Begriffsschrift and absolutely unrestricted quantification

We owe to Frege in Begriffsschrift our modern practice of taking unrestricted quantification (in one sense)  as basic. I mean, he taught us how to rephrase restricted quantifications by using unrestricted quantifiers plus connectives in the now familiar way, so that e.g. “Every F is G”  is regimented as  “Everything is such that, if it is F, then it is G” , and some “Some F is G”  is regimented as  “Something is such that it is F and it is G” — with the quantifier prefix in each case now running over everything. And we think the gain in formal simplicity in working with unrestricted quantifiers outweighs the small departure from the logical forms of natural language (and quirks to do with existential import, etc.).

But what counts as the “everything” which our unrestricted quantifiers run over? In informal discourse, we cheerfully let the current domain of quantification be set by context (“I’ve packed everything”,  “Everyone is now here, so we can begin”). And we are adepts at following conversational exchanges where the domain shifts as we go along.

In the context of using a formal first-order language, we require that the domain, i.e. what counts as “everything”, is fixed once and for all, up front: no shifts are then allowed, at least while that language with that interpretation is in force. All changes of what we want to generalize about are to be made by explicitly restricting the complex predicates our quantifiers apply to, as Frege taught us.  The quantifiers themselves stay unrestrictedly about the whole domain

What about Frege in Begriffsschrift, however? There’s nothing there explicit about domains. Is that because he thinks that the quantifiers are always to be taken as ranging, not over this or that domain, but over absolutely everything — over all objects that there are?

Some have taken this to be Frege’s view. In particular, when Dummett talks about Frege and unrestricted quantification in Frege: Philosophy of Language, he is firmly of the view that “When these individual variables are those of Frege’s symbolic language, then their domain is to be taken as simply the totality of all objects” (p. 529).

But it isn’t obvious to me that Frege is committed to an idea of absolutely general quantification, at least in Begriffsschrift. (Re)reading the appropriate bits of that book plus the two other contemporary pieces published in Bynum’s Conceptual Notation, and the two contemporary pieces in Posthumous Writings, there doesn’t seem to be a clear commitment to the view.

OK, Frege will write variations on: “\forall x(Fx \to Gx)” means that whatever you put in place of the “x”, “(Fx \to Gx)” is correct. But note that here he never gives daft instantiations of the variable, totally inappropriate to the e.g. arithmetic meaning of F and G.

This is not quite his example, but he does the equivalent of remarking that “\forall x(x is even \to x^2 is even)” isn’t refuted by “(1\ \textrm{is even} \to 1^2\ \textrm{is even})” because (given the truth-table for “\to”), that comes out true. But he never, as he should if the quantifiers are truly absolutely unrestricted, consider instances such as “The Eiffel Tower is even \to  The Eiffel Towe\textrm{r}^2 is even” — which indeed is problematic as the consequent looks nonsense.

Similarly, in PW, p. 27, Frege cheerfully writes “The numbers … are subject to no conditions other than \vdash n = n + 0, etc.”. There’s not a flicker of concern here about instances — as they would be if the implicit quantifier here were truly universal — such as “\vdash \textrm{Napoleon} = \textrm{Napoleon} + 0”. Rather it seems clear that here Frege’s quantifiers are intended to be running over … numbers! (Later, in Grundgesetze, Frege does talk about extending addition to be defined over non-numbers: but  it is far from clear that the Frege of Begriffsschrift has  already bitten the bullet and committed himself to the view that every function is defined for the whole universe.)

Earlier in PW, p. 13, Frege talks about the concept script “supplementing the signs of mathematics with a formal element” to replace verbal language. And this connects with what has always struck me as one way of reading Begriffsschrift.

