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Kant famously distinguished between analytic and synthetic a priori knowledge as two kinds of knowledge that can be acquired a priori, that is, without relying on experience. The distinction is that analytic knowledge is knowledge of ones own concepts. It is knowledge of how one concept includes or does not include another. For example, you know that all women are human because being human is a part of the concept of being a woman. Synthetic a priori knowledge is any a priori knowledge that is not analytic.

Frege set out to prove that arithmetic is analytic by using pure logic to construct arithmetic. However, I don't think he was using the same concept of analyticity as Kant. Kant's definition of analyticity was as a relationship purely between concepts, not a relationship between concepts and the things that fall under the concept. A person can have the concept of being gold, for example, without being able to identify gold.

Even in the abstract realm, someone can have the concept of being a legal move in chess without being able to identify all legal moves. I had the concept of being a legal move in chess for years before I knew that castling was such a move. And when someone told me what castling was, I still wasn't sure how to apply it to the queen side.

Numbers may be a bit different because identifying a number is a priori, but it seems to rely on an intuition. For example neo-logicists define the concept of number via a Principle of Abstraction:

Given two concepts A and B, the number of A's equals the number of B's iff there is a one-to-one correspondence between the A's and the B's.

I have no problem with the claim that this can be used as a definition of number--it is possible to derive all of natural-number arithmetic from it--but I have a big problem with the assumption that identifying a number given this definition is analytic.

So, my question is: neo-logicists seem to think that a judgment that object A falls under concept B can be an analytic judgment in Kant's sense. Is that correct, and is there any argument that Kant would have thought so?

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  • There's an intro here: plato.stanford.edu/entries/logicism/#HisBac
    – J D
    Commented Mar 20, 2023 at 20:14
  • Neo-logicists do not define the numbers by the Hume principle (or define them at all) nor do they think that it is analytic. They also discard Kant's archaic concept of "analytic", as already Frege did, because it is based on the syllogistic fragment of logic, which is expressively impoverished.
    – Conifold
    Commented Mar 20, 2023 at 20:14
  • See e.g. Heck, The Julius Caesar Objection:"The attractions of the genetic story told at the beginning of this section do not depend upon the claim that the various instances of Hume’s Principle are logical truths, analytic truths, or any such thing. Frege’s most fundamental thought — that our knowledge of the truths of arithmetic derives, in some sense, from our knowledge of Hume’s Principle — could well be true, even if it does not have the epistemological implications he had hoped it would."
    – Conifold
    Commented Mar 20, 2023 at 20:14
  • @Conifold, I just finished reading a couple of papers, one by a neologicist and one criticizing neologicism, both of whom seem to think neologisists do define numbers by the Hume principle. I've just started another paper by a neologicist where he claims that the method of abstraction is analytic, and if he doesn't mean Kant's notion of analytic, he hasn't said so so far. Your quote doesn't say what neologists believe, it just claims that Frege's work is important even without those positions.
    – David Gudeman
    Commented Mar 20, 2023 at 22:11
  • SEP:"The neo-Fregean no longer accepts Frege’s definition of numbers as classes of equinumerous classes. Instead, the numbers are vouchsafed as sui generis, courtesy of newly chosen abstraction principles". Numbers are undefinable, unless you mean in the practical sense that it is all they need or care about. Kant's syllogistic "analytic" is not much used these days by anybody, so it wouldn't be spelled out specifically. I doubt that after Russell many are even familiar with Kant's original notion, see Friedman, Kant's Theory of Geometry, p. 486ff.
    – Conifold
    Commented Mar 20, 2023 at 23:06

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You're right: the Fregean/neo-Fregean talk of analytic truths is not one-to-one (so to say) with Kant's such talk. The SEP article on the analytic/synthetic distinction notes:

Kant tried to spell out his “containment” metaphor for the analytic in two ways. To see that any of set II is true, he wrote, “I need only to analyze the concept, i.e., become conscious of the manifold that I always think in it, in order to encounter this predicate therein” (B10). But then, picking up a suggestion of Leibniz, he went on to claim:

I merely draw out the predicate in accordance with the principle of contradiction, and can thereby at the same time become conscious of the necessity of the judgment. (B11)

As Jerrold Katz (1988) emphasized, this second definition is significantly different from the “containment” idea, since now, in its appeal to the powerful method of proof by contradiction, the analytic would include all of the (potentially infinite) deductive consequences of a particular claim, many of which could not be plausibly regarded as “contained” in the concept expressed in the claim. For starters, Bachelors are unmarried or the moon is blue is a logical consequence of Bachelors are unmarried—its denial contradicts the latter (a denial of a disjunction is a denial of each disjunct)—but clearly nothing about the color of the moon is remotely “contained in” the concept bachelor. To avoid such consequences, Katz (e.g., 1972, 1988) went on to try to develop a serious theory based upon only the initial containment idea, as, along different lines, does Paul Pietroski (2005, 2018).

