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Erschienen in Ausgabe: No. 25 (2/2006) Letzte Änderung: 27.01.09

Some Fregean Considerations on Predicates and Their Reference

von Ari Maunu


The aim of this paper is (i) to defend Frege's view that the referents of predicates are certain kinds of functions, or "concepts", i.e. incomplete entities, and not their extensions (i.e. sets of objects described by those predicates); and (ii) to justify, by a natural augmentation of Frege's semantic theory with modal ingredients, Frege's position that the sameness between concepts, or property-sharing, turns only on the sameness of extensions. Several problems with the doctrine that a predicate's extension is its referent are presented, including the regress argument and an argument from the modern philosophy of language related to natural kind terms. In this connection, it is also pointed out that all referential expressions are in a sense rigid.

1. Introduction

The aim of this paper is (i) to defend Frege's view that the referents of predicates, such as '(is a) horse', are certain kinds of functions ─ or "concepts" (Begriffe) as Frege calls them ─ i.e. incomplete or "unsaturated" entities, and not their extensions (i.e. sets of objects described by those predicates); and (ii) to justify Frege's position that the sameness between concepts, or property-sharing, turns only on the sameness of extensions. After preliminary Sections 2 and 3, dealing with Frege's approach to concepts as functions and the notion of sharing a property, respectively, in Section 4 some arguments, including the traditional regress argument, are given against the view that the referent of a predicate is to be identified with its extension. In Section 5 I present a further argument from the modern philosophy of language related to natural kind terms. In this connection, it is pointed out that all referential expressions are in a sense rigid. In the final section I defend further Frege's view that the referent of a predicate is a function, and also substantiate, by a natural augmentation of Frege's semantic theory with modal ingredients, his position that property-sharing turns only on the sameness of extensions.

2. Frege's concepts as functions

For Frege, there is a fundamental, undefinable difference between Gegenstände, or objects, and functions. In his paper "Function und Begriff" (KS 125-42 / CP 137-56, 1891)i Frege explains this difference by means of an arithmetical example as follows: A function such as 2x³+x, where 'x' indicates an empty place, or is a place-holder, is incomplete or "unsaturated". For it does not designate an object ─ only after it is properly supplemented, we obtain an object, e.g. the number 132, when we supplement this function by the number four (i.e. when we apply this function to 4 as an argument).

That this notion of a mathematical function can be applied more generally is one of Frege's greatest ideas (especially with respect to the development of logic, for it leads directly to the introduction of quantifiers). For example, x²=4 may be regarded as a function as well, viz. the function that gives as the result the truth value the True for the arguments 2 and -2 and the truth value the False for all other arguments. Frege calls functions that return a truth value on application Begriffe, or concepts.ii Those objects that give the True as the result when a concept is applied to it, are said to fall under or to be subsumed under that concept.

Concepts may also be used outside mathematical discourse; for example, x is mortal is a function that returns the True when applied to mortals and the False for the rest, i.e. all and only mortals fall under being a mortal. I this connection, in particular, Frege's "plug-in" notion of concepts, or his view that objects and "gappy" concepts are "made for each other" (NS 193 / PW 178, 1906), is important: it appears that the classic question of the nature of the connection between individuals and whatever it is that is predicable of them ─ the question of "the unity of the proposition" ─ does not even arise.

Besides one-place or unary concepts (and other functions) there are, of course, also many-place concepts, or relations, such as x>y and x gives y to z. I shall call relations concepts as well. Frege often calls an expression of a concept, i.e. an expression that has a concept as a referent (Bedeutung), a Begriffswort ─ I shall use the word predicate for this purpose.

Concepts and other functions are non-objects ─ however, extensions or, in general, what Frege calls Wertverläufe, or courses of values, are objects that correspond to concepts or, in general, to functions. The extension of the concept being mortal, for example, is the set of mortal things; the extension of x²=4 is the set {-2,2}.iii The general notion of a course of values of a function may be seen as a generalization of that of extension of a concept: The course of values of a concept is its extension, while the course of values of a function that is not a concept is a logical object that is the same for any two functions which always return the same value for the same argument ─ thus for instance x²-1 and (x+1)(x-1) have the same course of values (see here especially GG1 §§9-10). For my present purposes I shall use the word extension in an extended sense to cover also functions that are not concepts: I say that two such functions have the same extension if their courses of values coincide.

