Please See Note On Viewing the Symbolism In This Paper


Hartley Slater 

There has been a lot of work done on Frege's views on concepts, but there is a good deal of work still remaining to be done, it would appear. My attention was drawn to this fact through reading Antony Kenny's recent book on Frege. Kenny is a very knowledgeable, highly skilled and widely respected philosopher, yet, remarkably, he finds ample justification for Frege's contention that the concept horse is not a concept (Kenny 1995, 125). This is in contrast to common opinion, and most previous commentators, who held that Frege's contention is clearly a paradox. Indeed, that it is considerably more than a paradox was seemingly even proved formally by Milton Fisk. For, with respect to what Frege said in 'On Concept and Object', Fisk said 'I wish to show that these principles, which constrain Frege to deny that the concept horse is a concept, lead to a contradiction' (Fisk 1968, 382). Clearly something fundamental has not been made clear, if Kenny can return, in the face of this, to a support for Frege.

I support previous, learned, as well as everyday opinion in this paper, revealing not only the fallacy in Kenny's present, and Frege's original reasoning, but going deeper to unearth an inadequacy in the common logical symbolism for concepts - the lack of second-order nominal expressions, like 'being a P'. Thus I start by showing that, since the predicate in a sentence is not in such a nominalised form, no reference to, merely expression of the associated concept is made there. The point has been suggested before (c.f. Kneale and Kneale 1962, 585-586, Wright 1983, 21), but re-working and extending it not only improves upon Dummett's treatment of Frege on concepts, it also shows why some traditional difficulties, like the problem of the unity of the proposition, the paradox of predication, and the Liar Paradox, are not present in natural language. I principally point out that, when we talk about a concept we nominalise a predicate, and so make the concept into a subject, even though the concept is not an object. On the way to correcting Kenny on Frege on concepts, I therefore also show why Frege's associated belief that numbers are not concepts but objects is mistaken.


As Dudman showed (Dudman 1976), there is considerable difficulty in supposing that predicates have referents. Frege held that the referent of a predicate is a concept, and while, undoubtedly, predicates have a semantic role in determining the truth value of sentences in which they occur - so they do have 'reference' in one sense - this is not sufficient to ensure that they have anything which compares with the paradigm referents, i.e. bearers of proper names, and similar first-order individuating expressions. The problem is that a predicate is not a nominal phrase: 'lived to 100', for instance, is not 'living to 100'. The participle refers to living to 100, but 'John lived to 100' does not refer to this, even if, in some way, the concept is involved in the proposition.

The point, although widely discussed at one time, has not always been taken note of recently, even by those discussing 'the unity of the proposition'. For the presupposition of one train of thought (see Linsky 1992) is that while 'Dobbin is a horse', for instance, cannot just list the object Dobbin and the concept of being a horse (for how can a list of names say anything?), instead what the sentence says is that the object is related in a certain way to the concept, i.e. is of the form

The object is an instantiation of the concept.

But this then itself would have to be analysed in the same way, which would generate Bradley's infinite regress, and most find that unacceptable, even if one recent thinker (Gaskin 1995) asserts that such a regress is just what is required to provide for propositional, or sentential, unity.

The trouble with this tradition, it ought to be remembered, is its starting point: the idea that the sentence 's is a P' is not only about s, but also, in some way or other, about being a P. What refers to the concept of being a P is a nominal expression, but 'is a P' is not a name, it is a description. The relation between a predicate and the associated concept, therefore, cannot be reference: it would be better called 'expression'. But in just that fact lies the everyday power of 's is a P' to say something, in opposition to 's, being a P'. For while no collection of names says anything, still a predicate, when appropriately completed, does. In other words, the non-referential, descriptional form is crucial, and is just what is forgotten in the tradition above. 'Dobbin is a horse' does not relate Dobbin to some other object, a concept; it simply conceptualises Dobbin in the way described by the given predicate, and so says something about him. Somebody who claims 'Plato is wise' has the concept of wisdom, but in that statement he is then applying the concept, not talking about it.

