# The material conditional and the logic textbooks (3)

So turning to Nick Smith’s long and discursive book, what line does he take on the relationship between everyday conditionals and the material conditional?

Smith, as usual, sets aside subjunctive conditionals; the issue then is the relation beween indicative conditional and the truth-functional ‘$\to$‘ (in his preferred notation). He explains that the material conditional is the only possible conditional-like truth-function and that there are indeed plausible arguments for the equivalence of not-A or B and of not-(A and not-B) with if A then B. Smith then notes that, despite those arguments, equating ‘if’ with ‘$\to$‘ leads to some apparently very unhappy results. For example,  “if this book [i.e. his book] is about pop music, it refers to the work of the logician Frege” seems “quite wrong”, yet this comes out true if we treat “if” as the material conditional. What to do?

One might conclude that for an indicative conditional to be true, it is not enough simply for it not to be the case that both the antecedent is true and the consequent is false: there must also be some sort of connection between the two. If we pursue this line of thought, we shall be led to the view that the indicative conditional is not truth functional … (p. 113)

However, Smith says nothing at this point about what such a view might look like, but instead immediately says

The alternative is to defend the view that indicative conditionals have the same truth conditions as material conditionals, by treating the problematic examples in a way that should now be familiar: we explain why these conditionals seem wrong in a way that is compatible with their being true. …

And off we go with the Comforting Story, presented in some detail, to arrive at the conclusion that “the existence of such a connection [as seems to be involved in nice conditionals] is an implicature: it is not required for the truth of the conditional. … We now have an explanation of why the conditionals [like the pop music conditional] seem wrong (i.e., we can imagine no situation in which we should want to utter them) that is compatible with their being true.” And that’s where the main text of the relevant section ends.

So at this point, although the idea that indicative conditionals are (always? characteristically? at least often?) non-truth-functional has been very briefly mooted, it hasn’t been taken at all seriously, and has been set aside in favour of what seems to be an  unqualified Gricean defence of the material conditional as getting the truth-conditions of the indicative condtional right. And although Smith doesn’t explicitly say so, this presumably means it that standard truth-functional logic gets the logic of arguments involving the indicative conditional right too (hooray!).

However, there is a long endnote attached to Smith’s discussion. And what was at least conversationally implied(!!) in the main text to be a splendid Comforting Story (“We have an explanation …”) is now officially demoted to being after all “just the opening moves from a long, ongoing debate”, and he refers the student reader to Bennett’s 2003 book for some of the details of that debate. However,  most of Smith’s ensuing long footnote in fact mentions further defences of the claim that indicative conditionals are material at heart — noting Lewis’s and then Jackson’s purported defences of the Adams Thesis that the assertibility of if A then B goes with the subjective probability of B given A, compatibly with if A then B having the truth-conditions of the material conditional. A student will get little sense that these efforts have been vigorously criticized in turn. And only in a couple of sentences at the very end of the footnote does Smith mention alternative lines again, namely — and really names are all we get — Stalkaner’s position that conditionals have non-truth-functional truth-conditions, and Edgington’s position that conditionals aren’t propositions with truth-conditions at all.

So the student — even if they delve into the endnote (and that’s going to be exceptional!) — might very well be left with the impression that yes, the view that the indicative conditional is at heart truth-functional is, though disputed, still a Best Buy. They might well be surprised then to find that Bennett, should they ever open his book, dismisses the Comforting Story as one that almost no-one (meaning no philosopher who has been thinking through the previous couple of decades of work on conditionals) still accepts as correct — or, we might add, still thinks is rescuable with relatively minor tweaks.

Fair enough: perhaps Nick Smith thinks some variety of truth-functionalism really is a Best Buy, that standard propositional logic gets the logic of indicatives conditionals basically right, and so is happy if students go away having formed that impression (having done his duty and pointed them to a discussion of other views). But it does mean that those of us who have been more convinced by (the arguments epitomized by) Edgington and Bennett, and so are no longer card-carrying truth-functionalists, can’t follow Smith’s line of presentation as a model of how to — honestly, without holding our noses — continue to sell the material conditional in elementary logic.

To be continued

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### 2 Responses to The material conditional and the logic textbooks (3)

1. Jon Awbrey says:

I begin to sense that the problem of conditionals is a symptom of a deeper lying difference of understanding about the purpose of logic itself.

2. Jan von Plato says:

The conditional in logic books:

Thank you Peter again for an interesting discussion; I almost missed it because of the vacations.

It is not that I am insouciant about the conditional in the “Elements of Logical Reasoning,” but it has to come in the right way for students to understand anything. Teach skills first, then reflect on their nature, that’s my order of things. Students learn natural deduction as a skill and learn through experience that the justification of A -> B comes in the canonical case from the existence of a logical argument that leads from the assumption of A to the conclusion of B. The discussion of this matter comes on p. 61 of my book, way after students know how to reason with conditionals. Next, on page 83, they learn that to assume A is different from assuming A provable. Here most logic books err, into the fallacy by which, if from the provability of A the provability of B follows, then the provability of A -> B follows. How many times have I read stupidities like “the deduction theorem fails in modal logic,” not one or two. The worst example is one rather famous book titled “Reasoning about Knowledge.”

I come to the semantics of classical propositional logic on page 96, and hold presumably the world record of postponing truth tables in a first logic book. My students learn to think that classical propositional logic is the logic of decidable atomic propositions. Page 97 contains a discussion of implication in classical logic. I found later that Russell made a colossal conceptual error with it: p -> q is just -p v q, so, wrote he, implication does nothing because either p is false and p -> q is never put to use, or q is true and what use is there to add the condition p. This implicit application of the disjunction property to classical logic would be a topic my students can discuss.

Finally, concerning the notion of a tautology, I explain by way of contrast what a constructive proof of an implication does p. 108). This is further backed up, as an optional matter in a supplementary chapter, by a brief explanation of the Curry-Howard correspondence.

My students may not be best prepared to “talking about logic,” but they understand the way conditionals work in first-order logic.