Frege contends that we cannot sensibly talk about identity between concepts, for identity can hold only between objects, and, as indicated, "concepts are non-objects". However, he says that a relation "corresponding to identity" holds between concepts whenever the extensions of these concepts coincide (see, for instance, KS 184 / CP 200 (1894), NS 131-3 / PW 120-2 (ca. 1892-5), and NS 197-8 / PW 182 (1906)). This is based on the following consideration (see especially NS 128, 197-8 / PW 118, 182): For singular terms 'a' and 'b' it is clear that they have the same referent if they are salva veritate (i.e. truth-preservingly) substitutable with each other in all fully extensional positions in all statements. The analogous criterion for the sameness of concepts turns on the sameness of extension, for two predicates (or expressions of functions) are salva veritate substitutable in all extensional positions just in case they share the extension. Thus, it is natural to hold that if we say that there is between concepts a relation corresponding to identity of objects, this holds whenever these concepts have the same extension.

3. Sharing a property

The question of the relation between concepts and properties arises. It is natural to say that an object may have a property, but to say that it may "have a concept" is unnatural, for a concept is a function from objects to truth values, and it does not make much sense to say that an object "has" such an entity. However, there is an obvious correspondence between the talk of concepts and of properties: To say that an object has the property of being a horse, for example, is to say that it falls under the concept being a horse. I shall speak rather freely of items such as being an F either as concepts or as properties.

In colloquial English, it is perfectly natural to say, "Berlin and London share some properties, for example, they are both cities". In Fregean terms this might be put as follows: Berlin and London share the property of being a city ─ the concept being a city is truly applied both in "Berlin is a city" and "London is a city". It may now be asked whether this intuitive talk of property-sharing is the same as Frege's extension-based "sameness" of concepts I explained above. If we assume this, Frege's extension criterion appears as too liberal when compared to the natural notion of sharing a property ─ for even though it for some reason were the case that, say, all and only left-handed persons were musical, and thus musicality and left-handedness were the "same" concept on Frege's standards, it is not natural to say, on the basis of Ann's being musical and Bill's being left-handed, that they share a property. However, as we shall see below, on a closer look into this matter of "same" property (concept), Frege's extension criterion turns out to be entirely natural.iv

4. The referent of a predicate is not its extension

The extension of a "natural" predicate, such as 'horse' or 'bald', varies from a possible world to another. By this it is of course not meant that the objects that happen to be called horses or bald by the denizens of some possible world may be different from what we call horses or bald, but that the extensions of our words 'horse' and 'bald', when used predicatively, vary with respect to possible worlds ─ that is, quite simply, that there might have been different set of horses and different set of bald persons from what there actually are.

The extension of an expression is often regarded as the referent of that expression. For singular terms, this is entirely natural: The extension of 'Kofi Annan' is Kofi Annan, which is also the referent of this name, and the extension of 'the actual shortest spy at New Year 1983, Berlin time' is (or was) a certain person that is (was) the referent of the singular term given. However, it is problematic to take the referent of a predicate as its extension, as is very often done. I shall argue that we should prefer the Fregean option, i.e. the identi­fication of the referent of a predicate with a concept,v appropriately extended to take modalities into account. As we shall see, this means that the referent of a predicate should be equated, not with its extension, but with its Carnapian intension (i.e. with a function from possible worlds to (tuples of) objects), for an object a belongs to the extension-in-the-world-w of the predicate 'being an F' (to utilize Carnapian terms) just in case (in extended Fregean conception) 'a is an F in the world w' refers to the True. (This may be surprising since Carnap (1947) wanted to identify his intensions with Fregean Sinne (senses) and not with Fregean Bedeutungen (referents).)

