## Thursday, May 5, 2011

### Indicative and material conditionals

I will use "pq" for the indicative conditional "if p, then q". I will use "pq" for the material conditional "(not p) or q". I will say that "indicatives are material" providing that pq and pq are logically equivalent for all p and q, where a and b are logically equivalent if and only if it is necessary that (a if and only if b). I will say that p entails q provided that it is necessary that pq.

Almost no philosopher thinks indicatives are material. There are very plausible counterexamples. For instance, suppose it is lightly raining in Seattle and Seattle is not having a drought. Let p be "Seattle is having heavy rain" and let q be "Seattle is having a drought". Then pq, since p is false. But it seems quite wrong to say that if Seattle is having heavy rain, then Seattle is having a drought, so pq doesn't seem to be true.

I am going to offer some arguments that indicatives are material. Say that → is non-hyperintensional provided pq and p*→q* are logically equivalent whenever p and p* are logically equivalent and q and q* are logically equivalent. Consider the following two theses:

1. For any possible world w: (p at w) → (q at w) if and only if (pq at w).
2. For any predicates F and G, from "Every F is a G" (where "x is an F" is more euphonious way of saying that x satisfies F) together with the assumption that c exists, it logically follows that if c is an F, then c is a G.
I will argue in S5, and using some fairly uncontroversial further premises:
1. If (1) is true and → is non-hyperintensional, then indicatives are material.
2. If (2) is true and → is non-hyperintensional, then indicatives are material.
Moreover, I will try to make plausible:
1. If (2) is true, then one has to assign the same truth value as the material conditional does to a number of paradoxical-sounding examples of indicative conditional sentences that are relevantly just like the standard alleged counterexamples to the thesis that all indicatives are material.
I think (1) and (2) are quite plausible. I don't know, however, how plausible it is that → is non-hyperintensional. My argument for (3) is very similar to more general arguments in Williamson. However, given (5), even if we don't assume that → is hyperintensional, there seems to be little advantage to denying the elegant and simple view that indicatives are material.

Argument for (5): Take my heavy rain and drought in Seattle case. Suppose that as it happens, there is no place where there presently is heavy rain. Let Fx say that x is having heavy rain. Let Gx say that x is having drought. Then all Fs are Gs. (If you think, with Aristotle, that "All Fs are Gs" requires there to be an F, then add the premise that on Venus somewhere right now there is a drought but a very, very brief heavy rain is currently occurring. I will leave out such modifications in the future.) Then by (2), we have to say that if F(Seattle), then G(Seattle):

1. If Seattle is having heavy rain, then Seattle is having drought.
And that is pretty much a standard alleged counterexample to the view that indicatives are material, of the false-antecedent sort. The case divides into two: we might suppose that Seattle is having neither drought nor heavy rain, in which case (6) is false-antecedent, false-consequent, or we might suppose that Seattle is having drought and (unsurprisingly) no heavy rain, in which case we have false-antecedent, true-consequent.

We can also use (2) to manufacture a true-antecedent, true-consequent case. Suppose that it is raining in both Seattle and the Sahara. Then the following is a standard alleged counterexample of the true-antecedent, true-consequent sort:

1. If it's raining in Seattle, then it's raining in the Sahara.
Let Fx say that x is a planet on which it is raining in Seattle, and let Gx say that x is a planet on which it is raining in the Sahara. Then every F is a G, since the only F is earth. By (2):
1. If earth is a planet on which it is raining in Seattle, then earth is a planet on which it is raining in the Sahara.
And that sounds about as paradoxical as (7). That completes my argument for (5).

Argument for (3): First we need a special case:

1. If p and q are non-contingent and → is non-hyperintensional, then pq is logically equivalent to pq.
To argue for (9), consider the following four sentences:
1. 2+2=4→2+3=5. (necessary, necessary)
2. 2+2=5→2+3=6. (impossible, impossible)
3. 2+2=5→ (2+2=5 or 1+1=2 or both). (impossible, necessary)
4. 2+2=4→2+2=5. (necessary, impossible)
If p and q are non-contingent, then they are respectively logically equivalent to the antecedent and consequent of exactly one of (10)-(13). By non-hyperintensionality of →, it follows pq must have the same truth value as the conditional in that line. But the truth values of (10)-(13) are just as the material conditional says they are: thus, clearly, (10)-(12) are (necessarily) true and (13) is (necessarily) false. So, pq must have the same truth value as pq, assuming p and q are non-contingent.

The argument for (3) is now easy. Observe that (p at w) and (q at w) are non-contingent, even if p and q are contingent. So,

1. (p at w) → (q at w) is logically equivalent to (p at w) ⊃ (q at w).
But, plainly:
1. (pq at w) is logically equivalent to (p at w) ⊃ (q at w).
From (1), (14) and (15) we conclude that:
1. (pq at w) is logically equivalent to (pq at w)
and hence indicatives are material.

Argument for (4): The most intuitive form of the argument is to assume theism, and let Fx say that x is an omniscient being that knows that p, and let Gx say that x knows that q. Then as long as pq, it will be the case that every F is a G (just think about the four possible truth-value combinations). Hence:

1. If God is an omniscient being that knows that p, then God knows that q.
Since God's existence and omniscience are necessary, the antecedent and consequent are logically equivalent to p and q, respectively, and so we get pq by non-hyperintensionality.

If we don't want to suppose there is a God, let's suppose that numbers and sets exist necessarily. Let P be the singleton set whose only member is p. Let Q be the singleton set whose only members is q. Then, let Fx say that x is greater than zero and x equals the number of truths in P. Let Gx say that x is greater than zero and x equals the number of truths in Q. Then, if pq, it is easy to see that all Fs are Gs, so:

1. If one is greater than zero and one equals the number of truths in P, then one is greater than zero and one equals the number of truths in Q.
But the antecedent and consequent are logically equivalent to p and q respectively, so by non-hyperintensionality we get pq, once again.

#### 1 comment:

Heath White said...

It’s been a while since I looked at this literature, but I’m not sure your sociological claim is true. You may have more company than you think.

On the argument for (5):

The most serious competitor view to the material indicatives view, I think, is that “If p then q” expresses the view that P(q|p) is high. Then (2) looks like this: Given that Every F is a G, and that c exists, it follows that P(c is a G|c is an F) is high.

In one sense yes, in another sense no. In one sense, the probability is 1. But one might want to say that the “givens” have to be incorporated into the probability measure: so that what follows is P(c is a G | c is an F & Every F is a G & c exists) is high; but it does not follow that P(c is a G | c is an F) is high absolutely. Or maybe one might want to resist the idea of “the” P() function, making it context-relative somehow, in which case the competing view of conditionals will not assign them a single meaning.