## Wednesday, March 19, 2008

### Indicative conditionals

On the material conditional interpretation, the propositional content of the indicative conditional "If p, then q" is pq, i.e., (not-p or q).

I claim that this is basically the right interpretation if "If p, then q" expresses a proposition whose truth-value is mind-independent (except for any mind-dependence in p and q themselves). You can take this as evidence that the material conditional interpretation is right—that is how I take it—or that English indicative conditionals do not express a mind-independent proposition.

The argument is simple. Suppose that p and q concern non-mental matters, and suppose that w is a world pq holds, i.e., p is false or q is true or both. Then there is a world w* which is very much like w, except that it contains two persons, A and B, conversing about p and q, neither of whom has any false or misleading or unjustified beliefs, and neither of whom has any beliefs giving significant evidence for any of the propositions p, q, not-p and not-q. We could then imagine A learning that either p is false or q is true or both, and that then the conversation turns to the subject of p and q. I claim that it would then be appropriate for A to say: "Well, I don't have any idea which if any of p and q is true, but I now know that if p holds, so does q." This seems quite right. Moreover, in saying this, A would not be saying anything false. Therefore, if "If p, then q" expresses a proposition, it expresses a true proposition in w*. But if the proposition it expresses is mind independent, it is also true in w, since the two worlds differ only in respect of mind-dependent stuff.

Hence, pq entails that if p, then q. The converse is easy. If pq is false, then p is true and q is false, and it is clear that then if p, then q isn't true. Therefore, necessarily, pq holds iff if p, then q does. Hence, the material conditional gets the truth conditions for the indicative "if... then..." right.

Could it be that there is still a difference in meaning? The only way I could see that would be if "If p, then q" said something additional, something entailed by pq, but nonetheless added on to it. But I just cannot see what that could be, unless it be something mind dependent.

But perhaps there is a difference here like that between "p or q" and "q or p"? Maybe there really is a difference in the proposition expressed by these claims, even though neither adds anything to the other. If there really is a difference in the propositions expressed by "p or q" and "q or p", then I guess there might be a difference between those expressed by "pq" and if p, then q. But if so, that difference is not very significant, it seems. Basically, the two say the same thing. Of course, even if there is no difference in proposition, there may be pragmatic differences.

What about standard counterexamples to the material conditional interpretation? For instance, could I say about a batch of cookies that I know to be poisoned
(*) "If George eats these cookies, he won't feel sick"
simply because I know that George won't eat them? Well, I think such counterexamples at most challenge the claim that the indicative conditional expresses a proposition, not the claim that if it expresses a proposition, the proposition it expresses either is or is basically the same as a material condition. Suppose that I don't know that the cookies were poisoned, but Patricia tells me: "An omniscient being either told me that George won't eat these cookies, or that he won't feel sick, but I can't remember which." It seems perfectly appropriate for me to utter (*), then. Suppose I later learn that the cookies are poisoned and that George won't eat them. Do I have any reason to say that I was mistaken when I uttered (*)? Surely not. I can say that what I said was misleading, but not that it was false. Whether (*) is appropriate to say depends on mind-dependent stuff. But if (*) expresses a proposition, then that proposition is mind-independent. Consequently, the intuitions about the appropriateness of saying (*) should not be taken as evidence about what propositional content (*) has if it has any.

Brandon said...

I claim that it would then be appropriate for A to say: "Well, I don't have any idea which if any of p and q is true, but I now know that if p holds, so does q." This seems quite right. Moreover, in saying this, A would not be saying anything false.

This seems to me to beg the question; one could just as easily claim that it is inappropriate and that A is saying something false, because he doesn't, in fact, know that "if p holds, so does q," unless he has learned something about an objective connection between p and q. But the material conditional interpretation doesn't involve any such connections, and doesn't ever require them. (That's why people take the material conditional paradoxes to be paradoxes.) Consider: if the moon is made of green cheese, the sea is salt-water. This is true on the material conditional interpretation. But it is false on any interpretation in which the antecedent is a condition for the consequent; even if the moon were made of cheese, we have no particular reason to regard this as relevant to the salt of the sea.

Further, note that the material conditional interpretation, strictly considered, puts a fairly significant chasm between the conditional and inference, despite the fact that most people think there is a close relation between the two. This is because on the material conditional interpretation, indicative conditionals do not represent anything as being a cause or sign of anything else, but simply the claim that it's false that p and not-q are both true.

Alexander R Pruss said...