This is a distinction which is not taken note of in current logical symbolism. Thus if we move to second-order logic, and start talking about concepts, then from 'Wp' one might try to deduce

($ F)Fp

or produce an equivalent

($ F)(F=W.Fp)

But there is something wrong with this way of writing the matter, since the intention is that in '($ F)' the 'F' is a nominal form - 'there is something/some property/a concept such that' - whereas in 'Fp' the 'F' is not nominal, but predicative. If 'F' is to be a nominal phrase throughout, then what is wanted is some 'de-nominaliser' which will take, for example, 'living to 100' into 'lived to 100', and more generally participles into appropriate finite forms of the verb. In semi-formal natural language a kind of pluralising is often used: if F is a concept, then maybe x Fs. Hence the sort of thing which is grammatical is

($ F)Fsp


($ F)(F=W.Fsp))

In these terms, if what is expressed when 'Ws' is used is the concept of being wise, then there is nothing wrong with deriving these from Wsp, or with it being equivalent to, for instance,

There is a characteristic Plato had, namely the characteristic of being wise


($ C)(Hs(p, C).C=W)

But no vicious, hierarchical regress ensues, since an explanation of how 'Wsp' says something was provided before, and this then applies to all higher-order predications.

Moreover, no paradox then arises akin to the standard paradox of predication, since, if we formulate the predicate

'x is a property which does not apply to itself',


'($ F)(F=x.Ø Fsx)',

then we still need that something is expressed by this predicate, for instance the concept G, in order to produce the propositional equivalence

($ F)(F=G.Ø FsG) º GsG,

which is a contradiction. By Reductio, no property corresponds to the linguistic predicate. Again, the temptation might be to try to abbreviate the predicate

'($ F)(F=x.Ø Hs(x, F))'


'Hs(x, G)',

from which a contradiction could be drawn, by taking 'x' to be 'G'; but that abbreviation equally makes the assumption that there is a property G. The basic difficulty is that, through not making the discrimination between expression and reference, the standard logical symbolism, when automatically writing

($ F)(F=x.Ø Fx) º Gx,

provides no way to say that no property is expressed by the predicate.

Comparable points hold for propositions, of course (c.f. Kneale 1972, 239f, Slater 2001), giving a resolution of other paradoxes. In the equivalence between 'that p is true' and 'p', we find the connection between reference to and expression of propositions: what is in one place referred to with the nominal phrase 'that p' is expressed and asserted in other places, with 'p'. So, given that it operates on a sentence, 'that' is a nominaliser: it operates just like a demonstrative to produce, with the sentence then immediately used, a linguistic phrase which refers to what is then expressed. In connection with the Liar Paradox, for example, one must then consider what proposition is expressed, for instance, by the sentence, 'what is expressed by this sentence is not true'. But one needs something of the form

what is expressed by that sentence is that p

in order to obtain the supposed proposition, before one can derive the contradiction

p º that p is not true.

It follows, by Reductio, that the sentence does not express a proposition.


Some of the above points about concepts were recognised by Dummett, in his lengthy treatment of such issues (Dummett 1981, Ch 7), but he did not investigate the full consequences for second-order logic. Thus in place of

... is a concept

which might seem to require '...' to be replaced by a first-order referring phrase, immediately leading to the absurdity that what satisfies it is both a concept and an object, Dummett exploited Frege's own way out of the paradox (at the time unpublished), and constructed what he took to be a second-order predicate

... is something which everything either is or is not,

(Dummett 1981, 216-217) which enabled a replacement of '...' with 'a horse'. Dummett realised that if we wrote this more formally:

($ F)(...=F.(x)(Fx v Ø Fx)),

then the concatenation 'Fx' would have to be shorthand for 'x is F', so he was aware of some of the problems with this symbolism. But the major problem with Dummett's analysis is that it has still totally avoided nominal phrases, like 'being F'. This is unfortunate, since the formal replacement for specifications like 'the concept horse', would be

iF(H=F.(x)(Fx v Ø Fx)),

or its equivalent 'iF(F=H)', with the fact that the concept horse is a concept then following from the evident truth that

($ G)(iF(F=H)=G).

