Gottlob Frege’s "On Concept and Object"

Gottlob Frege’s 1892 paper, “On Concept and Object” attempts, by clarifying what he means (and meant) by the terms ‘concept’ and ‘object’, to correct some misunderstandings of his position, particularly those exemplified in Benno Kerry’s reading of Die Grundlagen der Arithmetik (The Foundations of Arithmetic). Frege accuses Kerry, a student of Brentano, of misunderstanding his usage of the word ‘concept’ and therefore of misunderstanding the necessity to differentiate between concepts and objects. The paper attempts to clarify how and why we should maintain an absolutely rigorous distinction between the two. This is accomplished by explaining what concepts are and what they can not be; the nature of objects is only mentioned secondarily.

At the outset, Frege makes it clear that he will stick to “a purely logical use” (168) of the term concept, rather than a psychological or a logical-psychological one. He admits that such usages of the term are possible, though they do not fall within logic proper. Here he does not precisely characterize the difference between these positions, nor what makes his position purely ‘logical’, though this does become somewhat clearer later on.

In response to Kerry, he makes a very interesting statement about what it means to treat the concept as a ‘logically simple item’. He insists that any treatment of his remarks on concepts as definitional is misleading insofar as “One cannot require that everything shall be defined, any more than one can require that a chemist shall decompose every substance” (168). The logically simple, like true elementary particles in physics, “simply cannot have a proper definition” but must be shown or elucidated by means of “hints”. As I will discuss in more detail in the second half of this writeup, I find the notion of the logically simple somewhat confusing. What I take Frege to be saying here is that we cannot determine the logically simple by means of analysis in the same way that we might with more complex logical entities. But this does not mean that the logically simple is given to us at the outset; we can (and perhaps must) arrive at the simple through synthetic means. Frege’s point is simply that these synthetic means do not ‘determine’ or ‘define’ logically simple items in the same way that they define more complex items. They merely point us toward them. What exactly the logically simple is, what role it plays, and how precisely we are able to understand it seem to remain unclear in this paper.

Frege moves on to discuss some of Kerry’s direct challenges to his (Frege’s) usage of the term concept, and, more specifically, to his distinction between objects and concepts. In order to reject this distinction, Kerry employs a simile, arguing that maintaining this distinction is much like holding that the “relation of father and son is one that can not be further reduced, that a man can not be at once a father and a son” (169). So, Kerry seems to suggest with this simile that beneath the very distinction between concepts and objects lies a deeper logical similarity of some kind.

Frege wishes to fasten onto this simile himself in order to strengthen and clarify his own position. He writes that

If there were, or had been, beings that were fathers but could not be sons, such beings would obviously be quite different in kind from all men, who are sons. Now it is something like this that happens here. The concept (as I understand the word) is predicative. On the other hand, a name of an object, a proper name, is quite incapable of being used as a grammatical predicate. This admittedly needs elucidation; otherwise it might appear false (169).

So Kerry’s simile is misleading in the following sense. It allows us to think that because it happens that human beings can be both fathers and sons, the distinction between father and son is further reducible to some quality of these relations as such. But this has nothing to do with logic: there is nothing in the relation between father and son that guarantees that a father can be a son. Frege suggests we will see his point if we apply the simile more forcefully: it is indeed conceivable that there are beings who can be fathers but not sons. He pushes the initially misleading simile to a logical point. There is nothing about the relation between father and son that necessarily implies the reversibility of the relation, quite the contrary. It is much the same with the distinction between concept and object.

But, as Frege notes, this can appear false and requires elucidation. It needs elucidation just insofar as we can say of a thing that it is Socrates just as easily as we can say that it is old or snub-nosed. Our language makes sentences like “There is Socrates” and “There is a snub-nosed man” appear identical. But Frege believes the word ‘is’ plays two very different functions here. In the statement “there is a snub-nosed man” we use ‘is’ to predicate something of a thing (i.e., we use it to show that the thing falls under the concept ‘snub-nosed’). We can say ‘Socrates is snub-nosed’, clearly. But we cannot say anything like ‘snub-nosed is Socrates’. The predication is a one-way relation, it cannot be reversed. Here it seems clear that an object (or a proper name) like ‘Socrates’ cannot sensibly fulfill the role of a predicate, and the reverse is true as well. We cannot apply a predicate to a predicate (unless, as we’ll see later, the predicate is of a different level). We cannot say, for example, ‘snub-nosed is snub-nosed’. Concepts, as the references of grammatical predicates, must predicate something of some object. Objects, as the references of grammatical subjects, must be predicated by some concept.