My intuitions disagree here. Suppose I don't know anything about seas, salt, cheese or the moon, but I am told that either the sea is salt or the moon is not made of green cheese. I then think about this. It seems perfectly appropriate for me to say: "Oh, so if the moon is made of green cheese, then the sea is not salt." And if I learned additionally that there is no nomic or other ontological connection between the constitution of the moon and the constitution of the sea, I don't think I would be tempted to withdraw my assertion.

Think of what people do when they do various puzzles. Take one of those puzzles that used to be on the old GRE analytic: There are three people, Alf, Betty and Gammon, and among them they have the professions of mathematician, neurologist and pediatrician. No one has more than one profession. Alf is not a neurologist. And so on. The question, is of course, to match the people with their professions.

It seems to me that when we hear data like this, we (or maybe this is merely autobiographical--maybe you don't do this) immediately organize things into conditionals: "Oh, so if Alf is not a neurologist, then either Betty or Gammon is." Moreover, when we organize things into conditionals in this way, we don't care whether some of the facts are "conditions" for the others. When I say "If Alf is not a neurologist, then either Betty or Gammon is", I am not assuming a background on which Betty or Gammon is forced to be a neurologist by Alf's choosing not to be one, or anything like. In fact, we probably make no causal hypotheses at all. But my assertion "If Alf is not a neurologist, then either Betty or Gammon is" is one I inferred precisely from the fact that either Alf is a neurologist or Betty or Gammon is--in other words, I inferred it precisely from the material conditional "Alf is not a neurologist → Betty or Gammon is a neurologist".

It seems hard to imagine responding: "Wait, you're not justified in inferring that if Alf is not a neurologist, then Betty or Gammon is one, because you don't have any data about there being a connection, about some objective signhood or conditionhood being had by Alf's not being a neurologist."

Now maybe I am idiosyncratic in this respect, and that might be explained by my mathematical education which has taught me to treat "If ... then ..." as a material conditional (in a math paper, you can pretty much count on that being the meaning). But I think not. I suspect almost everybody who is any good at eliminative reasoning of the sort that these puzzles involve will start to organize the data into conditionals.

We can easily imagine this in forensic contexts. "We have presented conclusive evidence that George did not kill rob the store. The evidence also shows that George or Betty robbed the store. So if George didn't do it, Betty did." The last sentence seems perfectly appropriate, whether or not there is some mind-independent signhood or condition relation between George's not not robbing and Betty's robbing. It seems clear that to justify the conditional statement all you need to do is to justify the disjunctive claim that George or Betty did the robbery. This seems a staple of how we verbalize our reasoning.

Alexander R Pruss said...

I am afraid I don't see the point about inference at all. If I know that p or q is true, and I know that p is not true, then I can infer q. Likewise, if I know that not-p or q is true, and I know that p is true, then I can infer q. That is all that is going on in inferences from material conditionals.

Alexander R Pruss said...

Here's a context which yields something like your moon and sea conditional, if you grant me the claim that one can substitute logically equivalent statements in indicative conditionals.

Consider the following dialogue:

A: Frank told me he has two propositions, p1 and p2, of which at least one is true. He didn't tell me what these propositions are.

B: Ah, so if p1 isn't true, p2 is.

It seems that B's inference is precisely right. But now suppose that p1 is the proposition that the moon is not made of green cheese and p2 is the proposition that the sea is salty. That doesn't seem to undercut B's statement. Assuming we can intersubstitute logical equivalents, and assuming that names of propositions refer rigidly, it follows that if the moon is made of green cheese, the sea is salty.

Brandon said...

Yes, people do this because they treat indicative conditionals as statements of possible inference. This is a pretty common view. But on the material conditional interpretation, indicative conditionals are not statements of possible inference; they are statements of correlated truth values. We can, of course, infer things from them, but that's a different story. Nobody thinks "Either the sea is salt or the moon is not made of green cheese" indicates an inference; it's just a possible premise from which we may infer.

Think of "What the Tortoise Said to Achilles." The Tortoise manages to trap Achilles because neither of the two distinguishes between the two functions for which one might use conditional language: the longstanding use of the conditional to state paths of inference and the (at the time) very recent material conditional interpretation.

Neither your intuitions nor mine can be trusted in this matter because we've had our heads drilled with material conditionals for years and years. But if your intuitions were the general rule, rather than the exception, the nineteenth century Boolean logicians wouldn't have been so puzzled by some of the implications of this interpretation, we wouldn't have had long discussions in philosophy of logic literature about the paradoxes of material implication, and it wouldn't be so hard to get ordinary undergraduates to treat indicative conditionals as material conditionals (or truth functionally, for that matter).

B, it seems to me, is not doing anything but saying that p2 can be concluded from p1. No one I know of interprets "P2 can be concluded from P1" as a material conditional.

