Saturday, June 07, 2008

Tractarian formality

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. Read more

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.


Daniel Lindquist said...

"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."

Sullivan's reading of the TLP is similar to the resolute reading, at least based on What is the Tractatus About?. The TLP is supposed to be a criticism of transcendental idealism because it is a criticism of "transcendental philosophy" generally: of the attempt to understand how thought is "grounded". I should be surprised if the abandonment of "transcendental philosophy" left an interesting notion of logical formality in its wake; "the form of thoughts" certainly seems like something transcendental, and TLP 6.13 seems explicit enough. If we can't go outside of thought to speak of its "grounding", then I shouldn't think we could talk of its form, either. To abstract from all "material or conceptual content" is to abstract away from any particular thought. And so to claim that there is something inherent to thought's "form" that could be treated of by logic would be to make a claim from outside of any particular thought, i.e. from outside thought generally. Which is shown, on Sullivan's reading of the TLP, to be just the problem with transcendental idealism.

(Sullivan does explicitly claim that we should recognize that criticism of transcendental idealism is at the center of Wittgenstein's concerns in the TLP. So you had the right name, or at least one of them. I am probably reading Sullivan as closer to the resolute readers than he really is, but at least on the topic of transcendental idealism his reading of the TLP seems to jibe with Conant et al, whatever else they disagree on. The TLP rejects transcendental idealism as one of the bits of metaphysics it particularly wants to liberate us from.)

It strikes me as odd to think that a resolute reading of the TLP might be able to support a characterization of logic as such-and-such, generally. Conant, for instance, thinks that Early Wittgenstein is committed to some such thesis as "There is such a thing as the logical form of our language", but that this commitment is unwitting (and is rejected by Later Wittgenstein, once he notices it). So it would seem that the TLP can't offer a principled reason to accept views about "logical form"; Early Wittgenstein, on the resolute reading, doesn't have principled reasons for holding such views. And the same would seem to hold for 1- 2- and 3-formality, I would think. If there is no reason for thinking there is such a thing as "the logical form" of our language, it would be odd if there was still some clean way to demarcate "logical form" and "material content".

Now, the "standard" reason of the TLP does indeed attribute to Wittgenstein views such as 2- and 3-formality. (And, as I understand them, 1-formality, though this is less clear.) But the "standard" reading also claims that the TLP teaches transcendental idealism. (See chapter four of P.M.S. Hacker's "Insight and Illusion", "Empirical Realism and Transcendental Solipsism", for a typical treatment.) So I'm not sure that you aren't trying to have your cake and eat it, too. I'm not aware of any reading of the TLP that both claims Early Wittgenstein is not a transcendental idealist and that he gives us reasons for thinking such-and-such about the formal nature of logic.

It is thinkable that the TLP might offer reasons for holding 3-formality without appeal to transcendental idealism, like you suppose it might. But I think this would likely have to take the form of showing that you can get 3-formality out of the TLP without relying on Early Wittgenstein's transcendental idealism, rather than being able to take it for granted that anything that comes from the TLP won't be transcendentally idealistic. Which sounds like a harder task. On the standard reading, transcendental idealism in the TLP is pretty central. (And even in some non-standard/non-resolute readings, such as John Holbo's in his dissertation, Prolegomena to a Reading of the Tractatus, wherein it's argued that the TLP is basically a "logically purified" version of "The World as Will and Representation".) Though if you can get some "algebraic" notion of logical formality that ties in to the TLP, it very well might be able to come apart from transcendental idealism; the proof of that would have to be in the pudding.

Your recent posts are making it more and more mysterious to me why I have still not read MacFarlane's dissertation. I seem to not want to do things I will enjoy having done.

Shawn said...

I'm glad I at least got the name of one of the proponents of that reading correct.

