18 May 2013

Dummett's Frege and Rödl

I have not been good at blogging recently; I have neglected comments for about a year, and have written nothing. Apologies to those who I did not respond to (which by my records include Daniel Nagase, N.N., Charles Wolverton, Evan Kuehn, and Duck; I could make excuses for this bad behavior, but they would be of merely psychological importance, and that's a poor way to start a Frege post).

But if I cannot manage to blog well, then I should at least blog badly more often; here then are things I typed to try to get clear to myself on what Rödl is getting from the Dummett essay he leans on in the first chapter of "Categories of Temporality".

In "The Context Principle: Centre of Frege's Philosophy" Dummett claims that Frege tried to use the context principle to justify his "realism", his treatment of numbers as objects, while simultaneously using it to answer the question of how numbers can be given to us ("epistemologically", in Dummett's term). Dummett's idea is that Frege tried to answer this question (and establish his realism) by fixing the truth-values of every sentence in which a number-word appeared; in the Grundgesetze, this is fixing the truth-value of every sentence in which a term for a value-range appears.

The line of thought seems to be this:
1. Frege can fix the truth-values of sentences by (in part) stipulation, going through each possible combination of a value-range term with a primitive concept-term and assuring that it has a truth-value.
2. For Frege truth-values are the Bedeutungen of sentences.
3. The context principle: the Bedeutung of a subsentential term is determined only by the way it contributes to the Bedeutungen of the sentences it appears in.
Which gives Dummett's Frege the conclusion that by stipulating truth-values for each (atomic) sentence in which a value-range term appears, he has also settled what the Bedeutungen of value-range terms are.

Dummett contrasts this to a way of providing terms with Bedeutungen which would go against the context principle: first establish the domain over which the variables of the language can range, and then determine for each term of the language which needs a Bedeutung which of the items from that domain is to be its Bedeutung. As this initial domain-determination requires a grasp of the possible values of variables anterior to the securing of Bedeutungen to the sentences of the language in which those variables appear, it violates Dummett's version of the context principle.

Dummett spends some time on the objection that one might ascend to a metalanguage to avoid this violation of the context principle: if the domain-determination for the object language takes place in sentences of another language, then the context principle is not sinned against. But Dummett argues that Frege did not mean to be giving Bedeutungen to the terms of Begriffschrift which were already understood by anyone who could read his German: this would make some of Frege's prose a necessary element of his logic, which Frege clearly wants to avoid. The Begriffschrift is supposed to stand on its own, with the German prose serving only as a propaedeutic to its understanding; in particular, if the question of whether numbers are objects or how they may be given to us is to be solved by the Grundgesetze project, then it cannot rely on the German-reader's already knowing these objects and including them in the domains over which Begriffschrift variables are allowed to range.

Dummett's objection to Frege here is put rather tersely, and I am writing this post because I need to do some work to unpack it for myself: "The fallacy appeared at the very first step. The stipulations governing the primitive functors [what I called concept-words, above], including the criterion of identity for value-ranges embodied in Axiom V, could be determinate only if the domain, consisting wholly or largely of value-ranges, was determinate; but the domain was in the process of being determined by fixing the Bedeutungen of the value-range terms, and so the procedure went round in a circle." (p.18)

The circle seems to be this: Frege tries to establish that among the objects are value-ranges (the Bedeutungen of value-range terms) by stipulating truth-values for each atomic sentence which results from combining a value-range term and a primitive function of Begriffschrift. But the sentences which are here stipulated to have truth-values can only have truth-values if the domains over which their variables range is determinate; an indeterminacy in the domain means an indeterminacy in what can count as the sentence being true or false, and Frege's stipulations do not settle this question about domains. In fact, Frege wants to settle questions about domains by securing Bedeutungen for his primitive terms, so that he can settle the question of whether numbers are objects, and how numbers can be given to us, by making clear how sentences featuring number-words function in inference. So he both needs to settle the domain issue prior to his procedure and only by means of it, which is contradictory. This is why he is able to think he has given Bedeutungen to all of his Begriffschrift expressions such that all of his Basic Laws are true, when in fact they are jointly inconsistent (because of Basic Law V, whose Bedeutung was supposed to be settled by the procedure of securing Bedeutungen for value-range terms).

