Decaffeinated sets

It is a familiar enough point that while logic texts for beginners often fall into talking about sets (sets of premisses entailing conclusions, sets of objects being extensions of predicates, sets of objects being domains of quantification, etc.), this set talk is doing no substantive work at least in elementary contexts. It can be construed in a decaffeinated way, as talk about no more than virtual classes in Quine’s sense.

I found myself making a few remarks to this effect at scattered places in  IFL2, but doing so distracted a bit from the flow of exposition. So I’ve decided to gather together various remarks into one four-page chapter. Here it is:

What do people think? I’d very much welcome comments. I don’t want to avoid distractions of one kind by e.g. being thought distractingly misguided!

3 thoughts on “Decaffeinated sets”

  1. Let me put the content of your “very short word” aside. There’s a prior question.

    The prior question: who is your target reader? Is it someone who reacts to your “set” talk with a flinch, fearing that suddenly the lessons in logic are bringing on big ontological commitments? If not, then why bother with the “very short word”? If so (i.e., if the reader does flinch just so), I reckon that that reader will also flinch at talk of “pluralities” — fearing that they too, on reflection, bring on unbearable ontological commitments. And now you need yet another “very short word” to comfort the plurality-talk flinching. Etc.

    A different way to go is simply to say that set theory (or at least the very low-hanging bits) is often used to model important phenomena. One important phenomenon is logical consequence, which is a relation on our language — and, in turn, is modeled as such over a (“model”) language (with well-defined syntax, semantics, etc.). If a reader masters your book (i.e., masters the standard model of the relation of logical consequence), she can — or not — proceed to ask after the extent, if any, to which the reality of logical consequence matches the model (e.g., is it in fact a binary relation from a set to a sentence? etc.). But that’s a different topic for a different book.

  2. I must say that I am quite unsympathetic to the entire methodology lying behind the short chapter.

    It is a methodology of boot-strapping: when teaching logic, at least introductory logic, to students of philosophy, one should not make use of any notions or facts coming from outside logic – no substantive use of sets, nor of any kinds of number, even the positive integers and, if one is a real purist, no use of logic itself until the principles that one would like to use have been justified in the text.

    This boot-strapping methodology has, in my view, been dominant for far too long. As well as being very constraining, it is a vain enterprise, for there is no way of doing without abstract items of one kind or another. Even a single propositional variable is a highly abstract item, different from any particular mark on a piece of paper at a specific time and place; a derivation is a finite sequence (or tree) of arbitrary length, which is pretty abstract; a valuation is a function, etc. Avoiding use of sets, functions and numbers in the presentation still leaves us with abstract items such as propositional variables, derivations and valuations but without the means of clarifying them. And if one follows the purist road of not using any principles of logic until they have been justified in the logic text, one ends up either immobilized or cheating.

    In my view, one should feel perfectly free to use, without apology, whatever abstract notions and principles that are helpful in explaining the subject and already more or less familiar to the intended audience. Ontological and metaphysical doubts should be left aside to a stage when the basics of the discipline have been absorbed. That is what is done when teaching mathematics, physics, statistics and biology, and logic should be no exception. There are deep and fascinating questions in the philosophy of all these subjects, but students should not be deluded into taking positions on them before getting the hang of the discipline itself.

    In effect, the purpose of the chapter is to reassure the reader that although the text may be taking an occasional tipple of sets and other abstract items, it is really at heart on the wagon, at least as far as sets are concerned. My criticisms are that one should not feel at all guilty about taking a drink with meals, and that consumption ends up being hidden rather than halted.

  3. I think it should be treated more lightly. A four-page chapter is making too much of it. The chapter’s summary says pretty much what ought to be said, if the issue is raised at all. If your book used footnotes (or endnotes), I’d say that one of those, containing something like that summary, would be enough. Since the book doesn’t use such notes, however, there’s a problem of where to put the text. Perhaps a ‘box’ in the section of chapter where it first becomes awkward to avoid talk of sets or tuples?

    I think there are two errors in introductory texts, in opposite directions, that an author who knows a lot about a subject can fall into. (I’ve seen both quite a few times and have fallen into them myself.) One is that, because you know certain things don’t really matter, you’re loose or inconsistent in your wording. This confuses readers too new to the subject to know what matters and what doesn’t. Some uses of ‘set’ language can have this problem.

    The other is that, because you’re aware of certain problems and issues, you feel it would be misleading not to discuss them. This is a more subtle matter, I think, and harder to judge. The danger is that they’ll be given too much prominence, so that they compete for the reader’s attention with more important issues, or make something more complicated than it need be, and thus harder to understand. I think this may be what’s happening with the ‘very short word about sets’.

Leave a Comment

Your email address will not be published. Required fields are marked *

Scroll to Top