You are probably right that I was, inadvertently, trying to have my cake and eat it too. Sullivan is sort of in the resolute reading camp (There is also a third group situated roughly between the resolute and traditional readers that he is probably better assimilated to. I think Marie McGinn is an example.). As such, it is not licit for me to appeal to a reading of the TLP that rejects transcendental idealism to escape 1-formality as well as accepting other things it says that would let one attribute 3-formality to it. Probably illicit. I'm not sure how much of the stuff about logic is salvageable once one has drunk the resolute kool-aid. There are clearly some, let us say, metaphysical trappings, such as the scaffolding of the world stuff, that fall by the wayside. It's not clear that claim such as that tautologies are true no matter what the facts must similarly go. We're certainly tempted to throw out everything (or most everything) in the book after the late 6's and an approach that the resolute reading recommends. From what I can tell though, there are a few of the resolute commentators that are interested in the logical theory of the Tractatus. It is a flaw in the program that nothing has been written on how the resolute reading meshes with (or not) the logical views in the book. There have been some things here and there, but nothing sustained as far as I know.

After writing the post I realized that it is premised on a tendentious claim: that the TLP gives a unified philosophy of logic. It seems to me that there is such buried in it. I'm not sure that the resolute reading has tapped it yet, although I'm not sure anyone has. Sullivan, I think, comes close in a few places.

What parts of the TLP do you think support 1-formality? Like I said, I couldn't find anything.

N. N. said...

Which notions of formality does the philosophy of logic in the Tractatus exemplify? It certainly is 3-formal.

I agree. Cf. TLP 3.33: "In logical syntax the meaning of a sign ought never to play a role; it must admit of being established without mention being thereby made of the meaning of a sign; it ought to presuppose only the description of the expressions."

I disagree concerning '2-formality.' Anscombe puts it nicely when she writes (in "Ludwig Wittgenstein"),

"I was slow to realise that I had been wrong in assuming that the objects, the simples, spoken of in the Tractatus were uniform characterless atoms, whose arrangement alone produced the characters of familiar things. (These characters indeed Wittgenstein called 'external'.) The assumption was absurd - the internal characters of objects are not of the same logical form (2.0233) - in fact, it looks as if their logical form and their internal character were the same thing. The possibility of a given fact must be 'prejudged' in the things that can occur in such a fact. (2.012) This at least suggests that it is not possible for every simple object to occur in just any fact. Rather, as holders of their names too, the objects can only enter into certain compositions."

Concerning '1-formal,' I'm not sure how to understand 'normative.' The Preface does say that the work sets limits to thought (or the expression of thought).

Of course, as Daniel indicates, all of these questions are clouded by the current disputes about the Tractatus. An interesting thing about Conant: on his view, there is such a thing as the logical form of a proposition. That is, so long as we realize that the sense of the proposition depends on the string's context. In other words, there is such a thing as the logical form of this sentence in this context.

Daniel Lindquist said...

On 1-formality: I think that "normativity" is connected to the "ethical" aim of the book, which is not included within the book itself. My thought is roughly that the ethical point/perspective/whatever which W. wants the reader to grasp is connected with not treating illogical connections of signs as if they expressed thoughts. Now that I think about it for a moment, I'm not sure if this would be something that counts as a 1-normative conception of logic or not. I want to say that W. (on the standard reading) thinks that thoughts which are not connected logically are not connected in any rational manner -- a perspicuous notation shows what follows from what, and whatever is not thus shown to follow does not follow.

N.N.: In the Tractarian sense of "object", I agree that the TLP doesn't support 2-normativity. But as W. himself lated noted, in ordinary talk the relation between two objects is not a third object. In the sense of "object" which Anscombe originally misread as being W.'s in the TLP, I think the TLP does support 2-normativity (on the standard reading). Actually, come to think of it, I'm not sure what the distinction is between 2- and 3-formality -- it seems like 3-formality should imply 2-formality. If logic abstracts from all material content, then it seems like any distinctions between objects must also be abstracted away from. Again, I need to go back to McFarlane's dissertation.

That surprises me about Conant. Can I get a citation for that?

Shawn said...

The issue with 2-formality is a little trickier than I indicated originally. That objects are different and have some individual character or structure isn't sufficient for refuting 2-formality. Logic has to be sensitive to those differences. There is a bit of slack here in what it means for logic to be sensitive to the individual structure or character of objects. Would permuting the objects disrupt the status of logical sentences as logical? It might depend on whether there is some object a that can't appear in a fact involving F. I'd have to go back over the relevant parts of TLP and think about it for a while.
I'm not sure that the Anscombe quote settles the matter though.