10 Aug 2016

Priest (1.2) One. ‘Frege and the Unity of the Proposition,’ summary

by Corry Shores

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[The following is summary. All boldface, underlying and bracketed commentary are my own. Proofreading is incomplete, so please excuse my typos.]

Summary of

Graham Priest

One: Being an Investigation into the Unity of Reality and of its Parts, including the Singular Object which is Nothingness

Part 1: Unity

Ch.1: Gluons and their Wicked Ways

1.2 Frege and the Unity of the Proposition

Brief summary:

Frege’s theory of meaning on the one hand accounts for the unity of a proposition while on the other hand makes it impossible for there to be such unity when making statements about concepts and their important structures. A concept has the same structure as a function. This means there are two elements involved. {1} There is an unsaturated function part, in which there is a ‘gap’ or variable where the argument can go. And when a specific argument is put in that open place, then the concept has {2} an object. For example, ‘is unsaturated’ is a concept, and it has a gap (at the beginning) where we could place one or another  particular concept (which is something that structurally speaking is unsaturated) so that it takes ‘is unsaturated’ as its predicate. However, this structure does not allow such an expression about concepts to be made, despite the fact that the theory requires we do so and also despite the fact that our intuition tells us we should be able to do so. Were we to use a name for concepts, like ‘a concept is unsaturated’, then we have a false sentence. For, the name ‘a concept’ is not unsaturated, because it itself, as the name taking that predicate, is an object and not a concept. It is in this way that Frege has a problem of unity with regard to his notions of concept and proposition.


Priest notes that in philosophy, it is not always obvious if the problems we choose to deal with are really profound of if they are merely silly (Priest 5). He then says that one way we can determine that a problem is the profound kind is by seeing “how some important philosophical projects have run aground on the matter” (6). Priest will look at one such case, namely, Frege’s view on meaning and, in particular his account of the unity of the proposition” (6).

[Priest will have us consider the sentence ‘Sortes homo est’. I am not familiar with this formulation, but I would think it is the classic logic sentence, ‘Socrates is a man’ or ‘Socrates is mortal’. At this webpage it is stated that ‘Sortes’ was a contracted form of ‘Socrates’, as used in Medieval logic. Now let us recall from “On Sense and Reference” Frege’s notions of sense and reference. Here we learned that both names (signs, terms, etc.) and whole sentences can have both reference and sense: {a} the reference of a sign is the specific object(s) it designates; {b} the sense of a sign is its mode of presentation (the contextual elements of how it is to be grasped. The same object can be cognized using different cognizable contents); {c} the reference of a sentence is its truth value; and, {d} the sense of a sentence is its thought (that which is cognized when it is conceptualized, usually taking the structure of predication). The thought or sense of ‘Sortes homo est’, when articulated as a proposition, is that Socrates is a person. Now recall from “Function and Concept” how Frege distinguished two parts of a function (and a predicate proposition can be understood as a function). A function can be understood as having a variable part and a non-variable part. So he has us consider a function that when its variable part is assigned values gives us:

‘2∙13 + 1,’
‘2∙23 + 2,’
‘2∙43 + 4,’

These stand for the values 3, 18, and 132, respectively. As we can see, there is a part that varies consistently. We can think of these and all the other possible variations of the above form as being a structure with a gap in it, and that gap is filled by specific arguments.

‘2∙(   )3 +(   )’

We can also think of those gaps as letter variables.

‘2∙x3 + x

(See Frege, “Function and Concept”, pp.21-24)

