defined by replacing ‘‘G²’’ with ‘x’ and ‘H²’ with ‘Y²’ in the semantic clause for ‘G²’ to obtain ‘∀Z¹(x 2-applies to Z¹ ↔ Y²(Z¹)’.

38. Hale and Wright (2012: 104) use not (B) but ‘first-level functions can take only objects as arguments’. Since the functions are a subclass of the relations, this amounts to a claim about relational 1-concepts. And since it’s couched in terms of what these functions take as arguments, rather than what they map to the True, Hale and Wright’s claim is slightly more general than (B*). Assuming classical logic, a better regimentation of (a monadic version of) Hale and Wright’s principle is thus: (∀X¹ : 1-concept²(X¹))∀y((X¹(y) ∨ ¬X¹(y)) → object¹ (y)). I ignore this complication in the main text.

39. Objection: (B*) doesn’t capture (B)’s intended force; because it employs first-order quantification over objects, it doesn’t prevent 1-concepts from holding of non-objects that aren’t first-order aspects of reality, whereas (B) does. Reply: this goes the same way as for (O) and (C) in Section 4.1. If restrictivism about semantic evaluability holds, there are no contents attributing 1-concepts to non-first-order aspects of reality which need ruling out as false. If restrictivism fails, we can supplement (B*) with principles like the following to capture (B)’s intended force: (∀X¹ : 1-concept²(X¹))∀Y¹ (X¹(Y¹) → object¹(Y¹)). Either way, the objection fails. Similarly for the parallel objection to (C*).

40. The same goes for these generalisations of (A)–(C): ∀i(i-level predicates (i + 1)-refer to i-level concepts); ∀i(i-level concepts are had only by (i − 1)-level concepts). I don’t discuss these explicitly because they raise no new issues beyond those of (D)–(F).

41. Note that since the 1-reference predicate requires a singular term (or first-order variable) in its second argument position, no contents predicate, of any singular term, 1-reference to a concept (of any level). So no such contents need ruling out as false to capture the intended import of (E). Likewise for 2-reference to things other than 1-concepts, 3-reference to things other than 2-concepts, etc. See Section 4.1’s discussion of (O) and (C) for more.

42. Exception: each type’s semantic role involves some word-world relation, in line with the Criterion of Referentiality. Section 5.3 discusses this issue.

43. The proposal draws on Wright (2007: Section 2).

44. In the existential case, the relevant kind of argument forms will plausibly be: (i) from an instance to its existential generalisation; and (ii) from an existential generalisation to whatever follows from a suitably arbitrary instance.

45. When metalanguage doesn’t contain object-language, we can formulate similar true principles in which the second argument-position of ‘i-refers’ is occupied by a translation into the metalanguage of the expression named by the occupant of its first argument-position. For example: 2-refers(‘est un cheval’, is a horse). For further development of the connection between reference and disquotation, see Krämer (2013) and Trueman (2016a).

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