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

by Corry Shores

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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.

Summary

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.

(7)

From:

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).

.

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.

 

Summary

 

[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.

(200)

 

[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.

(202)

 

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.

(202)

 

 

 

From:

 

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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Priest (12.1) Beyond the Limits of Thought, ‘Frege, Concept and Object,’ 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

 

12.1 Frege, Concept and Object

 

 

 

Brief summary:

There is an aporia in Frege’s groundbreaking theory of meaning, namely, that the sense and reference of larger structures is compositionally based on that of its parts, but precisely how to conceptualize the way this compositional relation works is not entirely apparent. Frege’s notion of the unsaturated function structure is our best means for understanding this compositionality. For, the sense and reference of the whole does not result simply from the bare addition of the sentence’s parts. Rather, they must be combined and related by means of the operation of the function upon the terms in its argument place.

 

 

 

Summary

 

Priest explains that Frege’s major philosophical project was “to demonstrate that mathematics (or, at least, analysis) was logic” (Priest 197). [Nolt in section 14.1 of Logics mentions this and notes that it is called “logicism”.] The discovery of Russell’s paradox ruined hopes in this project. But there was a “subsidiary” part of Frege’s project, namely, a formulation for a systematic philosophy of language (Priest 197). Priest explains that “The clarity and power of its structure | had never before been achieved” (Priest 198). So even though this was just a side-project, it became “one of the major influences on twentieth-century philosophy” (198).

 

Priest’s main concern in Frege’s philosophy of meaning will be a particular aporia found especially in “Function and Concept,” “On Sense and Reference,” and “On Concept and Object” (198).

 

Priest then explains Frege’s important innovations. [It seems that before Frege there was a certain understanding of the subject-predicate relation, but Frege modifies it by replacing the category of subject with that of name. This means that a predicate can contain within it these names which are of a different category. So this also means that the category of name is wider than that of subject (since subjects cannot be parts of predicates). But also note that a subject can be an expression like “all men” with the predicate being something like “are mortal.” However, Frege’s category of name does not include such quantified expressions as “all men”, so in that sense the category of name is more limited than that of subject. (Recall from “On Concept and Object” Frege’s explanation for this. He shows that the proper negation of a quantified expression would belong to the quantified expression (that is, it would be positioned in front of the quantified expression), which actually means it belongs to the predicate. I am still not exactly sure what the reasoning is, but what I proposed in that summary as a possible explanation for the reasoning is that the negation term should always be affixed to the predicate (since we are denying the predication to some object), and since the proper place to put it is next to the quantifier in such quantified expressions as “all mammals are land dwellers”, thereby giving us, “not all mammals are land dwellers”, that means the “all” must belong to the predicate and not to the noun coming after it. He wrote:

It must here be remarked that the words ‘all,’ ‘any,’ ‘no,’ ‘some,’ are prefixed to concept-words. In universal and particular affirmative and negative sentences, we are expressing relations between concepts; we use these words to indicate the special kind of relation. They are thus, logically speaking, not to be more closely associated with the concept-words that follow them, but are to be related to the sentence as a whole. It is easy to see this in the case of negation. If in the sentence

‘all mammals are land-dwellers’

the phrase ‘all mammals’ expressed the logical subject of the predicate are land- dwellers, then in order to negate the whole sentence we should have to negate the predicate: ‘are not land-dwellers.’ Instead, we must put the ‘not’ in front of ‘all’; from which it follows that ‘all’ logically belongs with the predicate. On the other hand, we do negate the sentence ‘The concept mammal is subordinate to the concept land-dweller' by negating the predicate: ‘is not subordinate to the concept land-dweller.’