But Dummett would also have to say, as a result,


which, given that he wanted to substitute phrases like 'a P' for the capital letters, would imply that the concept horse was a horse!

The problem is that 'a horse' is not a referring phrase, so Dummett should have started from, for instance, a property which everything either has or has not,

from which he could have derived

($ F)(...=F.(x)(Fsx v Ø Fsx)),

in which the gap is to be filled by participles like 'being a horse'.

Despite this grammatical confusion, however, Dummett's thoughts here, about second-order logic, clearly show that he is trying to follow Frege in denying there is an overlap between the two categories of concept and object. But many others, perhaps through following Fisk's second way out of the Fregean Paradox (Fisk 1968, 387), have tried to distinguish between concrete and abstract objects, with concepts being in the latter class, but also, thereby, still clearly taken to be objects of a sort. As should be well known, Frege took numbers to be abstract objects, although he also sometimes took them to be second-order concepts, so, despite his avowed separation between objects and concepts, even he had inclinations the other way. To some extent the issue is a matter of linguistic taste, to be settled by some agreed convention, which, however, should be strictly adhered to. Either one says the only objects are concrete objects (like Julius Caesar), and then 'abstract objects' are dropped in favour of concepts of one kind or another; or one allows that some objects may be concepts - the abstract objects. The grammatical facts are that 'the concept of a P', 'the proposition that p', 'the number of Ps' etc. are all referring phrases, like 'Julius Caesar', so if referring phrases always denote objects then some objects are concepts. Alternatively one can say that some referring phrases do not denote objects, but simply, instead, concepts.

I will adopt the latter usage, for several reasons. The first is to negate the idea that talk about concepts can be replaced with talk about certain object correlatives associated with them: sets. Fisk, in fact, shows this to be impossible, as Frege understood such things (Fisk 1968, 389, n16). Still, higher-order logicians have commonly tried to distinguish first-order from second-order logics, by taking the semantics of the latter to be in terms of sets. Quantified property variables are then taken to range over sets, sets of sets, etc. (see, for instance, Gamut 1991, Vol I, 170-171). But any association between properties and sets obviously cannot hold in general - otherwise the Naive Abstraction Axiom would hold, and we would have Russell's Paradox. Wright has pointed out (Wright 1983, 2-3) that the association with sets is only plausible for predicates which determine sortal concepts i.e. count nouns, and mass terms, such as 'yellow things' ought to be handled differently. As Tiles said (Tiles 1989, 151), Frege's logical symbolism does not discriminate between those concepts which do, and those concepts which do not, determine a unit, suggesting that the concept of number is applicable to all concepts without restriction. So some symbolisation of the difference between 's is a P' and 's is P' is required before the area where sets arise can be identified.

A different discrimination needs to be drawn by those who try to analyse propositions in terms of the sets of possible worlds in which they are true. A proposition isn't the set of possible worlds in which it is true - because sets aren't the sort of things which can be true. The general form of a proposition is maybe that the actual world is a member of the set of worlds in which it is true. But propositions can only be given by being expressed—and there is no way to express sets. The proposition is, above all, that so-and-so, so, as before, propositions are displayed using the form 'that p'. Wright forgot this when he said

...what should count, for example, as the ability to identify the referent of an abstract singular term if it is 'appropriately presented'? Well, evidently, there is no presenting an abstract object save by the use of an expression of which it is the referent. So the ability to identify the referent of an abstract singular term, appropriately presented, cannot but be the ability to recognise the coincidence in reference of that term with another, used to 'present' the object to one. Knowledge of the reference of an abstract singular term, if taken to be a recognitional ability at all, thus has to be construed as the ability to recognise the truth or falsity of identity statements involving that term... (Wright 1983, 78).