In order to clarify this distinction, which remains a bit unclear at this point, offers an example which may at first appear to be a counterexample. On the purely linguistic (as opposed to the logical) level, the sentence ‘The morning star is Venus’ seems to be evidence of an object (‘Venus’) functioning as a predicate on another object (‘the morning star’). So isn’t Venus both a concept and an object here? Frege answers no, this is only an apparent confusion resulting from our logically impure language. On the logical level, what is going on here is quite in keeping with his distinction. What is really being said here is not exactly that the morning star and Venus are identical (which would imply a reversible equation, a=b and b=a). Rather, the word ‘Venus’ here really stands for the concept ‘no other than Venus’. It is not a reversible equation but an irreversible statement of predication about the object ‘the morning star’. It is being predicated of this object that it falls under the (admittedly limited) concept ‘no other than Venus’. Thus, ‘Venus’ here, despite linguistic appearances, does not function as a proper name, but as a portion of a predicate. ‘Venus’ is assuredly an object and cannot fulfill the role of a concept, but here the word ‘Venus’ does not stand for the object ‘Venus’ but, instead, falls within the broader concept ‘no other than Venus’.

Frege now attempts to deal with another potential counterexample to his distinction. Kerry offers up the sentence “the concept ‘horse’ is a concept easily attained170 as an obvious example of a concept (to whit: the concept ‘horse’) being used as an object. But, counterintuitive as it must have seemed to Kerry and others, Frege sticks to his guns even here. He writes:

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 (170-171).

This is, Frege thinks, an ineradicable clumsiness of our natural language which we must strain to resist. He writes that the fact that the concept ‘horse’ is not a concept while the city Berlin is a city is a

predicament of language that justifies the departure from custom. The peculiarity of our case is indicated by Kerry himself, by means of the quotation-marks around ‘horse’ … There was no reason to mark out the words ‘Berlin’ and ‘Vesuvius’ in a similar way. In logical discussions one quite often needs to assert something about a concept, and to express this in the form usual for such assertions, viz., to make what is asserted of the concept into the content of the grammatical predicate. Consequently, one would expect that the reference of the grammatical subject would be the concept; but the concept as such cannot play this part, in view of its predicative nature; it must first be converted into an object, or, speaking more precisely, represented by an object (172).

In a footnote, he makes the point in even simpler terms: “By the very act of explicitly calling it a predicate, we deprive it of this property” (Footnote 1, 172). Frege now makes the distinction a little clearer. He writes that a concept is “the reference of a predicate” (173), while an object can never be the complete reference of a predicate, though it can, as we saw with the ‘Venus’ example, partially fulfill the role of a subject. Referring back to his paper “Function and Concept,” Frege remarks that concepts are “just a special case” of the unsaturated or incomplete nature of functions generally. Concepts are a special kind of function, and as such can only be discussed if they are represented (by our use of the definite article) as objects. Just how an object is capable of ‘representing’ a concept is unclear and points to the essential difficulties Frege encounters in bringing logic into language (and perhaps at the difficulties of maintaining a strictly anti-psychologistic view of logic as well).

Thinking of concepts as functions is a helpful way of understanding the difficulties that they present for investigators attempting to provide us with a clear understanding of their nature. We of course have to conduct such analyses within the natural languages we are given (this is so even if we create a logical symbolism like Frege’s, just insofar as this symbolism itself must be explained and taken into meaningful discourse if it is to be understood). But these natural languages pose obvious problems for understanding the distinctively unsaturated or function-like quality of concepts.

This is made evident in the ‘Venus’ example above, but also in the fact that many (perhaps all) sentences can, with equal validity, be understood as assertions about concepts and as assertions about objects. The analysis, i.e., the separation of the logically basic parts of a sense (i.e., into functions and objects), is not determined at the multiply interpretable level of the linguistic meaning (or ‘about-ness’) of the sentence itself. Because the linguistic and logical levels are separable (even if natural languages fortuitously mirror logic on occasion), the same sentence can seem to be ‘about’ many things at once.