(Frege, “On Concept and Object”, 48)

But this is not an essential point in our current context.) Priest then says that names include proper names and definite descriptions. And Frege also will offer a fresh conception of the predicate. Priest says that instead we are to think of the concept-expression. [From the way Priest describes them, they seem to be the part of the function minus the argument, or the function properly speaking, or the unsaturated part of the function. I am not sure how to conceive it. Frege describes this structure in “Function and Concept”. Let us look first at a mathematical structure and then relate it to a sentential structure. In this text, Frege says that a function can be understood as having two main parts: a variable part and a non-variable part. He illustrates with a function that when its variable part is assigned three different values it produces these formulations:

‘2∙1³ + 1,’

‘2∙2³ + 2,’

‘2∙4³ + 4,’

The first stands for 3, the second, 18, and the third, 132. We ask, what part is variant and what part is not? We might think of the variant part as being like an opening or gap in the structure where the variant argument inputs can be placed. So we might display the structure as:

‘2∙(   )³ +(   )’

More conventionally we use letter variable symbols for those gaps. So we are more accustomed to seeing:

‘2∙x³ + x’

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

If the argument is not specified, as when there is just a gap or variable in the structure, then it is said to be ‘unsaturated’, Frege says. But when instead a specific argument is placed into the gap where it arguments are supposed to be placed, then we say that the function is ‘saturated’. This structure also applies to concepts which are expressed using sentence formulations. We can have the function (or as Priest will call it, the ‘concept-expression’) “... is the capital of England” (Frege will just write it as “the capital of England” but it is to be understood as a function and not as a name). Here we have marked the gap with an ellipsis. When we fill that argument place with ‘Sydney’ it is false, but it is true if the argument is ‘London’. For Priest it seems that the ‘name’ is to be understood as the argument in this function structure, and he calls the function part the ‘concept-expression’. But I might have this wrong. He says that the concept-expression is everything minus the names. But I wonder what that means for the ‘London is the capital of England’ example. ‘London’ and ‘England’ would seem to be names, and then “... is the capital of ...” would then seem to be the expression. Or is ‘capital’ also a name, and thus the concept expression would be ‘... is the ... of ...”? Or is ‘the capital of England’ all one name, and thus the concept-express would be ‘... is ...’? In other words, I am not sure how what Priest is calling the category of name corresponds to the ‘argument’ in a function, since I would think that names could appear in concept-expressions without being variable parts of the expression. The point here might be that “...is the capital of England” is all one concept, but it is itself a function composed of the concept/function “is the capital of ...”, which is a function/concept composed of “is the ... of ...”. I am just wondering. But that would make Priest’s definition of ‘name’ fit how I understand what Frege considers the argument of a function or concept.]

Frege takes the traditional distinction between subject and predicate, and refashions it for his own ends. Instead of the category of subject, Frege proposes the category name. This is wider than the traditional category, since it includes those noun-phrases that occur within the predicate as a grammatical object. But it is also narrower than the traditional category, since it excludes quantifier phrases such as ‘all men’. We are left with proper names and definite descriptions. Instead of the category of predicate, Frege proposes the category concept-expression. A concept-expression is what is left when names are deleted from a sentence. Thus, in ‘Oswald was framed for the murder of Kennedy’, ‘Oswald’ and ‘the murder of Kennedy’ are names and ‘was framed for’ is a concept- expression.

(Priest 198)

 

Priest next addresses another of Frege’s innovations, namely, his new way of conceiving the distinction between connotation and denotation, which are two notions for meaning. Frege’s terms are sense and reference (or Sinn and Bedeutung). The reference is the denotation. For a name the denotation is the object it refers to, and for concept-expression it is a concept. [Let me review the understanding we came to when summarizing Frege’s “On Sense and Reference.” The main distinctions we made there were between a name and a sentence and between sense and reference. The reference of a sign is the thing it designates, and its sense is the contextual conceptual contents that are evoked by the sign in its designation of some object. We can also think of it as the mode by which that object is presented by means of the sign. To illustrate how sense can vary for the same object, Frege offers three useful examples. The first in fact comes from the Begriffsschrift section 8, and it perhaps is not the ideal example for this; but let us begin with it to see what insights it provides in this regard. The basic idea is that we will have two points that begin by not coinciding, and then after a movement takes place, they later will coincide. Frege’s main point will be that when the points coincide, insofar as they occupy that place, they have no need of two different names. But, he argues, that does not mean we have reason to drop one of the two names. For, the way this point of correspondence is designated requires first a distinction not just of the names but also of the things they designate. So, it seems he is saying, when they do coincide, we should still distinguish them, because the mode by which that coincidence is presented requires separate names and entities. (Again, see Begriffsschrift section 8 for a fuller account of this example, along with illustration.) The two other examples come from “On Sense and Reference,” and they perhaps are more obviously illustrative of the difference between the sense and reference of a sign. In the first one, we are to think of a triangle where we might draw lines from any of its vertices to the opposite side’s midpoint.