Brandon said...

Here is a further argument. What you can modus ponens, you can modus tollens. So:

(1) If it is raining, the sidewalk is wet.

(2) But the sidewalk is not wet.
(Reason: I put a tarp over it.)

Therefore:

(3) It is not raining.

(1) is entirely the sort of conditional you keep pointing to. But if it were interpreted as a material conditional, rather than as a statement of a possible defeasible inference, it would not be possible to reject (3) given (2). But we do it all the time, which suggests that at least much of the time we are not treating indicative conditionals as material conditionals.

Alexander R Pruss said...

Remember, though, that I'm not arguing for the material conditional interpretation of indicatives. Rather, I am arguing that either the material conditional interpretation holds or else indicatives do not express a mind-independent proposition.

The data about how people reason suggests strongly:

1. Knowing that p or q is sufficient to make it appropriate for one to say "If not-p, then q", absent any further evidence.

Moreover, if (1) is true, then I think the following is plausible:

2. If "If not-p, then q" expresses a mind-independent proposition, then it expresses a proposition entailed by (p or q).

Conversely, modus ponens requires:

3. If "If not-p, then q" expresses a mind-independent proposition, then it expresses a proposition incompatible with (not p and not q).

From (2) and (3) it follows;

4. If "not-p, then q" expresses a mind-independent proposition, then it expresses a proposition logically equivalent to p → q.

I am having a bit of a hard time following your response. Are you denying (1), or are you denying that (1) makes (2) highly plausible?

A point worth making with respect to 19th century logicians, is that language shifts. It may well be that present-day usage of "if ... then ..." is affected by mathematics, logic and computer science teachers and practitioners who have absorbed material conditional interpretations, by discourse around the solution of puzzles, by forensic usages coming into ordinary language via detective and courtroom fiction, etc. Thus it is quite possible that the material conditional interpretation is correct with respect to a large amount of contemporary usage, but only correct with respect to a small amount of 19th century usage.

Alexander R Pruss said...

4. If "not-p, then q" expresses a mind-independent proposition, then it expresses a proposition logically equivalent to not-p → q.

Alexander R Pruss said...

Note that if you read something as a statement of a possible defeasible inference, then you're not reading it as an assertion of a mind-independent proposition. And so maybe we have no disagreement.

I have no problem with the idea that sometimes we do something other than assert a proposition when we affirm an indicative conditional. There may well be something to conditional probability interpretations, for instance, in many cases.

Brandon said...

I don't think the option about non-mind-independent proposition makes a difference; for one thing, there is nothing incoherent about taking possible inference as mind-independent, just as there is nothing incoherent about taking truths (and falsehoods) about possible creatures as independent of actual creatures. (Since I think propositions are mind-dependent, I think this is false; but it's not obviously false, and as long as it is not ruled out there is a way of jumping through the horns of your dilemma.) And it is clear that if conditionals have a relation to inferences it is to possible inferences, not actual ones.

On language shift, yes, it is quite possible but it is also implausible; contemporary logicians working in strict implication and relevant implication still point to the paradoxes of material implication. Your intuitions are at least not shared, and haven't been shared by people who have tried to work out forms of strict implication or relevant implication. At the very least, it is a common intuition that the material conditional interpretation, fails to identify some key feature of at least most indicative conditionals.

On your argument (1)-(4), I am arguing (as I said to begin with) that it begs the question. (1) is only true and relevant on the condition that "If not-p, then q" is read as a material conditional and "p or q" is read as a truth-functional disjunction. If the 'or' is not truth-functional, it is false, because it is not sufficient for the material conditional. If the conditional is not a material conditional it is not relevant. Since we are talking about indicative conditionals, and trying to decide whether they are material conditionals, we cannot assume that that it is a material conditional. (What is the 'absent further evidence' supposed to be doing? Perhaps the 'absent further evidence is supposed to be a qualification to deal with cases of defeasibility of modus ponens. But defeasibility of modus ponens is incompatible with the material conditional interpretation, for the reason I noted previously. If the conditional in (1) is a material conditional, further evidence is irrelevant.)

(2) doesn't seem to me to be plausible for indicative conditionals, whether they express mind-independent propositions or not. Again, this is the defeasibility problem: if (2) were true, there would be no defeasible natural language cousins of the rigorous modus ponens we all know and love. But this is obviously false; much reasoning with conditionals is defeasible, and thus there is no reason to suppose that the natural language conditionals actually used are entailed (rather than, for instance, suggested, or defeasibly implied) by their corresponding natural language disjunctions, and certainly not that the natural language condtiionals are entailed by the corresponding truth-functional disjunction. Indeed, I'm not sure whether any indicative conditionals that are not stipulated to be material conditionals meet the condition suggested by (2).

