112 On Concept and Object
quently different terms are required.
quently different terms are required.
Gottlob-Frege-Posthumous-Writings
102 On Concept and Object
[Footnote* top. 100 continued]
characteristic marks and mode of formation. When Kerry says 'By the number 4 we understand the result of additively combining 3 and 1', this is obviously meant to be a definition. A definition is always a logical identity. The sentence 'The number 4 is nothing other than the result of additively combining 3 and 1' would have the same sense. Thus this sentence does not claim to assert that the number 4 falls under the concept result o f additively combining 3 and 1, as does 'The number 4 is a result of additively combining 3 and 1'. This latter cannot be construed as a logical identity, as a definition, because it leaves it open whether there may not be something else which is also a result of this operation, and because it has a sense only if it has previously been established what the words 'the
number 4' designate. This can only be fixed by a logical identity to the effect that the same thing is to be understood by the words 'the num- ber 4', whose sense we know because the meanings of its parts and of the grammatical forms em- ployed are known to us. I t is, accordingly, clear that the definite article makes an essential con- tribution to the sense of our sentence and that the position here is conse- quently quite different from what it is in the case of 'The capital of the German Empire is the city of Ber- lin', where if you leave out 'the city of' you are only suppressing a quali- fication which is incidental to the sense. Thus I see no alternative but to deny that the words 'the result of additively combining 3 and 1' mean a concept, if one refuses to allow the possibility that concepts should be distinct from one another even through their characteristic marks
? ? On Concept and Object 103 [Footnote* top. 100 continued]
and mode of formation are the same. This alternative would be open to Kerry.
In fact Kerry seems to coun- tenance this possibility when he says on pp. 456 ff. 1 that the concept of a concept can contain the same characteristic marks as the concept itself. But there is no doubt that here again a property is being confused with a characteristic mark. I do not believe that anything of value is to be learned by taking this route or that logicians will have any inclination to tread the same path. On the contrary, it seems far and away more appropriate to regard the replacement of the indefinite by the definite article as signifying the transition from a concept to an object falling under it, where the case holds that there is only one such object. So I do not go along with Kerry's saying that the object is like the concept in having as characteristic marks those of the concept; I say instead that the object has the characteristic marks of the concept as properties. And here we have the distinction. For the relation of a characteristic mark to a concept is different from that of property to an object (cf. Grund- lagen, ? 53).
Another consideration shows that the words 'the result of additively corn bining 3 and 1' designate an object. We have already seen that the first sentence of the passage from Kerry quoted above is to be construed as a definition and thus as a logical identity. Now at the end we have: '"the" 4 is likewise the result of additively combining 3 and 1'. The author's view is that the role of
1 Vjschr. f wissensch. Philosophie 10 (1886) (ed. ).
? 104 On Concept and Object
[Footnote* top. 100 continued]
the grammatical subject is to desig- nate the concept-object and pre- sumably the definite article is enclosed in quotation marks to mark this object off from the concept. Obviously this last sentence has the sense of a logical identity too: "'the" 4 is nothing other than the result of additively combining 3 and 1'.
The view that this sentence is meant to express that an object falls under the concept result of etc. is immediately ruled out by the occur- rence of the definite article before 'result'. Otherwise the indefinite ar- ticle would have been used or the article would have to have been omitted. Now if the left-hand side of this identity is an object, the right- hand side must be an object too. Further, the right-hand side of the first identity is at the same time the right-hand side of the second: hence the two expressions to the left must designate the same thing: the num- ber 4 is nothing other than 'the' 4. I can see no significance in the use of quotation marks here. In my view the reason for the sorry state of affairs we find in Kerry, where the distinctions between concept and object, characteristic mark and pro- perty, are effaced is that logical and psychological questions and view- points are scrambled together, which greatly detracts from the value of his articles. He will speak now of a concept, then of the idea of a concept, now of an object, then of the idea of it, without its even being wholly clear whether it is the one or the other that is in question, whether we are engaged in a logical or psychological inquiry. Now it is no particular cause for wonder if we can find no way of distinguishing between the idea of a concept and
? ? can be the reference of a subject. It must here be remarked that the words 'all', 'any', 'no', 'some', are prefixed to concept-words. In uni- versal and particular affirmative and negative sentences, we are express- ing 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. I f 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
On Concept and Object 105
that of an object. Perhaps they are often even difficult to hold apart. But here Kerry has simply succum- bed to a widespread sickness. In- deed would not Locke's empiricism and Berkeley's idealism, and so much that is tied up with these philosophies, have been impossible if people had distinguished adequately between thinking in the narrower sense and ideation, between the parts of a content (concepts, ob- jects, relations) and the ideas we have? Even if with us men thinking does not take place without ideas, still the content of a judgement is something objective, the same for everybody, and as far as it is con- cerned it is neither here nor there what ideas men have when they grasp it. In any case these are subjective and will differ from one person to another. What is here being said of the content as a whole applies also to the parts which we can distinguish within it.
? ? 106 On Concept and Object
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 sen- tence 'The concept mammal is subordinate to the concept land- dweller' by negating the predicate: 'is not subordinate to the concept land-dweller'.
Ifwekeepitinmindthatinmy
way of speaking expressions like
'the concept F ' designate not con-
cepts but objects, most of [199]
Kerry's objections already collapse.
If he thinks (cf. p. 281) that I have
identified concept and extension of
concept, he is mistaken; I merely
expressed ny view that in the ex-
pression 'the number that applies to
the concept F is the extension of the
concept like-numbered to the
concept F' the words 'extension of words 'extension of the concept' the concept' could be replaced by
'concept'. Notice carefully that here
the word 'concept' is combined with
the definite article. Besides, this was
only an incidental remark; I did not
base anything upon it.
Thus Kerry does not succeed in filling the gap between concept and object. Someone might attempt, however, to make use of my own statements in this sense. I have said that to assign a number involves an
Thus Kerry in his account does not succeed in filling the gap between concept and object. Someone might attempt, however, to make use of my own statements in this sense. For example, I have said (Grund?
? The question whether one should simply put 'the concept' for 'the extension of the concept' is in my view one of expediency.
So if we hold on to the fact that the words 'the concept F' designate not a concept but an object, most of Kerry's objections don't begin to stand up. If he thinks (p. 281) that I have identified concept and exten- sion of a concept, this is simply not the case. I merely expressed my view that in the expression 'the number belonging to the concept F is the extension of the concept like- numbered to the concept F', the
could be replaced by the word 'concept'. N. B. the word 'concept' is here combined with the definite article. Besides this was only an incidental remark; I based nothing on it in order not to have to grapple with the misgivings to which it might give rise. So Kerry's opposition to it has no bearing at all on the real core of my position. ?
? ? On Concept and Object 107
assertion about a concept;* I speak
of properties asserted of a concept,
and I allow that a concept may fall
under a higher one. ** I have called
existence a property of a concept. that a concept may fall under a
How I mean this is best made clear
by an example. In the sentence
'there is at least one square root of
4', we have an assertion, not about
(say) the definite number 2, nor
about -2, but about a concept,
square root of 4; viz. that it is not
empty. But if I express the same
thought thus: 'The concept square
root of 4 is realized', then the first viz. that it is not empty but is
realized. But if I express the same thought thus: 'The concept square root of 4 is realized', then the first six words form the proper name of an object. In general a content can be analysed in a number of ways and language seeks to provide for this by having at its disposal dif- ferent expressions for the same content. The distinction between the active and passive form, for exam- ple, enables us to present different parts of the content as the subject. This is why it is almost invariably a
six words form the proper name of
an object, and it is about this object
that something is asserted. But
notice carefully that what is asserted
here is not the same thing as was
asserted about the concept. This will
be surprising only to somebody who
fails to see that a thought can be
split up in many ways, so that now
one thing, now another, appears as
subject or predicate. The thought
itself does not yet determine what is
to be regarded as the subject. If we
say 'the subject of this judgement',
we do not designate anything mistake to put the definite article
definite unless at the same time we indicate a definite kind of analysis; as a rule, we do this in connexion with a definite wording. But we must never forget that different sentences may express the same thought. For example, the thought we are con- sidering could also be taken as an assertion about the number 4:
'The number 4 has the property that there is something of which it is the square. '
* Grund/agen, ? 46. ** Grundlagen, ? 53.
before the word 'subject' in such expressions as 'the subject of the judgement', 'the subject of the con- tent of a possible judgement'; for no part of the content can be picked out in advance as the subject. Even the expressions 'singular content of pos- sible judgement', 'particular content of possible judgement' are not quite accurate in that they ascribe to the content itself an attribute which, strictly speaking, belongs to it only under a certain form-a certain way
lagen, ? 46) that a statement of number contains an assertion about a concept; I speak of properties asserted of a concept, and I allow
higher one (Grundlagen, ? 53). I have called existence a property of a concept. How I mean this is best made clear by an example.