The point of intersection at the center can be designated with just two of the lines. But the sense will differ depending on which two lines are chosen. He has us consider the point designated as being the intersection of lines a and b.

or as being the point of intersection between lines b and c.

So the same point can be designated by means of two different modes of presentation, each with its own sense. This is why I defined the sense of a sign as being the contextual conceptual material involved in its conception. And by this I do not mean to confuse it with what Frege calls an ‘idea’, which would be the subject associative mental content involved when one is conceptualizing something. The contextual conceptual material I refer to are objective in the sense that they are specified explicitly and would be common to all people conceiving the sense by means of that particular mode of designating the object. Different lines are used for each manner of designating the point of intersection, and so were we to form a concept of it by means of any of the various modes of presenting the object, there would be different contextual conceptual material in each case even though those different sets of contents would designate exactly the same object. The other example is the famous case of ‘the morning star’ and ‘the evening star’. Here, they have different senses, even though they designate the same object. Both of course are Venus, the third brightest heavenly body. But to conceive of Venus with the sense of it being the evening star means to conceive it for example as being the first star to appear at dusk. And to conceive it with the sense of it being the morning star means to conceive it for example as being the last star to disappear in the morning. One reason this example is excellent is that it highlights the connotational component of sense. And in this case of ‘the morning star’ and ‘the evening star’, that connotational difference takes on a poetic quality that is hard to miss. So Priest notes that the denotation (reference) of a name is an object, and the denotation of a concept-expression is a concept. Here is something in addition to what we determined, for we said that the denotation (reference) of a sentence is a truth value., but we did not in our summary mention the reference of the concept-expression. The difference here seems to be that the concept-expression is not a full sentence but is rather the sentence minus the name(s). Priest then says that the sense of a linguistic unit is in general what determines which thing (be it an object, concept, or truth value) is the correct referent. This is not something that I was able to discern in the texts, but I have come across similar such interpretations. For example, in Roger Vergauwen’s A Metalogical Theory of Reference he writes, “Informally stated, an intension is something which makes it possible, in any ‘possible world’, to recognize or determine the extension of a specific element” (Vergauwen 30). Also on this point are also course recordings of John Campbell’s “Theory of Meaning” class at UC Berkeley. In the first lecture he discusses Frege’s “On Sense and Reference”. At around 33 minutes he begins discussing this role of sense in connecting signs with things in the world (and the part we discuss in more detail comes at around 36 minutes). He makes the point that sense explains informativeness. So we know that “the morning star” and “the evening star” are two names for the same object. But ‘the morning star is the evening star’ is informative, because they have different senses. Suppose we say that another name for point A is point alpha (α), such that when we say point A we can if we want instead say or think point α. This means that, so far as we can tell, there is no difference in sense between the terms, and so there is nothing informative in making formulating the equality ‘a = α’. But the next step in Campbell’s reasoning, which will make this point that sense is what serves to pick out the reference, I do not follow very well (again see around 36 minutes). He says that because sense explains informativeness, sense fixes the reference of the sign, and in fact the sense uniquely determines the sign’s reference. He explains that sameness of sense makes the identity uninformative, therefore sameness of sense must guarantee sameness of reference. This should be easy to grasp, but I do not follow really. So in our example, we have ‘a’ and ‘α’. We assumed that both signs use the same conceptual contents when designating their objects. So they are both determined by the same means and with the same conceptual components (unlike the point determined by lines a and b and by b and c in the triangle example). The fact that they have the same sense means that all of these conceptual determinations that are bound up in its designation of the object are identical, in which case it would seem to be that they would have to designate the same object. So sameness of sense must guarantee sameness of reference. But still I am not sure that I followed that right. The next inference, which is based on this prior one, is that we can then conclude that sense determines reference. But I do not see exactly how we are supposed to make that inference, because two senses can pick out the same reference, as the morning star and evening star example illustrates. (Basically the difficulty in my understanding is that if you told me that two senses can pick out the same reference, then I would be inclined to think that we need instead to find something they have in common to explain why they pick out a common reference, and not use something that makes them distinct. This of course is just a failure in my logic abilities, so I would need a simplified elaboration that makes the steps of reasoning more explicit.) I would have thought that instead the reasoning for why the sense determines the reference is because the sense determines what qualifies as such a thing under this designation. So the reason ‘the morning star’ designates Venus is because the sense of ‘the morning star’ is being the thing that is the last star to disappear in the morning (or being the brightest star in the morning) which is Venus, and the reason that ‘the evening star’ designates Venus is because the sense of ‘the evening star’ is being the first star to appear at dusk (or being the brightest star at dusk) which is Venus. In other words, the sense of ‘the morning star’ picks out Venus in the night sky because it directs our attention to the appropriate thing by means of properties and determinations that are properties and determinations that belong to Venus. And depending on its temporal (and spatial) context, Venus has different properties and determinations, meaning that different senses can correspond to or belong to it (or to names designating it). At any rate, my point here in these comments is that I do not find Frege directly making this point that sense is what determines the correct referent of a sign or concept-expression. But I can see how it is implied by the way he defines sense in terms of mode of presentation of the designated thing. For, the mode of presentation is the means by which some sign is correlated to the determining features of some thing. Priest’s last point in this paragraph is that the objective thought expressed by a statement, that is, the proposition it expresses, is its sense.]