The claim,

(1) The referent of a predicate is its extension,

runs at least into the following difficulties:

(i) To say that the referent of a predicate, e.g. 'horse', is its extension, is to say that it is the set of things described by this predicate ─ e.g., in the case of 'horse', the set of horses. It is thus said that the referent of 'horse' is the same as the referent of 'the set of horses'. Now, since the predicate 'horse' appears in the expression 'the set of horses' as well such an explanation seems seriously circular.

Perhaps the "extensionist" is tempted to object at this point that if this is a problem it is a problem for Frege as well: it appears that for Frege the referent of 'horse' is the same as that of 'the concept being a horse'; "but the predicate 'horse' appears in the latter, etc." Without going into Frege's notorious "the concept horse problem" here,vi I note only that Frege denies the appropriateness of the expression 'the concept horse' (as an attempt to refer to a concept), and, furthermore, Frege has a justification for this denial, while the extensionist, it seems, cannot have any grounds for the claim that 'the set of horses' is somehow inappropriate. (Ultimately, however, perhaps it should it be admitted that (i) is not a particularly convincing argument against (1), for if Frege is allowed to say that the referent of 'horse' is "just a certain function" f such that f(x) returns the True just in case ..., the "extensionist" should be allowed to say just as well that it is "just a certain set" S such that x ∈ S just in case ....)

(ii) The true statement,

(2) Bucephalus is a horse (in the actual world α at the moment of time t),

is fully extensional. In fully extensional statements, when an expression is substituted with another expression having the same referent as the first expression, the truth value of the statement does not change. The claim (1) obviously means that the referent of 'horse' (e.g. in (2)) is the same as the referent of 'the set of horses'. Thus, on (2) and substitutivity,

(3) Bucephalus is the set of horses (in α at t),

─ or, perhaps, "Bucephalus the set of horses (in α at t)" ─ should be true, which it is not.

(iii) Perhaps it is now objected that such problems with substitutions are only to be expected since 'horse' (in (2)) is a predicate (general term) while 'the set of horses' is a singular term (i.e. Eigenname in Frege's terminology). However, far from serving as an objection to the point in (ii), this "objection" reinforces that point: 'the set of horses' is a singular term, 'horse' in (2) is not; ergo, the referent of 'horse' in (2) is not the set of horses (i.e. the extension of 'horse').

(iv) Of course, by the contention (1) it is usually not meant or implied that (3) says the same as (2) does, but rather that the "real form" of (2) is,

(4) The extension of 'horse' contains Bucephalus

(from now on, I leave the world-time indication "in α at t" as implicit), or,

(5) The set of horses contains Bucephalus.

(This is also what is codified in the standard predicate logic: The atomic sentence "a is an F" is true just in case the object that is the interpretation of 'a' is contained in the set that is the interpretation of 'F', i.e. the extension of 'F'). Inevitably, the question as to the referent of the predicate 'contains' in (4)-(5) arises. Under (1), it would be at this point rather odd, and, moreover, entirely unjustified, to acknowledge incomplete entities in the manner of Frege ─ i.e. acknowledge an incomplete entity such as x contains y as the referent of the 'contains' predicate ─ thus, it seems clear that the advocates of (1) must hold that (4) "really" says something like the following:

The extension of 'contains' contains the pair: the extension of 'horse', Bucephalus.

Unfortunately, this still has 'contains' as a predicate: We have an infinite regress (in such a way that at no stage an explication of the matter at hand is reached).