When the argument is not specified, and there is a variable or ‘gap’ placeholder of some kind, then it is unsaturated, in Frege’s terminology. But when the specific argument is supplied into the structure, then the function is saturated. It might not be obvious then how Frege’s notion of function applies to predications like ‘Socrates is a human’. It works like the following. In this example, the unsaturated function is the predicate “... is a human”. The argument place then can take any name, but depending on which name, the whole sentence will be true or false. So when it is a name of an animal, for example, then the sentence is false. Priest will divide the sentence into two parts, namely, the noun-phrase ‘Sortes’ (s) and the verb phrase ‘homo est’ (h). Frege does not give a name for the senses for each part of the function, so Priest supplies them as ‘object-senses’ and ‘concept-senses’. Priest’s philosophical concern is that on the one hand, there is a real structural unity to the proposition, perhaps because it constitutes a singular thought, but I am not sure. However, despite that unity, Frege insists that there is a fundamental internal difference. (In fact, recall that Frege even says that the argument “does not belong with the function, but goes together with the function to make up a complete whole; for the function by itself must be called incomplete, in need of supplementation, or ‘unsaturated’ ” (Frege, “Function and Concept”, p.24).  As Priest explains, for Frege, the radical difference between concepts (and concept-senses) and objects (and object senses) is that concepts (and concept senses) are unsaturated, meaning that they are radically incomplete. However objects (and object senses) do not have such a gap in them. Priest then raises a problem. We are saying that ‘homo est’, as a concept, has a gap in it, because it is unsaturated. This means more specifically that its sense has a gap in it. (We might for example try to conceive this concept ‘... is a human’. It in a way has conceptual content, but something is missing from that concept, namely, that which is being said to be human.) Now I am not sure I have Priest’s point right here, but he seems to be saying that the problem is that since ‘homo est’ has a sense with a gap in it, we can make the following formulation: The sense of ‘homo est’ has a “gap” in it.  And this is problematic, because on the one hand we are saying that the sense of ‘homo est’ is unsaturated, but the noun-phrase ‘the sense of “homo est” ’ is an object (which would seem to be nothing other than the sense of ‘homo est’), which means it is saturated (that it does not have a gap). So it would seem that ‘the sense of “homo est”’ has a sense which is both an unsaturated sense and a saturated sense. But Frege makes it clear that an expression cannot be both a concept and an object. Let me quote as I did not render his insight well enough.]

Consider the sentence ‘Sortes homo est’. The sentence is constituted by a noun-phrase, ‘Sortes’, and a verb phrase, ‘homo est’. According to Frege, the sentence has a sense. This is the proposition (thought) that it expresses: that Socrates is a person. The proposition is composed out of the senses of its two components, the sense of ‘Sortes’ (s), and the sense of ‘Homo est’ (h). But the proposition is not a plurality, a congeries of its two parts, s and h. Somehow these cooperate to form a unity. How?

For Frege, names (including definite descriptions) refer to objects, and predicates refer to concepts. He has no special names for the senses of the two grammatical categories, so let us just call them object-senses and concept-senses. Frege’s answer was that concepts and concept-senses are radically different from objects and object-senses. Unlike objects and object-senses, they are “unsaturated”, radically incomplete. The sense of ‘homo est’ has a “gap” in it, which is plugged by the sense of ‘Sortes’ to produce a single thing. Note the form of words here:

(*) The sense of ‘homo est’ has a “gap” in it.

The notions of being unsaturated, of having a gap, and so on, are of course metaphorical. This is not in itself a problem: literal explanation may well give out somewhere. What is a problem is that concept-senses are supposed to be unsaturated. But, the expression ‘the sense of “homo est” ’ is a noun phrase, and so refers to an object(-sense); and these, according to Frege, are not unsaturated.

(Priest 6)