This leads to apparent counterexamples to the rigid distinction between concept and object, but these counterexamples confuse surface level linguistic features (e.g., the differences between universal, particular, negative, and affirmative judgments) with underlying basic logical features (i.e., predicate/object relations). This becomes clearer with an example. Is the sentence, “there is one square root of four” about four, about ‘the’ square root of four’, about square roots, about the number one? Well, depending upon the circumstances, the sentence can bear any of these senses. The sentence itself, devoid of these circumstances, is ‘about’ all of these things, or none. But, once we give it a determinate sense, this ability to multiply interpret it falls away. There are determinate logical characteristics underlying all of these potential ‘senses’.

Logical analysis can only begin once a determinate sense has been given. And such a sense can equally well be given by any number of linguistically quite different sentences. When we analyze a ‘sense’ of a sentence, we find that no matter how we phrase that sense, no matter what its linguistic garb, it has the same logical structure. Frege offers the example “All mammals have red blood” (173); we could equally well say “whatever is a mammal has red blood” (173) or “if anything is a mammal, then it has red blood” (173). In each of these quite different sentences, the concept ‘red-blooded’ is being predicated of an object, viz. the class of mammals (however we want to sort that out). So, despite the difference between universal and particular judgments, for example, a determinate logical relation remains. This seems to be what a ‘sense’ is: the basic logical relation between the saturated and unsaturated portions of a ‘thought’ which makes disparate sentences logically identical.

So, while a single sentence may present us with multiple interpretations, and may appear to be ‘about’ an object and ‘about’ a predicate as well, Frege insists that what is asserted of these two things must be entirely separate in the logical sense. He writes that “what is… asserted about a concept can never be asserted about an object; for a proper name can never be a predicative expression, though it can be part of one” (175). So while we can smuggle in multiple unrealized meanings in our natural language, when we make the move to a single determinate sense in logic, such admixture becomes literally ‘senseless’. This is an important point. It is not, for Frege, ‘wrong’ or ‘false’ to assert something about an object that we normally assert of a concept, or vice-versa. It is, rather, senseless and even impossible to do so. To say ‘taller than Greg is snub-nosed’ is not ‘false’ but senseless; while to say ‘the man is snub-nosed man’ makes clear sense.

Here Frege briefly discusses what I take to be the basic structure of his system of logic. What can be asserted about one item in his logical ontology can never be asserted of an item of a different type. Thus, there is a class of assertions which can be applied to objects, a class for first-level concepts, a class for second-level concepts and so on. Frege also notes that the relation of ‘falling under’ that holds between an object and a concept is similar, though not identical with, the relation between a first- and a second-level concept. He writes that we might say that “An object falls under a first-level concept; a concept falls within a second-level concept … The distinction of concept and object thus still holds, with all its sharpness” 176. The fact that these similar relations must be held distinct is a result of their difference as regards ‘saturation’ or ‘completeness’. For Frege, these metaphorical ways of talking help to get more precisely at what is distinctive about the basic items of his ontology. For the sense of a sentence to be ‘fully saturated’ (as it must be to convey meaning), then it must have both a saturated and an unsaturated part. Functions (concepts) are unsaturated and cannot convey meaning or full sense simply on their own, they require something (a lower-level concept, or an object) to ‘complete’ them. This is an appealing picture, though one which Frege believes cannot be made anymore distinct because of the obfuscating quality of natural language in relation to basic elements of logic.

At the end of the paper, Frege writes something quite helpful about complete and incomplete thoughts (he thinks of a thought as the sense of a sentence, i.e., what is analyzed in logical analysis):

…not all the parts of a thought can be complete; at least one must be ‘unsaturated’, or predicative; otherwise they would not hold together. For example, the sense of the phrase ‘the number 2’ does not hold together with that of the expression ‘the concept prime number’ without a link. We apply such a link in the sentence ‘the number 2 fall sunder the concept prime number’; it is contained in the words ‘falls under’, which need to be completed in two ways—by a subject and an accusative; and only because their sense is thus ‘unsaturated’ are they capable of serving as a link. Only when they have been supplemented in this twofold respect do we get a complete sense, a thought (180).

So, by way of conclusion, Frege says the following of concepts. They are preceded by the indefinite article; they are ‘unsaturated’ or ‘incomplete’; in sentences they are the references of grammatical predicates; they are essentially predicative in nature; there are different levels of concepts and what can be asserted of one level cannot be asserted of another; what can be asserted of them can never be asserted of objects; in logic they are entirely and necessarily distinct from an object; they are logically simple and, as such, cannot be defined but only hinted at or gestured towards.

