Hi Neil:

Thanks for following along.

I'm not sure how to address your ruminations vis-a-vis analytic judgments just yet. Post-Quine, the analytic judgments don't really exist, so that's one way to say that they are not important. I'm thinking that Wittgenstein would disagree on this point. (As do I, since I think that Quine does not have it quite right.) That is, some argument can be explicative in regard to the concepts without being ampliative, introducing something new, and the explication can come about through a combination of logical derivations and posited definitions. Kant interprets all of this through the lens of Aristotelian syllogistic, which is naturally given to interpret things by containment and exclusion.

So, it's not surprising that he has an artificially restricted view of analyticity. If we think about some mathematical examples, it is not intuitively evident that the order of every subgroup of a finite group divides the order of the group. I don't see this latter as being enclosed in the concept of a finite group. Yet it follows, of course, from the axioms of a group, the restriction that the group be finite, and the notion of cosets partitioning the set of elements of a group. This is a simple, straightforward example.

OK, on the Kantian other hand, would we say that Lagrange's Theorem is intuitive? No, I think, of course not. Without the proof and the supplementary notion of cosets, it is inattainable through intuition. Thus, it follows from logical derivation, and is contained, to follow Kant's lead a bit, not in the concept of a finite group, but in all of these: The concept of a group, being a finite group, there being a coset, and being derivable by first-order logical rules of derivation from the axioms supporting the above concepts. Lagrange's Theorem is analytic, in that it follows from the concept of a finite group through a complex series of derivations following first-order quantificational logic.

This brings us back to Wittgenstein's remark in the Philosophische Bemerkungen. Wittgenstein is going to endorse Bolzano and refute both Kant and Frege.

Thanks, --Ron