Thursday, October 30, 2008

"If... then..." and material conditionals

I will argue that if the indicative "If p, then q" in English has mind-independent truth value (a somewhat vague phrase, admittedly), then this truth value is the same as that of (not-p or q) (i.e., the material conditional). The way I shall argue this is as follows. Assume that "If p, then q" has mind independent truth value. Now, I will show that (i) if (not-p or q) is false, then "If p, then q" is false, and (ii) if (not-p or q) is true, then "If p, then q" is true. Claim (i) is easy. For if (not-p or q) is false, then p is true and q is false, and it clearly cannot be the case that "If p, then q" (modus ponens would be violated).

I now argue for (ii). The easiest way to do this is to specialize to the case where p and q and their denials do not tell us anything about what beliefs and credences people have (the proposition that there are dogs satisfies this constraint; the proposition that nobody believes anything does not satisfy this constraint). If (ii) holds for propositions satisfying this constraint, it will hold in general, surely (assuming "If... then..." has mind-independent truth value). Suppose that you rationally assign a probability very close to 1/2 to p as well as to q, and neither believe nor disbelieve either of these, and rationally assign a probability very close to 1 to the claim that (not-p or q), and, moreover, you know this disjunction to be true. Given the constraint on p and q, it should be quite possible to have a set of evidence that makes one have these probability assignments, and having this set of evidence should not affect the truth values of p, q or the indicative "If p, then q".

You then reflect on the following valid argument:

  1. p (premise)
  2. not-p or q (premise)
  3. Therefore, q.
You want to summarize what you've learned from this argument. You know (2) to be true, and you assign very high probability to it. You don't know (1) or (3) to be true, and you in fact do not have a belief either way about either of these. It seems quite right to summarize your current position as: "If (1) holds, then (3) holds." After all, you've got a valid argument from (1) to (3) given an auxiliary premise, namely (2), which you know to be true. But once we agree that "If (1) holds, then (3) holds", surely we likewise have to agree that "If p, then q." Hence, if the disjunction (2) is known with probability close to 1, and neither p nor q is known or has high or low probability, then "If p, then q" is true. But if "If... then..." has mind-independent truth value, then the assumptions about knowledge and probability are irrelevant to its truth value, and hence we can simply conclude that if (2) holds, then "If p, then q."

The conclusion might be taken as a reductio of the claim that "If... then..." has mind independent truth value.


Brandon said...

(i) doesn't seem to me so easy, since it seems clear enough that English conditionals allow modus ponens to be defeasible. And, indeed, they do for the same reason your argument for (ii) trades on: English (like just about every natural language, I would imagine) allows you to suppress elements of the antecedent in summarizing reasoning. Modus ponens can be made defeasible simply by doing the same thing you do for (ii) in cases where the implicit element of the antecedent is less than ideal (i.e., where rationally assigning certainties or the knowledge involved is only sufficient for practical purposes). Material conditionals don't allow this slippage.

Brandon said...

I suppose in re-reading my previous comment that I need to clarify the non-ideal circumstances, since I made them sound mind-dependent. What I have in mind is this. Suppose an object A, with certain dispositions and propensities. I know that for the most part these dispositions and propensities are such that they result in either not-p or q. So I use an approach analogous to that which you use with regard to (ii) and conclude, "If p, then q". But since I'm only talking dispositions and propensities, the p-to-q branch of possible happenings is not the only branch possible; it could be, in this or that rare case, that something impedes it. This is a situation (1) for which we would be perfectly justified using an English conditional (indeed, I would argue that most of the time we use such an English conditional we are using it like this); (2) in which the conditional seems to have a mind-independent truth value; and (3) modus ponens is defeasible. Thus it is a situation in which (i) is false. Natural language conditionals are in general 'always or for the most part'; when 'always' obtains, they are like material conditionals, but when it's only 'for the most part', the two split, because material conditionals can't accommodate that.

Alexander R Pruss said...


Thanks for the criticism of (i). Could you give me a concrete example of a situation where (a) an indicative should be read as "for the most part" but is not explicitly so marked, and (b) that indicative is not something one would disclaim when one found out that that p and not-q? That would help me get clear on your objection.

Brandon said...


(W) If this substance is water, it boils at 100 degrees.

But it could very well be water, and false that it boils at 100 degrees; e.g., if we're not at standard pressure. But if you were to bring this up in response to W in ordinary English discussion, it would likely be dismissed as a quibble: that we aren't dealing with weird or unusual or nonstandard circumstances is an implicit part of the antecedent. And if we are, it doesn't count against W that p & not-q when those circumstances obtain.

To put it in other words: English conditionals allow things like implied defaults, which can affect truth values, that won't be recognized if we read them as straight material conditionals.

Alexander R Pruss said...

(W) isn't quite right as a counterexample to modus ponens, because one can take the stuff about standard conditions into the consequent: "If this is water, then it standardly boils at 100C."

What if the conditional is:
"If this is water and it is at 100C, then it is boiling."
Well, this works less well for your purposes. Let's tell a story. I am told that a black box contains a liquid at a certain temperature. I can't tell what is happening in the box. I say: "If this liquid is water and it is at 100C, then it is boiling." I then open the box and find that it contained water at 100C which was not boiling (maybe the pressure in the box was non-standard).

What would I then say about my earlier statement? I think I could say one of two things:
(1) I was wrong.
(2) I didn't mean it literally. I meant that if it is water and it is at 100C and it is in standard conditions, then it is boiling, or maybe that if it is water and it is at 100C, then it is probably boiling.

But I don't think I would baldly maintain that my earlier conditional, as stated, was correct.

I am, of course, perfectly fine with the idea that a conditional may have implicit conditions. I guess when I am talking of an "If p, then q" conditional, I assume the implicit conditions are included. If we don't assume inclusion of what is implicit, then just about no logical analysis of a piece of natural language will work. (Actually, on my weird view of language, there is no significant difference between the implicit and the explicit. But that's a different story.)

Brandon said...

(W) isn't quite right as a counterexample to modus ponens, because one can take the stuff about standard conditions into the consequent: "If this is water, then it standardly boils at 100C."

I think this comment may suggest that I did not convey the point I intended to convey. The point is not that any of this is a counterexample to modus ponens, any more than an enthymeme is a counterexample to syllogistic. Rather, the point is that natural language conditionals exhibit pragmatic features that simply aren't captured by material conditionals. Thus if we take an English conditional like W, we find that it allows modus ponens to be defeasible. This is not, strictly speaking, a counterexample to modus ponens, any more than enthymemes are counterexamples to term-logical validity: implicit antecedents can be added to save modus ponens, and it is entirely reasonable for practical purposes to take them as assumed. But because they are not explicit -- and, indeed, not usually in view, and therefore vague and indeterminate -- there is no perfect translation between English conditionals and their material conditional counterparts, because such a translation would require the assumption that all information relevant to the truth value is captured explicitly in the English conditional, or in some expanded form in which implicit information has been made explicit. But this can never be guaranteed except by stipulation.

On your black box example, it seems to me that your options are not exhaustive. If the conditional is better modeled as a default rule or as a ceteris paribus conditional (to take just two examples) than as a material conditional, then both (1) and (2) are false, without modifying any feature of the scenario. As far as I can see, (1) and (2) will only be exhaustive options for models of (literal) conditionals that make them isomorphic to material conditionals.