9 Aug 2016

Priest (12.2) Beyond the Limits of Thought, “The Concept Horse”, summary


by Corry Shores


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[The following is summary. Boldface and bracketed commentary are my own. You will probably encounter typos and other distracting errors, because proofreading is incomplete. I apologize in advance.]





Graham Priest


Beyond the Limits of Thought


Part 4 Language and Its Limits


Ch.12 The Unity of Thought


12.2 The Concept Horse




Brief summary:

Frege’s theory of meaning is based in part on concepts being structured like functions, where there is a concept part (the predicate) and an argument part (what is being predicated). The argument part is structurally distinct from the concept part. The concept itself has a gap that is either filled or not filled by some argument. This creates a problem for Frege, because this basic structure for concepts prevents him from saying anything meaningful about concepts. There are two basic attempts one can make using this structure in order to say something about concepts, and both attempts fail. The first attempt uses a name for the concept, as in, “‘The concept horse’ is a concept.” Here the concept is “...is a concept”. It can take one or another argument, in this case, it takes the argument ‘the concept horse’ which is a name referring to that concept itself (which would have to have the function structure that all concepts share). This strategy fails, because the predicate “... is a concept” would only be true if it is predicated to a concept. But in this case, it is predicated to a name, and since names are not concepts, “‘the concept horse’ is a concept” would be false under Frege’s theory, even though we intended it to be true. And in fact, similar sorts of expressions would need to be true in order to actually describe the properties of concepts as they are under this theory. The other attempt is to keep the function structure by placing a predicate in the argument place. This would give us “is a horse is a concept”, which is nonsensical, because we cannot conceptualize how ‘is a horse’ can fit be the subject of such a sentence (and it does not work to put quotes around it as in ‘is a horse’ is a concept”, because then it is a name and not a concept).  Michael Dummett proposed as a solution that we use second-order quantification, which will allow us to use a variable in place of a concept. This way it might somehow serve both as a predicate and as an object in certain sorts of formulations. But this strategy fails for a number of reasons. The main problem is that in order to say what we have in mind, we need the concept and the other concept it falls under (is predicated by) to both be of the same order, so by making one at a higher order, we have not resolved the issue. Thus Frege’s notion of the concept is fundamentally inexpressible within his theory of meaning, which is based on that very same notion of the concept.




[Recall some of the important points from the previous section 12.1. We saw that Frege created a way of understanding propositions under the form of a function, where there is the functional part that operates on the argument part. The concept is to be understood as belonging to the function or predicate, and the object is a variable part that can be placed into the argument position of a function or predicate. And we learned that although another concept (another function or predicate) can be found in an argument place, that does not mean for Frege that the same thing can be both a concept and an object. Instead, such a concept in an argument place is to be understood as having its own structure of concept and object. In other words, the distinction is absolute and not relative to its functioning under a certain perspective.]

As is clear, concepts and objects are quite different kinds of thing, and must be so, since concepts perform a quite different kind of function in the composition of thought.

(Priest 199)


But, Priest notes, the fact that concepts and objects are very different kinds of things (as “concepts perform a different function in the composition of thought”) leads to a profound problem for Frege. Since a concept is very different from an object, it cannot be named using a noun-phrase like objects can. However, it does appear that we actually can refer to concepts using noun-phrases. Frege insists however that the appearances are deceiving. Whenever we use a noun phrase to name a concept, we are actually thereby not designating a concept.

But this poses a nasty problem for Frege. For since concepts are not objects, they cannot be named, that is, referred to, by a noun-phrase. But it would appear to be clear that we can refer to a concept with a noun phrase; for example, to use Frege’s notorious example (due to Kerry): the | concept horse. Frege is well aware of the problem. His solution to it is to insist, quite consistently, that, despite appearances, such phrases do not refer to concepts. As he puts it (p. 45):

the three words ‘the concept “horse”’ ... designate an object; but on that account they do not designate a concept as I am using that word.