In the sentence 'There is at least one square root of 4', we have an assertion not about, say, the par- ticular number 2, nor about -2, but about a concept, square root of 4;
? ? ? 108 On Concept and Object
Language has means of present- of analysing it into subject and
ing now one, now another, part of the [200] thought as the subject; one of the most familiar is the distinction of active and passive forms. It is thus not impossible that one way of analysing a given thought should make it appear as a singular judge- ment; another, as a particular judge- ment; and a third, as a universal
judgement. It need not then surprise us that the same sentence may be conceived as an assertion about a concept and also as an assertion about an object; only we must observe that what is asserted is dif- ferent. In the sentence 'there is at least one square root of 4' it is impossible to replace the words 'square root of 4' by 'the concept square root of 4; i. e. the assertion that suits the concept does not suit the object. Although our sentence does not present the concept as a subject, it asserts something about it; it can be regarded as expressing the fact that a concept falls under a higher one. * But this does not in any way efface the distinction between object and concept. We see to begin with that in the sentence 'there is at least one square root of 4' the predicative nature of the concept is not belied; we could say 'there is something that has the property of giving the result 4 when multiplied by itself. ' Hence what is here asser- ted about a concept can never be asserted about an object; for a
* In my Grund/agen I called such a concept a second-order concept; in my work Function und Begriff I called it a second-level concept, as I shall do here.
predicate.
It is conceivable that the same content should appear in one form as singular, in another as particular. It need not then surprise us that we can discern in the same content an assertion about a concept and also an assertion about an object. We must only take note that
what is asserted of the one is dif- ferent from what is asserted of the other. In the sentence 'There is at least one square root of 4' it is im- possible to replace the words 'square root of 4' by 'the concept square root of4'; i. e. the assertion that suits the concept does not suit the object. All the same something is asserted of a concept in our sentence. We can even say that our concept is presented as falling under a higher one*-one whose sole characteristic mark is is realized, understanding this word in the sense in which we are using it in the present context. But this does not in any way efface
the distinction between concept and object. We see to begin with that in the expression 'There is at least one square root of 4', 'square root of 4' is being used predicatively: 'there is something that is a square root of 4'. In place of this we could say 'there is something that has the property of giving the result 4 when multiplied by itself. Hence what is here asser-
* In my Grundlagen I called such a concept a second order concept; in my Funktion und Begriff I called it a second level concept, as I shall do here.
? proper name can never be a pre- dicative expression, though it can be partofone. Idonotwanttosayitis false to assert about an object what is here asserted about a concept; I want to say it is impossible, sense- less, to do so.
The sentence 'there is Julius Caesar' is neither true nor false but senseless; the sentence 'there is a man whose name is Julius Caesar' has a sense, but here again we have a concept, as the indefinite article shows. We get the same thing in the sentence 'there is only one Vienna'. We must not let ourselves be deceived because language often uses the same word now as a proper name, now as a concept-word; in our example, the numeral indicates that we have the latter; 'Vienna' is here a concept-word, like ? metropolis'. Using it in this sense, we may say: 'Trieste is no Vienna'. If, on the other hand, we substitute 1201] 'Julius Caesar' for the proper name formed by the first six words of the sentence 'the concept square root of 4 is realized', we get a sentence that has a sense but is false; for the assertion that some- thing is realized (as the word is being taken here) can be truly made only about a quite special kind of objects, viz. such as can be designated by proper names of the form 'the concept F'. Thus the words 'the concept square root of 4' have an
ted of a concept can never be asserted of an object. This is not to saythatitisfalsetodothis,butim- possible: a sentence which tried to express such a thing would be absolutely devoid of sense; for it has no sense to use the name of an object predicatively. We have seen that even where the form of a sentence makes it look as if this is being done, the truth is that the object forms only part of what is asserted, since 'nothing other than' has to be added in thought. A sentence such as 'There is at least Julius Caesar' is senseless, although the sentence 'There is at least one man whose name is "Julius Caesar"' has a sense; but here again we have a concept, as the 'one' shows.
On Concept and Object 109
If, on the other hand, we sub- stitute the proper name 'Julius Caesar' for the proper name formed by the first six words of the sentence 'The concept square root of 4 is realized', we get a sentence that has a sense but is false; for the assertion that something is realized, as the word is meant here, is one we can only truly make of such objects as stand in a quite special relation to a concept. Thus the words 'the con- cept square root of 4' have an essentially different behaviour, as
? 110 On Concept and Object
essentially different behaviour, as regards possible substitutions, from the words 'square root of 4' in our original sentence; i. e. the reference of the two phrases is essentially
different.
What has been shown here in one
example holds good generally; the behaviour of the concept is essen- tially predicative, even where some- thing is being asserted about it; consequently it can be replaced there only by another concept, never by an object. Thus the assertion that is made about a concept does not suit an object. Second-level con- cepts, which concepts fall under, are essentially different from first-level concepts, which objects fall under.
regards possible substitutions, from the words 'square root of 4' in our original sentence; i. e. the meanings of the two properties are different, thoughrelated.
What has been shown here in one example holds good generally: the behaviour of the concept is essen- tially predicative even when some- thing is being asserted of it; conse- quently it can be replaced there only by another concept, never by an object; i. e. the assertion that is made about a concept does not fit an object at all. Moreover a concept (of second level) under which a concept falls is essentially different from a concept (of first level) under which objects fall.
In the sentence 'There is at least one square root of 4' we assert that the first level concept square root of 4 falls under a concept of second level, whereas in the sentence 'The concept square root o f 4 is realized' we assert that the object the concept square root of4 falls under the first level concept concept that is realized. We do indeed have the same thought in the two concepts, [sic. ] but this, being analysed dif- ferently, is construed in a different way.
The relation of an object to a concept that it falls under is dif- ferent from the admittedly similar relation of a concept to a concept (of second level). From now on we shall give parallel expression to the similarity and the difference by saying than an object falls under a concept, and a concept falls within a concept (of second level); for, strictly speaking, we have been making a mistake in using the same
The relation of an object to a first-level concept that it falls under is different from the (admittedly similar) relation of a first-level to a second-level concept. (To do justice at once to the distinction and to the similarity, we might perhaps say: An object falls under a first-level concept; a concept falls within a second-level concept. ) The dis- tinction of concept and object thus still holds, with all its sharpness.
? With this there hangs together what I have said (Grund/agen, ? 53) about my usage of the words 'property' and 'mark'; Kerry's dis- cussion gives me occasion to revert once more to this.
The words serve to signify relations, in sentences like ? (/J is a property of T' and '(/J is a mark of il'. In my way of speaking a thing can at once be a property and a mark but not of the same
thing.
words to cover both cases. From this we can see that the distinction between object and concept still holds in all its sharpness.
Connected with this is another distinction, namely that between property and characteristic mark. I have already explained this in ? 53 of my Grundlagen and at the time I thought I had made it sufficiently clear. Since, however, Kerry makes no use of it and falls into error as a result, I can only assume, alas, that he has not understood me; this time I will try to be more successful.
The words 'property' and 'characteristic mark' serve to designate relations such that something can at the same time be a property of r and a characteristic mark of L1. We express ourselves more precisely if we use the words 'characteristic mark' only in the phrase 'characteristic mark of a concept'. If a thing is a property of an object, then it is a characteristic mark of a first level concept. We may also speak of properties of a first level concept, and these will then be characteristic marks of second level concepts. I will call the concepts under which an object falls its properties.
If the object r has the properties (/J, X and '1', I may combine them into n; so that it is the same thing if I say that r has the property n, or, that r has the property (/J and r has the property X and r has the property '1'. I then call (/J, X, and 'I' marks of the concept n, and, at the same time, properties of r. It is clear that the relations of (/J to r and to n are
I call the concepts under which an object falls its properties; thus 'to be (/J is a property of T' is just another way of saying: 'T falls under the concept of a (/J'. If the object r has the properties (/), X, and '1', I may combine them into n; so that it is the same thing if I say that r has the [202] property n, or, that r has the properties (/), X, and '1'. I then call (/J, X, and 'I' marks of the concept n, and, at the same time, properties of r. It is clear that the relations of (/J to r and to n are quite different, and that conse-
On Concept and Object Ill
?
112 On Concept and Object
quently different terms are required. quite different; r falls under the r falls under the concept f/J; but n, concept f/J; but n, which is itself a which is itself a concept, cannot fall concept, cannot fall under the first under the first-level concept f/J; only level concept f/J; only to a second to a second-level concept could it level concept could it stand in a
stand in a similar relation. n is, on the other hand, subordinate to f/J.
