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 ‘$latex \to$’ (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 ‘$latex \to$’ 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

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

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 …

Tarski on disjunction

Before going off to Florence, I was reworking chapters on the material conditional for IFL2 (in fact I posted a couple of draft chapters here, which I then thought I could improve on,  and so I rapidly took them down again). While away, it occurred to me that it might be prudent/interesting/useful to take a look at how various standard logic texts over the years have handled the conditional in propositional logic. So here I am, starting to work quickly through a pile of some 25 introductory texts, from  Alfred Tarski’s Introduction to Logic and to the Methodology of the Deductive Sciences (1936/1941) to Jan von Plato’s Elements of Logical Reasoning (2013).

I’m writing some telegraphic notes for myself as I go along, but I’ve quickly realized it would take far too much time (and be far too distracting from what I am supposed to be doing) to work these up to give detailed and fair-minded stand-alone reports here. But let me say something about the first book I turned to. For I was surprised and intrigued when reading Tarski to discover  how rocky his arguments are (and indeed how unclear his position is about the relation of ordinary language and the connectives of the formal logician). But let’s not tangle now with what he says about the conditional; his preceding remarks about disjunction already show some of the problems he gets into. So here are some quick notes about those remarks.

Logic (etc.) books of the year?

It’s that time again when the weekend papers are full of their lists of books of the year.  I have to say that so many recommendations sound frankly quite unappealing — surely, there’s a lot of literary virtue-signalling going on! —  but that still leaves me wanting to read more  than I will ever have time to get round to. But one book (perhaps not exactly a philosophy book but certainly of philosophical interest) which has been warmly recommended a number of times is Sarah Bakewell’s At the Existentialist Café: Freedom, Being and Apricot Cocktails. I loved Bakewell’s book on Montaigne, How to Live. So, overcoming my analytical prejudices, I’ve just bought her new new book on Sartre and company as a holiday read. I’ll let you know what I think!

But what about logic books this year (or come to that, books on philosophical logic, philosophy of maths, or other topics broadly related to logic matters)? I have bought a number of older books in the last twelve months, but my haul of recent publications in logic, even broadly construed, seems to have been very modest. I’ve mentioned here two collections of essays, the  Cambridge Companion to Medieval Logic, edited by Catarina Dutilh Novaes and Stephen Read (not quite what I’d hoped for) and  Kurt Gödel, Philosopher-Scientist, edited by Gabriella Crocco and Eva-Maria Engelen (a very mixed bag, and pretty disappointing). But I balked at the price of another collection, Gödel’s Disjunction, edited by Leon Horsten and Philip Welch, as the papers again looked likely to be a pretty mixed bag: so I can’t comment on that. The only logic monograph I bought was  the significantly expanded second edition of Alex Oliver and Tim Smiley’s Plural LogicOtherwise, my purchases seem to have been more skewed towards pure mathematics — the most accessible and fun read being Barry Mazur and William Stein’s Prime Numbers and the Riemann Hypothesis

So I’m not really in a position to recommend a logic (etc.) Book of the Year. All suggestions about what I’ve been missing out on will therefore be very welcome!

Update. I guess I should have mentioned the second edition of Hartry Field’s Science Without Numbers as of interest for its new fifty-page preface. But with this post now having already been read well over 800 times and with a dearth of new suggestions offered, perhaps my impression that 2016 hasn’t been a rich year for new books on logic/philosophy of maths is right!

Another postcard from Florence

img_2432Some wonderful sunny days. Flights to Pisa and hotels in Florence both have enough spaces in December to risk last minute bookings when you’ve seen the weather forecast — but that’s not our way: so this has been sheer good luck. So we took advantage of the blue skies to “do” the Roman site at Fiesole — which perhaps in itself isn’t very exciting, but which does have a very attractively presented small archeological museum which is certainly worth the bus-trip up from Florence.

However, it’s not been all galleries, churches and sites. A certain amount of rather terrific food and drink has been consumed (for mid-culture sustenance, there is Eataly: and after a tough day we can recommend again Olio e Convivium, and Il Santo Bevitore, and a new discovery Il Desco). None will break the bank. And it is a tad depressing that a place as rich and cosmopolitan as Cambridge hasn’t anywhere to touch them.

But no, I shouldn’t say that’s “depressing” even in jest. What’s depressing is the evolving ghastly political news from America. And the continuing profoundly damaging mess that Brexit looks certain to be, thoughts of which are bound to nag away as you wander the side streets of old Europe.

Postcard from Florence

img_2385smBack to Florence for five days. Lovely in the winter sun. Does it perhaps seems a little busier with groups of Chinese tourists than this time last year?  But of course there are still nothing like the dire crowds of summer. One high point has been seeing the restructured Botticelli rooms in the Uffizi which were opened in mid October. The improvement on the familiar Room 10-14 is simply stunning. The old large square room  has been partially divided, to make more wall space. So now the Birth of Venus and Primavera are both beautifully isolated and quite wonderfully well lit too. They can never have looked better.

Here though is a painting from the Uffizi that we’d never noticed before, the Madonna of the Well (c. 1510). Not Raphael as you might think at a first glance, but by one Francesco Cristofano Guidicis, known as Franciabigio. Lovely though.

IFL2 news

I have just heard that CUP are definitely going to offer me a contract for a shiny new edition of my Introduction to Formal Logic.

If the press had never mentioned the idea, I would have probably never taken a hard look at the book again (since I’m no longer teaching from it, and haven’t done for six years) and so I would not have fretted about it.  But once the seed was sown of the idea of a new edition, I of course found myself re-reading the book with a critical eye. Not very happy with what I found! And so then, of course, I did indeed want to try to do a better job. Hence I’m very relieved and pleased that I will get the chance.

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