Dummett's verdict here is despair. His closing paragraph:
"The realist interpretation could be jettisoned without abandoning the context principle itself, but only if that principle, as here understood, can be shown to be coherent; and this remains in grave doubt. And yet it is hard to see how it can be abandoned, so strong is the motivation for it. The alternative is an apprehension of objects, including abstract objects, underlying, but anterior to, an understanding of reference to them, or, indeed, a grasp of thought about them; and this is a form of [what Putnam calls] external realism too coarse to be entertained. I am therefore forced to conclude without either endorsing the central feature of Frege's philosophy or rejecting it; I can do no more than to say lamely that the issue if one whose resolution is of prime importance to philosophy." (p.19)

This is where Rödl intervenes: he claims that the context principle can be saved (and should be saved), but only by rejecting the idea that a logic such as the Begriffschrift can be said to present us with the form of thought as such. As Dummett argued, a coherent version of Frege requires something other than Begriffschrift to settle the question of what objects Begriffschrift is about; Begriffschrift cannot take care of itself, but needs the "pinch of salt" Frege infamously asks his readers for. The distinguishing characteristic of Begriffschrift Rödl picks out for blame here is that Begriffschrift expressions are characterized only by their inferential structure: a Begriffschrift expression is to have its meaning fixed solely by determining how it figures in lines of a Begriffschrift proof. This is what Frege really fixes by his procedures: how Begriffschrift expressions are to be used in constructing Begriffschrift proofs. But Rödl claims that this fails to settle the question of how Begriffschrift expressions relate to their objects, for the reasons Dummett gave: and this is why Frege fails to notice that his logic cannot take care of itself, as part of what determines the thoughts expressible in Begriffschrift is the relation of these thoughts to their objects, and not merely the relations of these thoughts to each other, and this is why Frege fails to determine thoughts expressible in Begriffschrift in the way he believed he had. So the form of thought as such cannot simply be an inferential order, but must already determine the relation of thoughts to their objects in a way that Frege's logic did not.

The option Dummett's despair overlooks is that of determining the truth-values of sentences not in a way anterior to their relatedness to their objects, but only by already having in view these sentences' relatedness to their objects. Dummett's Frege erred in trying to secure relatedness to objects only indirectly, by means of securing the inferential functions of thoughts, and Dummett as sees the only alternative to secure the relatedness to objects of thoughts to be that of grasping thoughts and objects independently and then bringing them together (in some medium other than thought, which yet stands in need of relatedness to objects). Rödl's excluded third alternative is to promote a logic which determines thoughts only as already related to objects which are given by these thoughts: such a logic is what Kant called "transcendental logic". Kant distinguished this from the sort of logic he infamously claimed to have been settled since Aristotle, which he characterized as a "general logic" that abstracts from all objects of thought and deals with modes of inference independently of the relatedness of thoughts to any objects (which is why for Kant general logic can lead to transcendent metaphysical thoughts by means of fallacious inferences, but transcendental logic does not give any "Sinn oder Bedeutung" to the "thoughts" of transcendent metaphysics).

