Category Archives: Logic

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.

Moreover:

(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.)]