Conditionals again

Here are two draft chapters on conditionals for the second edition of my Introduction to Formal Logic (to replace chapters 14 and 15 of the current edition). I’ve got to the point that I’d very much welcome comments on these. Note, there will be added exercises which will further explore e.g. the biconditional and further oddities of equating ‘if’ and ‘⊃’.

The main changes? I no longer endorse Jackson’s theory in the way I used to do.  So what positive line do I take? How do I sell the blasted material conditional?

… even if it turns out that ‘⊃’ is not a close analysis of ordinary ‘if’, we can still adopt it to serve as an easily managed, elegantly simple, substitute in formal languages for the messier vernacular conditional. We hereby do so!

In fact, this is exactly how the material conditional was introduced by Frege, the founding father of modern logic, in his Begriffsschrift. Frege’s aim was to construct a formal language in which mathematical reasoning, in particular, could be represented entirely clearly and unambiguously – and for him, such clarity requires departing from “the peculiarities of ordinary language” as he calls them, while capturing some essential logical content. Choice of notation apart, the central parts of Frege’s formal apparatus including the material conditional, together with his basic logical principles (bar one), turn out to be exactly what mathematicians need.

That’s why modern mathematicians – who do widely use logical notation for clarificatory purposes – often introduce the material conditional in text books, and then cheerfully say (in a Fregean spirit) that this tidy notion is what they are officially going to mean by ‘if’. It serves them perfectly in formally regimenting their theories (e.g. in giving axioms for formal arithmetic or set theory). And the rules that the material conditional obeys – like (MP) and (CP) – are just the rules that mathematicians already use in reasoning with conditionals. Much more about this in due course.

This gives us, then, more than enough reason to continue exploring the material conditional. For we will want to investigate what happens when we adopt ‘⊃’ as a ‘clean’ substitute for the conditional in our formal languages, one which serves the central purposes for which we want conditionals, at least in contexts such as mathematics.

For more, do please have a look at the two quite short chapters (I guess anyone teaching or indeed learning logic will have views on the material conditional — I’m trying to be pretty anodyne, so would like to know if I upset too many readers!). As I say, all comments will be most gratefully received.

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Serendipitous distractions

So the CUP Book Sale is over for another twelve months — and with changed rules after last year’s unseemly scrums, this year’s Sale was a very much more enjoyable and civilised affair. After a few pretty abstemious visits, I still came away with a dozen books in all, including to my suprise a couple that were on my wish-list from CUP books published in 2016 — George Herbert: 100 Poems (a beautiful book in form and content!) and Bart Jacobs’ Introduction to Coalgebra (for its promise of categorial interest).

Books are put on the sale shelves in a completely random order. So half the pleasure is making serendiptous finds of titles that I could not usually justify buying (even with my press author’s discount and my level of self-indulgence). At £3 for a paperback — only a few pennies more than the Saturday newspaper — how could I resist e.g. a little music handbook on The Goldberg Variations?  And I’ve been inspired by the excellent recent BBC film To Walk Invisible to start doing some re-reading of the Brontës; so The Cambridge Companion to The Brontës looks fascinating.

However, the book which I sat down with, a glass or two in hand, and devoured in a sitting later the very day I got it was G. H. Hardy’s A Mathematician’s Apology (with a long introduction by C. P. Snow)I’m not sure that I’ve read this cover-to-cover since I was a schoolboy, and if I ever had a copy it has long since gone astray. It is a strange book in some ways, and a sad one too. But this resonated for me: “When the world is mad, a mathematician may find in mathematics an incomparable anodyne.”  Perhaps not incomparable: there’s always Bach. But losing myself thinking through elegant mathematics, trying to get something really clear in my own mind, and perhaps trying to explain it as best I can to others, certainly works for me. Hardy also wrote, astringently, that “Exposition, criticism, appreciation, is work for second-rate minds.” Perhaps so: but it can keep us second-rate minds happily distracted just for a while from the world’s current madness!

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Frege on “if”

I’ve been looking at the passage early in the Begriffsschrift where Frege introduces the material conditional — not, of course, using that label, and not of course with our notation. He notes that A \supset B can be affirmed when A is denied or when B is affirmed, and in those cases “there need not exist a causal connection between the two contents” (the content of A and the content of B). One can also

make the judgment A \supset B without knowing whether A and B are to be affirmed or denied. For example, let denote the circumstance that the Moon is in quadrature [with the sun] and B the circumstance that it appears as a semicircle. In this case A \supset B can be translated with the aid of the connective ‘if’: ‘If the moon is in quadrature, then it appears as a semicircle’. The causal link implicit in the word ‘if’, however, is not expressed by our symbols, although a judgement of this kind can be made only on the basis of such a link.