They immediately continue (and then per Conifold's comment):

One reason Kant may not have noticed the differences between his different characterizations of the analytic was that his conception of “logic” seems to have been confined to Aristotelian syllogistic, and so didn’t include the full resources of modern logic, where, as we’ll see, the differences between the two characterizations become more glaring (see MacFarlane 2002).

The ensuing section, on Frege, contrasts the containment imagery's applicability with various "intuitively analytic(al)" propositions. I'm not sure Kant would have been moved more by lists of alleged stock examples of analytical truth/knowledge than by his technical concerns, but at any rate, if we do credit our "intuition" here, then we can take ourselves to be addressing the same property dichotomy that Kant did, and then the point is that valid disjunctive reasoning is not seen to depend on the interplay of the laws of identity and noncontradiction so much, so whereas Kant did not see anything else in saying that conceptual analysis proceeded from the identity of the subject to the "unfolded" identity of the predicate (in a subject-predicate case), for Frege, et. al., disjunctive reasoning does partake of noncontradiction without reducing to the kind of simple identifications Kant points to, here, and so if analytic(al) truth/knowledge involves the positive and negative conditions of the first two Aristotelian laws, it still will go beyond them, then.

Another angle to look at it from: Kant is using a subject-predicate logic (for the most part; he makes some prescient remarks about conditionals in e.g. the Transcendental Analytic, though). Frege explicitly is not, although I did see, I believe in a focused SEP article on Frege's notation/writing, that Frege was willing to countenance having complex expressions as subjects and "is a fact" as a universal predicate, or something along that line. (I'm not sure how exactly this reflects on Frege's "prosentential theory" of the use of the word "truth" in normal discourse, though by outward theme that theory seems to allow for "it is a fact that" to be the intended counterpart, which will then take that-clauses which might be subject-predicate in form, on pain of redundancy ("It is a fact that the princess's becoming an aardvark is a fact").) Depending on how actually important, or not, such a distinction is, then if we want to retain even an identitarian method of conceptual analysis, we will have to do something besides "extract predicates from subjects" to get at Frege-format analytic(al) truth/knowledge.

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  • I find Katz's critique unconvincing on two points. First, that Kant's remarks about contradiction were intended to constitute a second definition rather than merely a consequence of the original definition, and second that his appeal to logic appeals to a particular class of logics, which themselves are subject to criticism on the very point he exploits for his argument.
    – David Gudeman
    Commented Mar 20, 2023 at 22:19
  • KB, so, Kant relies on the intuition of semantic relations of syllogisms expressed either in the language of the metaphor of containment or through identity and contradiction as adequately characterizing 'analytical', but Frege in his Platonic and absurd antipsychological quest to show meaning is somehow objective decided to show that 'analytical' was tantamount to formal systems inhabitated by non-referential logical operators thus creating semantic notions of syntax instead leading early LW down the wrong path, only for late LW to recant Fregean 'analiticity' for meaning as a language-game?
    – J D
    Commented Mar 20, 2023 at 22:24
  • Then Quine comes along and shows the original analytical-synthetic dichotomy Kant proposes as a useful approximation at best under limited circumstances ultimately killing the notion that human-intuitional normativity can be stripped from language using magic symbols of Leibnitz's universal characteristic exposing the absurd Platonic notion that somehow meaning is ultimately derived from a dualistic plane of transcendental existence the empirically unvalidated farce that it is leaving bodily experience as the ultimate root of all human meaning?
    – J D
    Commented Mar 20, 2023 at 22:27
  • @DavidGudeman yeah I've always taken the containment-noncontradiction definitions to go hand-in-hand, with using noncontradiction to extract a negative predicate from a subject then just being the containment metaphor over negative identity. Apparently there's some hand-wringing about psychologism that went on, here, then, but then Frege was concerned to show that basic arithmetic is true always and everywhere in every possible relation, and Kant was not concerned to show this, so I don't see why Kant would've wanted to strengthen his talk of analytical knowledge in the Fregean way.
    – Kristian Berry
    Commented Mar 20, 2023 at 22:37
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    @JD my opinions/feelings on Frege are mixed (and not well-developed). Of all things, I heard that he was antisemitic (I heard Brouwer was, too, though, or at least got along with the eventual Nazi administration in the area or something), which makes me question his intelligence very much. I almost completely disagree with defining the empty set intensionally, as some sort of containment of "anti-identity," but it's still poetic, and his multidimensional notation intrigues me. Frege's "third realm," IDK, I guess I can countenance it only so far as it might be of a piece with the Kantian will.
    – Kristian Berry
    Commented Mar 20, 2023 at 22:46

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