28 Jul 2016

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


by Corry Shores


[Search Blog Here. Index-tags are found on the bottom of the left column.]


[Central Entry Directory]

[Logic & Semantics, Entry Directory]

[Graham Priest, entry directory]

[Priest’s Beyond the Limits of Thought, entry directory]


[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.






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.









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:


Course listed at:





No comments:

Post a Comment