The material conditional and the logic textbooks (1)

What is the relationship between the ordinary language conditional and the material conditional which standard first-order logic uses as its counterpart, surrogate, or replacement? Let’s take it as agreed for present purposes that there is a distinction to be drawn between two kinds of conditional, traditionally “indicative” and “subjunctive” (we can argue the toss about the aptness of these labels for the two kinds, and argue further about where the boundary between the two kinds is to be drawn: but let’s set such worries aside). Then, by common consent, the material conditional is at best a surrogate for the first kind of conditional. The issue is how good a surrogate it is.

Once upon a time, versions of the following story were more or less enthusiastically endorsed by various writers of introductory logic textbooks:

Given $latex \neg(A \land \neg B)$ we can infer if A then B, and vice versa. Similarly, from$latex (\neg A \lor B)$ we can infer if A then B, and vice versa. So ordinary language indicative conditionals really are (in their core meaning) material conditionals. True, identifying ordinary if with $latex \supset$ leads to some odd-looking or downright false-looking results; but we can explain away these apparent problems with treating ordinary ifs as material conditionals by appealing to Gricean points about general principles of conversational exchange.

A classic example is Richard Jeffrey’s wonderful Formal Logic: Its Scope and Limits (2nd edition, 1981). Jeffrey is frank about the prima-facie problems in identifying the indicative conditional with the material conditional as leading to a number of “astonishing inferences” (giving some memorable examples). But in his §4.7, Jeffrey goes on to argue that “Grice’s implicature ploy seems to work, and the astonishing inferences seem explicable on the truth-functional reading of the conditionals in them.” This indeed is a Comforting Story — comforting for the writers of logic textbooks, I mean: the truth-functional logic they teach the students gets it right about the logic of the (indicative) conditional.

But most philosophers interested in conditionals have long since stopped believing the Comforting Story. Over twenty years ago, Dorothy Edgington wrote a 94 page State of the Art essay “On Conditionals” for Mind (1995) which has its own agenda and in the end pushes a particular line, but which takes it as by then a familiar thought that the Comforting Story is a non-starter. And over a dozen years ago, Jonathan Bennett wrote A Philosophical Guide to Conditionals (OUP, 2003) and can say of the Comforting Story “Some philosophers have [in the past] accepted this account of what the conditional means, but nearly everyone now rejects it” (p. 2).

Why the wholesale rejection? This sort of thought looms large. Here in the bag of lottery balls are 990 white balls, and 10 coloured balls with 9 blue ones and a single jackpot red ball. You dip your hand into the bag, mix the balls around, and pull one out (without yet showing me). Let P = you have pulled out a coloured ball, Q = you have pulled out a red ball. My confidence in not-P is very high (99% in fact!). So, being a rational chap, my confidence in the truth of not-P or Q is at least as high (99.1% in fact). And my confidence level in not-P or not-Q only slightly different (99.9%). On the other hand, my confidence in if P then Q is very low (just 10%), and very different from if P then not-Q (90%). But if if P then Q indeed is equivalent to not-P or Q, I’d be guilty of two radically different confidence levels in the same proposition — and, as a rational chap, I protest my innocence of this confusion! And if if P then not-Q indeed is equivalent to not-P or not-Q, then (in the given circumstances) my confidence levels in if P then Q and if P then not-Q should be almost the same — and I protest that it is rational to have, as I do, very different levels of confidence in them. As Edgington puts it

… we would be intellectually disabled without the ability to discriminate between believable and unbelievable conditionals whose antecedents we think are unlikely to be true. The truth-functional account [even with Gricean tweaks] deprives us of this ability: to judge A unlikely is to commit oneself to the probable truth of $latex A \supset B$.

There are other troubles with the Comforting Story: but that’s a major one to be going on with.

Of course, there is little agreement about what the Comforting Story should be replaced by (quite a few are tempted by the line pushed by Edgington, that the root mistake we have made about the conditionals is in supposing them to be aiming to be fact-stating at all — but tell that to the mathematicians!). But I’m not concerned now with what the right story is, but rather what to say in our logic texts about the material conditional if that’s agreed to be, in general, the wrong story about indicative conditionals. Given that faith in the Comforting Story waned among philosophers interested in conditionals at least a quarter of a century ago, and given that many elementary logic textbooks are written by philosophers, you might have expected that recent logic texts would have other stories (maybe less Comforting) to tell about what they are up to in using the material conditional. So what do we find (ignoring my own earlier efforts!)?