[Priest will explain that Frege’s solution is to emphasize that the sense of ‘homo est’ is saturated. Priest further develops his critical examination of this problem in Frege in Priest’s Beyond the Limits of Thought, section 12.1 and section 12.2. The point Priest makes there is that in order for Frege to express his theory of meaning, he needs this notion of the concept having this form of the function where there is an open place for an argument to be placed. This allows for predicates to take objects, which is the main way of expressing propositional sense. And, since the concept (which is defined by this open structure) is something we need to describe and characterize in this theory of meaning, we need to predicate the notion of concept. However, that very same conceptual structure which is essential to what it is as a meaning-expressing entity is also what prevents it from expressing any meaning about concepts themselves. For, there is no way to predicate a concept (as nothing but a concept) by means of another concept. Priest mentioned a variety of ways to try to do so, and none of them are ultimately successful. The first is to use a name for the concept and put it in the argument place of a predicate that says something about concepts. This is what we saw happening above with the example. So we might say, “‘A concept’ is unsaturated”. But here, the expression for concept is an object, and thus is saturated, and therefore the expression is false. For, the predicate does not hold for the object that it predicates. But we need this expression of the unsaturatedness of the concept to be true. Another strategy is to articulate the expression of the concept in such a way that it maintains its unsaturated form. So we might write: ‘Is a concept is unsaturated’. This could be deemed true. However, it does not seem to make much literal sense. It would make more sense if we put quotes around the expression in the argument place, as “‘Is a concept’ is unsaturated”. But then we have the same problem as with the first proposed solution, where the sentence becomes false. ‘Is a concept’, when it takes that position in the sentence, is saturated, and thus the sentence is false. The third main solution Priest explored was to use higher-order quantification. This approach involves us using a variable in place of the concept that we want to predicate. This way it might somehow serve both as a predicate (in that it could be filled-in secondarily by other predicates) and as an object (as it takes the argument place of another predicate) 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. I return now to the present Priest text. Here Priest refers to the defense of his theory that Frege gives in On Concept and Object, where he deals with an example involving the concept horse. Here Frege was dealing with an objection (by someone named Kerry) that had been leveled against Frege’s notion of concept. Kerry argues (contrary to Frege) that there is not an absolute distinction between the content of the concept and concept-object, for, an expression can be dually both a concept and a concept-object. Frege clarifies that the concept belongs to the predicate of a sentence and the concept-object to the subject of the sentence. On this matter, Frege wrote:

There are, indeed, cases that seem to support his view. I myself have indicated (in Grundlagen, §53, ad fin.) that a concept may fall under a higher concept – which, however, must not be confused with one concept’s being subordinate to another. Kerry does not appeal to this; instead, he gives the following example: ‘the concept “horse” is a concept easily attained,’ and thinks that the concept ‘horse’ is an object, in fact one of the objects that fall under the concept ‘concept easily attained.’ Quite so; the three words ‘the concept “horse” ’ do designate an object, but on that very account they do not designate a concept, as I am using the word. This is in full accord with the criterion I gave – that the singular definite article always indicates an object, whereas the indefinite article accompanies a concept-word.

(Frege, On Concept and Object, 45)

Priest will quote the passage where Frege asks the reader just to understand what he really means, even though his theory does not allow him to say it properly. Priest says that this does not resolve the issue of the proposition supposedly being something unified but in cases of predicating concepts it cannot be unified.]

Frege was well aware of the matter. His solution was simply to reiterate the claim that the sense of ‘homo est’ is indeed saturated. But he was aware that this put him in a difficult situation. He says in the infamous concept horse passage:2 |

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 thoughts; I mention an object when what I intend is a concept[-sense]. 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 me a pinch of salt.

But Frege underestimated (or understates) the problem. If he is right in his insistence that the description refers to an object, this undercuts his whole explanation of the unity of the proposition. Merely reflect for a moment on (*). This is now simply false. We seem forced into the view that concept-senses are objects, even though they cannot be.

[Footnote 2 (quoting): Geach and Black (1952), p. 54. Frege actually addresses the matter for reference (bedeutung) rather than sense (sinn). However, he takes matters to be similar for sense. For our purposes, they are not. Frege takes a concept to be a function which applies to an object to form a truth value. Now, whatever a truth value is, the object is not part of it. (In the same way, the referent of ‘John’ is not a part of the referent of ‘the father of John’.) Of course, there is a very good question about how one is to understand the relationship between an object and a concept (property) which applies to it. This is the question of how to understand instantiation; we will come to it in due course. Note that if Frege had taken the referent of a sentence, more naturally, to be a state of affairs (either existent or nonexistent), of which the object and the concept are a part, the situation concerning reference would be exactly the same.]

(Priest 6-7)

