## Monday, February 2, 2009

### Material conditionals

Say that a proposition p is weakly earthly provided that for every pair of worlds w1 and w2 which exactly match one another in respect of all states of affairs localized within a thousand lightyears of earth and all entities and events capable of causally affecting states of affairs localized within a thousand lightyears of earth, p has the same truth value at w1 and at w2. It is very plausible that just about all propositions used in everyday speech are weakly earthly. A sentence is weakly earthly provided it expresses a weakly earthly proposition.

Now, suppose that "If p, then q" is weakly earthly. I shall argue that "If p, then q" has the same truth value as the material conditional. First, observe that if the material conditional is false, so is "If p, then q", since otherwise modus ponens wouldn't work.

For the converse, suppose the material conditional pq is true. Now imagine a possible world w* which is just like our world, but which also contains a one-way causally isolated island universe u, such that events in our universe can affect events in u but not conversely, and where u contains Frizzy, a being that knows the truth values of all weakly earthly propositions (maybe God has told them all to him), and that believes no contradictions. Moreover, Frizzy has the odd property that he always speaks sentences in pairs. First, he utters a claim with no regard for its truth. After that, if the first claim he had uttered turns out to be something he believes to be true, he utters a second claim that he believes to be true; otherwise, he utters another claim with no regard for its truth.

Now suppose Frizzy utters the pair of propositions a and b (in this order), and suppose a and b are propositions Frizzy knows the truth values of. Then, if a is true, so is b. And, hence, it is true that "if a, then b". But now as long as the material conditional pq is true, i.e., as long as we do not have both p and not-q, it is coherent with the above description of w* that Frizzy utters the pair p and q. So let us suppose that. Then, as noted above, it follows that it is true in w* that "if p, then q". But "if p, then q" is weakly earthly. Hence, if it is true in w*, it is true in the actual world.

What I have shown is that for any weakly earthly indicative conditional, the truth value of the conditional is equal to the truth value of the corresponding material conditional. Moreover, this argument can be run in any possible world (in some possible worlds, of course, the claim is close to trivial because there are no contingent weakly earthly claims). Therefore, necessarily, a weakly earthly indicative conditional holds iff the corresponding material conditional does. Now, assuming indicative conditionals have truth value it would, I think, be very unlikely that there would be something special in this way about weakly earthly indicatives. (The assumption is needed, because if indicatives don't have truth value, there are no weakly earthly indicatives.)

So, we have very good reason to think that either indicatives lack truth value or else indicatives are logically equivalent to material conditionals. I don't know which disjunct to choose.

Brandon said...

I don't quite see this. What you've shown is that there is at least one type of indicative conditional (namely, the sort that you use to describe the Frizzy rules, which are truth functional) are logically equivalent to the corresponding material conditional (or has no truth value) under certain conditions (namely, (a) that modus ponens is an indefeasible rule of inference for that particular kind of conditional; and (b) that the conditional describes a situation precisely analogous in structure to Frizzy's belief system); this is far and away removed from showing that all indicative conditionals are logically equivalent to their corresponding material conditionals (or lack truth values) simpliciter. And, indeed, I think it is a very complicated way of saying something that is trivially true; Frizzy's an elaborate way to guarantee that the indicative conditional in question simply describes the material conditional truth table. To put it in other terms: if we took indicative conditionals not to be material conditionals but instructions for constructing truth tables (which is one possible way to treat them in constructivist terms), all you would have shown is that indicative conditionals construct a material conditional truth table when combined with the rules governing Frizzy's belief system; but that doesn't mean that all indicative conditionals with truth values are logically equivalent to material conditionals. It would just mean that there are possible cases where they would be (which isn't very surprising).

Brandon said...

First, observe that if the material conditional is false, so is "If p, then q", since otherwise modus ponens wouldn't work.

Oh, and I also wanted to suggest that this is perhaps misleading. For modus ponens to work, there is only two conditions that have to obtain on the truth table, namely that Tp,Tq is true and Tp,Fq is false. But this is consistent with more than one truth table, only one of which belongs to the material conditional. Even if you meant implicitly to include modus tollens, that adds only one other line of the truth table, Fq yields Fp. But this is consistent with two different truth tables: one in which Fp,qT is false and one in which it is true. Only the latter is the material conditional. The former is the biconditional. (And the biconditional is not logically equivalent to the material conditional; so if there are indicative conditionals that are logically equivalent to biconditionals, there are indicative conditionals that are not logically equivalent to material conditionals.)

Alexander R Pruss said...

Brandon:

I am nto sure your first set of comments works given the structure of the argument. The structure is this: Take a weakly earthly indicative conditional. Move to a world w* that contains Frizzy. Show that in w*, the truth-conditions for the conditional match the material conditional. I think you may be agreeing with me here. Then move back to the actual world, and note that the truth-conditions had better be the same because of the weakly-earthly condition.

Brandon said...

Right, but this works for a general claim about indicative conditionals only if there really are Frizzy beliefs, in some Frizzy belief system, that correspond to every indicative conditional in every possible world. What Frizzy does is show that you can guarantee a truth condition match between indicative conditionals and material conditionals if there are systems that automatically correlates the two by guaranteeing that every indicative conditional has a corresponding Frizzy-like material conditional with the same truth conditions. (All this is assuming, of course, that modus ponens is indefeasible for all types of indicative conditionals.) But a constructivist about indicative conditionals would likely agree with this; Frizzy-like systems are just a way of constructing material conditional truth tables. But on a constructivist view like the one mentioned previously, not all indicative conditionals need have a complete truth table in every use (although they would perhaps have a truth value in every use), and those that don't (1) can't be logically equivalent to material conditionals, which would require a complete truth table; and (2) can be matched with any other Tp,Tq=T truth table by positing a different Frizzy system with different rules. I suppose you could be assuming that we shouldn't be constructivists about indicative conditionals; I think that would make the argument tighter, although it adds another disjunct to the disjunction. Or do you think there is some reason it would be impossible to have a Frizzy system isomorphic to a biconditional or a TFTF (e.g.) truth table?

Alexander R Pruss said...

Well, let's suppose Frizzy has the property that he utters sentences in pairs such that that he either believes both or he believes neither. In that case, the truth of the material conditional is not sufficient to ensure that Frizzy can utter the two sentences in order.