That’s Michael Beaney’s translation, with notation changed: but other translations don’t differ in relevant ways. In particular, they all use the word ‘causal’ in rendering Frege’s remarks. And this is what caught my eye.

For Frege seems to be intending to make general claims here. To judge  A \supset B we need not suppose that there is a causal connection or link between A and B, it suffices (of course) to be in a position to deny A or assert B. By contrast, however, a judgement if A then B can only be made on the basis of a causal link. And doesn’t that strike us as an odd line for him to take, given that Frege’s first interest is in the language of arithmetic and the  language of analysis, where causation doesn’t come into it? True arithmetical ‘if’s aren’t causal ‘if’s — or so many of us English-speaking analytic philosophers would be inclined to say (not least because we have read our Frege!).

We might wonder, then about the shared translation here. But the relevant German is “ursächlicher Zusammenhang” and “ursächliche Verknüpfung”; and according to the dictionary ‘ursächlich’ means ‘causal’. So it seems that the translations are right.

Though this sets me musing. In English (or at least, in my corrupted-by-philosophy English) there is something of a disconnect between ‘cause’ and ‘because’. If we have A true and this fact causes B to be true, then I am happy to say B, because A. But this doesn’t reverse: in particular, in mathematical cases where  I am happy to say something of the form B, because A, I’d usually balk at talking about causation. For example, I’m quite happy to say of a particular function that it is computable because it is primitive recursive, but would balk (wouldn’t you?) at saying that its being primitive recursive causes it to be computable.

Now I suppose English could have had the notion of becausal link, i.e. some connection or other that holds when B, because A is true (not necessarily causal in the narrow sense). And then we could imagine the view that “a becausal link is implicit in the word ‘if'” (however exactly we are to spell out ‘implicit’ here).

So that raises a question: when Frege talks about ‘if’s and causal connections, does he in fact mean anything stronger than becausal connections (assuming that a ‘because’ need not be causal ‘because’). How are things in philosophical German? Does “ursächliche Verknüpfung” definitely connote a causal as opposed to, more generally, becausal link?

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Scott Aaronson on P ?= NP

As of course is the way with these things, no sooner had I put online TYL2017 than something appears which I would like to add to the Guide. Scott Aaronson has put together a 120 pages survey article on the state-of-play on the question P ?= NP, written with his customary zip and clarity. This would make a very nice addition to the reading in my §6.3.2 on Computational Complexity. It may be for enthusiasts, but you don’t have to follow every detail to get something of the big picture.

There’s a little about the background to the piece and some responses to comments on Aaronson’s blog here.

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Tom Leinster’s Basic Category Theory

9781107044241I just love the opening two sentences of Tom Leinster’s 2014 introductory book, which still seem about as good a minimal sketch of what category theory is up to as you could hope for:

Category theory takes a bird’s eye view of mathematics. From high in the sky, details become invisible, but we can spot patterns that were impossible to detect from ground level.

Taking a bird’s eye view, and thereby seeing better how things hang together, is the sort of intellectual enterprise that appeals to philosophers. Hence — according to me — the interest of category theory  for those interested in the philosophy of mathematics.

I have praised Leinster’s book here before (I think it is terrific, and it worked well with a student reading group of mathematicians last year). The reason for mentioning it again is that he has now, with the kind agreement of CUP, made the book freely available at this arXiv page.

(If you are new here, and this post is of interest, then you’ll also want to look at my category theory page.)

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Teach Yourself Logic 2017

The Teach Yourself Logic 2016 Study Guide was viewed an astonishing 51K times and download 3K times from my page last year. It was also downloaded another 1.4K times from this Logic Matters site. I guess (or at least, hope!) that some people, somewhere, have found it useful.

Taking a quick look at last year’s version, I haven’t found myself moved to make significant changes right now (partly that’s because my mind, or at any rate the logical bit of it, is so taken up with thinking about IFL2). So the Teach Yourself Logic 2017: A Study Guide (find it on by preference, or here) is only a very modest “maintenance upgrade”.

But I must eventually give some thought as to whether it is best to continue with the Guide as a single long document. On the one hand, despite the friendly signposting, 90 pages could seem very daunting. On the other hand, different readers will come with such different backgrounds, interests, and levels of mathematical agility that it is might still seem best just to plot out long routes through the material, all in one place, and (as I do) invite people to get on and off the bus at whatever stops suit them.

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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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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. (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 \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

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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 \neg(A \land \neg B) we can infer if A then B, and vice versa. Similarly, from(\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 \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 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 …

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The Pavel Haas Quartet and Harriet Krijgh play Schubert

The Pavel Haas Quartet with Harriet Krijgh play the Adagio from the Schubert Quintet. Not the best quality video ever, but oh, the playing is beyond words …

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