(299-200, bracketed text mine. Here and in following instances, Priest’s parenthetical text citations are for the Geach Frege text, listed at the end of this post.)


The problem is not just in trying to explain how the appearances are deceiving. There is another even more profound problem as well that is related to this. [I might not express this idea right. It seems to be the following, but I am not sure. We begin with Frege’s notion that the noun phrase, “the concept horse” is an object and not a concept. Now, we say that an object falls under a concept. So if we have the concept “... has four legs” then we can say that a horse falls under this concept. But if we literally write “falls under”, then we are using falling under as the concept, that is, as the predicate. It seems to me that it is a two-place predicate, where we have as one argument some object in question, and we have as the other argument some concept (predicate) in question. So the concept of the whole structure is ‘falls under’. Let us take as the first argument the famous race horse Phar Lap, as in Priest’s example. And we will say that the second argument is ‘is a horse’. Now, Priest writes, “For example, consider the claim that Phar Lap falls under the concept horse. Falling under is a relationship between an object and a concept. Hence, this statement is false”. I am not certain I follow, but perhaps Priest is saying that Par Lap, the horse, cannot be said to fall under the concept horse, when ‘the concept horse’ is regarded as an object and not as a predicate. So perhaps we might write for this example: “Phar Lap falls under ‘the concept horse’.” This would make it clear that it is false, because the only thing an object can fall under is a concept and not an object like ‘the concept horse’. But I am not sure. (If I am following this right) Priest then acknowledges that we can still express what we really mean to say in that sentence in a way that preserves its truth, namely, we can simply say, “Phar Lap is a horse”. But we cannot find a correct way to say what we mean when we write, “the concept horse was considered by Frege”. I am not sure, but perhaps we should say that this is a false sentence as it is written here, but I am not sure why. Maybe it is because what was considered by Frege was a concept and not an object. I also am not sure why it cannot be written in a correct way. I suppose it is because any effort to do so would involve making the concept be an object. Conceptually speaking, we might think that the concept here is: the concept horse falls under the concept of things considered by Frege. Perhaps the problem is that in order for us to know that we are talking about the concept horse and not some horse itself, we need to specify that it is a concept, but by doing so, we have transformed it into an object. I probably have all of this wrong, so let me quote.]

This raises a more serious problem, however. If ‘the concept horse’ does not refer to a concept, a number of claims about concepts would appear to be problematic. For example, consider the claim that Phar Lap falls under the concept horse. Falling under is a relationship between an object and a concept. Hence, this statement is false. On this and similar occasions, what one would normally want to express by this sentence might be expressed in other ways. For example, one can say simply: Phar Lap is a horse. But on other occasions this is impossible. For example, there is no similar paraphrase of: the concept horse was considered by Frege.

(Priest 200)


[Recall the notion of definite descriptions from Priest’s Logic: A Short Introduction, chapter 4 and from Nolt’s Logics section 6.3. Nolt noted that sentences with definite descriptions, like “The present King of France is bald” has a structure that is more complicated than the normal subject-predicate relation. We are saying that there is at least one thing that is the King of France, that it is the only thing that is the king of France, and that thing is bald. Priest might be saying that “the concept horse” is a definite description. But he adds that the use of definite descriptions is not the source of the problem. Rather the problem is with the predicate “is a concept”. I am not sure if Priest is saying that our sentence is something like “ ‘the concept horse’ is a concept”, or if he is saying that the noun-phrase ‘the concept horse’ is to be understood as containing a concept and object. Kerry gave the example sentence, “the concept “horse” is a concept easily attained’” (Frege, “On Concept and Object”, p.45). So maybe Priest is saying that the predicate “is a concept” is found in the grammatical subject of that sentence, and the concept of the whole sentence is “is a concept easily attained”. At any rate, in cases like these, the phrase ‘is a concept’ needs to apply to concepts. However, since it is a predicate, it must apply to objects. The final point I may not follow well. He says that if we join the predicate ‘is a concept’ to a phrase that refers to a concept, we obtain non-sense. Priest gives the example, “is a horse is a concept”. I guess here “is a horse” refers to a concept. His point might be that we have no way to express this idea under Frege’s restrictions. If we formulate the sentence with a noun phrase, then we have a sentence that makes sense, but we are missing the idea, because we have expressed a concept as an object, thereby stripping it of its essential structure or basic mode of self-presentation. However, if we instead try to formulate the sentence by placing in it a grammatical structure that expresses a concept (namely, a predicate structure), then we have expressed the concept correctly, but we have created a non-sense sentence. Let me quote, as I might have this wrong.]