Let us consider an example. Instead of saying:
'2 is a positive number' and '2 is a whole number' and '2 is less than 10'.
we may also say
'2 is a positive whole number less than 10'.
Here
appear as properties of the object 2, and also as marks of the concept
positive whole number less than 10.
This is neither positive, nor a whole number, nor less than 10. It is indeed subordinate to the concept whole number , b u t d o e s n o t fall under it.
Let us now compare with this what Kerry says in his second article (p. 424). 'By the number 4 we understand the result of additively combining 3 and 1. The concept object here occurring is the numerical individual 4; a quite definite number in the natural number-series. This object obviously bears just the marks that are named in its concept, and no others besides-provided we refrain, as we
similar relation.
to be a positive number, to be a whole number, to be less than 10,
? ? surely must, from counting as propria of the object its infinitely numerous relations to all other individual numbers; "the" number 4 is likewise the result of additively combining 3 and 1. '
We see at once that my dis- tinction between property and mark is here quite slurred over. Kerry distinguishes here between the num- ber 4 and 'the' number 4. I must confess that this distinction is in- comprehensible to me. The number 4 is to be a concept; 'the' number 4 is to be a concept-object, and none other than the numerical individual 4. It needs no proof that what we have here [203] is not my distinction between concept and object. I t almost looks as though what was floating (though very obscurely) before Kerry's mind were my dis- tinction between the sense and the reference of the words 'the number 4'. * But it is only of the reference of the words that we can say: this is the result of additively combining 3 and 1.
Again, how are we to take the word 'is' in the sentences 'the num- ber 4 is the result of additively combining 3 and I' and '"the" number 4 is the result of additively combining 3 and I'? Is it a mere copula, or does it help to express a logical equation? In the first case, 'the' would have to be left out before 'result', and the sentences would go like this:
'The number 4 is a result of addi- tively combining 3 and I';
* Cf. my essay 'On Sense and Reference' (cited above).
On Concept and Object 113
? 114 On Concept and Object '"The" number 4 is a result of
additively combining 3 and 1. "
In that case, the objects that Kerry designates by
'the number 4' and "'the" number 4' would both fall under the concept result o f additively combining 3
and 1.
And then the only question would be what difference there was be- tween these objects. (I am here using the words 'object' and 'concept' in my accustomed way. ) I should express as follows what Kerry is apparently trying to say:
'The number 4 has those properties, and those alone, which are marks of the concept: result ofadditively combining 3 and 1. '
I should then express as follows the sense of the first of our two sen- tences:
'To be a number 4 is the same as being a result of additive com- bination of 3 and 1. '
In that case, what I conjectured just now to have been Kerry's intention could also be put thus:
'The number 4 has those properties, and those alone, which are marks of the concept a number 4. '
[204] (We need not here decide whether this is true. ) The inverted commas around the definite article in the words '"the" number 4' could in that case be omitted.
But in these attempted in- terpretations we have assumed that in at least one of the two sentences the definite articles in front of 'result'
? and 'number 4' were inserted only by an oversight. It we take the words as they stand, we can only regard them as having the sense of a logical equation, like:
'The number 4 is none other than the result of additively combining 3 and 1. '
The definite article in front of 'result' is here logically justified only if it is known (i) that there is such a result; (ii) that there is not more than one. In that case, the phrase designates an object, and is to be regarded as a proper name. If both of our sen- tences were to be regarded as logical equations, then, since their right sides are identical, it would follow from them that the number 4 is 'the' number 4, or, if you prefer, that the number 4 is no other than 'the' number 4; and so Kerry's dis- tinction would have been proved untenable. However, it is not my present task to point out contra- dictions in his exposition; his way of taking the words 'object' and 'con- cept' is not properly my concern here. I am only trying to set my own usage of these words in a clearer light, and incidentally show that in any case it differs from his, whether that is consistent or not.
I do not at all dispute Kerry's right to use the words 'concept' and 'object' in his own way, if only he would respect my equal right, and admit that with my use of terms I have got hold of a distinction of the highest importance. 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
On Concept and Object
115
? 116
On Concept and Object
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 upon a reader who would be ready to meet me half-way-who does not begrudge a pinch of salt.
Somebody may think that this is an artificially created difficulty; that there is no need at all to take account of such an unmanageable thing as what I call a concept; that one might, like Kerry, regard an object's falling under a concept as a relation, in which the same thing could occur now as object, now as concept. [205] The words 'object' and 'concept' would then serve only to indicate the different positions in the relation. This may be done; but anybody who thinks the difficulty is avoided this way is very much mistaken; it is only shifted. For not all the parts of a thought can be complete; at least one must be 'un- saturated', 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 falls under 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 ac- cusative; 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. I say that such words or phrases stand for a relation. We now get the same
? difficulty for the relation that we were trying to avoid for the concept. For the words 'the relation of an object to the concept it falls under' designate not a relation but an object; and the three proper names 'the number 2', 'the concept prime number', 'the relation of an object to a concept it falls under', hold aloof from one another just as much as the first two do by themselves; however we put them together, we get no sentence. It is thus easy for us to see that the difficulty arising from the 'unsaturatedness' of one part of the thought can indeed be shifted, but not avoided. 'Complete' and 'un- saturated' are of course only figures of speech; but all that I wish or am able to do here is to give hints.
It may make it easier to come to un understanding if the reader com- pares my work Function und BegrifJ. 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 inap- propriateness of linguistic ex- pression; and that there is nothing for it but to realize this and always take it into account.
On Concept and Object 117
? ? [Comments on Sense and Meaning] 1 [1892-1895]
In an article (Uber Sinn und Bedeutung) I distinguished between sense and meaning in the first instance only for the case of proper names (or, if one prefers, singular terms). The same distinction can also be drawn for concept? words. Now it is easy to become unclear about this by confounding the division into concepts and objects with the distinction between sense and meaning, so that we run together sense and concept on the one hand and meaning and object on the other. To every concept-word or proper name, there corresponds as a rule a sense and a meaning, as I use these words. Of course in fiction words only have a sense, but in science and wherever we are concerned about truth, we are not prepared to rest content with the sense, we also attach a meaning to proper names and concept-words; and if through some oversight, say, we fail to do this, then we are making a mistake that can easily vitiate our thinking. The meaning of a proper name is the object it designates or names. A concept-word means a concept, if the word is used as is appropriate for logic. I may clarify this by drawina attention to a fact that seems to weigh heavily on the side of extensionalist as against intensionalist logicians: namely, that in any sentence we can substitute salva veritate one concept-word for another if they have the same extension, so that it is also the case that in relation to inference, and where the laws of logic are concerned, that concepts differ only in so far as their extensions are different. The fundamental logical relation is that of an object's falling under a concept: all relations between concepts can be reduced to this. If an object falls under a concept, it falls under all concepti with the same extension, and this implies what we said above. Therefore just as proper names can replace one another salva veritate, so too can concept? words, if their extension is the same. Of course the thought will alter when such replacements are made, but this is the sense of the sentence, not ita meaning. * The meaning, which is the truth-value, remains the same. For thia reason we might easily come to propose the extension of a concept as the
* Cf. my article Uber Sinn und Bedeutung.
1 These comments were not composed before 1892, the year in which the article Ober Sinn und Bedeutung appeared. They are part of a bundle of papers entitled 'Schrodersche Logik', which existed in a complete form prior to the destruction of the NachlajJ. The first part of these constituted a draft of Frege's article Kritischl Beleuchtung einiges Punkte in E. Schroder's Vorlesungen iiber die Algebra der
Logik (ed. ).
? [Comments on Sense and Meaning] 119
meaning of a concept-word; to do this, however, would be to overlook the fact that the extensions of concepts are objects and not concepts (Cf. my essay Funktion und Begrifl). Nevertheless there is a kernel of truth in this position. In order to bring it out, I need to advert to what I said in my mono- graph on Funktion und Begriff. On the view expressed there a concept is a function of one argument, whose value is always a truth-value. Here I am borrowing the term 'function' from Analysis and, whilst retaining what is essential to it, using it in a somewhat extended meaning, a procedure for which the history of Analysis itself affords a precedent. The name of a func- tion is accompanied by empty places (at least one) where the argument is to go; this is usually indicated by the letter 'x' which fills the empty places in 4uestion. But the argument is not to be counted as belonging to the function, und so the letter 'x' is not to be counted as belonging to the name of the function either. Consequently one can always speak of the name of a function as having empty places, since what fills them does not, strictly speaking, belong to them. Accordingly I call the function itself unsaturated, or in need of supplementation, because its name has first to be completed with the sign of an argument if we are to obtain a meaning that is complete in itself. I call such a meaning an object and, in this case, the value of the function for the argument that effects the supplementing or saturating. In the cases we first encounter the argument is itself an object, and it is to these that we shall mainly confine ourselves here. Now with a concept we have the special case that the value is always a truth-value. That is to say, if we complete the name of a concept with a proper name, we obtain a sentence whose sense is a thought; and this sentence has a truth value as its meaning. To acknowledge this meaning as that of the True (as the True) is to judge that the object which is taken as the argument falls under the concept. What in the case of a function is called unsaturatedness, we may, in the case of a concept, call its predicative nature. ? This comes out even in the cases in which we speak of a subject-concept ('All equilateral triangles are equiangular' means 'If anything is an equilateral triangle, then it is an equiangular triangle').