While Frege himself presents, in effect, this traditional regress argument in his paper "Über Begriff und Gegenstand" (KS 177-8 / CP 193, 1892), it is sometimes claimed that such an argument applies to Frege's plug-in account of "the unity of the proposition" as well. More precisely, some commentators have implied that since 'Bucephalus is a horse', for instance, just means, for Frege, that Bucephalus falls under the concept being a horse, or that being a horse applies to Bucephalus, the regress problem can be raised for 'falls under' or 'applies'. However, as just indicated Frege gives the relevant regress argument (against views without plug-in) which suggests that he himself, at least, thinks it is not applicable to his treatment. Indeed, on Frege's view 'falls under' and 'applies' are entirely superfluent ─ and even misguided, in view of what Frege says about 'the concept horse' ─ ways of stating the plug-in in 'Bucephalus is a horse'; the connection is built into what the predicate refers to, and though we may use the "falling under" description it is not essential to introduce this further concept (relation). (See, for instance, NS 193 / PW 178 (1906), where Frege explicitly denies that "the relation of subsumption is a third element supervenient upon the object and the concept".) In contradistinction, in the extension view ─ as well as in some other well-known views, which utilize terms like instantiation and participation ─ no such explanation is available; in these accounts the relevant connection must be described, so to speak, "from the outside", which is precisely what creates the problem.

All in all, it seems that Frege is correct in his insistence that it is absolutely essential to recognize incomplete or "unsaturated" ingredients, i.e. functions.

5. Further argument from the modern philosophy of language

Within the confines of the theory of direct reference it may seem tempting ─ and many writers have been so tempted ─ to characterize natural kind terms, like just 'horse' (when used predicatively), by saying that they are rigid in the sense of Kripke (1971, 1972) and others. The rigidity of a term is customarily taken to mean that the referent of that term stays constant from one possible world to another. There is, however, a problem with this, noted, for instance, by S. P. Schwartz (1977, esp. 37-8): If we view the referent of a natural kind term such as 'horse' as the extension (in various possible worlds) then 'horse' (or any other natural kind term) appears not to be rigid any more than any other predicate. Thus it seems again, at least prima facie, that the referent of a predicate cannot be identified with its extension, since this extension may be different in distinct worlds. Schwartz (1977, 37) even holds that "explaining exactly what the reference of a rigid natural kind term is" is one of "severe difficulties [... the theory of direct reference] must face."vii A natural solution to this difficulty will be presented below.

6. Property-sharing again

In fact, all referential expressions appear to be rigid, if rigidity means referential constancy. The basic reason for this lies in the following fact: Since our language and our conceptions are under consideration, when we talk and think about something by means of a referring expression, it is the same something we are thinking and talking about, regardless of the possible world, whether actual or nonactual, we are relating to. Thus, the property of being bald, for instance, is the same no matter which world we are considering ─ and it is no less the same in the case of 'bald' than it is in connection with natural kind terms such as 'horse'.viii This claim of the constancy of reference (through worlds) is particularly striking in the case of definite descriptions, but it is still defensible, for what is meant in ordinary speech by a definite description such as 'the quickest horse' is, I submit, either the genuine singular term 'the quickest horse of the world w' (where 'w' usually refers to the actual world) or the functional expression 'the quickest horse of a world x', and these expressions are both rigid (i.e. have the same referent with respect to every world). That we always remain within our own language is the natural reason for the rigidity of all (referential) expressions (on the assumption that rigidity means referential constancy), including predicates and definite descriptions.

Perhaps this view sounds so unconventional that it needs further elaboration. Turning, again, to Frege's writings, we find in KS 313-4 / CP 329-30 (1906) the following highly interesting passage:

Now a real sentence [eigentlicher Satz] expresses a thought. The latter is either true or false: tertium non datur. Therefore that a real sentence should obtain under certain circumstances [Umstände] and not under others could only be the case if a sentence could express one thought under certain circumstances and a different one under other circumstances. This, however, would contravene the demand that signs be unambiguous [...]. A pseudo-sentence [uneigentlicher Satz] does not express a thought at all; consequently, we cannot say that it obtains. Therefore it simply cannot happen that a sentence obtains under certain circumstances but not under others, whether it be a real sentence or a pseudo-sentence. [...] A sentence that holds only under certain circumstances is not a real sentence. However, we can express the circumstances under which it holds in antecedent sentences and add them as such to the sentence. So supplemented, the sentence will no longer hold only under certain circumstances but will hold quite generally.