Frege also reshaped the traditional distinction between two notions of meaning: connotation and denotation. He distinguished between the sense (sinn), of a linguistic unit and its referent (bedeutung). According to Frege, all linguistic units have both a sense and a reference (denotation). The denotation of a name is an object; the denotation of a concept-expression is a concept. The denotation of a statement is a truth value (true or false). The sense of a linguistic unit is, in general, that which determines which object/concept/truth value is the correct referent. In the case of a statement, this is the (objective) thought expressed by it (the proposition expressed by it).

(Priest 198)

 

Priest’s next point about Frege’s theory of meaning is that it involves compositionality in that

the meaning of a compound linguistic expression is, in some sense, a function of the meanings of its parts. Frege thought that, by and large, the referent of an expression was a function of the referents of its parts, and the sense of an expression was a function of the senses of its parts. 

(Priest 198)

 

Priest now wonders, given that meaning in larger complex structures operates compositionally, how do the senses or the referents of the parts contribute to or produce the sense or references of the whole? Priest observes a problem with forming this conception. The meaning of a whole, like a whole sentence, is a unity of sorts. So for instance, “the thought that Brutus killed Caesar, for example, is a single thought” (Priest 199). This means that the meaning of a whole sentence is not the bare combination of the parts. “It is not, therefore, a mere congeries of the meanings of its parts: <the sense of ‘Brutus’, the sense of ‘killed’, the sense of ‘Caesar’>” (Priest 199). Frege explains the way this compositionality works with respect to reference, so Priest will give that account, but Priest says that “a parallel story is to be told for sense” (199).

 