Frege on Understanding Logic

In “On Concept and Object” Frege makes some interesting (though brief) remarks about the difficulty that language seems to pose for any investigation into the nature of concepts, or, more broadly, into anything logically simple. In this paper I would like to clarify why Frege thinks that such difficulties are “essential” or “necessary”. The paper will be divided into two sections; in the first I will outline Frege’s reasons for seeing this difficulty as an ‘essential’ one. Near the end of this section I will pose a number of questions which I find particularly unsettling about Frege’s attempt to meaningfully discuss things like ‘concepts,’ ‘functions’ and ‘the laws of logic’. In the second section I will, with some reference to Wittgenstein’s development of this problem, attempt to describe what seems most confusing about Frege’s attempt to overcome the unavoidable barriers that language (and, I take it, thought more generally) seems to pose for any clear understanding of logic.

1. Logically simple items

In this short article, written in response to Benno Kerry’s misunderstandings of his earlier work, Frege makes four remarks which seem to bear on the relationship between his logical investigations and the nature of language.

  1. On the introduction of a name for something logically simple, a definition is not possible; there is nothing for it but to lead the reader or hearer, by means of hints, to understand the words as is intended (168-169).
  2. It must indeed be recognized that here we are confronted by an awkwardness of language, which I admit cannot be avoided, if we say that the concept horse is not a concept, whereas, e.g., the city of Berlin is a city and the volcano Vesuvius is a volcano. Language is here in a predicament that justifies the departure from custom (172).
  3. I admit that 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 (179).
  4. …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 (180).

I think in these altogether too brief remarks, Frege shows us something of his understanding of the nature of language, and of language’s relation to logic. In what follows I’ll try to make clearer exactly how this is so.

He writes of an essential or unavoidable difficulty in analyzing logically simple items (e.g., functions, objects) by means of language. The particular difficulty Frege has here is in defining such terms and in illustrating the necessity of maintaining the absolute distinction between concept (or function) and object that forms the basis of his logical ontology. He is forced, of course, to discuss these things by means of language, but language itself, by its very nature, makes what he says seem absurd (e.g., ‘The concept ‘horse’ is not a concept’) or, even further, makes it impossible for him to say what he really means. He is forced, then, to use language as a sort of diversionary tool that only allows him to gesture towards what he really means without his being able to firmly encapsulate it with meaningful speech. The literal meaning of his expressions thus misses his intended target, but it nevertheless hints at what he cannot say.

This is indeed a peculiar sort of difficulty. But why does it arise? Why can we talk of Berlin and Vienna as cities so easily, without any slippage or awkwardness in meaning, while discussions of ‘the function F(x)’ seem to go completely awry? I think this becomes clearer when we see what it is that Frege is prevented from doing.

In the first of the remarks quoted above, Frege rebuts Kerry’s objection to his ‘definition’ of a concept. Frege responds by arguing that we cannot be forced to define a logically simple item like the concept. It is not merely that this is a difficult thing to do, but it is, indeed, impossible. We can only be led to see, by hints or gestures, what is meant by ‘function’ or ‘concept’.

Now, why is defining what is logically simple impossible? I think the answer lies in Frege’s understanding of the relationship between language (or, more generally, thought) and logic. As I understand Frege, which is admittedly not terribly well, he sees logic as providing the laws of truth, as determining what can possibly be true, let’s say. Thoughts all have as their object truth-values, i.e., the true or the false. And language is a particularly robust form of expressing our thoughts with some amount of precision. Thus, logic in a sense determines which thoughts are possible (or, more correctly perhaps, it determines what thought can be) by legislating the realm of truth. And, in turn, it limits what is sayable to those things which are possibly true and false.

Why then does language, when turned upon the basic items of logic itself (i.e., functions), seem to lose its capacity to express meaningful thought?

The answer seems to be one that Wittgenstein found so fascinating and troubling in the Tractatus and elsewhere: logic legislates truth, but is not (and can not possibly be) legislated by truth. The simple items of logic (however we wish to cash them out) are neither true nor false. They are, in a sense, tautologous, even meaningless. It is the ‘meaninglessness’ of the logically simple that I would like to fasten onto.

