3.7 A puzzle
4.1.2 What does Kant mean by “general”?
What does Kant mean by the “general” in “general logic”? General logic, Kant says, “. . . treats of understanding without any regard to difference in the objects to which the understanding may be directed” (A52/B76; cf. A54/B78; JL:12, 50-51; GMS:387). This suggests that by “general” he means “indifferent to the particular identities of objects,” i.e., 2-formal.8
However, this kind of generality does not distinguish logic from arithmetic and algebra. Kant holds that arithmetic and algebra, too, “. . . [abstract] completely from the properties of the object (Beschaffenheit des Gegenstandes) that is to be thought in terms of. . . a concept of magnitude” (A717/B745).9 An arithmetical truth like “5+7=12” is applicable to cows,
7In the JL, having just said that transcendental logic considers the object “as an object of the
mere understanding,” whereas general logic “. . . deals with all objects in general,” Kant defineslogic
as “. . . a science a priori of the necessary laws of thought, not in regard to particular objects, however, but toall objects in general. . . ” (JL:16, my emphasis).
8For the different notions of formality, see chapter 3, above. For the connection between these
notions and the generality of logic, see section 3.5.
9Friedman 1992:108 argues convincingly that although only algebra is explicitly mentioned in
this passage, the point applies to arithmetic as well, since both “[construct] magnitude as such (quantitas)” (A717/B745). In saying that algebra abstracts from the constitution (Beschaffenheit) of the object, Kant is not alluding to the fact that it operates with variables instead of determinate
murders, inches, and triangles alike: it considers objects asunits, abstracting entirely from their distinguishing characteristics. Kant does not think of the numerals “5,” “7,” “12” as names of objects. On his view, mathematics is about forms of empirical objects, not about “mathematical objects.”10 But while geometry is limited in its application to a special domain of such objects—the objects capable of being given in space—arithmetic and algebra have no corresponding limitation (Friedman 1992:113). As “techniques of calculation,” they are “independent of the specific nature of the objects whose magnitudes are to be calculated” (113). Pure algebra and arithmetic do not assume that the magnitudes to which they apply are spatial, or even temporal (114-5).11 On Friedman’s view, the role of pure intuition in arithmetic and algebra is not to provide these sciences with objects, but to give content to the concept of magnitude—that is, of successive iteration of units—on which they are based (122). Time is “a universal source of representations for the number series,” not (as on Parsons’ view) “a universal source ofmodels for the numbers” (122 n. 46, emphasis added; cf. Parsons 1969:140). The concept of magnitude must get its content from the iteration of operations in time, because the successive iteration of units cannot even be represented using the logic available to Kant. We post-Fregeans can represent the indefinite extendibility of the number series by saying “for every number, there is a successor.” Kant, quantities, but to the fact that it concerns itself with magnitude as such (quantitas), rather than particular magnitudes (quanta), like the spatial magnitudes considered in geometry. In this respect arithmetic and algebra are alike. Friedman also suggests, more contentiously, that algebra for Kant is distinguished from arithmetic primarily by its concern with irrational magnitudes (and not, in the first instance, by its use of variables) (108-112).
10See KrV:A239-40/B298-9, B147, A224/B271, Thompson 1972-3:338-342, and Friedman
1992:101.
11“The application of the science ofquantity, unlike that of the science of geometry, is therefore not
limited to the specific—that is, spatial—character of our intuition; in this sense it provides us with the concept of a thing in general” (114). In a letter to Schultz, Kant callsquantity“a concept of a thing in general by determination of magnitude” (quoted in Parsons 1969:134). In view of KVR:A719/B747, Parsons connects this phrase with Kant’s characterization of the categories as “concepts of an object in general” (A51/B75, A93/B126, B128) and with the “intellectual synthesis” of B151 (134-5; cf. Thompson 19723:338). “Thing in general” is used in the Amphiboly of Concepts of Reflection for a thing thought in abstraction from the conditions of intuition (A279/B335, A283/B339). Friedman suggests that it might be more correct to speak of “the concept ofan object of intuition in general” (114 n. 34), since magnitude has to do with “the synthesis of the homogeneous in an intuition in general. . . ” (B162). Even with this qualification, however, the concept of quantity has not been limited to specifically spatiotemporal objects. See also Pr:§§39, 45 and KrV:A53/B51.
lacking the apparatus of iterated quantifiers, is forced to appeal to the intuitive idea of iteration in time to represent the same idea:
The concept of magnitude in general can never be explained except by saying that it is the determination of a thing whereby we are enabled to think how many times a unit is posited in it. But this how-many-times is based on successive repetition, and therefore on time and the synthesis of the homogeneous in time. (KrV:A241/B300)
If Friedman is right, then logic cannot really be distinguished from algebra and arith- metic on the grounds of its 2-formality. Algebra and arithmetic are 2-formal, too: they abstract just as surely as logic from particular differences between objects. (This is not to say that they yield anyknowledge of objects beyond sensory experience. But neither does logic. Such knowledge is not possible at all, on Kant’s view.)
These considerations tell decisively against construing Kant’s “generality” as 2-formality. For in characterizing logic as general, Kant clearly intends to be distinguishing it from mathematics. In theJ¨asche Logic, for instance, Kant defines general logic as the science of thenecessaryrules of the understanding, “without which no use of the understanding would be possible at all,” contrasting these with various contingent rules of the understanding, “without which a certain determinate use of the understanding would not occur” (JL:12). One of his examples of such a “particular, determinate use of the understanding” is the “use of the understanding in mathematics” (ibid.). Arithmetic and algebra, then, are not “general” in the sense that logic is. Though they may not have their own special objects, and though their concepts may not distinguish between the particular identities of objects, their laws are applicable only to a “particular, determinate use of the understanding,” not to all uses of the understanding. Because the content of the concept of magnitude depends on sensibility, while thought is intelligible apart from sensibility, the norms governing this concept (i.e., the laws of arithmetic and algebra) cannot be norms for thought as such. They cannot be laws of “. . . the general . . . employment of the understanding” (A52/B76). It appears, then, that when Kant speaks of the “generality” of logic, he means its 1- formality (constitutive normativity for thought as such), not its 2-formality. This is borne
out by his characterization of general logic as the science of “. . . the absolutely necessary rules of thought without which there can be no employment whatsoever of the understand- ing,” as opposed to “. . . the rules of correct thinking as regards a certain kind of objects” (KrV:A52/B76; cf. JL:13). Kant is explicit that he does not mean rules by which the un- derstandingdoes proceed, but rules governing how it ought to proceed (JL:14). The point is that no activity that is not held accountable to these rules cancount as thought, and not that there cannotbe thought that does not conform to these rules.12