Frege states here very candidly his view that sentences, properly understood as "real", are stable with respect to truth value (which is really a consequence of such stability of the respective thoughts). In view of Frege's well-known aversion to modal (alethic) issues (in the sense of 'modal' familiar to us) it might be claimed that Frege's "circumstances" are here just different contexts of the actual world. However, the passage does seem to contain counterfactual considerations, with the use of 'Umstand' as a sort of pretheoretic proxy for possible world. It is thus, arguably, not too far-fetched to construe the central idea of this passage as follows: When it is said, "The quickest horse in the world is quicker than Bucephalus", the content of what is said involves, on the most natural interpretation, the actual world ─ the fully (or at least more fully) explicated or "supplemented" sentence might in this case be taken to be "The actual quickest horse in the world is actually quicker than Bucephalus" (in any case, it is certainly not "No matter with respect to which possible world (circumstance) we think about it, the quickest horse ...").

Now, we see that Frege's extension criterion for the "sameness" of concepts (and other functions) is entirely natural. The customary prima facie counter-example to Frege's extension criterion is that even though the (actual) extensions of the concepts x is a creature with a heart and x is a creature with kidneys coincide (or let us suppose so), these are not by any means the "same" concept (or property) because, first, there certainly could be a heartless creature with kidneys (or a creature with a heart but without kidneys), and secondly, one can consistently believe that a certain given creature has a heart but not kidneys (or vice versa). The answer to the first objection is that the concepts we are dealing with here are in fact the following relations: x is a creature with a heart in a world y and x is a creature with kidneys in y. If there is a world w with a heartless object having kidneys these relations do not share the extension ─ thus, according to Frege's extension criterion they are not the "same" relation either. The standard (and correct) answer to the second objection is that Frege is particularly famous for his theory of senses (and thoughts); and these are certainly epistemic notions for Frege. Thus, if it is said that the extension-sharing ─ and thus reference-sharing ─ predicates such as 'groundhog' and 'woodchuck' are epistemically to be separated, Frege's natural account for this is that although they have the same referent (i.e. a certain function from world-object pairs to truth values), they do not share a sense.

Thus, we can retain the Fregean position that the referents of predicates, such as 'horse', are concepts, i.e. functions from objects (or tuples of objects) to truth values. When modalities are taken into account, several distinct concepts may be considered in connection with a predicate such as 'horse', e.g. the concept x is a horse in the actual world and the relation (two-place concept) x is a horse in y. Here, we may say that x is a horse in (a world) y is the overall concept of being a horse, while, for a given possible world w, x is a horse in w, is a relational world-indexed concept, a concept-in-w, so to speak. What is constant, for all predicates, is the overall concept.

All told, Frege's account of predicates and their reference is more feasible than it, perhaps, seems at first sight ─ in any case, it is much more plausible than the view that the referent of a predicate is its extension.


Carnap, R. (1947), Meaning and Necessity, Chicago: University of Chicago Press.

Deutsch, H. (1995), "Intension/extension", in Kim, J. & Sosa, E. (ed., 1995), A Companion to Metaphysics, Oxford: Blackwell, pp. 158-60.

Dummett, M. (1973), Frege: Philosophy of Language, London: Duckworth. (Second revised edition 1981.)

(GL) Frege, G. (1884), Die Grundlagen der Arithmetik: Eine logisch mathematische Untersuchung Über den Begriff der Zahl, Breslau: W. Koebner. Reprinted in 1961, Hildesheim: G. Olms. English translation (with the German text) in Frege, G. (trans. J.L. Austin, 1950), The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, Oxford: Blackwell. (Second revised edition 1953.)

(GG1) Frege, G. (1893), Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet, Band I, Jena: H. Pohle. Reprinted in 1962, Darmstadt: G. Olms. Partial English translation in Frege, G. (ed. & trans. M. Furth, 1964), The Basic Laws of Arithmetic: Exposition of the System, Berkeley: University of California Press, pp. 1-126.

(KS) Frege, G. (ed. I. Angelelli, 1967), Kleine Schriften, Darmstadt: G. Olms.