And since concepts have the form of functions, the same problem arises in the context of functions. [Priest first notes that in Fregean semantics, both the senses and the referents of predicates are functions. I am not exactly sure I know what is meant. Suppose we have an unsaturated predicate. Its sense I guess is the thought involved in the concept expressed in that predicate. So the sense of ‘is a horse’ might be something like the notion of horseness, or something like that. The reference I would think is the set of all things that would make the predicate true. But I am not sure. Perhaps we can think of there being a function that assigns to the predicate the sense that belongs to it and also a function that assigns to the predicate its reference, being the set of items in its extensions. Now suppose instead we are speaking of a saturated predicate, like “Phar Lap is a horse.” Here the sense is the thought of Phar Lap being a horse, or having horseness, and the reference is the truth value. We can again perhaps think of there being a function that assigns to the sentence its sense and reference. At any rate, Priest has us consider the unsaturated predicate ‘the father of’. It refers to a function that “maps each person to their father (and each non-person to something else), and has a corresponding sense”. I do not know what is meant by “and each non-person to something else”. Does he mean simply each animal is assigned its male parent? Or does he mean that things which do not have fathers are assigned something else altogether? But I would not know what. Or would things without obvious fathers be assigned something like whatever is responsible for how they came about? At any rate, the sense of this function has a sense that Priest calls the ‘function-sense’. He gives the example, ‘the father of Frege’. This refers to a singular person, Frege’s father. He says further that its sense is an object-sense, a unitary thing. As I understand, here the function sense is the object, Frege’s dad. Then Priest explains the problem here, but I do not follow it very well. He says that the reason ‘the father of Frege’ has the sense of Frege’s dad is because there was a gap in ‘father of (x)’, and since ‘Frege’ filled that gap, it received the sense of Frege’s dad. (Something that I find a little confusing here is that if Frege’s dad is the sense of the function ‘father of Frege’, then what is the reference? I would think that the reference is Frege’s dad, and the sense is something more conceptual, or is something like Frege’s idea of the mode of presentation of Frege’s dad. For, Frege’s dad could have been presented as the husband of Frege’s mother, for example.) The part that I do not follow is the next sentence which says that ‘the father of’ is according to Frege’s criterion an object(-sense). I do not know what criterion would make it an object. Let me quote.]

It is not just the unity of propositions which generates this situation for Frege. In Fregean semantics, both the senses and referents of predicates are functions, and the same problem arises for normal cases of functional application. Thus, for Frege, ‘the father of ’ refers to the function that maps each person to their father (and each non-person to something else), and has a corresponding sense. Call this a function-sense. So ‘the father of Frege’ refers to Herr Frege sr., and its sense is an object-sense — a unitary thing. It has this because the sense of ‘Frege’ fills the gap in the sense of ‘father of ’. But the sense of ‘father of ’, according to Frege’s criterion, is an object(-sense), and so it does not have a gap at all. The situation is exactly the same.

(Priest 7)

[Priest then summarizes the problem. It is cast in a light that is a shade different from Beyond the Limits of  Thought. There the problem was more a matter of expressibility, but here it seems more to be an issue of unity. To explain the unity of the proposition, Frege has the structure of the concept-senses, by which a predicate is completed by some function. As such, concept-senses cannot be objects. For, objects are completed or closed structures, but concepts are open or incompleted ones, as they can take one or another or no particular argument. However, as we have seen, concept-senses are objects (since we need to place them into the argument place, and our intuition tells us we should be able to predicate them). Priest says that this is not merely a problem with meaning but rather is a deeper issue regarding how “parts cooperate to form a unity of any kind”. Priest in the following section will go into this problem in more detail.]

Frege’s problem, then, is this. If concept-senses and function-senses are to play their role in accounting for the unity of complexes, they cannot be objects. But they are. One might avoid Frege’s problem simply by rejecting his account of meaning. The situation in which Frege finds himself is, however, but an example of a much deeper problem which cannot be avoided in this way. At root, the problem is not about meaning at all. It is about how parts cooperate to form a unity of any kind. Let me spell this out.



Priest, Graham. One: Being an Investigation into the Unity of Reality and of its Parts, including the Singular Object which is Nothingness. Oxford: Oxford University, 2014.

Or if otherwise noted:

Frege, Gottlob. “On Sense and Reference”. Transl. P.T. Geach. In Translations from the Philosophical Writings of Gottlob Frege. Eds. P.T. Geach and Max Black. Oxford: Basil Blackwell, 1960, second edition (1952 first edition), pp.56-78.

Frege, Gottlob. “Function and Concept.” Transl. P.T. Geach. In Translations from the Philosophical Writings of Gottlob Frege. Eds. P.T. Geach and Max Black. pp. 21–41. Oxford: Basil Blackwell, 1960, second edition (1952 first edition).


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