Alexander R Pruss said...

Brandon:

I guess I'm confused by what you mean by "possible inference". It seems to me that what inferences are possible and what are not depends on your background knowledge, and hence is mind-dependent.

Alexander R Pruss said...

The "absent further evidence" was meant to strengthen (1)--but now I realize that (1) was ambiguously phrased. What I meant was that no further evidence beyond knowing "p or q" is needed to assert "If p, then q".

Here's another argument. Suppose that one can intersubstitute logically equivalent claims within "If ..., then ...". Suppose I have two objects, x and y, and a property P. Suppose the only thing I know about x, y and P is that P is a property and at least one of x and y has P. It seems to me that:

1*. I am entirely within my epistemic rights at this point, with no further information about x and y, and with complete certainty, to say: "If x doesn't have P, then y does."

If this is right, then the following seems right:
2*. If "If... then..." expresses a mind-independent proposition, and if x, y and P are such that at least one of x and y has P, then if x doesn't have P, y has P.

To get the claim about material conditionals that I want, let x and y be propositions, and let P be the property of being true. :-)

I expect you will say that (1*) begs the question. Maybe this is a limitation on my part, but I have a hard time imagining a smart speaker of English not being willing to say under the circumstances that if x doesn't have P, then y has P.

On the other hand, it may be that I am laying too much stress on forensic kinds of contexts as well as contexts of people talking about their reasoning in doing logic puzzles. I am fairly sure that there are many problem-solving contexts where material conditional interpretations are pretty much exactly right. It just may be that such contexts are more prominent in my life than in the lives other people. :-)

Let me offer a few such cases and see what you think.

"I know I have some Dickens novels somewhere, and all my books are either in the bedroom or office. So if there no Dickens novels in my bedroom, there will be some in my office."

"At least one of them was involved. If she wasn't in it, he was."

"I've narrowed down the bug to be either in the init or the cleanup. So if StartApplication() is bug-free, there is a bug in StopApplication()."

I do suspect these contexts are different from the ones where there is defeasibility. If I find that StopApplication() is bug free, I must either conclude that StartApplication() has a bug or that I was wrong about having narrowed down the bug.

If this is right, then we're getting something mildly interesting. There are ordinary contexts where the indicatives really are material conditionals. And there are ordinary contexts where they're not. Moreover, we may get biases based on which contexts figure more in one's life. I would be interested in seeing a more precise characterization here.

Brandon said...

what inferences are possible and what are not depends on your background knowledge, and hence is mind-dependent

It's true that what inferences are possible to me, i.e., what inferences I can make at a given point in time, depends on the information I have. But that doesn't tell me what inferences are possible or not, because there are inferences I can't make (not having the background knowledge) that are possible. And if there were no minds, someone could argue, it is still the case that there are possible inferences because there are possible minds, and thus things that could make those inferences if they existed. (Of course other things would have to be assumed given what we're interested in, e.g., we are restricting the possible inferences to rational inferences, and so forth.)

As far as I can see, 1* and 2* don't give you the material conditional interpretation (MCI, because I'm tired of writing it out) when x and y are propositions and P is the property of being true. What it does give you is a family of interpretations of conditionals of which MCI is only one member. To get the material conditional directly you have to assume that MCI is the only acceptable interpretation that allows the inference. For instance, here is no reason why you couldn't have some form of relevance interpretation of indicative conditionals and allow the inferences you are using as examples (because such an interpretation accepts that MCI identifies a necessary condition, namely truth-functionality, but rejects MCI because indicative conditionals are not material conditionals, i.e., not merely truth-functional).

Moreover, I think that "at least one of x and y is true" carries more baggage than it looks like it would carry. Because indicative conditionals are regularly used in normal conversation where the falsehood of the consequent and the truth of the antecedent are jointly possible, but the conditional is taken as true because that possibility is under the circumstances extremely improbable. But if MCI were true this would be impossible. So, someone might say that if it's raining, the sidewalk will be wet; if you point out that it is raining and the sidewalk is not wet, because someone put a tarp down, that would usually be dismissed as irrelevant. Ceteris paribus conditionals cannot be material conditionals because they can be true under conditions material conditionals can't. But many and arguably most indicative conditionals are ceteris paribus conditionals.

Alexander R Pruss said...

Interestingly, Sextus Empiricus attributes what seems to be a material conditional understanding of conditionals to the Stoics. So there is a history here...