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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11 Responses to Conditionals again

  1. Jon Awbrey says:

    The matter of conditionals is one of those places where we find ourselves needing to pause and reflect on the purpose we have in mind for logic. And here folks differ. One job might be to describe and rationalize our native linguistic intuitions. Another task is a bit more normative, to discover optimal guidelines for conducting our reason, whatever form of language or medium it might take to achieve that end. Those two aims may be compatible, complementary, or mutually exclusive in the long run. I do not know.

    • Peter Smith says:

      “[O]ne of those places where we find ourselves needing to pause and reflect on the purpose we have in mind for logic.” I agree: though there is only so much pausing and reflecting of that kind one can do in an intro book with a limited page budget!

  2. Nissim Francez says:

    I think that any serious attempt to introduce logic that presents in detail the material
    conditional should also present in details some of its alternatives: strict, relevant, connexive etc. In the era of flourishing non-classical logics, sticking to the material conditional only is a mis-service to the student.

    • Peter Smith says:

      Would you say “In the era of flourishing non-classical analysis, sticking to classical analysis only [in a first course] is a mis-service to the student”? I suspect not!

      But maybe I should add a sentence near the end of §16.5, making it clear that when I talk about a policy of adopting the material conditional in our formal languages, I mean the formal languages here in this introductory book (allowing for other conditional-like connectives in other languages).

      • Francez Nissim says:

        There is no comparison to non-classical analysis, which remains a curiosity of interest to some mathematicians only; non-classical logics solves many of the disturbing difficulties that classical logic faces in non-mathematical reasoning and is seen by many as a viable substitute.
        You do not have to endorse yourself such a logic, just present the reader with the problems, of which he presumably is aware anyway, and the advantages of non-classical it yet in solving those problems.

  3. David Auerbach says:

    I am entranced by the monosyllabic succinctness of ‘bar one’, given the amount of ink spilled over the *one*.

  4. Chris Hutchinson says:

    This is slightly off topic, but when do you expect the new edition to be on sale?

    • Peter Smith says:

      The plan is delivery this autumn, and then hopefully speedy publication (since I’ll be producing LaTeX files which can be used, hopefully more or less as is, for production). So maybe just over a year from now till the book is available?

  5. Tim says:

    I, too, struggled for some time with explaining the material conditional and still don’t know what works best. But in my experience something like the spoiler alert in 15.7.b) works surprisingly well. I don’t know what you have in mind here, but I always end my discussion of the material conditional with pointing out that the material conditional will come in handy when we start analysing sentences like “All frogs are green”, “all unicorns are green” and so on. Hint: There’s a material conditional hidden in them! That sparks students’ curiosity and they are more willing to accept the usefulness of the material conditional as a working hypothesis if they know that it will be tested again when we discuss general statements, or so it seems to me.

    • Peter Smith says:

      Yes, that’s an important observation — that if we are going to regiment Every A is a B as Everything is such that if it’s a A then it’s a B we’ll need a material conditional to do the job. That’s certainly going to be highlighted later when we come to doing quantificational logic. I’m still wavering about whether to have a pointer ahead to this thought at this stage.

      • The interaction of “if”-clauses with quantifiers also brings with it one of the crucial problems with material conditionals. If one says “Nothing is B if it is A”, the “if” obviously cannot be material.

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