Such being the essence of a concept, there is now a great obstacle in the way of expressing ourselves correctly and making ourselves understood. If I want to speak of a concept, language, with an almost irresistible force, compels me to use an inappropriate expression which obscures-! might nlmost say falsities-the thought. One would assume, on the basis of its 1malogy with other expressions, that if I say 'the concept equilateral triangle' I am designating a concept, just as I am of course naming a planet If I say 'the planet Nepture'. But this is not the case; for we do not have anything with a predicative nature. Hence the meaning of the expression 'the
? The words 'unsaturated' and 'predicative' seem more suited to the sense than the meaning; still there must be something on the part of the meaning which corresponds to this, and I know of no better words.
Cf. Wundt's Logik.
? 120 [Comments on Sense and Meaning]
concept equilateral triangle' (if there is one in this case) is an object. We cannot avoid words like 'the concept', but where we use them we must always bear their inappropriateness in mind. ? From what we have said it follows that objects and concepts are fundamentally different and cannot stand in for one another. And the same goes for the corresponding words or signs. Proper names cannot really be used as predicates. Where they might seem to be, we find on looking more closely that the sense is such that they only form part of the predicate: concepts cannot stand in the same relations as objects. It would not be false, but impossible to think of them as doing so. Hence, the words 'relation of a subject to a predicate' designate two quite different relations, according as the subject is an object or is itself a concept. Therefore it would be best to banish the words 'subject' and 'predicate' from logic entirely, since they lead us again and again to confound two quite different relations: that of an object's falling under a concept and that of one concept being subordinated to another. The words 'all' and 'some', which go with the grammatical subject, belong in sense with the grammatical predicate, as we see if we go over to the negative (not all, nonnulli}. From this alone it immediately follows that the predicate in these cases is different from that which is asserted of an object. And in the same way the relation of equality, by which I understand complete coincidence, identity, can only be thought of as holding for objects, not concepts. If we say 'The meaning of the word "conic section" is the same as that of the concept-word "curve of the second degree"' or 'The concept conic section coincides with the concept curve o f the second degree', the words 'meaning of the concept-word "conic section"' are the name of an object, not of a concept; for their nature is not predicative, they are not unsaturated, they cannot be used with the indefinite article. The same goes for the words 'the concept conic section'. But although the relation of equality can only be thought of as holding for objects, there is an analogous relation for concepts. Since this is a relation between concepts I call it a second level relation, whereas the former relation I call a first level relation. We say that an object a is equal to an object b (in the sense of completely coinciding with it) if a falls under every concept under which b falls, and conversely. We obtain something corres- ponding to this for concepts if we switch the roles of concept and object. We could then say that the relation we had in mind above holds between the concept ([> and the concept X, if every object that falls under ([> also falls under X, and conversely. Of course in saying this we have again been unable
to avoid using the expressions 'the concept ([>','the concept X', which again obscures the real sense. So for the reader who is not frightened of the concept-script I will add the following: The unsaturatedness of a concept (of first level) is represented in the concept-script by leaving at least one empty place in its designation where the name of the object which we are saying falls under the concept is to go. This place or places always has to be filled
? I shall deal with this difficulty.
? ? [Comments on Sense and Meaning] 121
in some way or other. Besides being filled by a proper name it can also be filled by a sign which only indicates an object. We can see from this that the sign of equality, or one analogous to it, can never be flanked by the designation of a concept alone, but in addition to the concept an object must also be designated or indicated as well. Even if we only indicate concepts schematically by a function-letter, we must see to it that we give expression to their unsaturatedness by an accompanying empty place as in C/>( ) and X( ). In other words, we may only use the letters (cl>, X}, which are meant to indicate or designate concepts, as function-letters, i. e. in such a way that they are accompanied by a place for the argument (the space between the following brackets). This being so, we may not write cp = X, because here the letters cp and X do not occur as function-letters. But nor may we write C/>( ) = X( }, because the argument-places have to be filled. But when they are filled, it is not the functions (concepts) themselves that are put equal to one another: in addition to the function-letter there will be something else on either side of the equality sign, something not belonging to the function.
These letters cannot be replaced by letters that are not used as function- letters: there must always be an argument-place to receive the 'd. The idea might occur to one simply to write cp = X. This may seem all right so long as we are indicating concepts schematically, but a mode of designation that is really adequate must provide for all cases. Let us take an example which I have already used in my paper on Funktion und Begriff.
For every argument the function x 2 = 1 has the same (truth-} value as the function (x + 1}2 = 2(x + 1) i. e. every object falling under the concept less by 1 than a number whose square is equal to its double falls under the concept square root of1 and conversely. If we expressed this thought in the
way that we gave above, 1 we should have
(al=1)~ ((a+1}2=2(a+1))
What we have here is that second level relation which corresponds to, but should not be confused with, equality (complete coincidence) between objects. If we write it-0-~(a2 = 1) = ((a + 1)2 = 2(a + 1}}, we have expressed what is essentially the same thought, construed as an equation between values of functions that holds generally. We have here the same second level relation; we have in addition the sign of equality, but this does not suffice on its own to designate this relation: it has to be used in combination with the sign for generality: in the first line we have a general statement but not an equation. In e(e2 = 1) = il((a + 1)2 = 2(a + 1}} we do have an equation, but not between concepts (which is impossible} but between objects, namely extensions of concepts.
Now we have seen that the relation of equality between objects cannot be
1 It may be that the notation used in the following formula, which Frege has not explained above, was introduced in the lost first part of the manuscript (see footnote lop. II8)(ed. ).
? ? 122 [Comments on Sense and Meaning]
conceived as holding between concepts too, but that there is a correspond- ing relation for concepts. It follows that the word 'the same' that is used to designate the former relation between objects cannot properly be used to designate the latter relation as well. If we try to use it to do this, the only recourse we really have is to say 'the concept ([> is the same as the concept X' and in saying this we have of course named a relation between objects,? where what is intended is a relation between concepts. We have the same case if we say 'the meaning of the concept-word A is the same as that of the concept word B'. Indeed we should really outlaw the expression 'the meaning of the concept-word A', because the definite article before 'meaning' points to an object and belies the predicative nature of a concept. It would be better to confine ourselves to saying 'what the concept word A means', for this at any rate is to be used predicatively: 'Jesus is, what the concept word "man" means' in the sense of 'Jesus is a man'.
Now if we bear all this in mind, we shall be well able to assert 'what two concept-words mean is the same if and only if the extensions of the corresponding concepts coincide' without being led astray by the improper use of the word 'the same'. And with this statement we have, I believe, made an important concession to the extensionalist logicians. They are right when they show by their preference for the extension, as against the intension, of a concept that they regard the meaning and not the sense of words as the essential thing for logic. The intensionalist logicians are only too happy not to go beyond the sense; for what they call the intension, if it is not an idea, is nothing other than the sense. They forget that logic is not concerned with how thoughts, regardless of truth-value, follow from thoughts, that the step from thought to truth-value-more generally, the step from sense to meaning-has to be taken. They forget that the laws of logic are first and foremost laws in the realm of meanings and only relate indirectly to sense. If it is a question of the truth of something-and truth is the goal of logic-we also have to inquire after meanings; we have to throw aside proper names that do not designate or name an object, though they may have a sense; we have to throw aside concept-words that do not have a meaning. These are not such as, say, contain a contradiction-for there is nothing at all wrong in a concept's being empty-but such as have vague boundaries. It must be determinate for every object whether it falls under a concept or not; a concept word which does not meet this requirement on its meaning is
meaningless. E. g. the word 'wiJ). v' (Homer, Odyssey X, 305) belongs to this class, although it is true that certain characteristic marks are supplied. For this reason the context cited need not lack a sense, any more than other contexts in which the name 'Nausicaa', which probably does not mean or name anything, occurs. But it behaves as if it names a girl, and it is thus assured of a sense. And for fiction the sense is enough. The tkought, though it is devoid of meaning, of truth-value, is enough, but not for science.
? These objects have the names 'the concept ([>'and 'the concept X'.
? ? ? [Comments on Sense and Meaning] 123
In my Grundlagen and the paper Uber formale Theorien der Arithmetik I showed that for certain proofs it is far from being a matter of indifference whether a combination of signs--e.