Actually, the problem is not so much with the use of a definite description, as with the predicate ‘is a concept’ itself, which allows us to form the description. To say what it needs to say, it must apply to concepts; yet, like all (first-order) predicates, it must apply to objects; if we join it to a phrase that refers to a concept, nonsense results, for example: is a horse is a concept.



[Priest then notes the problematic ramifications of this situation. Frege’s theory of meaning is based on his notion of a concept. This means that in order to explain his theory, he needs to discuss concepts in a way that is true to what they are, at least as what they are according to the theory. But as we just saw, there is no proper way to do so. Priest then gives some examples it seems of a particular ways that Frege would be unable to express basic ideas of the theory. Suppose that one idea in the theory is that all concept-words denote concepts, or in other words, that “for every concept-word there is a concept that it denotes”. But, as we saw, we here have a predicate, “... is a concept”. Since it is a predicate, it is filled-out by an object and not a concept. Since there are no concepts that can take this predicate, that statement is false. But we needed it to be true. The next example I do not understand. The claim is “that concepts are unsaturated”. For some reason, this will be contradicted. That contradiction arises because “Anything that satisfies ‘is a concept’ is an object, and so saturated.” I think the idea might be the following. Concepts are said to be unsaturated because they can take any number of arguments, and only when one is provided is it saturated. Another way we looked at this is by distinguishing the saturated from the unsaturated parts. The function part is unsaturated, and the argument is the saturated part. Priest might be saying that because some particular concept can take the argument place of ‘... is a concept’, that means that concepts are saturated (as they do the saturating). But we stated at first that they are unsaturated. I am not sure however if that is what Priest is saying. The third example involves Frege’s claim that “a concept is a function whose value is always a truth value”. (Here the important point seems to be about the concept being a function and not about its value). But as we saw, a concept can be an argument, and thus it is not always a function. Priest then ends this paragraph by writing, “To
say what he needs to say, Frege needs a predicate that applies to concepts, and this is just what he cannot have.” I am not certain, but I think he is saying that we need to be able to predicate concepts in such a way that we designate them as being concepts. However, any attempt to do so only at best designates them as being objects of concepts (rather than being concepts in their own right) or in other words as being arguments to functions (and thus not as functions themselves.) Restating: the core notion of Frege’s theory of meaning, namely, the concept-structure, cannot itself be expressed in the theory, because there is no expressive structure allowed in his system that can to do justice to the meaning of a concept (especially with regard to its important structural limitation, namely, that it can never be both a concept and an object).]

The ramifications of this are clear. Frege needs to be able to talk about concepts in order to express his own theory. Yet he cannot do so (meaningfully) by his own theory. For example, consider the claim that all concept-words denote concepts, i.e., for every concept-word there is a concept that it denotes. Whatever satisfies ‘is a concept’ is an object. Hence this is false. Or consider the claim that concepts are unsaturated. Anything that satisfies ‘is a concept’ is an object, and so saturated. One more example, amongst many, from Frege’s own words: he says (p. 30) ‘a concept is a function whose value is always a truth value’. Whatever satisfies ‘is a concept’ is an object, and so not a function. To say what he needs to say, Frege needs a predicate that applies to concepts, and this is just what he cannot have.

(Priest 200)


Priest then notes Frege’s admission of this problem.