First, let’s get clearer about why these items are meaningless, and then we’ll try and figure out what this ‘means’ (oddly enough) for logic and language. A brief recapitulation. The laws of logic are the laws of truth; they determine what can possibly be true. Thought is always a thought of some truth-value (the true, the false). Frege calls a thought the ‘sense’ of a sentence, its meaning. Meaning (sense) occurs as a result of a thought, which is a thought of some truth-value. So, if the laws of logic establish what can possibly be true, but can be neither true nor false themselves, neither can they be meaningful insofar as meaning is a result of thoughts which can only occur within the realm of possible truth-values delineated by those very laws. It is perhaps misleading to say that logical items or laws are ‘meaningless’ because the term ‘meaningless’ implies something that these items lack, as if they are capable of meaningfulness but somehow fail to be meaningful. This is misleading. Perhaps it is better to say that logic cannot be concerned with meaning, or that it is unconcerned with meaning.

So, if we cannot give the laws of logic or the items of logic any truth-value, how then can Frege ‘hint’ at them? How can he ‘show’ us the meaning (as he certainly seems to want to do) of something which is by its very nature exempt from the possibility of meaningfulness? Frege admits that he misses his mark with such ‘verbal gestures’ but how can he even be said to have a target? What can he have formulated? Is he thinking—are we all thinking—of something which cannot be thought when we think of the laws of logic?

Though these remarks are tangential to the main thrust of the article, they obviously have great import for Frege’s aim of clarifying what he means by a concept or a function. If these are logically simple items, then what is their status for us? They seem to be either ineffable or nonsensical (or ineffable because nonsensical perhaps).

2. Is understanding logic just gesturing at hintables?

In the Tractatus Wittgenstein takes up this problem more directly, though perhaps with equally mysterious results. At the outset of that work he writes that “the aim of the book is to draw a limit to thought, or rather—not to thought, but to the expression of thought: for in order to be able to draw a limit to thought, we should have to find both sides of the limit thinkable (i.e. we should have to be able to think what cannot be thought)” (Tractatus, preface, p. 3). He also writes that the propositions of the Tractatus are “elucidatory in this way: he who understands me finally recognizes them as senseless, when he has climbed out through them, on them, over them. (He must so to speak throw away the ladder, after he has climbed up on it.) He must surmount these propositions; then he sees the world rightly” (Tractatus, 6.54). In that work he famously concludes that “Whereof one cannot speak, thereof one must be silent” (Tractatus, 7).

Now it might appear that Wittgenstein is doing something very different than Frege here, but I think they are up to essentially the same thing, though Wittgenstein perhaps more openly. Frege attempts to point us towards an understanding of logic which can only appear clumsy and confused if we remain at the level of literal meaning. He suggests that what he wants to say cannot possibly be said but can, nevertheless be hinted at. So, once the hint is ‘grasped’ or understood, the nonsensical character of the literal ‘meaning’ of his statements ceases to be of any importance. This nonsensical level should, once we’ve gotten to what he really means (but can’t say), be recognized as elucidatory but ultimately empty of meaning. When we grasp what Frege ‘actually’ means by these seemingly absurd claims we see the logically simple correctly.

A similar process occurs in the Tractatus, the whole of the book is said, in the concluding sections, to have been nonsense, and all that we are left with after going through these empty gestures is a more ultimately meaningful silence that can only ever be gestured towards. We see the world correctly by first seeing it incorrectly, by seeing how the basic error of our picture of the world directs us to what the world ‘really’ is.

I am unsure whether we can properly understand what Frege and Wittgenstein are up to in their respective works given the terms they offer us. It seems to me that if we accept the idea that logic legislates and, in a sense, makes possible thought, then the notion of ‘understanding’ logic itself seems to make little sense. If it cannot be thought, it cannot be understood. Logic seems to occupy a strange epistemological ground: it is unthinkable and meaningless, but, at the same time, we can think it and understand what it means for us. So, in the end, I’m entirely uncertain whether we can make any sense of the notion of logic as prior to thought, and, even if we can, it is equally uncertain how this is to be done without recourse to some ineffable process that can never be explained or even pointed to. Just how we hint at logic seems inexplicable. It seems clear that Frege can talk about logical simples, and that we can understand his talk, but I find it difficult to mesh this intelligibility with his talk of logic as legislative for thought.


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