(NS) Frege, G. (ed. H. Hermes, F. Kambartel & F. Kaulbach, 1969), Nachgelassene Schriften, Hamburg: Felix Meiner. (Second revised edition 1983.)

(BW) Frege, G. (ed. G. Gabriel et al., 1976), Wissenschaftlicher Briefwechsel, Hamburg: Felix Meiner.

(PW) Frege, G. (ed. H. Hermes, F. Kambartel & F. Kaulbach, trans. P. Long & R. White, 1979), Posthumous Writings, Oxford: Blackwell.

(PC) Frege, G. (ed. G. Gabriel et al., trans. H. Kaal, 1980), Philosophical and Mathematical Correspondence, Oxford: Blackwell.

(CP) Frege, G. (ed. B. McGuinness, trans. H. Kaal et al., 1984), Collected Papers on Mathematics, Logic, and Philosophy, Oxford: Blackwell.

Kripke, S. (1971), "Identity and Necessity", in Munitz, M.K. (ed., 1971), Identity and Individuation, New York: New York University Press, pp. 135-64. References are to the reprint in Schwartz, S.P. (ed., 1977), Naming, Necessity, and Natural Kinds, Ithaca: Cornell University Press, pp. 66-101.

Kripke, S. (1972), "Naming and Necessity", in Davidson, D. & Harman, G. (ed., 1972), Semantics of Natural Language, Dordrecht: Reidel, pp. 253-355, 763-9. References are to the reprint, with some additions and revisions, in Kripke, S. (1980), Naming and Necessity, Cambridge: Harvard University Press.

Perry, J. (1998), "Semantics, Possible Worlds", in Craig, E. (ed., 1998), Routledge Encyclopedia of Philosophy, Vol. 8, pp. 662-9.

Schwartz, S.P. (1977), "Introduction", in Schwartz, S.P. (ed., 1977), Naming, Necessity, and Natural Kinds, Ithaca: Cornell University Press, pp. 13-41.

Schwartz, S.P. (2002), "Kinds, General Terms, and Rigidity: A Reply to LaPorte", Philosophical Studies 109, pp. 265-277.

Wright, C. (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, pp. 239-63.

iAbbreviations 'KS', 'CP', etc., of Frege's writings are given in the References section below.

iiI shall follow ─ with considerable hesitation ─ the custom of using for Frege's 'Begriff' its direct translation 'concept'. The reason for my hesitation is that Frege's 'Begriff' does not ─ while, arguably, sense (Sinn) does ─ correspond what is standardly called concept, either in the pre-Fregean tradition or in post-Fregean vocabulary (apart from Frege literature, of course).

iiiIt is surprising to find in a basic philosophical companion a statement such as, "many meaningful expressions lack extension. For example, the predicate 'cat with nine lives' (literally speaking) [... has] this property" (Deutsch 1995, 158). As Frege tells us, the extension of a predicate like 'cat with nine lives' is, in reality, the empty set, not "nothing" (see especially KS 193-210 / CP 210-28, 1895).

ivIn GL §68n Frege first makes the remark that to regard the "sameness" of concept as parallel to the identity of extension is open to the objection that "concepts can have identical extensions without themselves coinciding", and then assures us of his confidence that this objection can be rebutted.

vNothing is gained (but confusions may well be produced) by statements such as the following: "[according to] a version of Frege's choices for Bedeutung", the "extension of an n-place predicate is the set of n-tuples of objects of which the predicate is true" (Perry 1998, 663).

viSee, for example, KS 169-74 / CP 184-9 (1892); NS 103, 106-8, 117-20 / PW 93, 97-9, 107-10 (ca. 1891-2); NS 130-3 / PW 119-22 (ca. 1892-5); BW 218-9, 224, 229 / PC 135-7, 141-2, 146 (1902); Dummett 1973, 211-22; Wright 1998.

viiAgain, from Schwartz (2002, 265) we learn the following: "With general terms there is no obvious candidate for what is to stay the same" "in every possible world"; "In the case of general terms [...] the formal semantics of rigid designation has never been clarified."

viiiCf. here Schwartz 2002, 269, 272-3.

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