Friday, September 01, 2006

An argument for conditional excluded middle

(cross-posted from Theories n Things)

Conditional excluded middle is the following schema:

if A, then C; or if A, then not C.

It's disputed whether everyday conditionals do or should support this schema. Extant formal treatments of conditionals differ on this issue: the material conditional supports CEM; the strict conditional doesn't; Stalnaker's logic of conditionals does, Lewis's logic of conditionals doesn't.

Here's one consideration in favour of CEM (inspired by Rosen's "incompleteness puzzle" for modal fictionalism, which I was chatting to Richard Woodward about at the Lewis graduate conference that was held in Leeds yesterday).

Here's the quick version:

Fictionalisms in metaphysics should be cashed out via the indicative conditional. But if fictionalism is true about any domain, then it's true about some domain that suffers from "incompleteness" phenomena. Unless the indicative conditional in general is governed in general by CEM, then there's no way to resist the claim that we get sentences which are neither hold nor fail to hold according to the fiction. But any such "local" instance of a failure of CEM will lead to a contradiction. So the indicative conditional in general is governed by CEM

Here it is in more detail:

(A) Fictionalism is the right analysis about at least some areas of discourse.

Suppose fictionalism is the right account of blurg-talk. So there is the blurg fiction (call it B). And something like the following is true: when I appear to utter , say "blurgs exist" what I've said is correct iff according to B, "blurgs exist". A natural, though disputable, principle is the following.

(B) If fictionalism is the correct theory of blurg-talk, then the following schema holds for any sentence S within blurg-talk:

"S iff According to B, S"

(NB: read "iff" as material equivalence, in this case).

(C) The right way to understand "according to B, S" (at least in this context) is as the indicative conditional "if B, then S".

Now suppose we had a failure of CEM for an indicative conditional featuring "B" in the antecedent and a sentence of blurg-talk, S, in the consequent. Then we'd have the following:

(1) ~(B>S)&~(B>~S) (supposition)

By (C), this means we have:

(2) ~(According to B, S) & ~(According to B, ~S).

By (B), ~(According to B, S) is materially equivalent to ~S. Hence we get:

(3) ~S&~~S

Contradiction. This is a reductio of (1), so we conclude that

(intermediate conclusion):
No matter which fictionalism we're considering, CEM has no counterinstances with the relevant fiction as antecedent and a sentence of the discourse in question as consequent.


(D) the best explanation of (intermediate conclusion) is that CEM holds in general.

Why is this? Well, I can't think of any other reason we'd get this result. The issue is that fictions are often apparently incomplete. Anna Karenina doesn't explicitly tell us the exact population of Russia at the moment of Anna's conception. Plurality of worlds is notoriously silent on what is the upper bound for the number of objects there could possibly be. Zermelo Fraenkel set-theory doesn't prove or disprove the Generalized Continuum Hypothesis. I'm going to assume:

(E) whatever domain fictionalism is true of, it will suffer from incompleteness phenomena of the kind familiar from fictionalisms about possibilia, arithmetic etc.

Whenever we get such incompleteness phenomena, many have assumed, we get results such as the following:

~(According to AK, the population of Russia at Anna's conception is n)
&~(According to AK, the population of Russia at Anna's conception is ~n)

~(According to PW, there at most k many things in a world)
&~(According to PW, there are more than k many things in some world)

~(According to ZF, the GCH holds)
&~(According to ZF, the GCH fails to hold)

The only reason for resisting these very natural claims, especially when "According to" in the relevant cases is understood as an indicative conditional, is to endorse in those instances a general story about putative counterexamples to CEM. That's why (D) seems true to me.

(The general story is due to Stalnaker; and in the instances at hand it will say that it is indeterminate whether or not e.g. "if PW is true, then there at most k many things in the world" is true; and also indeterminate whether its negation is true (explaining why we are compelled to reject both this sentence and its negation). Familiar logics for indeterminacy allow that p and q being indeterminate is compatible with "p or q" being determinately true. So the indeterminacy of "if B, S" and "if B, ~S" is compatible with the relevant instance of CEM "if B, S or if B, ~S" holding.)

Given (A-E), then, I think inference to the best explanation gives us CEM for the indicative conditional.

[Update: I cross-posted this both at Theories and Things and Metaphysical Values. Comment threads have been active so far at both places; so those interested might want to check out both threads. (Haven't yet figured out whether this cross-posting is a good idea or not.)]


Robbie said...