Priest notes how for Frege, a concept can be understood as a function “that maps an object to a truth value”. [I do not follow these points very well. Priest is discussing a “problem”, which as I understand it is the problem of explaining how the sense or reference of the parts of a sentence compositionally contribute to the sense or reference of the whole. He notes that a mathematical function, like sin, maps a number to another number. For example, sin maps the  the value π to the value zero (see Suppes’ explanation of functions in terms of relations and ‘mapping’ in his Intro section 11.1). And as we noted, Priest says that a concept is a function that maps an object to a truth value. I am not exactly sure what the “object” is in this case. I suppose it would be the argument. So for Frege’s example of the concept-function “... is the capital of England”, it maps “London” to the True and “Sydney” to the false. Another interpretation I suppose could be that the function maps the whole sentence to a truth value, but that would seem strange that the function would include or reference itself in that way. But I am not sure. I then do not understand so well the next point. Priest says, “This does not solve the problem, since exactly the same problem arises with respect to a function and its arguments: ‘sin(π)’ is an expression referring to a single entity (the number zero); it is quite different from <sin, π>” (Priest 199). In the Suppes section I mentioned, we have a function also described as a binary relation whose extension is a set of ordered couples, where the first member of a couple is like the argument of the function and the second member of the couple is like the output value of the function. So for example, the function f(x) = x² can be understood as relation whose extension is the set including {<1,1>, <2,4>, <3,9>...} and so on. So I was a little confused at first by Priest’s formulation <sin, π>. I think now that he is not using the ordered n-tuple structure, but rather I think he means, like he seemed to above with <the sense of ‘Brutus’, the sense of ‘killed’, the sense of ‘Caesar’> to be listing the bare parts of the function sin(π), which would be <sin, π>. With that in mind, I think Priest’s point is that if we only think in terms of the composition in this simple manner of raw combination or concatenation, we still cannot explain how on the basis of combining the parts {sin, π} that we can explain why the function’s value is zero. We need something in addition to that. Frege’s answer is that the function is unsaturated, in that it has gaps. I am not sure how this concept of saturation explains the compositionality, however. So we have the parts {sin, π}. But this is not a function. It is just a set of parts which could make up a function. Instead, a function has more of a structure of something like: value gap, and operation on the value in that gap. I still am not sure how to articulate how this explains compositionality. As I understood from Frege’s analysis in “On Sense and Reference”, the compositionality was a matter of sentences with multiple clauses, and he showed how in the qualifying cases, the value of the whole was determined by the value of the component clauses. I am not sure how this would work for saying that a singular function’s sense or reference is somehow based on the sense and reference of its argument and its functional component. I wonder if perhaps one way to understand this would be the following. We have the function-concept, “the capital of England”. It has an extension (a reference), which includes just one member, namely, the city London. Then we consider two names, “London”  and “Sydney”. In Sydney’s extension (its reference) is the city Sydney, and in London’s extension (its reference) is the city London. Now, when we combine the references of “Sydney” and “capital of England” in a raw way, we get, <{Sydney}, {London}> or something like that. But when we combine it for “London”  and “capital of England”, we get <{London}, {London}>. We say that “London is the capital of England” has as its reference the True, and as its sense, the concept of London being the capital of England. So perhaps we might say that the reference of the whole saturated concept-function is built compositionally on the basis of the references of the parts by means of some sort of additional evaluative function which somehow assigns to couplings where there values are identical (or where the first term is found in the set coming in the second place), like <{London}, {London}>, the value true and ones where they are not identical (or where the first term is not found in the set of members in the set comprising the second term) like <{Sydney}, {London}> the value false. But of course we are not saying that the falsity of the whole sentence “Sydney is the capital of England” is somehow based on the truth values of the parts, because they have no truth values. Rather, the references (or extensions) of the parts are still determinative of the reference of the whole sentence, if we can apply an additional evaluative procedure which assigns truth when the argument is included in the relation’s set. But what about sense? We said that the sense of a term is a matter of its mode of presentation, which we specified as the conceptual determinations (some of which being contextual and arbitrarily related, like the contextual elements bound up in the sense of “the evening star”) that pick out the proper reference to the term. And the sense of a sentence is the concept it expresses, often taking the form of predication. We also had the idea of the sense of a concept-expression (a function), which I think is a more general sort of conception, like “being the capital of England” or just “being the capital of”.  So how might the sense of the whole expression be composed of the sense of its parts? I can only guess; I am sorry. But perhaps the idea is like the following. The argument term as a name has a sense, which is the conceptual material by which the reference is determined as the sign’s proper object. So the sense of “London” is whatever conceptual determinations being used to adequately point us to the city in question. Perhaps this includes geographical co-ordinates, or descriptions of distinguishing features of the city, or something like that. And we also in our example have the concept of “being the capital of England”. I would suspect that contained somehow in this sense are implied criteria for what would qualify as a proper argument. So for example, part of what is implied in the sense of that expression is that the city is geographically located somewhere within English territory. And also implied in its sense is that the city functions the way a capital city functions in a nation of the sort that England is. And so on. So I am just making wild guesses, but perhaps we can say that the sense of the whole sentence is based compositionally on the parts, when we again have some evaluative function or operation which detects whether the determining conceptual material that picks out the city London also picks out an object that fulfills all the criteria implied in the concept. At any rate, somehow the notions of gaps and saturation in the function explain how the sense and reference of the whole structure is based on the sense and reference of the parts. Priest then says that these terms Frege is using are just metaphors, but they are the best means we have at the moment for conceptualizing this notion. And Priest further says that we have reached “bedrock”. Perhaps he means that since we have no better means for understanding the compositionality, that we can really not go too much further in our analysis and understanding. Or perhaps he means something like the point Frege makes about the fact that the argument and the concept cannot be defined, because they are logically simple in that they do not contain any parts (see “On Concept and Object” pp.42-43).]