To present a transcendental logic is to present a logic which includes within it an account of the objects which can be given to thought; non-transcendental logic can omit this only because it treats of thoughts without inquiring into their relatedness to objects. If a logic is not to leave the question of the relatedness of thougts to their objects outside of itself (as a topic for something other than logic), i.e. if a logic is to take care of itself, to be able to present the form of thought as such, then the objects which are given in thoughts must be treated of by logic itself: thus transcendental logic must be metaphysics. This is just what we find in Kant: the Transcendental Logic is just where Kant establishes the principles of his "metaphysics which is to come forth as a science", that all appearances are substances undergoing lawful changes in mutual interaction etc. As general logic determines the forms of thoughts which are capable of figuring into inference, so transcendental logic determines the forms of thoughts of objects: and so it determines the ways in which objects may be given to us, the way in which objects which can be given to us may (or must) be. And unless there is a distinction made between objects which can be given to thought and objects which can be given to our thought, transcendental logic will not have as a consequence transcendental idealism: if the metaphysics of transcendental logic determines how objects must be to be given to thought, then to speak of objects which are not (or might not be) thus is to speak of something contradictory, or to put forward a thought which has no relation to any object: thus it cannot have as its object any "thing in itself" which is unknowable by thought. Kant's transcendental idealism arises because of his accounting our form of sensibility as not the only logically possible one: hence his transcendental logic does not present the form of objects which can be given to thought, but only the form of objects which can be given to spatiotemporally-formed thinkers; the "thing in itself" thus remains as something which (as far as logic allows) might be given to thought, but cannot be given to our thought, and so is for us unknowable and undetermined by Kant's metaphysics. If Kant's forms of sensibility can be shown to be the only logically possible ones, to be demanded by transcendental logic and not merely an addition from a "transcendental aesthetic", or if Kant can be shown to have erred in his claiming that the (logically contingent) forms of space and time are the forms of our sensibility, then Kant's transcendental idealism can be excised from his system. This is part of Rödl's project, in line with earlier German Idealists such as Ficthe and Hegel: to carry out transcendental logic without an independent transcendental aesthetic, so as to avoid transcendental idealism and the "thing in itself". They do not reject the division between transcendental logic and transcendental aesthetic because of ignorance of the importance of a transcendental aesthetic, of an account of the form of objects which can be given to thought, but because they seek to have transcendental logic alone provide for it, as it should be able to if transcendental logic is a logic which can take care of itself.


Daniel Nagase said...

Hey, first of all, don't worry about replying quickly. I myself am not the fastest "replier" around, and, as you said to me before, this kind of exchange is best when it's not made in a hurry.

In any case, I have a simple question about Rödl. I haven't read his book yet (though I'm very interested in it!), but, from your description of it, it seems he opposes a certain strand in the philosophy of logic, which has its roots in Gentzen (and Dummett), and which tries to characterize the "province" of logic as being (basically) proof theory; according to this strand, logical constants, for example, would be completely characterized by their introduction and elimination rules. Does Rödl explicitly address this tradition? Or is more like a background discussion?

Daniel Lindquist said...

From what I've seen, his discussion of the philosophy of logic is pretty strictly limited to Frege, Quine, some Wittgenstein, and some Kant commentators. He only looks at Dummett qua reader of Frege.

But he does explicitly (on occasion) set aside topics as being things he just doesn't need to discuss for his purposes, and I think proof-theoretical stuff falls under that (or at least he would put it there). Insofar as "logic" designates something other than the study of the form of thought as such, which Roedl calls "metaphysical logic", he looks at it only to distinguish it from "metaphysical logic".

Chapter one of "Categories of the Temporal" is where to look for this stuff; my post is pretty much just me trying to get clear on what Roedl's arguments in the last few pages of that chapter are. I'm only just now *finally* reading Roedl's books in their entirety; there's a reading group here for "Self-Consciousness", and I've now reread the first chapter of "Categories of the Temporal" enough times over the past six months that I feel like I can probably move on to the next chapter after one more go. (It's a dense book! But very good so far.)

Ben W said...

IIRC, the introduction to Kategorien des Zeitlichen addresses (what Rödl sees as) the impoverishment of the concept of "logic" in latter days. The discussion isn't very in-depth, though.

S. A. Lee said...

The third chapter of Markus Gabriel's book "Transcendental Ontology" has a fantastic discussion of Hegel's ontological logic and late Schelling's response to that. You should check it out. Also, this was a brilliant post, as usual.

S. A. Lee said...

The third chapter of Markus Gabriel's book "Transcendental Ontology" has a fantastic discussion of Hegel's ontological logic and late Schelling's response to that. You should check it out. Also, this was a brilliant post, as usual.

Daniel Lindquist said...

Glad you enjoyed it, Lee.