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 …