[Footnote* top. 100 continued]
characteristic marks and mode of formation. When Kerry says 'By the number 4 we understand the result of additively combining 3 and 1', this is obviously meant to be a definition. A definition is always a logical identity. The sentence 'The number 4 is nothing other than the result of additively combining 3 and 1' would have the same sense. Thus this sentence does not claim to assert that the number 4 falls under the concept result o f additively combining 3 and 1, as does 'The number 4 is a result of additively combining 3 and 1'. This latter cannot be construed as a logical identity, as a definition, because it leaves it open whether there may not be something else which is also a result of this operation, and because it has a sense only if it has previously been established what the words 'the
number 4' designate. This can only be fixed by a logical identity to the effect that the same thing is to be understood by the words 'the num- ber 4', whose sense we know because the meanings of its parts and of the grammatical forms em- ployed are known to us. I t is, accordingly, clear that the definite article makes an essential con- tribution to the sense of our sentence and that the position here is conse- quently quite different from what it is in the case of 'The capital of the German Empire is the city of Ber- lin', where if you leave out 'the city of' you are only suppressing a quali- fication which is incidental to the sense. Thus I see no alternative but to deny that the words 'the result of additively combining 3 and 1' mean a concept, if one refuses to allow the possibility that concepts should be distinct from one another even through their characteristic marks
? ? On Concept and Object 103 [Footnote* top. 100 continued]
and mode of formation are the same. This alternative would be open to Kerry.
In fact Kerry seems to coun- tenance this possibility when he says on pp. 456 ff. 1 that the concept of a concept can contain the same characteristic marks as the concept itself. But there is no doubt that here again a property is being confused with a characteristic mark. I do not believe that anything of value is to be learned by taking this route or that logicians will have any inclination to tread the same path. On the contrary, it seems far and away more appropriate to regard the replacement of the indefinite by the definite article as signifying the transition from a concept to an object falling under it, where the case holds that there is only one such object. So I do not go along with Kerry's saying that the object is like the concept in having as characteristic marks those of the concept; I say instead that the object has the characteristic marks of the concept as properties. And here we have the distinction. For the relation of a characteristic mark to a concept is different from that of property to an object (cf. Grund- lagen, ? 53).
Another consideration shows that the words 'the result of additively corn bining 3 and 1' designate an object. We have already seen that the first sentence of the passage from Kerry quoted above is to be construed as a definition and thus as a logical identity. Now at the end we have: '"the" 4 is likewise the result of additively combining 3 and 1'. The author's view is that the role of
1 Vjschr. f wissensch. Philosophie 10 (1886) (ed. ).
? 104 On Concept and Object
[Footnote* top. 100 continued]
the grammatical subject is to desig- nate the concept-object and pre- sumably the definite article is enclosed in quotation marks to mark this object off from the concept. Obviously this last sentence has the sense of a logical identity too: "'the" 4 is nothing other than the result of additively combining 3 and 1'.
The view that this sentence is meant to express that an object falls under the concept result of etc. is immediately ruled out by the occur- rence of the definite article before 'result'. Otherwise the indefinite ar- ticle would have been used or the article would have to have been omitted. Now if the left-hand side of this identity is an object, the right- hand side must be an object too. Further, the right-hand side of the first identity is at the same time the right-hand side of the second: hence the two expressions to the left must designate the same thing: the num- ber 4 is nothing other than 'the' 4. I can see no significance in the use of quotation marks here. In my view the reason for the sorry state of affairs we find in Kerry, where the distinctions between concept and object, characteristic mark and pro- perty, are effaced is that logical and psychological questions and view- points are scrambled together, which greatly detracts from the value of his articles. He will speak now of a concept, then of the idea of a concept, now of an object, then of the idea of it, without its even being wholly clear whether it is the one or the other that is in question, whether we are engaged in a logical or psychological inquiry. Now it is no particular cause for wonder if we can find no way of distinguishing between the idea of a concept and
? ? can be the reference of a subject. It must here be remarked that the words 'all', 'any', 'no', 'some', are prefixed to concept-words. In uni- versal and particular affirmative and negative sentences, we are express- ing 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. I f 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
On Concept and Object 105
that of an object. Perhaps they are often even difficult to hold apart. But here Kerry has simply succum- bed to a widespread sickness. In- deed would not Locke's empiricism and Berkeley's idealism, and so much that is tied up with these philosophies, have been impossible if people had distinguished adequately between thinking in the narrower sense and ideation, between the parts of a content (concepts, ob- jects, relations) and the ideas we have? Even if with us men thinking does not take place without ideas, still the content of a judgement is something objective, the same for everybody, and as far as it is con- cerned it is neither here nor there what ideas men have when they grasp it. In any case these are subjective and will differ from one person to another. What is here being said of the content as a whole applies also to the parts which we can distinguish within it.
? ? 106 On Concept and Object
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 sen- tence 'The concept mammal is subordinate to the concept land- dweller' by negating the predicate: 'is not subordinate to the concept land-dweller'.
Ifwekeepitinmindthatinmy
way of speaking expressions like
'the concept F ' designate not con-
cepts but objects, most of [199]
Kerry's objections already collapse.
If he thinks (cf. p. 281) that I have
identified concept and extension of
concept, he is mistaken; I merely
expressed ny view that in the ex-
pression 'the number that applies to
the concept F is the extension of the
concept like-numbered to the
concept F' the words 'extension of words 'extension of the concept' the concept' could be replaced by
'concept'. Notice carefully that here
the word 'concept' is combined with
the definite article. Besides, this was
only an incidental remark; I did not
base anything upon it.
Thus Kerry does not succeed in filling the gap between concept and object. Someone might attempt, however, to make use of my own statements in this sense. I have said that to assign a number involves an
Thus Kerry in his account does not succeed in filling the gap between concept and object. Someone might attempt, however, to make use of my own statements in this sense. For example, I have said (Grund?
? The question whether one should simply put 'the concept' for 'the extension of the concept' is in my view one of expediency.
So if we hold on to the fact that the words 'the concept F' designate not a concept but an object, most of Kerry's objections don't begin to stand up. If he thinks (p. 281) that I have identified concept and exten- sion of a concept, this is simply not the case. I merely expressed my view that in the expression 'the number belonging to the concept F is the extension of the concept like- numbered to the concept F', the
could be replaced by the word 'concept'. N. B. the word 'concept' is here combined with the definite article. Besides this was only an incidental remark; I based nothing on it in order not to have to grapple with the misgivings to which it might give rise. So Kerry's opposition to it has no bearing at all on the real core of my position. ?
? ? On Concept and Object 107
assertion about a concept;* I speak
of properties asserted of a concept,
and I allow that a concept may fall
under a higher one. ** I have called
existence a property of a concept. that a concept may fall under a
How I mean this is best made clear
by an example. In the sentence
'there is at least one square root of
4', we have an assertion, not about
(say) the definite number 2, nor
about -2, but about a concept,
square root of 4; viz. that it is not
empty. But if I express the same
thought thus: 'The concept square
root of 4 is realized', then the first viz. that it is not empty but is
realized. But if I express the same thought thus: 'The concept square root of 4 is realized', then the first six words form the proper name of an object. In general a content can be analysed in a number of ways and language seeks to provide for this by having at its disposal dif- ferent expressions for the same content. The distinction between the active and passive form, for exam- ple, enables us to present different parts of the content as the subject. This is why it is almost invariably a
six words form the proper name of
an object, and it is about this object
that something is asserted. But
notice carefully that what is asserted
here is not the same thing as was
asserted about the concept. This will
be surprising only to somebody who
fails to see that a thought can be
split up in many ways, so that now
one thing, now another, appears as
subject or predicate. The thought
itself does not yet determine what is
to be regarded as the subject. If we
say 'the subject of this judgement',
we do not designate anything mistake to put the definite article
definite unless at the same time we indicate a definite kind of analysis; as a rule, we do this in connexion with a definite wording. But we must never forget that different sentences may express the same thought. For example, the thought we are con- sidering could also be taken as an assertion about the number 4:
'The number 4 has the property that there is something of which it is the square. '
* Grund/agen, ? 46. ** Grundlagen, ? 53.
before the word 'subject' in such expressions as 'the subject of the judgement', 'the subject of the con- tent of a possible judgement'; for no part of the content can be picked out in advance as the subject. Even the expressions 'singular content of pos- sible judgement', 'particular content of possible judgement' are not quite accurate in that they ascribe to the content itself an attribute which, strictly speaking, belongs to it only under a certain form-a certain way
lagen, ? 46) that a statement of number contains an assertion about a concept; I speak of properties asserted of a concept, and I allow
higher one (Grundlagen, ? 53). I have called existence a property of a concept. How I mean this is best made clear by an example.
