## Wednesday, November 30, 2016

### Material conditionals and quantifiers

From:

1. Every G is H
it seems we should be able to infer for any x:

1. If x is G, then x is H.

This pretty much forces one to read “If p, then q” as a material conditional, i.e., as q or not p. For the objection to reading the indicative conditional as a material conditional is that this leads to the paradoxes of material implication, such as that if it’s not snowing in Fairbanks, Alaska today, then it’s correct to say:

1. If it’s snowing in Fairbanks today, then it’s snowing in Mexico City today

even if it’s not snowing in Mexico City, which just sounds wrong.

But if we grant the inference from (1) to (2), we can pretty much recover the paradoxes of material implication. For instance, suppose it’s snowing neither in Fairbanks nor in Mexico City today. Then:

1. Every truth value of the proposition that it’s snowing in Fairbanks today is a truth value of the proposition that it’s snowing in Mexico City today.

So, by the (1)→(2) inference:

1. If a truth value of the proposition that it’s snowing today in Fairbanks is true, then a truth value of the proposition that it’s snowing today in Mexico City is true.

Or, a little more smoothly:

1. If it’s true that it’s snowing in Fairbanks today, then it’s true that it’s snowing in Mexico City today.

It would be very hard to accept (6) without accepting (3). With a bit of work, we can tell similar stories about the other standard paradoxes. The above truth-value-quantification technique works equally well for both the true⊃true and the false⊃false paradoxes. The remaining family of paradoxes are the false⊃true ones. For instance, it’s paradoxical to say:

1. If it’s warm in the Antarctic today, it’s a cool day in Waco today

even though the antecedent is false and the consequent is true, so the corresponding material conditional is true. But now:

1. Every day that’s other than today or on which it’s warm in the Antarctic is a day that’s other than today or on which it’s cool in Waco.

So by (1)→(2):

1. If today is other than today or it’s warm in the Antarctic today, then today is other than today or today it’s cool in Waco.

And it would be hard to accept (9) without accepting (7). (I made the example a bit more complicated than it might technically need to be in order not to have a case of (1) where there are no Fs. One might think for Aristotelian logic reasons that that case stands apart.)

This suggests that if we object to the “material conditional” reading of “If… then…”, we should object to the “material quantification” reading of “Every F is G”. But many object to the first who do not object to the second.

#### 1 comment:

I'm an undergrad and a fan. Thanks so much for your blog. I learn from it daily.
I'm wondering if the paradoxes of the material conditional can be resolved the following way:

P = It's snowing in Fairbanks today
Q= It's snowing in Mexico City today

Fact (F): If ~P, then it's true to say 'If P, then Q'.

But using the material conditional, F can be symbolized as:

1. ~P → (P → Q)

(1) is equivalent to (2) by Exportation

2. (~P • P) → Q

Consider:

3. If P, then Q

Hypothesis:
When material conditionals with false antecedents are judged to be paradoxically true, the antecedent of the conditional is being construed as true and not also false. To construe a false antecedent as 'true and not also false' is to entertain a counterfactual, not a material conditional.

So, to think (3) is paradoxically true is to confuse indicative conditionals with subjunctive conditionals.

Argument:

Either the operator in (3) is a material conditional or it's not. If it is, then (3) is not paradoxically true [it's just conforming to the principle of explosion]. If it's not, then (3) is a false subjunctive and so not paradoxically true. So, (3) is not paradoxically true. So, C is false.