5.3 The Quantitative Forms of Judgment
5.3.1 Universal and Singular Judgments
Longuenesse’s interpretation of Kant’s notion of extension leads to at least one problem. How can Kant say that, in a singular judgment, “a concept that has no sphere at all is enclosed, merely as a part, under the sphere of another”, or – as he puts it in the Critique – that singular judgments “have no domain [Umfang] at all” (A71/B96)? An example of a singular judgment Kant provides is ‘Caius is mortal’ (A322/B378). If the extension of a concept consists in the representations thought under that concept, it seems impossible to say that the concept ‘Caius’ has no extension. ‘Caius’, after all, applies to at least one sensible representation: the person Caius. If we assume the extension of a concept consists of the representations thought under that concept, then Kant’s claim that the subject concept of a singular judgment has no extension becomes incomprehensible.
17
See Thompson, 1989: 170n.
18“Now in space there is nothing real which can be simple; points, which are the only
simple things in space, are merely limits, not themselves anything that can as parts serve to constitute space” (B419, quoted by Thompson, 1989: 170n.)
Thompson’s interpretation of extension does not suffer from this prob- lem. According to Longuenesse’s interpretation, the singular judgment ‘Ais
B’ expresses that to the one objectx we think under a conceptA, the con- ceptB applies. If we follow Thompson’s interpretation of extension, we can say that the singular judgment ‘A isB’ expresses that the conceptA under which we happen to think no other concepts is contained in the extension of
B.19 If we adopt Longuenesse’s notion of extension it becomes incomprehen- sible why Kant says that the subject concept of a singular judgment has no extension. If we follow Thompson’s interpretation, however, Kant’s remark can be understood. Under the subject concept of a singular judgment we do not think any concepts, only an object.20 For this reason, this concept has no extension.
How, then, should we understand singular judgments? To understand this, it will prove helpful to consider howgeneral judgments should be inter- preted. Thompson, we saw in chapter 3, thinks that the universal judgment ‘AllA’s areB’ expresses that any object x that falls under the concept A, also falls under the concept B (1989: 179). I think this interpretation of general judgments is correct. A great advantage of this interpretation over Longuenesse’s or Frede and Kr¨uger’s interpretation, we saw, is that this interpretation enables us to do justice to Kant’s idea that judgments are rules: combinations of concepts expressing necessary relationships among representations.
Our analysis of general judgments also enables us to analyse singular judgments. I think the singular judgment ‘A is B’ – just like general judg- ments – expresses ‘Any object x that falls under the concept A, also falls under the concept B’.21 The only difference between singular and general concepts is that the subject concept of a general judgment does, and the subject concept of a singular judgment does not have an extension. This analysis of singular judgments enables us to understand a striking remark 19This is not Thompson’s own interpretation of singular judgments. Thompson (1972)
provides an extensive analysis of singular judgments. If I understand Thompson correctly, he thinks we should regard the singular judgment ‘A is B’ as a judgment of the form ‘There is a uniqueA, andAisB’. Thompson says: ‘The general logic required by Kant’s transcendental logic is thus at least first order quantificational logic plus identity but minus proper names or other singular terms that are in principle eliminable. A proper name represents an empirical concept used with an existence and a uniqueness claim and is hence eliminable in favour of a predicate expression.’ (334-5) I think this analysis of singular judgments is incompatible with Kant’s remarks in Reflection 3068 which I will discuss below.
20
Or: multiple objects. I will say more about this later.