In the sentence 'There is at least one square root of 4', we have an assertion not about, say, the par- ticular number 2, nor about -2, but about a concept, square root of 4;
? ? ? 108 On Concept and Object
Language has means of present- of analysing it into subject and
ing now one, now another, part of the [200] thought as the subject; one of the most familiar is the distinction of active and passive forms. It is thus not impossible that one way of analysing a given thought should make it appear as a singular judge- ment; another, as a particular judge- ment; and a third, as a universal
judgement. It need not then surprise us that the same sentence may be conceived as an assertion about a concept and also as an assertion about an object; only we must observe that what is asserted is dif- ferent. In the sentence 'there is at least one square root of 4' it is impossible to replace the words 'square root of 4' by 'the concept square root of 4; i. e. the assertion that suits the concept does not suit the object. Although our sentence does not present the concept as a subject, it asserts something about it; it can be regarded as expressing the fact that a concept falls under a higher one. * But this does not in any way efface the distinction between object and concept. We see to begin with that in the sentence 'there is at least one square root of 4' the predicative nature of the concept is not belied; we could say 'there is something that has the property of giving the result 4 when multiplied by itself. ' Hence what is here asser- ted about a concept can never be asserted about an object; for a
* In my Grund/agen I called such a concept a second-order concept; in my work Function und Begriff I called it a second-level concept, as I shall do here.
predicate.
It is conceivable that the same content should appear in one form as singular, in another as particular. It need not then surprise us that we can discern in the same content an assertion about a concept and also an assertion about an object. We must only take note that
what is asserted of the one is dif- ferent from what is asserted of the other. In the sentence 'There is at least one square root of 4' it is im- possible to replace the words 'square root of 4' by 'the concept square root of4'; i. e. the assertion that suits the concept does not suit the object. All the same something is asserted of a concept in our sentence. We can even say that our concept is presented as falling under a higher one*-one whose sole characteristic mark is is realized, understanding this word in the sense in which we are using it in the present context. But this does not in any way efface
the distinction between concept and object. We see to begin with that in the expression 'There is at least one square root of 4', 'square root of 4' is being used predicatively: 'there is something that is a square root of 4'. In place of this we could say 'there is something that has the property of giving the result 4 when multiplied by itself. Hence what is here asser-
* In my Grundlagen I called such a concept a second order concept; in my Funktion und Begriff I called it a second level concept, as I shall do here.
? proper name can never be a pre- dicative expression, though it can be partofone. Idonotwanttosayitis false to assert about an object what is here asserted about a concept; I want to say it is impossible, sense- less, to do so.
The sentence 'there is Julius Caesar' is neither true nor false but senseless; the sentence 'there is a man whose name is Julius Caesar' has a sense, but here again we have a concept, as the indefinite article shows. We get the same thing in the sentence 'there is only one Vienna'. We must not let ourselves be deceived because language often uses the same word now as a proper name, now as a concept-word; in our example, the numeral indicates that we have the latter; 'Vienna' is here a concept-word, like ? metropolis'. Using it in this sense, we may say: 'Trieste is no Vienna'. If, on the other hand, we substitute 1201] 'Julius Caesar' for the proper name formed by the first six words of the sentence 'the concept square root of 4 is realized', we get a sentence that has a sense but is false; for the assertion that some- thing is realized (as the word is being taken here) can be truly made only about a quite special kind of objects, viz. such as can be designated by proper names of the form 'the concept F'. Thus the words 'the concept square root of 4' have an
ted of a concept can never be asserted of an object. This is not to saythatitisfalsetodothis,butim- possible: a sentence which tried to express such a thing would be absolutely devoid of sense; for it has no sense to use the name of an object predicatively. We have seen that even where the form of a sentence makes it look as if this is being done, the truth is that the object forms only part of what is asserted, since 'nothing other than' has to be added in thought. A sentence such as 'There is at least Julius Caesar' is senseless, although the sentence 'There is at least one man whose name is "Julius Caesar"' has a sense; but here again we have a concept, as the 'one' shows.
On Concept and Object 109
If, on the other hand, we sub- stitute the proper name 'Julius Caesar' for the proper name formed by the first six words of the sentence 'The concept square root of 4 is realized', we get a sentence that has a sense but is false; for the assertion that something is realized, as the word is meant here, is one we can only truly make of such objects as stand in a quite special relation to a concept. Thus the words 'the con- cept square root of 4' have an essentially different behaviour, as
? 110 On Concept and Object
essentially different behaviour, as regards possible substitutions, from the words 'square root of 4' in our original sentence; i. e. the reference of the two phrases is essentially
different.
What has been shown here in one
example holds good generally; the behaviour of the concept is essen- tially predicative, even where some- thing is being asserted about it; consequently it can be replaced there only by another concept, never by an object. Thus the assertion that is made about a concept does not suit an object. Second-level con- cepts, which concepts fall under, are essentially different from first-level concepts, which objects fall under.
regards possible substitutions, from the words 'square root of 4' in our original sentence; i. e. the meanings of the two properties are different, thoughrelated.
What has been shown here in one example holds good generally: the behaviour of the concept is essen- tially predicative even when some- thing is being asserted of it; conse- quently it can be replaced there only by another concept, never by an object; i. e. the assertion that is made about a concept does not fit an object at all. Moreover a concept (of second level) under which a concept falls is essentially different from a concept (of first level) under which objects fall.
In the sentence 'There is at least one square root of 4' we assert that the first level concept square root of 4 falls under a concept of second level, whereas in the sentence 'The concept square root o f 4 is realized' we assert that the object the concept square root of4 falls under the first level concept concept that is realized. We do indeed have the same thought in the two concepts, [sic. ] but this, being analysed dif- ferently, is construed in a different way.
The relation of an object to a concept that it falls under is dif- ferent from the admittedly similar relation of a concept to a concept (of second level). From now on we shall give parallel expression to the similarity and the difference by saying than an object falls under a concept, and a concept falls within a concept (of second level); for, strictly speaking, we have been making a mistake in using the same
The relation of an object to a first-level concept that it falls under is different from the (admittedly similar) relation of a first-level to a second-level concept. (To do justice at once to the distinction and to the similarity, we might perhaps say: An object falls under a first-level concept; a concept falls within a second-level concept. ) The dis- tinction of concept and object thus still holds, with all its sharpness.
? With this there hangs together what I have said (Grund/agen, ? 53) about my usage of the words 'property' and 'mark'; Kerry's dis- cussion gives me occasion to revert once more to this.
The words serve to signify relations, in sentences like ? (/J is a property of T' and '(/J is a mark of il'. In my way of speaking a thing can at once be a property and a mark but not of the same
thing.
words to cover both cases. From this we can see that the distinction between object and concept still holds in all its sharpness.
Connected with this is another distinction, namely that between property and characteristic mark. I have already explained this in ? 53 of my Grundlagen and at the time I thought I had made it sufficiently clear. Since, however, Kerry makes no use of it and falls into error as a result, I can only assume, alas, that he has not understood me; this time I will try to be more successful.
The words 'property' and 'characteristic mark' serve to designate relations such that something can at the same time be a property of r and a characteristic mark of L1. We express ourselves more precisely if we use the words 'characteristic mark' only in the phrase 'characteristic mark of a concept'. If a thing is a property of an object, then it is a characteristic mark of a first level concept. We may also speak of properties of a first level concept, and these will then be characteristic marks of second level concepts. I will call the concepts under which an object falls its properties.
If the object r has the properties (/J, X and '1', I may combine them into n; so that it is the same thing if I say that r has the property n, or, that r has the property (/J and r has the property X and r has the property '1'. I then call (/J, X, and 'I' marks of the concept n, and, at the same time, properties of r. It is clear that the relations of (/J to r and to n are
I call the concepts under which an object falls its properties; thus 'to be (/J is a property of T' is just another way of saying: 'T falls under the concept of a (/J'. If the object r has the properties (/), X, and '1', I may combine them into n; so that it is the same thing if I say that r has the [202] property n, or, that r has the properties (/), X, and '1'. I then call (/J, X, and 'I' marks of the concept n, and, at the same time, properties of r. It is clear that the relations of (/J to r and to n are quite different, and that conse-
On Concept and Object Ill
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112 On Concept and Object
quently different terms are required. quite different; r falls under the r falls under the concept f/J; but n, concept f/J; but n, which is itself a which is itself a concept, cannot fall concept, cannot fall under the first under the first-level concept f/J; only level concept f/J; only to a second to a second-level concept could it level concept could it stand in a
stand in a similar relation. n is, on the other hand, subordinate to f/J.
Let us consider an example. Instead of saying:
'2 is a positive number' and '2 is a whole number' and '2 is less than 10'.
we may also say
'2 is a positive whole number less than 10'.
