New APPS

This semester I'm teaching Stuart Brock and Edwin Mares' generally excellent Realism and Anti-Realism. If you want to quickly introduce students to a lot of canonical debates in analytic philosophy, it would be hard to do better. The first part of the book discusses generalized anti-realist positions such as Kantianism, and the second part goes over localized anti-realisms (color, morality, science, modality, etc.).

The general excellence of the book sets in bold relief the manner in which the authors thoroughly botch their discussion of Michael Dummett. One can't really blame them though, as their botching does cite a lot of canonical secondary literature, where the mistakes continue to linger sixteen years after Neil Tennant pointed them out in The Taming of the True.

On the traditional account of Dummett, his argument for intuitionism goes like this:

Verifiability is a necessary condition on the truth of claims (Verificationism). But some claims are such that neither they nor their negations are verifiable (Single Sentence Undecidability). But then by two applications of Modus Tollens, such claims will be neither true nor false. Thus we must reject bivalence and adopt intuitionistic logic, which has the added virtue of being justifiable by a constructivist semantics that equates truth and verifiability.

This the argument that Brock and Mares attribute to Dummett, and they follow standard secondary literature in doing so.

The problem is, this can't be Dummett's argument, since one can in intuitionist logic derive a contradiction from the claim that a sentence is neither true nor false. Even more damning, one can in intuitionist logic show Verificationism to be inconsistent with Single Sentence Undecidability. For the former, we need avail ourselves of Wright's Lemma (so-called because it is a key claim for the argument in Truth and Objectivity).

Claim:
~TP –> T~P

Proof:
1. | ~TP
2. || P
3. || TP…………2 T intro.
4. || #………….1,3 ~ elim.
5. |~P…………..2-4 ~ intro.
6. |T~P………….5 T intro.
7. ~TP –> T~P……1-6 –> intro.

This says that if it's not true that P, then it's true that it is not the case that P (Wright gets a lot of argumentative weight out of the fact that the conditional is not in general true for assertibility).* Note that no classical negation rules are used. As long as the intuitionist is committed to what we are calling "T intro." which is just one half of the disquotationalist schema for the truth predicate, nothing is amiss.

 So now if we consider a sentence that is neither true nor false, we can use Wright's Lemma as an inference rule (WL) to derive a contradiction in intuitionist logic.

1. ~TP & ~T~P
2. ~TP………..1 & elim.
3. ~T~P……….1 & elim.
4. T~P………..2 WL
5. T~~P……….3 WL
6. ~P…………4 T elim.
7. ~~P………..5 T elim.
8. #………….6,7 ~ elim.

So this is already enough to conclude that either Dummett has an inconsistent set of philosophical commitments, or that he had a different argument in mind for intuitionistic revision. Nobody, on pain of contradiction can say they are an intuitionist because some sentences fail to be true or false.

But it gets worse. Verificationism and Single Sentence Undecidability on their own intuitionistically entail a contradiction. So, if Dummett's argument is as it is standardly reported, then it would be, as Tennant writes, "a non-sequitur of numbing grossness." In what follows, I'll treat Verificationism as an inference rule V that allows us to go from the truth of a claim to its knowability. I won't bother with doing a higher order existential elimination on the claim that there exists an undecidable sentence, but will just allow myself to suppose an arbitrary instance of such a claim on the first premise (since the conclusion is a contradiction, in which the arbitrary instance doesn't occur, this is kosher).

1. ~<>KP & ~<>K~P…Single Sentence Undecidability
2. ~<>KP………………..1 & elim.
3. ~<>K~P………………1 & elim.
4. | TP
5. | <>KP…………………3 V.
6. | #………………………2,5 ~elim.
7. ~TP…………………….4-6 ~intro.
8. |T~P
9. |<>K~P………………..8 V.
10.| #………………………3,9 ~elim.
11.~T~P……………………8-10 ~intro.
12. ~TP & ~T~P…………..7,11 & intro.
13. #…………………………12, by previous proof using Wright's Lemma

But sixteen years after Tennant points this out in an Oxford University Press book, people still attribute the elemtary error to Dummett without realizing that it is an elementary error.

Tennant goes on to pretty brilliantly reconstruct Dummett's argument around the undecidability of a discourse, in the sense of no effective decision procedure existing for that discourse. I criticize this reconstrual in "Manifest Invalidity: Neil Tennant's New Argument for Intuitionism," Synthese, 134.3 (2003), pp. 353-362.**

My own reconstruction of Dummett's argument can be found in "The Logic of Logical Revision: Formalizing Dummett's Argument," The Australasian Journal of Philosophy, 83.1 (2005), pp. 15-32. I looked at the broader theory of meaning debates in which Dummett was a key player to show that Dummett's key premise is that the correctness of bivalent truth conditional semantics as part of a theory of meaning entails the possible existence of an undecidable sentence. Then, as long as one takes Verificationism to be a necessary truth, it follows in a very weak modal logic (higher order K, if I remember right) that bivalent truth conditional semantics are incorrect. I argued that the future of the debate has got to concern how to make sense of the relevant modal notions such that the proof doesn't equivocate.***

In either case, neither Tennant nor myself read Dummett as so logically naive as what continues to be the null hypothesis about his argument. So don't blame us.

[Notes:

*Wright's actual arguments in this area are also generally misconstrued so as to license Strawman attacks. I won't go into this here, but if you are interested see my "Logical Revision Re-revisited: The Wright/Salerno Argument for Intuitionism," Philosophical Studies, 60.1 (2000), pp. 5-12.

**Given all the personal kindness and philosophical illumination Tennant gives to his students (I would have almost certainly washed out of graduate school if not for him), I very quickly came to loathe the title and tone of that piece.

I should note that it was the first thing I got accepted after being out of graduate school for two and a half years. I was at that point extraordinarily frustrated with the whole process. I had good ideas, but just couldn't seem to master the rhetorical art involved in writing competent analytical philosophy. This is no doubt due in part to dyslexia and in part to reading a lot of continental philosophy, which leaches into your writing style in a way succeptible to external reviewers mocking you to editors (the dyslexia thing isn't such a big problem now, but this other problem has only gotten worse). After two and a half years of rejection, the frustration boiled into the style of the piece on Tennant's argument as well as the previously noted one on Wright and Salerno (and, after a dry spell of two and a half years, both angry pieces promptly got accepted).

Of course Tennant was a mensch about the whole thing. I also think I've to some extent paid my pennance to the karma gods here. For a rousing defense of Tennant on Fitch's paradox, see my "Paradox Lost," Canadian Journal of Philosophy, 34.2 (2004), pp. 195-216. The central claim of that piece is that if Tennant is wrong about his reaction to Fitch's paradox, then the standard solution to the paradox of the stone does not work. For any philosophers of religion reading this, my modus tollens might be your modus ponens. Given the connection I establish in my piece, a very easy publication would be to take criticisms of Tennant on Fitch by Kvanvig, Williamson, and others and apply them to the standard solution to the paradox of the stone. Just trying to be helpful here. This is low hanging publication fruit.

***I still think this is correct, and still think that Dummett is relevant. If one take as primary notions about what can be done, as opposed to epistemic notions, then a lot of Dummett's arguments translate and shed new light on realist debates about modality. There's actually a relevant historical background here that has been effaced by the canonical debates about modality in analytic philosophy. This tradition goes through German Idealism and leads up to Nicolai Hartmann's fantastic book. Hartmann actually had a (to me much superior) form of two-dimensionalism in the 1920s. In a decade or so, when I'm up to speed on German Idealism, I'm going to return to this as a research project. Unlike the previous footnote, the fruit is not low hanging here. It's very difficult stuff.]