Kant makes in Reflection 3068:
In general judgments, thesphere[sphaera] of a concept is wholly enclosed within the sphere of another; in the particular, a part of the former is enclosed under the sphere of the other; in the singular a concept that has no sphere at all is enclosed, merely as a part then, under the sphere of another. Therefore, the
iudicia singularia are to be assessed as like the universalibus, and, conversely, a iudicium universale is to be assessed like a singular judgment with respect to thesphere. Plurality in as far as in itself, it is only one. (Reflection 3068 (1776-89), 16: 640)22 In this reflection, Kant says that singular and general judgments are equiv- alent. Not only is it possible to understand a singular judgment as a general one. Kant explicitly says that, “with respect to the sphere”, general judg- ments can be understood as singular ones. My interpretation of singular and universal judgments enables us to regard these judgments as equiva- lent. What Reflection 3068 shows, I think, is that from the perspective of general logic, it is irrelevant whether the subject concept of a general con- cept has an extension. Both in singular and general judgments the subject concept is regarded as “only one”. The general judgment ‘AllA’s areB’ and the singular judgment ‘A is B’ both express that any object falling under
Afalls under B.23 The fact that the subject concept of a general judgment does, and the subject concept of a singular concept does not have an exten- sion only becomes relevant if we compare the quantity [Gr¨oße] of cognition these judgments express:
If, on the contrary, we compare a singular judgment with a gen- erally valid one, merely as cognition, with respect to quantity 22
Im allgemeinen Urteile wird diesphaeraeines Begriffs ganz innerhalb derSphaeraeines andern beschlossen; im particularen ein theil des ersteren unter die Sph¨are des andern; im einzelnen ein Begrif, der gar keine Sphaeram hat, mithin blos als Theil, unter die
sphaeram eines andern beschlossen. Also sind die iudicia singularia den vniversalibus gleich zu schatzen, und Umgekehrt ist einiudicium vniversaleals ein einzelnes Urtheil in Ansehung dersphaera zu betrachten. Vieles, so fern es an sich nur eines ist.
The translation of the first sentence is a modification of Michael Young’s translation of
§21 theLogic (Ak. 9: 102). The translation of the remaining of the passage partly relies on this translation.
23
An interesting point to note, is that Kant himself does not always formulate general judgments by means of sentences of the form ‘All A’s are B’. Geometrical axioms, for instance, Kant regards as general judgments (see A164/B205). Examples of such judg- ments he formulates as follows: “between two points only one straight line is possible” (A163/B204) en “two straight lines do not enclose a space” (A163/B204).
[Gr¨oße], then the former relates to the latter as unity relates to infinity, and is therefore in itself essentially different from the other. (A72/B97)
A general judgment ‘All A’s are B’ contains infinitely many cognitions, because it contains infinitely many other judgments. Under A infinitely many concepts can be thought. For each of these conceptsC, the judgment contains the judgment ‘All C’s are B’ (or ‘C is B’). A singular judgment does not contain any other judgments. It contains, therefore, one cognition only. In this sense, the quantity of cognition a singular judgment provides relates to the quantity of cognition a general concept provides like a “unity relates to infinity”.24
Interpreting singular judgments as universal ones enables us to do jus- tice to Kant’s remarks about these judgments. It enables us, moreover, to regard these judgments as expressions of necessary relations. The judgment ‘Caius is mortal’ does not express that a certain individual object is mortal. Rather, it expresses that the concept ‘mortal’ applies to every object I bring under the concept ‘Caius’. The judgment expresses a necessary relationship between the concept ‘Caius’ and the concept ‘mortal’. Also, one should note that according to this interpretation of singular judgments, it is – strictly speaking – not necessary that a singular judgment applies to one object only. A judgment is singular when its subject concept does not have an exten- sion. If the subject concept of a judgment has no extension, usually this will mean that the judgment applies to at most one object. This, for instance, is the case when the subject concept is a name, like in ‘Caius is mortal’. What is interesting, however, is that Kant regardsarithmetical judgments as singular judgments too. The reason these judgments are singular, Kant says, is that a number concept like ‘seven’ “is possible only in a single way” (A165/B205). In this respect, the concept differs from a geometrical con- cept like ‘triangle’, Kant says (A164-5/B205). The concept ‘triangle’ does have an extension, because there are various kinds of triangles. The con- cept ‘seven’ doesnot have an extension, as there is only one kind of seven. Nevertheless, the concept ‘seven’ can be applied to various objects. The ex- ample of arithmetical judgments suggests that Kant does indeed think that the singularity of singular judgments is due to the concepts thought under their subject concept, not to the objects thought under them.
24This explanation I derive from Thompson, 1989: 170. As I interpret singular judg-