Here
appear as properties of the object 2, and also as marks of the concept
positive whole number less than 10.
This is neither positive, nor a whole number, nor less than 10. It is indeed subordinate to the concept whole number , b u t d o e s n o t fall under it.
Let us now compare with this what Kerry says in his second article (p. 424). 'By the number 4 we understand the result of additively combining 3 and 1. The concept object here occurring is the numerical individual 4; a quite definite number in the natural number-series. This object obviously bears just the marks that are named in its concept, and no others besides-provided we refrain, as we
similar relation.
to be a positive number, to be a whole number, to be less than 10,
? ? surely must, from counting as propria of the object its infinitely numerous relations to all other individual numbers; "the" number 4 is likewise the result of additively combining 3 and 1. '
We see at once that my dis- tinction between property and mark is here quite slurred over. Kerry distinguishes here between the num- ber 4 and 'the' number 4. I must confess that this distinction is in- comprehensible to me. The number 4 is to be a concept; 'the' number 4 is to be a concept-object, and none other than the numerical individual 4. It needs no proof that what we have here [203] is not my distinction between concept and object. I t almost looks as though what was floating (though very obscurely) before Kerry's mind were my dis- tinction between the sense and the reference of the words 'the number 4'. * But it is only of the reference of the words that we can say: this is the result of additively combining 3 and 1.
Again, how are we to take the word 'is' in the sentences 'the num- ber 4 is the result of additively combining 3 and I' and '"the" number 4 is the result of additively combining 3 and I'? Is it a mere copula, or does it help to express a logical equation? In the first case, 'the' would have to be left out before 'result', and the sentences would go like this:
'The number 4 is a result of addi- tively combining 3 and I';
* Cf. my essay 'On Sense and Reference' (cited above).
On Concept and Object 113
? 114 On Concept and Object '"The" number 4 is a result of
additively combining 3 and 1. "
In that case, the objects that Kerry designates by
'the number 4' and "'the" number 4' would both fall under the concept result o f additively combining 3
and 1.
And then the only question would be what difference there was be- tween these objects. (I am here using the words 'object' and 'concept' in my accustomed way. ) I should express as follows what Kerry is apparently trying to say:
'The number 4 has those properties, and those alone, which are marks of the concept: result ofadditively combining 3 and 1. '
I should then express as follows the sense of the first of our two sen- tences:
'To be a number 4 is the same as being a result of additive com- bination of 3 and 1. '
In that case, what I conjectured just now to have been Kerry's intention could also be put thus:
'The number 4 has those properties, and those alone, which are marks of the concept a number 4. '
[204] (We need not here decide whether this is true. ) The inverted commas around the definite article in the words '"the" number 4' could in that case be omitted.
But in these attempted in- terpretations we have assumed that in at least one of the two sentences the definite articles in front of 'result'
? and 'number 4' were inserted only by an oversight. It we take the words as they stand, we can only regard them as having the sense of a logical equation, like:
'The number 4 is none other than the result of additively combining 3 and 1. '
The definite article in front of 'result' is here logically justified only if it is known (i) that there is such a result; (ii) that there is not more than one. In that case, the phrase designates an object, and is to be regarded as a proper name. If both of our sen- tences were to be regarded as logical equations, then, since their right sides are identical, it would follow from them that the number 4 is 'the' number 4, or, if you prefer, that the number 4 is no other than 'the' number 4; and so Kerry's dis- tinction would have been proved untenable. However, it is not my present task to point out contra- dictions in his exposition; his way of taking the words 'object' and 'con- cept' is not properly my concern here. I am only trying to set my own usage of these words in a clearer light, and incidentally show that in any case it differs from his, whether that is consistent or not.
I do not at all dispute Kerry's right to use the words 'concept' and 'object' in his own way, if only he would respect my equal right, and admit that with my use of terms I have got hold of a distinction of the highest importance. 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
On Concept and Object
115
? 116
On Concept and Object
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 upon a reader who would be ready to meet me half-way-who does not begrudge a pinch of salt.
Somebody may think that this is an artificially created difficulty; that there is no need at all to take account of such an unmanageable thing as what I call a concept; that one might, like Kerry, regard an object's falling under a concept as a relation, in which the same thing could occur now as object, now as concept. [205] The words 'object' and 'concept' would then serve only to indicate the different positions in the relation. This may be done; but anybody who thinks the difficulty is avoided this way is very much mistaken; it is only shifted. For not all the parts of a thought can be complete; at least one must be 'un- saturated', 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 falls under 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 ac- cusative; 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. I say that such words or phrases stand for a relation. We now get the same
? difficulty for the relation that we were trying to avoid for the concept. For the words 'the relation of an object to the concept it falls under' designate not a relation but an object; and the three proper names 'the number 2', 'the concept prime number', 'the relation of an object to a concept it falls under', hold aloof from one another just as much as the first two do by themselves; however we put them together, we get no sentence. It is thus easy for us to see that the difficulty arising from the 'unsaturatedness' of one part of the thought can indeed be shifted, but not avoided. 'Complete' and 'un- saturated' are of course only figures of speech; but all that I wish or am able to do here is to give hints.
It may make it easier to come to un understanding if the reader com- pares my work Function und BegrifJ. 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 inap- propriateness of linguistic ex- pression; and that there is nothing for it but to realize this and always take it into account.
On Concept and Object 117
? ? [Comments on Sense and Meaning] 1 [1892-1895]
In an article (Uber Sinn und Bedeutung) I distinguished between sense and meaning in the first instance only for the case of proper names (or, if one prefers, singular terms). The same distinction can also be drawn for concept? words. Now it is easy to become unclear about this by confounding the division into concepts and objects with the distinction between sense and meaning, so that we run together sense and concept on the one hand and meaning and object on the other. To every concept-word or proper name, there corresponds as a rule a sense and a meaning, as I use these words. Of course in fiction words only have a sense, but in science and wherever we are concerned about truth, we are not prepared to rest content with the sense, we also attach a meaning to proper names and concept-words; and if through some oversight, say, we fail to do this, then we are making a mistake that can easily vitiate our thinking. The meaning of a proper name is the object it designates or names. A concept-word means a concept, if the word is used as is appropriate for logic. I may clarify this by drawina attention to a fact that seems to weigh heavily on the side of extensionalist as against intensionalist logicians: namely, that in any sentence we can substitute salva veritate one concept-word for another if they have the same extension, so that it is also the case that in relation to inference, and where the laws of logic are concerned, that concepts differ only in so far as their extensions are different. The fundamental logical relation is that of an object's falling under a concept: all relations between concepts can be reduced to this. If an object falls under a concept, it falls under all concepti with the same extension, and this implies what we said above. Therefore just as proper names can replace one another salva veritate, so too can concept? words, if their extension is the same. Of course the thought will alter when such replacements are made, but this is the sense of the sentence, not ita meaning. * The meaning, which is the truth-value, remains the same. For thia reason we might easily come to propose the extension of a concept as the
* Cf. my article Uber Sinn und Bedeutung.
1 These comments were not composed before 1892, the year in which the article Ober Sinn und Bedeutung appeared. They are part of a bundle of papers entitled 'Schrodersche Logik', which existed in a complete form prior to the destruction of the NachlajJ. The first part of these constituted a draft of Frege's article Kritischl Beleuchtung einiges Punkte in E. Schroder's Vorlesungen iiber die Algebra der
Logik (ed. ).
? [Comments on Sense and Meaning] 119
meaning of a concept-word; to do this, however, would be to overlook the fact that the extensions of concepts are objects and not concepts (Cf. my essay Funktion und Begrifl). Nevertheless there is a kernel of truth in this position. In order to bring it out, I need to advert to what I said in my mono- graph on Funktion und Begriff. On the view expressed there a concept is a function of one argument, whose value is always a truth-value. Here I am borrowing the term 'function' from Analysis and, whilst retaining what is essential to it, using it in a somewhat extended meaning, a procedure for which the history of Analysis itself affords a precedent. The name of a func- tion is accompanied by empty places (at least one) where the argument is to go; this is usually indicated by the letter 'x' which fills the empty places in 4uestion. But the argument is not to be counted as belonging to the function, und so the letter 'x' is not to be counted as belonging to the name of the function either. Consequently one can always speak of the name of a function as having empty places, since what fills them does not, strictly speaking, belong to them. Accordingly I call the function itself unsaturated, or in need of supplementation, because its name has first to be completed with the sign of an argument if we are to obtain a meaning that is complete in itself. I call such a meaning an object and, in this case, the value of the function for the argument that effects the supplementing or saturating. In the cases we first encounter the argument is itself an object, and it is to these that we shall mainly confine ourselves here. Now with a concept we have the special case that the value is always a truth-value. That is to say, if we complete the name of a concept with a proper name, we obtain a sentence whose sense is a thought; and this sentence has a truth value as its meaning. To acknowledge this meaning as that of the True (as the True) is to judge that the object which is taken as the argument falls under the concept. What in the case of a function is called unsaturatedness, we may, in the case of a concept, call its predicative nature. ? This comes out even in the cases in which we speak of a subject-concept ('All equilateral triangles are equiangular' means 'If anything is an equilateral triangle, then it is an equiangular triangle').