Just a few further thoughts about my premises; specifically (B) and (C).

(B) is a substantive position on what fictionalism consists in. Other versions would e.g. say that "blurgs exist" was false (after all, there are no blurgs!) and that it's only assertible because we assume an "according to" operator is tacitly invoked.

A more interesting way of denying this would be to restrict the equivalence somehow. Daniel Nolan suggests doing this, in response to Rosen, in his Stanford encyclopedia article on modal fictionalism.

(C) is again substantive. Not everything hangs on the claim that "according to fiction" should quite generally be analyzed as an indicative conditional (Cian Dorr suggests that a fictionalism about mereology should be cashed out in terms of a counterfactual conditional: I don't think he intends that to be part of a general analysis of fiction. FWIW I think that the indicative analysis does better than Cian's counterfactual analysis as part of a formulation of mereological fictionalism considered on its own merits.)

But it'd be nicer if "according to" in general could be given this analysis. And I don't think it's crazy to think this is the case. Stefano Predelli gave an interesting talk in Leeds this year, where he pointed to some odd (monstrous!) semantic behaviour of "according to the fiction". What's interesting about that observation is that indicative conditionals seem to exhibit exactly the same, monsterous, behaviour.

Greg Restall said...

Interesting. As someone who's generally not a fan of CEM for a wide range of conditionals, I'm not convinced by the conclusion. But where do things go wrong?

There are a number of possibilities. Rejecting fictionalism as the right analysis of blurg talk comes to mind, but let's not go that way.

What strikes me as an interesting fact is the following. Suppose that you're right, and that for the ZF case we have the the following choice of disjunct for CEM: that according to ZF, GCH holds.

Now consider the theory ZF+~GCH. As Cohen has shown, this is perfectly consistent if ZF+GCH is consistent (and, I think we've assumed that it is consistent, provided that you're prepared to believe that according to ZF, all of the axioms of ZF are true: that is, identity for according-to). Now, what holds according to ZF+~GCH? Clearly ~GCH holds (by identity again), as does each axiom of ZF. ZF+~GCH is a properly stronger theory than ZF, but what happens according to it is, in general, incomparable with what happens according to ZF. Some things hold according to ZF (namely, GCH) that don't hold to according to ZF+~GCH, and some things hold according to ZF+~GCH that don't hold according to ZF.

You must reject the thesis that if T2 is a stronger theory than T1 (in the sense that T2 entails more things than T1 does) then (at least for claims p pertaining to both theories), if p holds according to T1, then it holds according to the stronger theory T2 too. At least, when it comes to mathematical theories, a reading of according to satisfying CEM is at the very least highly nonstandard, and mathematically opaque. How do I begin to evaluate a claim such this:

According to T, for every x and y, (x * y = y * x)

where T is the very small theory that states that * is a binary operator on some unspecified domain? I know that this claim isn't entailed by the theory (and that neither is its negation) but I have no idea what it would mean to take a stand on one disjunct in the instance of CEM for "according to."

Furthermore, the account seems infelicitous when it comes to everyday fictions too, when an author writes a sequel to the story (given that they have a choice to claim that p is the case, or that ~p is the case for some p that is salient to the original story) then one of these choices actually contradicts what was the case according to the original story. Even though this was completely opaque to the readers, the author, and everyone else.

I think that this at least counts as a cost of the view rather than a benefit, and helps articulate why I find it odd.

I hope this helps.

Robbie said...

Hi Greg,

Thanks for this. Interesting stuff!

Nice point about extensions of the "fiction". "If F1 is stronger than F2, then F1 entails more things than F2": that'll be at best indeterminate for me (when stronger is read as you stipulate).

I don't see the problem though... for a start, is it non-standard? "Stronger" isn't a mathematical concept in any ordinary sense: how do we determine the standard views about it? And I don't see why it'd be mathematically opaque. After all, we can divide props into three classes: those that the theory entials, those that it entails the negation of, and those about which it does neither. Those are all mathematically tractable, and correspond to what is supertrue/superfalse/indeterminate by the lights of the "according to" operation.

Similarly, in the "very small" theory you mention, why do we have to get involved in the issue of what it would be to "take a stand on whether [p] holds" according to that theory? The relevant p I take it is indeterminate: so I reject it. On one precisification, it is true; but what follows from that?

Is this an objection to (e.g.) supervaluationism per se or to Stalnaker's application of it in particular? An analogy: is it legitimate to object to the idea that Fine's predicate "nice number" is indeterminate in extension between (e.g.) "greater than 5" and "greater than 7", by asking how one would begin to evalute the claim "6 is a nice nunber"?

