Monthly Archives: December 2009

Motivating material conditionals

Soul physics has a post that raises a vexed issue: how to say something to motivate the truth-table account of the material conditional, for people first encountering it.

They give a version of one popular strategy: argue by elimination for the familiar material truth table. The broad outline they suggest (which I think is a nice way to divide matters up) goes like this.

(1) Argue that “if A, B” is true when both are true, and false if A is true and B is false. This leaves two remaining cases to be consider—the cases where A is false.

(2) Argue that none of the three remaining rivals to the material conditional truth table works.

I won’t say much about (1), since the issues that arise aren’t that different from what you anyway have to deal with for motivating truth tables for disjunction, say.

(2) is the problem case. The way Soul Physics suggests presenting this is as following from two minimal observations about the material conditional (i) it isn’t trivial (i.e. it doesn’t just have the same truth values as one of the component sentences) and it’s not symmetric—“if A then B” and “if B then A” can come apart.

In fact, all of the four options that remain at this stage can be informatively described. There’s a truth-function equivalent to A (this is the trivial one); the conjunction of A&B; the biconditional between A and B (these are both symmetric); and finally the material conditional itself.

But there’s something structurally odd about these sort of motivations. We argue by elimination of three options, leaving the material conditional account the winner. But the danger is, of course, that we find something that looks equally as bad or worse with the remaining option, leaving us back where we started with no truth table better motivated than the others.

And the trouble, notoriously, is that this is fairly likely to happen, the moment people get wind of paradoxes of material implication. It’s pretty hard to explain why we put so much weight on symmetry, while (to our students) seeming to ignore the fact that the account says silly things like “If I’m in the US, I’m in the UK” is true.

One thing that’s missing is a justification for the whole truth-table approach—if there’s something wrong with every option, shouldn’t we be questioning our starting points? And of course, if someone raises these sort of questions, we’re a little stuck, since many of us think that the truth table account really is misguided as a way to treat the English indicative. But intro logic is perhaps not the place to get into that too much!

So I’m a bit stuck at this point—at least in intro logic. Of course, you can emphasize the badness of the alternatives, and just try to avoid getting into the paradoxes of material implication—but that seems like smoke and mirrors to me, and I’m not very good at carrying it off. So if I haven’t got more time to go into the details I’m back to saying things like: it’s not that there’s a perfect candidate, but it happens that this works better than others—trust me—so let’s go with it. When I was taught this stuff, I was told about Grice at this point, and I remember that pretty much washing over my head. And its a bit odd to defend widespread practice of using the material conditional by pointing to one possible defence of it as an interpretation of the English indicative that most of us thing is wrong anyway. I wish I had a more principled fallback.

When I’ve got more time—and once the students are more familiar with basic logical reasoning and so on, I take a different approach, one that seems to me far more satisfactory. The general strategy, that replaces (2), at least, of the above, is to argue directly for the equivalence of the conditional “if A, B” with the corresponding disjunction ~AvB. And if you want to give a truth table for the former,  you just read off the latter.

Now, there are various ways of doing this—say by pointing to inferences that “sound good”, like the one from “A or B” to “if not A, then B”. The trouble is that we’re in a similar situation to that earlier—there are inferences that sound bad just nearby. A salient one is the contrapositive: “It’s not true that if A, then B”  doesn’t sound like it implies “A and ~B”. So there’s a bit of a stand-off between or-to-if and not-if-to-and-not.

My favourite starting point is therefore with inferences that don’t just sound good, but for which you see an obvious rationale—and here the obvious candidates are the classic “in” and “out” rules for the conditional: modus ponens and conditional proof. You can really see how the conditional is functioning if it obeys those rules—allowing you to capture good reasoning from assumptions, store it, and then release it when needed. It’s not just reasonable—it’s the sort of thing we’d want to invent if we didn’t have it!

Given these, there’s a straightforward little argument by conditional proof (using disjunctive syllogism, which is easy enough to read off the truth table for “or”) for the controversial direction of equivalence between the English conditional “if A, B” and ~AvB. Our premise is ~AvB. To show the conditional follows, we use conditional proof. Assume A. By disjunctive syllogism, B. So by conditional proof, if A then B.

If you’ve already motivated the top two lines  of the truth table for “if”, then this is enough to fill out the rest of the truth table—that ~AvB entails “if A then B” tells you how the bottom two lines should be filled out. Or you could argue (using modus ponens and reasoning by cases) for the converse entailment, getting the equivalence, at which point you really can read off the truth table.

An alternative is to start from scratch motivating the truth table. We’ve argued that ~AvB entails “if A then B”. This forces the latter to be true whenever the former is.  Hence the three “T” lines of the material conditional truth table—which are the controversial bits. In order that modus ponens hold, we can’t have the conditional true when the antecedent is true and the consequent false, so we can see that the remaining entry in the truth table must be “F”. So between them, conditional proof (via the above argument) and modus ponens (directly) fix each line of the material truth table.

Now I suspect that—for people who’ve already got the idea of a logical argument, assumptions, conclusions and so on—this sort of idea will seem pretty accessible. And the idea that conditionals are something to do with reasoning under suppositions is very easy to sell.

Most of all though, what I like about this way of presenting things is that there’s something deeply *right* about it. It really does seem to me that the reason for bothering with a material conditional at all is its “inference ticket” behaviour, as expressed via conditional proof and modus ponens.  So there’s something about this way of putting things that gets to the heart of things (to my mind).

But, further, this way of looking at things provides a nice comparison and contrast with other theories of the English indicative, since you can view famous options as essentially giving different ways of cashing out the relationship between conditionals and reasoning under a supposition. If we don’t like the conditional-proof idea about how they are related, an obvious next thing to reach for is the Ramsey test—which in a probabilistic version gets you ultimately into the Adams tradition. Stalnakerian treatments of conditionals can be given a similar gloss. Presented this way, I feel that the philosophical issues and the informal motivations are in sync.

I’d really like to hear about other strategies/ways of presenting this—-particular ideas for how to get it across at “first contact”.