Such being the essence of a concept, there is now a great obstacle in the way of expressing ourselves correctly and making ourselves understood. If I want to speak of a concept, language, with an almost irresistible force, compels me to use an inappropriate expression which obscures-! might nlmost say falsities-the thought. One would assume, on the basis of its 1malogy with other expressions, that if I say 'the concept equilateral triangle' I am designating a concept, just as I am of course naming a planet If I say 'the planet Nepture'. But this is not the case; for we do not have anything with a predicative nature. Hence the meaning of the expression 'the
? The words 'unsaturated' and 'predicative' seem more suited to the sense than the meaning; still there must be something on the part of the meaning which corresponds to this, and I know of no better words.
Cf. Wundt's Logik.
? 120 [Comments on Sense and Meaning]
concept equilateral triangle' (if there is one in this case) is an object. We cannot avoid words like 'the concept', but where we use them we must always bear their inappropriateness in mind. ? From what we have said it follows that objects and concepts are fundamentally different and cannot stand in for one another. And the same goes for the corresponding words or signs. Proper names cannot really be used as predicates. Where they might seem to be, we find on looking more closely that the sense is such that they only form part of the predicate: concepts cannot stand in the same relations as objects. It would not be false, but impossible to think of them as doing so. Hence, the words 'relation of a subject to a predicate' designate two quite different relations, according as the subject is an object or is itself a concept. Therefore it would be best to banish the words 'subject' and 'predicate' from logic entirely, since they lead us again and again to confound two quite different relations: that of an object's falling under a concept and that of one concept being subordinated to another. The words 'all' and 'some', which go with the grammatical subject, belong in sense with the grammatical predicate, as we see if we go over to the negative (not all, nonnulli}. From this alone it immediately follows that the predicate in these cases is different from that which is asserted of an object. And in the same way the relation of equality, by which I understand complete coincidence, identity, can only be thought of as holding for objects, not concepts. If we say 'The meaning of the word "conic section" is the same as that of the concept-word "curve of the second degree"' or 'The concept conic section coincides with the concept curve o f the second degree', the words 'meaning of the concept-word "conic section"' are the name of an object, not of a concept; for their nature is not predicative, they are not unsaturated, they cannot be used with the indefinite article. The same goes for the words 'the concept conic section'. But although the relation of equality can only be thought of as holding for objects, there is an analogous relation for concepts. Since this is a relation between concepts I call it a second level relation, whereas the former relation I call a first level relation. We say that an object a is equal to an object b (in the sense of completely coinciding with it) if a falls under every concept under which b falls, and conversely. We obtain something corres- ponding to this for concepts if we switch the roles of concept and object. We could then say that the relation we had in mind above holds between the concept ([> and the concept X, if every object that falls under ([> also falls under X, and conversely. Of course in saying this we have again been unable
to avoid using the expressions 'the concept ([>','the concept X', which again obscures the real sense. So for the reader who is not frightened of the concept-script I will add the following: The unsaturatedness of a concept (of first level) is represented in the concept-script by leaving at least one empty place in its designation where the name of the object which we are saying falls under the concept is to go. This place or places always has to be filled
? I shall deal with this difficulty.
? ? [Comments on Sense and Meaning] 121
in some way or other. Besides being filled by a proper name it can also be filled by a sign which only indicates an object. We can see from this that the sign of equality, or one analogous to it, can never be flanked by the designation of a concept alone, but in addition to the concept an object must also be designated or indicated as well. Even if we only indicate concepts schematically by a function-letter, we must see to it that we give expression to their unsaturatedness by an accompanying empty place as in C/>( ) and X( ). In other words, we may only use the letters (cl>, X}, which are meant to indicate or designate concepts, as function-letters, i. e. in such a way that they are accompanied by a place for the argument (the space between the following brackets). This being so, we may not write cp = X, because here the letters cp and X do not occur as function-letters. But nor may we write C/>( ) = X( }, because the argument-places have to be filled. But when they are filled, it is not the functions (concepts) themselves that are put equal to one another: in addition to the function-letter there will be something else on either side of the equality sign, something not belonging to the function.
These letters cannot be replaced by letters that are not used as function- letters: there must always be an argument-place to receive the 'd. The idea might occur to one simply to write cp = X. This may seem all right so long as we are indicating concepts schematically, but a mode of designation that is really adequate must provide for all cases. Let us take an example which I have already used in my paper on Funktion und Begriff.
For every argument the function x 2 = 1 has the same (truth-} value as the function (x + 1}2 = 2(x + 1) i. e. every object falling under the concept less by 1 than a number whose square is equal to its double falls under the concept square root of1 and conversely. If we expressed this thought in the
way that we gave above, 1 we should have
(al=1)~ ((a+1}2=2(a+1))
What we have here is that second level relation which corresponds to, but should not be confused with, equality (complete coincidence) between objects. If we write it-0-~(a2 = 1) = ((a + 1)2 = 2(a + 1}}, we have expressed what is essentially the same thought, construed as an equation between values of functions that holds generally. We have here the same second level relation; we have in addition the sign of equality, but this does not suffice on its own to designate this relation: it has to be used in combination with the sign for generality: in the first line we have a general statement but not an equation. In e(e2 = 1) = il((a + 1)2 = 2(a + 1}} we do have an equation, but not between concepts (which is impossible} but between objects, namely extensions of concepts.
Now we have seen that the relation of equality between objects cannot be
1 It may be that the notation used in the following formula, which Frege has not explained above, was introduced in the lost first part of the manuscript (see footnote lop. II8)(ed. ).
? ? 122 [Comments on Sense and Meaning]
conceived as holding between concepts too, but that there is a correspond- ing relation for concepts. It follows that the word 'the same' that is used to designate the former relation between objects cannot properly be used to designate the latter relation as well. If we try to use it to do this, the only recourse we really have is to say 'the concept ([> is the same as the concept X' and in saying this we have of course named a relation between objects,? where what is intended is a relation between concepts. We have the same case if we say 'the meaning of the concept-word A is the same as that of the concept word B'. Indeed we should really outlaw the expression 'the meaning of the concept-word A', because the definite article before 'meaning' points to an object and belies the predicative nature of a concept. It would be better to confine ourselves to saying 'what the concept word A means', for this at any rate is to be used predicatively: 'Jesus is, what the concept word "man" means' in the sense of 'Jesus is a man'.
Now if we bear all this in mind, we shall be well able to assert 'what two concept-words mean is the same if and only if the extensions of the corresponding concepts coincide' without being led astray by the improper use of the word 'the same'. And with this statement we have, I believe, made an important concession to the extensionalist logicians. They are right when they show by their preference for the extension, as against the intension, of a concept that they regard the meaning and not the sense of words as the essential thing for logic. The intensionalist logicians are only too happy not to go beyond the sense; for what they call the intension, if it is not an idea, is nothing other than the sense. They forget that logic is not concerned with how thoughts, regardless of truth-value, follow from thoughts, that the step from thought to truth-value-more generally, the step from sense to meaning-has to be taken. They forget that the laws of logic are first and foremost laws in the realm of meanings and only relate indirectly to sense. If it is a question of the truth of something-and truth is the goal of logic-we also have to inquire after meanings; we have to throw aside proper names that do not designate or name an object, though they may have a sense; we have to throw aside concept-words that do not have a meaning. These are not such as, say, contain a contradiction-for there is nothing at all wrong in a concept's being empty-but such as have vague boundaries. It must be determinate for every object whether it falls under a concept or not; a concept word which does not meet this requirement on its meaning is
meaningless. E. g. the word 'wiJ). v' (Homer, Odyssey X, 305) belongs to this class, although it is true that certain characteristic marks are supplied. For this reason the context cited need not lack a sense, any more than other contexts in which the name 'Nausicaa', which probably does not mean or name anything, occurs. But it behaves as if it names a girl, and it is thus assured of a sense. And for fiction the sense is enough. The tkought, though it is devoid of meaning, of truth-value, is enough, but not for science.
? These objects have the names 'the concept ([>'and 'the concept X'.
? ? ? [Comments on Sense and Meaning] 123
In my Grundlagen and the paper Uber formale Theorien der Arithmetik I showed that for certain proofs it is far from being a matter of indifference whether a combination of signs--e.