Lastly, on the ordinary fictions case. What's maybe disturbing is that the fiction is said to represent things as being indeterminate; and the successor fiction represents them as being determinate (one way or the other).

There's some sense in which the later fiction contradicts the earlier one (though not in the familiar--synctactic, semantic---senses of "contradict", of course). But personally i don't find it disturbing at all. Maybe I'm not getting it, but I woudl think that it'd be fine to use the word "contradict" so that representing p as indeterminate is consistent with representing p as determinately true. Is there some reason why this is a bad way to use the term? (If not, I just don't see why we should be worried about the result you notice).

andrew said...

(B) seems wrong to me. We don't endorse:

Sherlock Holmes is a detective iff According to Holmes-fictions, SH is a detective.

Rather, we replace sentences apparently of the form of the LHS with sentences of the form of the RHS when it matters. This is different from saying that the two sentences have the same truth-conditions. (for example, the LHS arguably only expresses a truth-conditional content if 'SH' refers; there seems less such controversy with the RHS.)

Greg Restall said...

Hi Robbie,

Strength for theories is a completely standard mathematical notion: T2 is stronger than T1 if what's entailed by T2 is entailed by T1. (So, PA is stronger than Robinson's arithmetic, ZFC is stronger than ZF, etc.)

As to the rest, I did not have the supervaluationist in mind -- which was rather neglectful of me. So what I had in mind was not an objection to supervaluationism, but an objection to CEM that ignored supervaluationism.

So is the account that what is determinately true according to the theory (or story) is what is entailed by that theory (or story)? I suppose that this gives rather a lot of acceptable evaluations in the case of rather incomplete mathematical theories. It seems ... curious.

Robbie said...

Hi Andy;

Right: dropping the claim of semantic equivalence is a natural suggestion (I guess one natural elaboration is the error-theoretic fictionalism that I mentioned above).

In the "fictionalism about semantic properties" part of my thesis, I endorsed this move; and my opinion then was that if we wanted some kind of equivalence, we should restrict it to sentences "that matter", a la Nolan. But this is apt to seem pretty ad hoc. In my thesis, I tried to provide some motivation for it in general when doing fictionalist metaphysics. It'd be interesting if it turned out to be that we already should do this for the ordinary fiction case.

Anyway, so I don't think it's unreasonable at all to reject that premise. But I guess I think it's at least interesting to explore the consequences of the stronger fictionalism where the biconditional holds. I can think of interesting projects where it'd argubly be this thing we want. E.g. suppose what you wanted wasn't a semantic analysis of possibilia-talk, but a story about what "possibilia propositions" hold in virtue of. (The idea behind this being that while possibilia exist, they don't exist in fundamental reality; so how they "emerge" from fundamental reality needs explaining). It seems to me that something like the biconditional follows quite naturally.

Of course, I don't want the argument to rest on you buying all of this project, but I hope it gives a sense of why it might be interesting to explore the consequences of (B).

Robbie said...

Hi Greg!

Ok, right: the supervaluationism is totally crucial to my being prepared to endorse CEM.

Of course, in the original post I was being explicitly non-committal about what the relevant conditional was, so it was a bit unfair of me to start going on about one particular account (Stalnaker's) as if it were canonical. If you combined Stalnaker's defence of CEM with an epistemicism about vagueness, for example, the results you mention would hold. (FWIW, it was the superval+Stalnaker conditional I had in mind when writing the post).

I guess it does illustrate that some objections to the status will turn on, not the logical behaviour, but the semantic behaviour, of the conditional.

I withdraw what I said about "stronger"... not sure why I said that! Maybe what I meant to write was that "according to" isn't a mathematical notion, and so I didn't see why it would be somehow non-standard to drop the link between strength and what's true according to the theories.

On the last thing you say: I was thinking that "if PA then q" would only be true if q is a consequence of PA. If so, then I think that the equivalence you mention holds. But maybe that's not what we have to say: maybe we want to say that "if PA then g" is true (where g is the Godel sentence for PA). Certainly "if PA then T(PA)" looks pretty good, for T a truth predicate, and that might already get us beyond the deductive consequences. I haven't thought through what happens if you do this --- maybe the account would overall be less attractive. (Thinking back, I deliberately chose GCH rather than the Godel sentence so as to get a more intuitively "robust" case of incompleteness than the Godel ones.)