# 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”.

### 14 responses to “Motivating material conditionals”

1. Daniel Elstein

A supplement to your way, which I like of course, is to directly prove the things the students find most bizarre: that ~P entails P -> Q, and that Q entails P -> Q. The problem is that although those proofs revolve around CP, they involve other sophisticated ideas too (arguably trickier ones than disjunctive syllogism). I find that ultimately more satisfying than the or-to-if argument, because it tackles the craziness head on. Maybe do things this way if you can get them good at natural deduction? I suppose the tricky case is what to do if we aren’t going to go the whole hog on natural deduction. It does seem feasible to explain both CP and disjunctive syllogism in an informal way, and maybe we should be teaching the inference ticket idea (and hence CP and MP) even if we don’t do natural deduction. Hmmm. Ok, I’ve convinced myself that you’re right.

2. The approach I take in my book and in my course, is two-fold. I do use the elimination approach given above, while making the assumption that we’re looking for a truth functional connective explicit. That is a part of the explanation.

The second argument is an intuitive explanation as to why one can validly infer from “if p then q” to “~(p & ~q)” and the reverse. This, of course, uses conditional proof for the natural language “if … then …” and modus ponens for the material conditional.

Then, I round off with a discussion of problem cases (like “if I’m dead, I’m alive”, etc.), explaining why conditional proof in cases like that are exactly what one wouldn’t want if we wanted the conditional to be counterfactually robust, and this is a part of the motivation for (a fairly simple treatment of s5) modal logic.

The point is that the naive “add the antecedent to your assumptions and then try to derive the consequent” allows for lots of room to wiggle: what if the antecedent you wish to add is inconsistent with your other assumptions?

In my experience, this approach works well enough to make clear the virtues of the material conditional (truth functionality, straightforward proof rules, it works in mathematical reasoning, etc.), while exploiting the intuitive worries that students have as springboards to all sorts of interesting discussions in more advanced logic (in my treatment, both Grice and truth/assertibility; and modal/counterfactual logics).

The cost, of course, is that this takes a while to get across. You get the students to live with it for a while (a week or so), focussing on what works, while then promising to get to how (or whether) it works once they get the details under their belt.

3. I use a strategy similar to the Soul Physics one, but focusing on inferences throughout. I point out that what we most want for the conditional is that it (a) make modus ponens and modus tollens valid, and not trivially valid (i.e., there are *possible* sound arguments of that form), and (b) make affirming the consequent and denying the antecedent invalid.

Line 2 comes from validity and non-triviality of MP; line 4 from validity and non-triviality of MT. Line 3 is needed for invalidity of affirming the consequent; line 1 for denying the antecedent. If we’re going to be truth-functional about things, each line is needed to secure some well-entrenched inferential role for the conditional.

4. Lee Walters

Jason suggests a nice a way of presenting an elimination strategy, but as an elimination strategy it faces the obvious objection, why assume truth functionality?

In any case, forgetting the top two lines, one could argue for alternative truth functional treatments by reasoning that there are cases of ~A&B and ~A&~B where the conditional is clearly false (?) and so if truth functional, the conditional is equivalent to conjunction.

The most convincing strategy to my mind, is the one that Robbie hits on last – use MP and CP. This could be supplemented by Edgington’s tetralemma: if you know B, you know If A, B. If you know A and B equivalent you know If A, B, if you know A&~B you know ~If A, B.

5. Lee: I wasn’t wanting to argue that the indicative of English was the material conditonal, because I don’t believe it, and don’t want my students to. (Well, they can if they want, but I don’t want them to on the basis that their logic tutor told them so. For my part, I don’t think the English indicative supports CP using unrestricted side premises; I’ll leave the tetralemma for another day.) Rather, I’ve already explained how we’ll be focusing on purely truth-functional connectives in the module, and present the question as what the best truth-functional approximation to the ordinary indicative is. Then I use the argument to argue for the particular truth-function we use rather than any other. I was reading the Soul Physics question as along those lines; apologies if I’ve construed it too narrowly.

6. Daniel Elstein

Jason, I don’t think that using CP as a justification is to assume that the indicative conditional is the material conditional. I take Robbie’s point to be that we can motivate CP independently of truth-functionality, not as something we want for *the* (indicative) conditional, but as something we want for *a* conditional when we’re doing logic. That is, we motivate CP via (an informal explanation of) the deduction theorem. The inference ticket behaviour is useful, even if it isn’t strictly the behaviour of the indicative conditional. (Because when we use the indicative conditional we aren’t reasoning purely deductively – there’s something non-monotonic going on and this shows up in the failure of the inference from Q to If P then Q).

We might stipulate truth-functionality, and that will be good enough for some students, or we might try to justify truth-functionality, which might satisfy more of them. Maybe the ones who still aren’t satisfied will be helped by the CP argument. And it does seem as though thinking that way helps to get at the fundamental difference between the material conditional and the indicative conditional. It sets us for the question: what’s the difference between CP and the Ramsey test? (And in what contexts would you want a conditional obeying one rather than the other?)

