The material conditional and the logic textbooks (2)

Continuing from the previous post, I’ll consider five elementary textbooks aimed at philosophers, all either first published, or with new editions, well after e.g. Edgington’s State of the Art article. The first three texts I’ve chosen to look at because they are so widely used and are often recommended. The fourth is by such a careful philosopher that one would hope for good things. And I have chosen the fifth because it is the most recent major introduction to logic and has many admirable features.

Let’s start, then, with Bergmann, Moor and Nelson, The Logic Book (I’m looking at the sixth edition, 2014). After introducing the material conditional, they have a §2.4 ‘On non-truth-functional uses of connectives’. They there note that the material conditional can’t be used to paraphrase subjunctive conditionals. But our authors also offer the following reason for supposing it doesn’t serve to render some indicative conditionals either:

But when an English conditional is based on a scientific law, paraphrasing that conditional as a material conditional can be problematic. An example is

If this rod is made of metal, then it will expand when heated.

A simple law of physics lies behind this claim: all metals expand when heated, and the conditional is in effect claiming that if the rod in question is made of metal then heating it will cause it to expand. A paraphrase of this causal claim as a material conditional does not capture this causal connection.

But this seems to confuse what “lies behind” the conditional claim with its literal content. After all suppose the rod is made of metal. And suppose that, when it is heated it happens to expand but not because of the heat but because of some accidentally present other cause. Then what I actually say is true by accident even if the heating doesn’t cause the expansion. (The problem here, then, is not specifically about paraphrasing an explicitly causal claim as a material conditional but is already there when we paraphrase a causal claim as a bare conditional: content can be lost.)

There seems to be little else about the relationship between indicative conditionals and material conditionals in The Logic Book. Grice and the Comforting Story are nowhere mentioned, let alone post-Gricean discussions. So let’s move on.

We’ll next look at Language, Proof and Logic by Barker-Plummer, Barwise and Etchemendy (second edition, 2011). As in my IFL, this book first introduces negation, conjunction, disjunction and explores their logic, before turning to conditionals later. At their first pass, having mentioned an example with a subjunctive conditional and explained why the material conditional can’t be used to render it, our authors give an interim summary “While the material conditional is sometimes inadequate for capturing subtleties of English conditionals, it is the best we can do with a truth-functional connective. But these are controversial matters,” with a promise to return to these matters in §7.3. That later section is entitled “Conversational Implicatures”. It introduces Gricean ideas and use them to explain first why we might hold that the implications of exclusiveness in some uses of disjunctions are generated conversationally (so we don’t have to suppose that “or” has a special exclusive sense). Then the Gricean ideas are used to explain why we often hear “only if”s as “if and only if”s, and explained why “unless” shouldn’t be equated with “if and only if”. But very oddly, despite their promise, our authors do not return to discuss plain “if”, and don’t elaborate the Comforting Story, let alone criticize it. So the student is left pretty unclear how “these controversial matters” impact on the logic of arguments involving ordinary conditionals.

Let’s next consider Gensler’s Introduction to Logic (second edition, 2010). I don’t really know this book but I have taken a look since I’ve repeatedly seen it recommended as working well with students. Gensler first notes (p. 123)

Our truth table can produce strange results. Take this example:

If I had eggs for breakfast, then the world will end at noon. $latex (E \supset W)$

Suppose I didn t have eggs, and so E is false. By our table, the conditional is then true … This is strange. We d normally tend to take the conditional to be false since we d take it to claim that my having eggs would cause the world to end. So translating if-then as $latex \supset$ doesn t seem satisfactory. Something fishy is going on here.

Well, yes. And the treatment of the conditional is left in that very fishy state for over 250 pages (which I would have thought most students would find pretty unsatisfactory). Eventually, we reach an apparently optional chapter on Deviant Logics, and here at last we meet Grice and the Comforting Story as a conservative alternative to a revisionary relevant logic. However, the section is a bit of a mess (and there is no engagment with the post-Grice literature).

So far then, so unimpressive. Fourthly, then, we move on to consider Deductive Logic (2003) by that most careful of philosophers, Warren Goldfarb. His §7 is on conditionals. It starts somewhat unhappily:

In common practice, if someone asserts a statement of the form “if p then q” and the antecedent turns out to be false, the assertion is simply ignored, and the question of its truth or falsity is just not considered. In a sense, we ordinarily do not treat utterances of the form “if p then q” as statements, that is, as utterances which may always be assessed for truth-values as wholes. Our decision as logicians to treat conditionals as
statements is thus something of a departure from everyday attitudes …

What Goldfarb says is common practice isn’t. If I assert “if you do that again, I’ll stop your pocket money” and as a result the child desists, the antecedent of the conditional is false. But the assertion hasn’t been ignored: on the contrary! And the child may wonder whether my threat was an idle one and whether I was really speaking the truth. Again, if I use modus tollens to infer that a conditional assertion has a false antecedent, I don’t ignore the assertion — I may use it, precisely, to draw an important conclusion.

Leaving that aside, oddly, only a page after talking of logicians “decision” to treat conditionals as statements (as if it is a useful dodge), Goldfarb is talking of the “analysis” of conditionals as material conditionals. So which is it? Decision or analysis?

Once he has mentioned subjunctive conditionals and set them aside, Goldfarb says “we intend the material conditional as an analysis only of indicative conditionals”. And he then considers some objections to the analysis which he fends off with a very rough-and-ready version of …. the Comforting Story (without mentioning Grice). But then a page later we are seemingly back not with a defensible analysis but a decision: “We adopt the material conditional as a rendering of “if… then” because it is useful.” Goldfarb’s vacillating discussion is brief and perhaps we shouldn’t be too stern about it: but still, this is disappointing.

So let’s turn to our fifth book, Nick Smith’s admirable Logic: The Laws of Truth (2012). Smith at least is aware of the philosophical literature on conditionals: so how does this impact on his story about what is going on with the conditionals in our official first-order logic?

To be continued

2 thoughts on “The material conditional and the logic textbooks (2)”

  1. Does finding some exceptions mean Goldfarb is wrong about common practice? He could be talking about what’s usually the case, rather than always.

    Re “if you do that again, I’ll stop your pocket money”, it could be argued that if the child desists, the antecedent isn’t yet false because the child still might do it again; and it’s that possibility that keeps the conditional in play.

    (BTW, in most programming languages, “if” is a control structure, so that it’s truth or falsity as a whole isn’t considered; and even in (most?) languages where there are “if” expressions, “if P then Q” isn’t evaluated for truth or falsity: there has to be an “else” clause that says what to evaluate if P is false.)

    Re “decision or analysis?”, how about both: we’ve decided to analyse (interpret, treat) it that way.

    1. The important observation here is the one about programming languages: “if” behaves differently from a boolean connectives. Arguably that’s a good clue to how “if” behaves in ordinary language too — the trouble with logicians’ treatment of “if” comes from trying to assimilate to a propositional connective. Which is (coming at it the long way around) where e.g. Edgington ends up.

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