We see that the effect of Frege’s view is to put much, including his own theory, beyond the limit of the expressible. Frege recognises this, and is obviously embarrassed by it (p. 54):7 |

I admit that there is a quite peculiar obstacle in the way of an understanding with my reader. By a kind of necessity of language, my expressions, taken literally, sometimes miss my thought: I mention an object when what I intend is a concept. I fully realize that in such cases I was relying on the reader who would be ready to meet me half-way – who does not begrudge a pinch of salt.

But he was not embarrassed enough. It is one thing for mystics, such as, perhaps, Cusanus, to hold views that they also hold to be ineffable; it is quite another for a man of science, such as Frege was.

(Priest 200-201, bracketed text mine)

[Footnote 7: See also the last paragraph on p. 55.]

[Additional quotation, from Frege p.55:

It may make it easier to come to an understanding if the reader compares my work Function [sic] und Begriff. For over the question what it is that is called a function in Analysis, we come up against the same obstacle; and on thorough investigation it will be found that the obstacle is essential, and founded on the nature of our language; that we cannot avoid a certain inappropriateness of linguistic expression; and that there is nothing for it but to realize this and always take it into account. (Frege 55)]


Priest then notes one of Frege’s solutions to the problem that he proposed later on. He proposes that instead of merely stating the concept that we place it within a ‘what’ clause. [The idea seems to be that the ‘what’ serves to point us to the conceptual content of the concept, while still using a name for it. Let me quote.]

In a later and, at the time, unpublished, essay (c1892) Frege comes back to the issue and offers a solution to it. He suggests that we may refer to a concept with a ‘what’ clause, as in ‘what “is a horse” stands for’. The point of such clauses is that they can themselves be used predicatively. Consider, for example, the sentence: Frege is what I am, a philosopher-logician. In this way, we can say, for example, Phar Lap is what ‘is a horse’ stands for.

(Priest 201)


[The important point here seems to be that the what-clauses are used predicatively, meaning it seems that they themselves are structurally speaking predicates, and they take some object. So in the example “Phar Lap is what ‘is a horse’ stands for,” here the predicate is, “... is what ‘is a horse’ stands for,” and the object is Par Lap. Priest’s next point might be that were it really the case that the what-clause is a predicate, then we still have the same problem we saw before.]

The construction does strain the ear somewhat. But in any case, the trick cannot do the required job. For if a what-clause must be construed predicatively then the claim, for example, ‘what “is a horse” stands for was considered by Frege’ is nonsense. Ditto the claim: ‘what “is a horse” stands for is not an object’, etc. And so we cannot paraphrase away all the things we need to say.

(Priest 201)


