I spent some time last year reading through Dummett on non-classical logics. One aim was to figure out what sorts of arguements there might be against combining a truth-value gap view with intuitionistic logic. The question is whether in an intuitionist setting it might be ok to endorse ~T(A)&~T(~A) (The characteristic intuitionistic feature, hereabouts, is a refusal to assert T(A)vT(~A)—which is certainly weaker than asserting its negation. Indeed, when it comes to the law of excluded middle, the intuitionist refuses to assert Av~A in general, but ~(Av~A) is an intuitionistic inconsistency).

On the motivational side: it is striking that in Kripke tree semantics for intuitionistic logic, there are sentences such that neither they nor their negation are “forced”. And if we think of forcing in a Kripke tree as an analogue of truth, that looks like we’re modelling truth value gaps.

A familiar objection to the very idea of truth-value gaps (which appears early on in Dummett—though I can’t find the reference right now) is that asserting the existence of truth value gaps (i.e. endorsing ~T(A)&~T(~A)) is inconsistent with the T-scheme. For if we have “T(A) iff A”, then contraposing and applying modus ponens, we derive from the above ~A and ~~A—contradiction. However, this does require the T-scheme, and you might let the reductio fall on that rather than the denial of bivalence. (Interestingly, Dummett in his discussion of many-valued logics talks about them in terms of truth value gaps without appealing to the above sort of argument—so I’m not sure he’d rest all that much on it).

Another idea I’ve come across is that an intuitionistic (Heyting-style) reading of what “~T(A)” says will allow us to infer from it that ~A (this is based around the thought that intuitionistic negation says “any proof of A can be transformed into a proof of absurdity”). That suffices to reduce a denial of bivalence to absurdity. There are a few places to resist this argument too (and it’s not quite clear to me how to set it up rigorously in the first place) but I won’t go into it here.

Here’s one line of thought I was having. Suppose that we could argue that Av~A entailed the corresponding instance of bivalence: T(A)vT(~A). It’s clear that the latter entails ~(~T(A)&~T(~A))—i.e. given the claim above, the law of excluded middle for A will entail that A is not gappy.

So now suppose we assert that it is gappy. For reductio, suppose Av~A. By the above, this entails that A is not gappy. Contradiction. Hence ~(Av~A). But we know that this is itself an intuitionistic inconsistency. Hence we have derived absurdity from the premise that A is gappy.

So it seems that to argue against gaps, we just need the minimal claim that LEM entails bivalence. Now, it’s a decent question what grounds we might give for this entailment claim; but it strikes me as sufficiently “conceptually central” to the intuitionistic idea about what’s going on that it’s illuminating to have this argument around.

I guess the last thing to point out is that the T-scheme argument may be a lot more impressive in an intuitionistic context in any case. A standard maneuver when denying the T-scheme is to keep the T-rules: to say that A entails T(A), for example (this is consistent with rejecting the T-scheme if you drop conditional proof, as supervaluational and many-valued logicians often do). But in an intuitionistic context, the T-rule contraposes (again, a metarule that’s not good in supervaluational and many-valued settings) to give an entailment from ~T(A) to ~A, which is sufficient to reduce the denial of bivalence to absurdity. This perhaps explains why Dummett is prepared to deny bivalence in non-classical settings in general, but seems wary of this in an intuitionistic setting.

The two cleanest starting points for arguing against gaps for the intuitionist, it seems to me, are either to start with the T-rule, “A entails T(A)” or with the claim “Av~A entails T(A)vT(~A)”. Clearly the first allows you to derive the second. I can’t see at the moment an argument that the second entails the first (if someone can point to one, I’d be very interested), so perhaps basing the argument against gaps on the second is the optimal strategy. (It does leave me with a puzzle—what is “forcing” in a Kripke tree supposed to model, since that notion seems clearly gappy?)

What do you mean by prove the second entails the first? I think you can if you have allow more axioms for T, like A entails ~T~A (I think…) But if you don’t have more axioms for T you could prevent me stipulating that T~A=A and TA=~A so you’d get the second principle but not the first one.

Hi Andrew,

Yes, I was interested in what you have to add to get from one to the other, intuitionistically. A|=~T~A seems like a decent proposal. And then you’d argue from A to Av~A to T(A)vT(~A). And then from A again to ~T(~A), and by disjunctive syllogism to T(A). Am I getting your thinking right? And the entailment you mention is pretty plausible!

I guess one question is: can we get from denial of bivalence and “A entails ~T~A” to contradiction directly, without routing through the LEM-to-bivalence principle? I’m guessing not, since interpreting T as box, that entailment holds, but presumably “denial of bivalence” (i.e. on that interpretation, assertion of contingency) is unproblematic.

Thanks Robbie,

Yes, that’s what I was thinking. Regarding your second question – I guess not, or like you say things would be pretty bad for the intuitionist + modal logic!

By the way I was interested by the thing you said about forcing being gappy. I’m not sure what the argument is that forcing (wrt a node) in the intended Kripke tree is gappy. But here’s my worry:

For starters, Dummett explicitly says, when he defines Beth trees, he doesn’t assume that “truth at a node” obeys LEM (elements of intuitionism, p138.) So bearing that in mind I’m not sure how to argue that truth at a node is gappy.

I’d like to think about this some more, but I reckon you could come up with a tree+ that has no truth gaps, where a tree+ is like a tree but we define it in terms of species instead of sets, etc, and the metalanguage is all intuitionistic.

Here’s the kind of thing I mean but for three valued Heyting algebras. Let L be a language whose only atomic sentence is p. Let L+ be a language whose atomic sentences are p, |phi|=0, |phi|=1/2, |phi|=1 whenever phi belongs to L. Suppose the semantic value of each atomic sentence of L+ is 1/2. Then we, as meta-linguists speaking L+, shouldn’t assert (~|phi|=1 ^ ~|~phi|=1) for any |phi| in L. (Of course, we shouldn’t assert the T-schema either, but that’s the issue.)

Oops, I missed out the crucial thing that in L+ schemata like the following come out with semantic value 1: “if |phi|=x and |psi|=y then |phi and psi|=min(x,y)” etc…