Some are cheerfully insouciant about the whole business. Jan von Plato, for example, in his intriguing Elements of Logical Reasoning (CUP 2013) doesn’t even mention the material conditional truth-function as such. Volker Halbach, in The Logic Manual (OUP, 2010/2015), after noting some problems, optimistically says “For most purposes, however, the arrow is considered to be close enough to the if …, then … of English, with the exception of counterfactuals.” Close enough for what? He doesn’t say. Not, if Edgington is right, close enough for use when we need to discriminate between believable and unbelievable conditionals, which you might suppose that logicians might want to do! Still, von Plato’s book is unrelentingly proof-theoretic in flavour, and Halbach’s is very short and brisk. So let’s now turn to rather more discursive books which do come closer to addressing our issue.

To be continued …

10 thoughts on “The material conditional and the logic textbooks (1)”

  1. Somewhere between argument and cheerleading is the material conditional’s usefulness in representing All A’s are B’s. And that ends up being a pretty major use of it. (And yes, All unicorns are green.) (My students had no difficulty grasping that my evasive ‘Everyone who came to my party had a good time’ is true…)

    1. Yes, absolutely — and I think this important point goes right back to early Quine. And oddly, I can’t recall it being highlighted at all in e.g. Edgington or Bennett. What’s interesting me here (just today!) is the disconnect between the wealth(?!?) of modern philosophical discussion of conditionals and the poor(!?!) offerings of the intro logic texts. And, of course, how the heck to do any better in my very constrained page budget in IFL2!

    2. Could a different conditional also make “all As are Bs” work? The peculiarities of the material conditional don’t seem essential to me there, but I could well be wrong.

  2. A professional rendition of an Ayckbourn comedy or a Shostakovitch quartet carries innumerably many nuances that are not explicit in the script or score, and may actually vary from it, but the written trace remains an essential basis for the live performance.

  3. I notice that you used principles of conversational exchange against Tarski and a “once-popular” view in the post on “or”, but now principles of conversational exchange have the opposite role: they’re part of a once-popular story that’s presumably is no longer thought to be right.

    1. BTW, and in line with my comment on “all As are Bs” above, one reservation I have about resorting to “principles of conversational exchange” to solve a problem with an interpretation of “or” or “if” is that such principles might also be able to rescue a different version of the connective, so we still don’t know that the version we’re defending in that way is the best one.

    2. I’m a fan of Grice’s work: but the question is about what can and what can’t be explained using it. Qn.1: are the implications of exclusiveness involved in many uses of disjunctions due to “or” having a different literal meaning in those cases from the inclusive cases, or are the implications in fact generated by context, background beliefs, and/or general principles of conversational exchange? Qn.2: can the apparent divergences between everyday indicative conditionals and mere material conditionals be explained by invoking general principles of conversational exchange? It is evidently quite consistent to answer the first question by “yes” and the second question by “no”: and I’d say more, i.e. that there are strong reasons for the split verdict.

      1. Is it your view that English doesn’t have an xor?

        I think it’s pretty clear that it does, and your “fake news” example in the “or” post even includes one. (We’re left with two claims; we think one’s false and one’s true, but we don’t know which. The “or” in that “P or Q” looks like it’s an xor, because we’re not saying that both might be true.)

        1. English has an xor: it is pronounced “[Either] … or … but not both”.

          The suggestion under consideration is this. (1) The meaning of (the first) “or” doesn’t change between “[Either] … or … but not both” and “[Either] … or … or both”. (2) The semantics each time of the shared “[Either] … or …” component is given by the standard (inclusive) truth table (a suggestive point: this makes the negation “Neither … nor …” as we’d want). (3) Conversational context, as in the example you mention, might adjoin an unspoken “but not both” to a plain “… or …” claim: but consistently with that, the literal content of what is actually said (as opposed to contextually implied) is inclusive.

          1. It looks like you’re avoiding saying that “or” can mean xor by saying that “or” can have an implicit “but not both”. That doesn’t look interestingly different to me. What’s the problem with just saying that “or” can mean xor and that, since “or” can also mean inclusive-or, something about the context tells us which one is meant? That’s how it normally works with ambiguous words, so why not with “or”?

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