Priest also notes that this solution does not address the more basic problem with expressing the notion of concept in his system, which is namely that he needs predicates to apply to concepts, but the theory prevents him from doing so. Priest then notes Dummett’s solution. Dummett says that ‘is a concept’ is more of a pseudo-expression, and we should not try to use it. Instead, we should use a second-order predicate, for example, a quantifier phrase, that can be combined with a first order predicate to make a proper sentence. [I do not follow this solution very well. It seems to be the following, but I will be sure to quote for your interpretation. First recall what Nolt said about higher-order logic, in section 14.1 of Logics. Let us begin by considering first-order logic. Here we have quantifiers that range over variables that stand for individuals. But in second-order logic, we can have variables that range over predicates rather than variables. Nolt gave the following example of an argument that can be expressed using second-order predicate variables. In this example, we will have constants for individuals, and the variable X for predicates: “Al is a frog. Beth is a frog. Therefore, Al and Beth have something in common.” Nolt said that we can write this symbolically as: ‘Fa, Fb ⊢ ∃X(Xa & Xb)’. So in this case, we have the predicate, ‘... is a frog’. Then we make the inference that both individuals share one same unspecified predicate, here written in English as ‘share something in common.’ But we can think of that also as being, ‘share some predicate’. This is symbolized with the X, and that variable is ranged over by the existential quantifier. Let us return now to Priest, Frege, and Dummett. Again, I cannot offer an interpretation that I am confident in. But let us try to work through the ideas. Dummett’s proposal seems to be that we first craft a second-order formula that could be understood as meaning that a concept is unsaturated. So in English we would write first, “Everything either is ... or is not ...’, with the ‘...’ being the ‘gap’ for some predicate. It would be written symbolically as: ∀y(Xy∨¬Xy). So it seems already that we have made the predicate be variable. Apparently Dummett suggests that we try then, on the basis of this formula, to construct other formula that can express Frege’s notion of the concept. Priest says that for the notion that every concept is unsaturated, using Dummett’s formulation, we would write, ∀X(∀y(Xy∨¬Xy) → X is unsaturated). The idea here might be that we are saying for any given predicate (i.e., concept), it can be said for every term that it either belongs to that predicate or it does not. So perhaps what is expressed here is maybe something like the following. If we can say that any given predicate (or concept) either predicates some given term or it does not, then we are saying that a term can either be said to saturate it or not. But, this means that on its own, it is not itself saturated. For, its saturation is something that can be under variance, as some supplied argument might hold for it and others not, and possibly many might hold for it. That is probably not right, but I am not sure how else to understand the reasoning behind that formula. Priest says that this formula can be written more simply as ∀X(X is unsaturated). How then might this possibly avoid the problem Frege encountered? Perhaps the idea here is that just by using first-order logic, we cannot predicate a concept and say for example that ‘the concept horse is unsaturated’. However, perhaps we can render the concept into a variable, and quantify over it using a higher-order quantifier. So we might say, ‘any concept is unsaturated.’ Here, what is taking the argument place is a variable for a predicate and not a name for a predicate. Priest then says that “This raises the question of the intelligibility of second-order quantification in the present context”. But I am not sure why, and he sets that matter aside anyway. Maybe it has something to do with the fact that what we say about concepts we would want to apply to concepts on all orders, and maybe further we cannot do so this way. But that cannot be right. At any rate, his point is that “we still have the problem of the predicate ‘is unsaturated’, which is an ordinary first-order predicate; and so this sentence is nonsense”. I am not sure what is meant here. The issue might be that ‘is unsaturated’ is a first-order predicate that can only predicate given specific first-order predicates, and it cannot predicate variables for predicates. And so we cannot, as in our example, say all predicates X are unsaturated. For, “are unsaturated” can only be saturated by a first order specific predicate. I will quote now.]

More importantly, Frege’s suggestion does not address the fundamental problem, which was, as we saw, that he has at his disposal no predicate that applies to concepts. Dummett, in his discussion of the problem ((1973), pp. 211–22) suggests that ‘is a concept’ should be eschewed as a pseudo-expression. We need, instead, an appropriate second- order predicate, i.e., a phrase, like a quantifier phrase, that fits together with an ordinary predicate to make a sentence. His suggestion is ‘Everything either is . . . or is not . . . ’. If we use upper-case variables for concepts, this is ∀y(Xy∨¬Xy). The suggestion will not do the job, however – if the job is to express Frege’s theory. Consider, for example, the claim that every concept is unsaturated. This becomes: ∀X(∀y(Xy∨¬Xy) → X is unsaturated), or more simply ∀X(X is unsaturated). This raises the question of the intelligibility of second-order quantification in the present context. But setting that aside, we still have the problem of the predicate ‘is unsaturated’, which is an ordinary first-order predicate; and so this sentence is nonsense.

(Priest 201, citing Dummett M. (1973), Frege: Philosophy of Language, Duckworth.)