But there is one form of referential term which is significantly different from those which Wright seemingly has in mind: that which involves the demonstrative 'that' as in 'that p'. For then the referent is not only referred to, but actually present, by being expressed. The idea that one can get no closer to 'abstract objects' than terms which refer to them is perhaps one reason why they are sometimes taken to be 'abstract'; but there is nothing hidden, distant or non-concrete about an expressed sentence. And by being not just mentioned but used, it is not at all like an external, physical object; it is more like a tool, in both Heidegger's, and Wittgenstein's, senses.

The point applies to concepts as well as propositions. We can see the planets, but why cannot we see the number of them? Is it that the number of them is still an object, but one which is invisible, except maybe 'to the mind's eye'? Well, determining the number of the planets, like determining any other property or quality of them is a perfectly straightforward physical process. We can, in particular, count the number of them, and so see that they are nine in number, but that they are nine in number is not a further object which is then directly seen, since 'seeing that' is an indirect speech locution, and refers instead to some propositional judgement. If 'seeing that' is 'seeing in the mind', all right, but there is nothing abstract, or other-worldly, about this activity.

Wright, amongst many others, has discussed the epistemological problem with access to abstract objects (Wright 1983, 84f). There are difficulties for Realism on this score; but Empiricism seems no more promising on these matters (Wright 1983, 1f). The fact that Mathematics and Logic, for instance, are skills, and that one's knowledge of these subjects only comes through practise with symbols and words is the basis for a more satisfying Anti-Realism. But, by the nature of our relation with our skills, one has to be satisfied with a less articulate form of philosophy in support of it.


With the above discriminations between concepts and various objects made, we now start to see why it would, after all, be better to follow Frege's main thought, and distinguish concepts categorically from objects - although clearly we cannot follow Frege all the way, since when talking about concepts we must then not think we are talking about these concepts' 'object correlates', as Frege tried to have it. Instead, when we are talking about concepts we are merely talking about them as subjects - which simply involves nominalising the associated predicates. But because of that we now have a decisive way of separating out concepts from objects such as Julius Caesar, for instance. It is not that he is or was a concrete object rather than an abstract one, or that he is the possible subject only of first-order predications, and it is categorically different objects which are the possible subjects of second-order predications. For the crucial reason why such higher-order predications are to be distinguished from predications about objects is now apparent: Julius Caesar can be presented independently of language, whereas the higher-order subjects cannot.

Now Frege was plainly not too clear about some of these discriminations. For Frege had his notorious difficulty with Julius Caesar, and he believed that numbers were objects, even though he could formulate no clear way of separating them from other objects (Wright 1983, 111). Indeed, Julius Caesar, the concept horse, the number of the planets, and John's living to 100 all may be spoken about, i.e. be subjects, but they are subjects all of different orders, and only the first is an object. But it was Frege's identification of numbers with objects rather than concepts which supported the specific reasoning which led him to separate objects from concepts, since he thought of the former as 'saturated' while the latter were not. So we must now inspect his reasons for that identification.

In 'Function and Concept' (Geach and Black 1980, 24-5), Frege says:

The two parts into which a mathematical expression is thus split up, the sign of the argument and the expression of the function, are dissimilar; for the argument is a number, a whole complete in itself, as the function is not... For instance, if I say 'the function 2.x3 + x', x must not be considered as belonging to the function; this letter only serves to indicate the kind of supplementation that is needed; it enables one to recognise the places where the sign for the argument must go in.

But, by Frege's own grammatical criterion, the expression 'the function 2.x3 + x' must denote an object, even though, of course, such a function is just what would be quantified over in a modern functional calculus, showing that, despite Frege, that function is indeed a function. Moreover, Frege elsewhere, as was mentioned before, took numbers to be second-order concepts, so they, like functions, must be incomplete: 'there are seven' for instance, needs a substantive added to make a complete thought, even though we can also say things like '7 + 5 = 12'. The arithmetical statement means that seven things plus five things equals twelve things, although the fact that 'there are seven things' is only fully completed in sentences like 'there are seven horses' is often forgotten in talk about the more abstract statement.