According to Frege, a concept is a function, like the mathematical function, sin, which maps a number to another number. A concept is a function that maps an object to a truth value. This does not solve the problem, since exactly the same problem arises with respect to a function and its arguments: ‘sin(π)’ is an expression referring to a single entity (the number zero); it is quite different from <sin, π>. Frege's solution to the problem is that a function is, in some sense, inherently ‘gappy’. Objects (the arguments of the function) may fill those gaps, giving completion. As he puts it (p. 24):

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’. And in this respect functions differ fundamentally from numbers [i.e. , objects].

The words ‘incomplete’, ‘unsaturated’, etc. are, of course, metaphors. Frege realised this, but could do no better; neither can I. At this point we seem to have reached bedrock.

(199)

 

 

 

 

 

From:

 

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

 

 

Or if otherwise noted:

 

Gottlob Frege. “Begriffsschrift (Chapter 1)”. 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).

 

Gottlob Frege. “Function and Concept.” 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).

 

Gottlob Frege. “On Concept and Object”. 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).

 

Gottlob Frege. “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).

 

 

Roger Vergauwen. A Metalogical Theory of Reference: Realism and Essentialism in Semantics. Lanham / New York / London: University Press of America, 1993.

 

John Campbell. “Lecture 1” of Philosophy 135: Theory of Meaning. Recorded course of UC Berkeley. On youtube at:

https://youtu.be/vN_yGxabNFw

Course listed at:

http://www.openculture.com/freeonlinecourses

 

 

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Priest (12.0) Beyond the Limits of Thought, ‘Introduction [to chapter 12],’ summary

 

by Corry Shores

 

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Graham Priest

 

Beyond the Limits of Thought

 

Part 4 Language and Its Limits

 

Ch.12 The Unity of Thought

 

12.0 Introduction

 

 

Brief summary:

There are contradictions that lie at the limits of thought, and in this chapter we look at such contradictions as they manifest in Frege’s and (early) Wittgenstein’s theories of meaning.

 

 

Summary

 

Priest notes that the main issue of this book is the limits of thought. He recalls that “We have seen how contradictions at the limits of thought that we have been concerned with take a very sharp form in the inclosure contradictions of self-reference” (197). We saw in the book’s second part that for Kant it was a matter of the limits of reason. But in the twentieth century, language takes the center stage in philosophical debates. So we look now at how the problem of the limits of thought has manifested in twentieth century philosophy in terms of language (197).

 

Priest says that “It is natural for a theory of language to have implications about what can and what cannot be expressed” (Priest 197). Priest says that in fact modern theories of language are somehow able to “render some very important things – usually themselves – beyond the limit of expression” (197). [I am not sure, but perhaps the “usually themselves” part means that these very theories or systems themselves are somehow self-referentially able to express themselves within themselves.] Priest says that the “contradictions at the limits of thought” will “appear in a new guise” in this context. So now we will examine some modern accounts of meaning, looking for these contradictions at the limits of thought. Priest will not try to evaluate the theories; rather, he will “examine their consequences” (197).

 

Priest will begin with Frege then move to Wittgenstein’s Tractatus, all the while focusing on the unity of thought (197).

 

 

 

 

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

 

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