Priest then says that we might think that we could ‘rejig’ the predicate [‘is unsaturated’ for example] so to make it a “a suitable second-order predicate” (Priest 201). Priest, however, doubts that this will work. He then says that anyway there will still be other worse problems. He has us consider the “claim that concept-words refer to concepts, i.e., ∀x(x is a concept-word → ∃Y x refers to Y)” (Priest 201). [Here perhaps the formula is simply saying that if something is a concept-word, perhaps like, ‘the concept horse’, then it refers to some concept. He says we have the same problem with the predicate ‘refers to’. Perhaps the problem here is that ‘refers to’ should refer to a first-order predicate but instead in this formulation refers to a second-order one. But I am not sure. He then says that we cannot rejig it, but I do not follow why. The idea might be that ‘refers to’ is something that must apply at least to names and objects they refer to, so in that sense it must at least be a first-order predicate. He then seems to be saying that it cannot also be a second-order predicate. Maybe the reason is that it cannot be both at the same time, and it must at least be first-order, therefore it cannot be second-order. But I am not sure. Priest addresses an objection that seems to be that we must distinguish two sorts of reference relations, namely ones for names and ones for concept-words. Priest then quotes Frege where he seems to be saying that there should be one reference relation for both concept-words and names. Thus this objection does not apply in Frege’s case. Priest also says that if we make this distinction of references, it contradicts Frege’s point that “the reference of a complex linguistic expression is a function of the references of its parts”. I am not exactly sure why, but perhaps the idea is that for the reference of a complex expression to be compositionally based on the reference of its parts, that means the parts, namely, the names, must have the same sort of reference as that of the whole expression, in which perhaps the concept-name would be found. Let me quote, as I am uncertain in my understanding.]

Maybe some way could be found to rejig the predicate as a suitable second-order predicate (though I doubt it). But worse is in store. Consider the claim that concept-words refer to concepts, i.e., ∀x(x is a concept-word → ∃Y x refers to Y). The same problem arises with respect to the predicate ‘refers to’ ; and this time there is certainly no way of | rejigging it, simply because it must be legitimate to say that names refer to objects, and so ‘refers to’ must be a predicate that is first order in both its arguments. If it be retorted that the reference relations for names and concept-words must be distinct, then we cannot say with Frege ((c1892), p. 118):

To every concept-word, or proper name, there corresponds as a rule a sense and a meaning [reference], as I use these words.

or that the reference of a complex linguistic expression is a function of the references of its parts.

(Priest 201-202)


Priest then notes other cases where Frege’s claims about meaning are really non-sense, for reasons similar to those above. [In this case, Priest seems to be saying that we cannot make any generalization that holds for both concepts and objects. I am not sure why, but the reasoning might again be that by doing so we are applying the same predicate to both objects and concepts when as we have seen that cannot be done. I am guessing. Priest then notes that since we cannot make a generalization for both concepts and objects, we cannot, as Frege tries to do, even say that they are different. Let me quote.]

In fact, we cannot make any generalisations over objects and concepts, as Frege often does, or even say that objects and concepts are different; for example, ((c1892) p. 120):

From what we have said it follows that objects and concepts are fundamentally different and cannot stand in for one another.

Any attempt to express this which paraphrases ‘is a concept’ as a second-order predicate, results in nonsense.



Priest then wraps up this section by saying that we will fail if we try to solve the problem in Frege’s system either by simply making a minor modification to the system hoping it does not change the main philosophical thinking behind it or by removing some purportedly false and unnecessary part of the system.

We see that the Frege/Dummett repair will not solve Frege's fundamental problem. It might be thought that some minor modification of Frege’s views would dispose of it, whilst leaving their essence intact; or that the problem is generated by some quirky and false Fregean doctrine, which should be disposed of anyway. Neither thought is correct; but let us leave the issue there for the time being and move on to Wittgenstein.







Graham Priest. Beyond the Limits of Thought. Cambridge: Cambridge University, 1995.


Priests parenthetical page citations for Frege come from:


Gottlob Frege. Translations from the Philosophical Writings of Gottlob Frege. Eds. P.T. Geach and Max Black. Transl. P.T. Geach. Oxford: Basil Blackwell, 1960, second edition (1952 first edition).



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