In fact, when numbers are thought of as abstract objects, the omission of 'thing' is surely crucial. For there is no problem about being acquainted with a number of things, as there is with being acquainted with a number; and one's acquaintance with the number, also, is no great problem, once it is realised that that is a matter of one's practised activity with numerals and the like, in such processes as counting, addition, multiplication, etc. According to Wright, however (Wright 1983, 11), there were three more specific considerations which were involved in Frege's judgement that numbers were abstract objects. One was the use of definite descriptions like 'the number of the planets', which we saw above could also be used for functional expressions. Another was the currency of numerical identities, like '5+7=12', but there are identities at all levels, as we have seen; for instance, 'what John feared was what Mary feared', and 'what 'lived to 100' expresses is the concept of living to 100'. Neither of these considerations is, therefore, decisive.

The remaining consideration Frege appealed to was the contrast between, for instance 'the number of planets is 9' and 'there are 9 planets'. The latter could be given a quantificational analysis, with '9' emerging as a numerical quantifier, thus denoting a second-order concept. The former, by contrast, represents '9' as a singular term. Now certainly if we write

Nx:Fx = 9

then the singular term on the right might induce us to think it is not functional. But it is equivalent to 'the number of F's' on the left, which reveals the general functionality of numbers, and in

($ 9x)Fx

the place of the number 9 amongst second-order concepts is clear. The other, equational form was the basis for Frege's formal analysis of Arithmetic, but, as Wright himself acknowledges, a construction of Arithmetic starting from numerical quantifiers has been developed by Bostock (Wright 1983, 36f). Wherein lies the priority of the seemingly non-adjectival form of expression, then? Frege's use of this form of expression, of course, is in his generation of the number series, starting with his definition of zero:

Nx: x ¹ x = 0

and here he is supplying a specific function 'Fx' to fill the gap in 'Nx: ...', which means that the latter function is saturated. But that does not mean that '0' is not itself a function, since it still leaves the truth value of


a function of 'F'. Whatever priority Frege's preferred form of expression therefore has in the development of pure Arithmetic, that does not show that in their application numbers are not concepts. What things are talked about is irrelevant in pure Arithmetic, but that does not mean they are irrelevant tout court.

The point, therefore, is that while functions and numbers are unsaturated, they still may be spoken about, i.e. singular subject expressions may be formed which make reference to them, though not as objects. So some grammatical subjects do not denote objects, and Frege's grammatical criterion for designating objects must be modified: it is first-order definite descriptions, and the like, which alone identify them (c.f. Higginbotham 1998, 6). The concept of seven things is then definitely a concept, simply because 'thing' makes it unsaturated. Not only Kenny seems to have been unaware of this, but Kenny's discussion in particular, we can now see, has evidently not improved much upon Frege's in this area.


Kenny, in fact, does not take any note of the material Dummett cites as Frege's own way out of his paradox; he treats merely the material published in Frege's lifetime, notably 'On Concept and Object'. There, Frege held there was merely an 'awkwardness of language' in saying 'the concept horse is not a concept'. He pointed out (Geach and Black 1980, 46):

A similar thing happens when we say as regards the sentence 'this rose is red': The grammatical predicate 'is red' belongs to the subject 'this rose'. Here the words 'The grammatical predicate "is red"' are not a grammatical predicate but a subject. By the very act of explicitly calling it a predicate, we deprive it of this property.

Taking heart from this, Kenny asks us to consider (Kenny 1995, 125):

(18) the verb 'swims' is not a verb,

(19) 'the verb "swims" ' is not a verb,

(20) the verb ' "swims" ' is not a verb,

arguing that the third form is the key to understanding Frege's contention that the concept horse is not a concept. For certainly, as Kenny says,

' "swims" ' is not a verb.

