In MacFarlane’s thesis, he distinguishes three related but distinct notions of formality that have been important in the evolution of the conception of logic, 1-formality, 2-formality, and 3-formality. The first is defined as being normative for thought as such. The second is defined as being insensitive to distinctions amongst objects, usually cashed out in terms of permutation invariance. The third is defined as abstracting from all conceptual or material content. In Kant these three notions are equivalent due to his other commitments, notably theses connected to his transcendental idealism. In Frege, the first and third come apart and Frege thinks the second does not characterize logic. Tarski and those writing after him focus mainly on the latter two and it seems that the second has been given pride of place since it admits of such a crisp mathematical formulation.

It is unfortunate that MacFarlane’s thesis does not cover the Tractatus. The omission is entirely understandable since one can only cover so much in a dissertation and this one already covered a great deal. MacFarlane’s dissertation covers in detail Kant’s, Frege’s, and Tarski’s (although ‘Tarskian’ may be a better way of putting it) views about logic. The appendices go on to discuss developments in ancient logic. Fitting the Tractatus into MacFarlane’s story would be interesting and surely fill it out. The philosophy of logic found in the Tractatus is a bridge from Frege to the Vienna Circle and their descendants. Granted, by the time we get to the Vienna Circle, especially post-Logical Syntax, Tarski’s work has been absorbed. However, the Tractatus was skipped over, so we jump from Frege straight to Tarski, Carnap, and Quine.

Which notions of formality does the philosophy of logic in the Tractatus exemplify? It certainly is 3-formal. The rejection of Frege’s claims that logic has a certain subject matter, namely the logical constants, is one of its notable features. To supply a quote, 6.124 says: “The logical propositions describe the scaffolding of the world, or rather they present it. They ‘treat’ of nothing.” It takes 3-formality as one of the essential features of logic, not a consequence of some other essential feature. Thus it breaks with Kant as well. I think it adopts 2-formality, although I am not sure if this is a definitional feature or not. In the 4.12’s Wittgenstein says that variables are the signs of formal concepts, which present a form that all its values possess. Although, I’m not sure how correct it is to say that logic in the Tractatus is all that much involved with that sort of form. The logical sentences, the tautologies, are certainly going to be insensitive to variation in the names of objects appearing in them. In that sense one could say that the Tractatus is 2-formal. Nowhere, as far as I remember, does Wittgenstein talk about permuting objects in those terms, although he does talk about the range of possibilities of the existence and non-existence of facts. One is always dealing with the same objects although they may be arranged differently. (I haven’t yet figured out where 6.1231 fits in. It says: “The mark of logical propositions is not their general validity. To be general is only to be accidentally valid for all things. An ungeneralized proposition can be tautologous just as well as a generalized one.” I suspect it is dismissing a spurious notion of formality. In context it is criticizing Russell.)

The big question, after reading MacFarlane, is whether the Tractarian view claims that logic is 1-formal. Kant does. Frege does. MacFarlane does. Wittgenstein does not seem to. There is not any talk of the normativity of logic, as far as I could discern from my quick check. When he talks about logic showing that something follows from something else, it seems to be entirely in descriptive language without any normative aspect. Despite this, logic seems to be constitutive of thought and concept use, as in 5.4731 “What makes logic a priori is the impossibility of illogical thought.” In a few other places he talks about the impossibility of illogical thought. There doesn’t seem to be any normative dimension to this talk at all. I’m not sure where to look for anything more normative. Although, it would not be surprising for there not to be any normative element to the philosophy of logic expressed in the Tractatus since Wittgenstein left the ethical out of the book for the most part, excepting a few mentions in the late 6’s. I will hesitantly conclude that the view in the Tractatus does not claim 1-formality.

The view of the Tractatus then is an example of 2- and 3-formality. I don’t remember if there is an earlier example of this combination pointed out by MacFarlane. I expect that the reasons motivating this combination in Wittgenstein differ from those in any earlier examples if such exists.

Incorporating the Tractatus into the story told by MacFarlane would be interesting for other reasons. For one, it includes some criticisms of Frege, in particular on Frege’s doctrines that logic’s subject matter is the logical constants and that the laws of logic govern all thought. There is also a reading of the Tractatus, supported by Sullivan I think, that takes the book to be a criticism of transcendental idealism. If that is on track, then it would provide an example of a principled acceptance of 3-formality independent of transcendental idealism, whose doctrines Kant appeals to in order to support 3-formality. This, by itself, would be neat. Finally, MacFarlane does not claim that his three notions of formality exhaust the possibilities. This is a wild conjecture (that might be generous; perhaps wild hunch is better), but there might be another sort of formality available in the (sort of) algebraic approach found in the Tractatus and also found in Schroeder and Peirce apparently. I haven’t worked out any of the details, but it seems plausible that there might be something there.

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