CEM journalism

The literature on the linguistics/philosophy interface on conditionals is full of excellent stuff. Here’s just one nice thing we get. (Directly drawn from a paper by von Fintel and Iatridou). Nothing here is due to me: but it’s something I want to put down so I don’t forget it, since it looks like it’ll be useful all over the place. Think of what follows as a bit of journalism.

Here’s a general puzzle for people who like “iffy” analyses of conditionals.

  • No student passes if they goof off.

The obvious first-pass regimentation is:

  • [No x: x is a student](if x goofs off, x passes)

But for a wide variety of accounts, this’ll give you the wrong truth-conditions. E.g. if you read “if” as a material conditional, you’ll get it coming out true if all the students goof and succeed! What is wanted, as Higgenbotham urges, is something with the effect:

  • [No x: x is a student](x goofs off and x passes)

This seems to suggest that under some embeddings “if” expresses conjunction! But that’s hardly what a believer in the iffness of if wants.

What the paper cited above notes is that so long as we’ve got CEM, we won’t go wrong. For [No x:Fx]Gx is equivalent to [All x:Fx]~Gx. And where G is the conditional “if x goofs off, x passes”, the negated conditional “not: if x goofs off, x passes” is equivalent to “if x goofs off, x doesn’t pass” if we have the relevant instance of conditional excluded middle. What we wind up with is an equivalence between the obvious first-pass regimentation and:

  • [All x: x is a student](if x goofs off, x won’t pass).

And this seems to get the right results. What it *doesn’t* automatically get us is an equivalence to the Higgenbotham regimentation in terms of a conjunction (nor with the Kratzer restrictor analysis). And maybe when we look at the data more generally, we’ll can get some traction on which of these theories best fits with usage.

Suppose we’re convinced by this that we need the relevant instances of CEM. There remains a question of *how* to secure these instances. The suggestion in the paper is that rules governing legitimate contexts for conditionals give us the result (paired with a contextually shifty strict conditional account of conditionals). An obvious alternative is to hard-wire in CEM into the semantics, as Stalnaker does. So unless you’re prepared (with von Fintel, Gillies et al) to defend in detail fine-tuned shiftiness of the contexts in which conditionals can be uttered then it looks like you should smile upon the Stalnaker analysis.

[Update: It’s interesting to think how this would look as an argument for (instances of) CEM.

Premise 1: The following are equivalent:
A. No student will pass if she goofs off
B. Every student will fail to pass if she goofs off

Premise 2: A and B can be regimented respectively as follows:
A*. [No x: student x](if x goofs off, x passes)
B*. [Every x: student x](if x goofs off, ~x passes)

Premise 3: [No x: Fx]Gx is equivalent to [Every x: Fx]~Gx

Premise 4: if [Every x: Fx]Hx is equivalent to [Every x: Fx]Ix, then Hx is equivalent to Ix.

We argue as follows. By an instance of premise 3, A* is equivalent to:

C*. [Every x: student x] not(if x goofs off, x passes)

But C* is equivalent to A*, which is equivalent to A (premise 2) which is equivalent to B (premise 1) which is equivalent to B* (premise 2). So C* is equivalent to B*.

But this equivalence is of the form of the antecedent of premise 4, so we get:

(Neg/Cond instances) ~(if x goofs off, x passes) iff if x goofs off, ~x passes.

And we quickly get from the law of excluded middle and a bit of logic:

(CEM instances) (if x goofs off, x passes) or (if x goofs off, ~ x passes). QED.

The present version is phrased in terms of indicative conditionals. But it looks like parallel arguments can be run for CEM for counterfactuals (Thanks to Richard Woodward for asking about this). For one of the controversial cases, for example, the basic premise will be that the following are equivalent:

D. No coin would have landed heads, if it had been flipped.
E. Every coin would have landed tails, if it had been flipped.

This looks pretty good, so the argument can run just as before.]

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