7. Rich Woodward

Hey guys. I’m teaching this stuff next term so I thought I’d outline how I’m approaching it. Dialectically, my plan is to set it up as a kind of paradox. On the one hand, it seems like no truth-table can be right — that’s what’s suggested by the paradoxes of material implication — but on the other hand it seems like some truth-table *is* right — that’s what we seem to get if we spot MPP and CP. It’s important to note that we don’t need to assume truth-functionality in order to get the second thing: MPP and CP jointly seem to give us the basis of a good argument for truth-functionality, *and* also a good argument for the conclusion that a specific truth-table is right. And I do think that setting things up as a paradox allows us to segway nicely into the Ramsey test, as Daniel points out.

8. Daniel (and Rich): you’re right, you can use the CP argument to plump for truth-functionality independently. I was running those together.

I very much like the idea of arguing for the material conditional using the CP+MP arguments. (Notice, incidentallty, that you also need disjunctive syllogism to get the equivalence.) But it looks too pedagogically hard: I’ve got a difficult enough time getting my students to understand CP (and DS, if I’m using a system where that’s an assumption-discharging rule) once they’re off and doing proofs, and they’re doing truth-tables before proofs. They’ll have had enough informal logic to recognize simple (non-assumption-involving) inferences as valid or invalid, but anything involving assumptions just gets them totally befuddled at that stage. (And it’s even worse if I’m not teaching a natural deduction system at all, going for tableaux or something instead.) I find it’s just a whole lot easier pedagogically to announce early on that we’re going to be using only truth-functional connectives, big up the merits of this (especially the benefits for the students, since they get to avoid the complexities of non-truth-functional connectives), and then convince them the horseshoe is the only truth function that has even got a shot of modeling the inferential behavior we care about for conditionals.

Incidentally, I don’t present the issue as “We’re assuming that ‘If, …, then’ is truth-functional, and…” I present it as “We’re creating a formal language, and in that language we’re going to have a truth-functional counterpart to “If, …, then”, which may, or may not, be equivalent to that expression in English but will model its core inferential behavior.” I’ve only been doing this for about three years, but I’ve never had a single student express any uneasiness about this approach.

9. Rich Woodward

@Jason. I guess it all depends how much time you’ve got and the context in which you’re teaching. I agree that running things via CP might be a bit awkward if we’re in the context of a basic logic class — we don’t want to rely on things that are just as unclear to them as the thing we’re trying to explain! And I also agree that in that context setting things up along the lines you sketch — we’re creating a formal language, etc — seems a totally good way to go. FWIW, the lectures I’m giving are to level 3 students who’ve done alot of logic before, so in that context I’ve got a bit more time and can rely on certain things being in their lockers (CP, DS, etc.). In *that* context, I do think the CP-style arguments for materialism look the best way to go, partially because it sets up the Ramsey test nicely (which one wouldn’t mention to first years).

10. Robbie

Hi guys, thanks for the helpful discussion! I have only used the CP/MP motivation at a 3rd year level, and I do think it’s tricky for intro logic. (When Jason and I cotaught intro logic, he handled the truth table lectures, and the students appeared to get it, so I can only say good things about his methods!)

Just on the presentation: if I was to try something like this with intro class, I wouldn’t write it as a natural deduction proof. I’d just give an informal schematic argument that exploited conditional proof without making a song and dance out of it (like the one on p40 of Greg’s intro book, which I just looked up!). That wouldn’t be what I’d do with the advanced class—I’d be wanting to bring out the connection with supposition. But it still seems like a pretty decent argument to give—perhaps as a supplement to an argument by elimination of the alternatives.

Just a note on how I was thinking of things: disjunctive syllogism is needed to argue from ~AvB + A to B. That’s the direction I was proposing to give them; you’d need reasoning by cases (+modus ponens) for the reverse, but it’s probably easiest to do that just by looking at the impact of modus ponens on the truth table directly. But I was thinking one needn’t explicitly call it disjunctive syllogism—it’s pretty intuitively compelling in its instances, no? And it is, essentially, what we’re using in arguing by eliminating alternatives.

(@Rich—-“arguments for materialism”. Heh.)

11. (Oops, sorry, I said DS earlier, I meant reasoning by cases — got my rules mixed up!)

Yeah, I was thinking in the first-year context the whole time. This year in the 3rd-year seminar, I did the CP argument, the one from Jackson’s “passage principle”, and Gibbard’s from exportation. The students seemed most taken with Jackson’s, for some reason (most of the essays were on that!); I find Gibbard’s the most difficult to find something wrong with, though.

12. Hi Jason,

If I recall correctly, one thing I liked about Gibbards was that the contraposition of the inference principle he uses (one of importantion/exportation) doesn’t look obviously bad, but the contraposition of or-to-if looks really ugly. That’s what makes me really suspicious about or-to-if on its own. Though of course, the CP argument is basically deriving or-to-if on other grounds, so you might think of it as a supplementation rather than rival.

13. will1987

I don’t get any of this but I’ve gotta start chatting with you guys. I’ve been thinking about all kinds of metaphysical and epistemological and ontological questions for years but like Jack Kerouac’s Dean Moriarty in On The Road I’m trying to get “in there with all the terms and lingo.” I think if I could get up on it I’d probably make a decent philosopher.

14. will1987

That is I get it only very generally and it’s all mainly a linguistic and referential problem at the moment