But Kenny's explanation of how the third form (20) is to apply to the concept case has no backing in Frege. Kenny says

The expression 'the concept...' is really meant to serve the same purpose with regard to concepts which quotation marks serve in relation to predicates (Kenny 1995, 125),

whereas Frege merely relevantly says

The peculiarity of our case is indicated by Kerry himself, by means of the quotation-marks around 'horse'; I use italics to the same end (Geach and Black 1980, 46).

So Frege took there to be only one pair of quotation marks, in

The concept 'horse' is not a concept,

even if Kenny's device would produce the two pairs needed to parallel (20).

But, in addition, Frege draws no proper parallel with Kenny's (19), since, while, undoubtedly

'The grammatical predicate "is red" ' is not a grammatical predicate

the controversial claim is not that

'The concept "horse" ' is not a concept.

The relevant parallel in Kenny's collection to remarks not about 'the concept "horse" ', or ' "horse" ', but about the concept 'horse', is neither (19) nor (20), but (18)—unfortunately for Frege. For then, while certainly

the words 'the concept "horse" ' are a subject

and so, in Frege's terms,

'the concept "horse" ' denotes an object,

it is still the case that

The grammatical predicate 'is red' is a grammatical predicate,

and so, likewise, using Frege's own parallel, it must be that

The concept 'horse' is a concept.

That means that Frege's first treatment of the issue is condemned out of his own mouth, and Kenny's attempt to defend him has lead instead to a clarification of just why he was wrong. The full explication of what concepts are, and how to handle them formally, without contradiction or paradox, must extend Frege's own amended treatment, as we saw in Dummett. But that extension must include second-order nominal expressions, which have yet to be incorporated into the public symbolism.



Dudman, V.H. 1976, 'Bedeutung for Predicates', in M. Schirn, (ed.), Studien zu Frege III: Logik und Semantik, Fromann-Holzboog, Stuttgart, see also H. Sluga, (ed.), The Philosophy of Frege, Vol 4, Garland, New York, 1993.

Dummett, M. 1981, Frege: Philosophy of Language, 2nd Ed., Duckworth, London.

Fisk, M. 1968, 'A Paradox in Frege's Semantics', in E.D. Klemke (ed.), Essays on Frege, University of Illinois Press, Chicago 1968, see also H. Sluga, (ed.), The Philosophy of Frege, Vol 3, Garland, New York, 1993.

Gamut, L.T.F. 1991, Logic, Language and Meaning, University of Chicago Press, Chicago.

Gaskin, R. 1995, 'Bradley's Regress, the Copula and the Unity of the Proposition' Philosophical Quarterly, 45.

Geach, & Black, M. (eds) 1980, Translations from the Philosophical Writings of Gottlob Frege, 3rd Ed., Blackwell, Oxford.

Higginbotham, J. 1998, 'On Higher-Order Logic and Natural Language', Proceedings of the British Academy, 95.

Kenny, A. 1995, Frege, Penguin, Harmonsworth.

Kneale, W. 1972, 'Propositions and Truth in Natural Languages', Mind, 81.

Kneale, W. and Kneale, M. 1962, The Development of Logic, Clarendon Press, Oxford.

Linsky, L. 1992, 'The Unity of the Proposition', Journal of the History of Philosophy, 30.

Slater, B.H. 2001, 'Prior's Analytic Revised', Analysis.

Tiles, M. 1989, The Philosophy of Set Theory, Blackwell, Oxford.

Wright, C. 1983, Frege's Conception of Numbers as Objects, Aberdeen University Press, Aberdeen.

Copyright © 2000 Minerva. All Rights Reserved.

Dr. Hartley Slater is a Senior Research Fellow at the Department of Philosophy, University of Western Australia.

Mail to: Hartley Slater

NOTE: Readers using some of the earlier versions of Netscape Navigator/Communicator may experience difficulties in viewing the symbolism contained in this paper. To rectify this, it is suggested that you download the latest version of Netscape Communicator here, or MS Internet Explorer here.

Return to Minerva (Volume 4) Main Page                          Go to Top of This Page