  1.  Yes it is all purpose, in that the logical apparatus can be added to any suitable base language (the signs of mathematics, the signs of chemistry, etc. and as we get cleverer and unify more science, some more inclusive languages too). And then too we have the resources to do conceptual analysis using that apparatus (e.g. replace informal mathematical notions with precisely defined versions) — making it indeed a concept-script. But what the quantifiers, in any particular application, quantify over will depend on what the original language aimed to be about: for the original language of arithmetic or chemistry or whatever already had messy vernacular expressions of generality, which we are in the course of replacing by the concept script.
  2. Yes, the quantifiers will then unrestrictedly quantify over all numbers, or all chemical whatnots, or …, whichever objects the base language from which we start aims to be about (or as we unify science, some more inclusive domain set by more inclusive language).
  3. And yes, Frege’s explanation of the quantifiers — for the reasons Dummett spells out — requires us to have a realist conception of objects (from whichever domain) as objects which determinately satisfy or don’t satisfy a given predicate, even if we have no constructive way of identifying each particular object or of deciding which predicates they satisfy. Etc.

But the crucial Fregean ingredients (1) to (3) don’t add up to any kind of commitment to conceiving of the formalised quantifiers as absolutely unrestricted. He is, to be sure, inexplicit here — but it not obvious to me that a charitable reading of Begriffsschrift at any rate has to have Frege as treating his quantifiers as absolutely unrestricted.

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9 Responses to Begriffsschrift and absolutely unrestricted quantification

  1. S. Motta says:

    Great post, as usual. Don’t you think that a consequence of your reading of Frege is to render unintelligible the notion of “absolutely unrestricted quantification”? We understand what it means to quantify in a given grammar (or if the term “grammar” is too wittgensteinian, in a given language/domain) (e.g. the grammar/language/domain of numbers, the grammar/language/domain of chemical elements, etc.), and we understand – as you do in your post – what it means that, given this language/domain, we quantify with no restriction over its “objects”. What would it mean to quantify over “absolutely everything” (i.e. “all the objects”) when the very concept of “object” is an indissociable part of the apparatus of quantification and when our understanding of this apparatus is always integrated in one grammar (or another)? One has to have a metaphysical notion of “object” to do the trick when logic is indeed not here to give us such a metaphysical notion!

    • Peter Smith says:

      I agree that the notion of “absolutely unrestricted quantification” is problematic — and yes, one reason is the reason you outline. But I do find the bundle of issues about absolute generality to be very obscure. I did comment on some of the papers in the Rayo/Uzquiano volume in 25(!!) posts here (and there’s a very short version of those comments as a published review here). But as I recall, though this might be quite unfair as it is almost ten years ago, none of the contributors really discussed the sort of issue you rightly raise.

      • David Auerbach says:

        I’m relying on memory too here, but didn’t Cartwright ur-article on this implicitly address this point?

    • Rosety Moid says:

      There’s no restriction over “objects” in a natural language such as English. We can talk about numbers, chemical elements, whatever. Yet we still use quantifiers.

      • Peter Smith says:

        But numbers, chemical elements etc. are objects in Frege’s (generous, logical) sense.

        • Rosety Moid says:

          I’m finding it hard to understand how your reply fits with mine. Perhaps I’m missing something, or perhaps my comment was too brief to be sufficiently clear. So I’ll try to explain.

          I thought the account of Frege’s views that you and S. Motta were giving went something like this: Quantifiers aren’t (absolutely) unrestricted. They don’t range over absolutely everything (over all the objects that there are). Instead, we start with a “base language” that has a restricted domain (the language of arithmetic, perhaps, or the language of chemistry), and when we bring in the “logical apparatus”, quantification is over (restricted to) the objects that the base language aims to be about. You both describe that as unrestricted over those objects (and syntactically the quantifiers are written without a restriction), but quantification is in effect restricted to those objects nonetheless. (If the language is that of arithmetic, for example, the objects the quantifiers range over do not include paintings, say, or cats.)

          That seems plausible as a way to address such specialised, limited-domain, languages, but S. Motta seemed to be saying that we could understand quantification only in such cases. My point was that a natural language such as English is not restricted in the way those languages are, yet we still use quantifiers (and at least seem to understand them).

          Your suggestion for natural languages (“informal discourse”) seemed to be that we implicitly use many different domains of quantification and shift among them as we go. I think there may be some problems with that idea. For a start, it looks like the domain must be able to shift with each quantification, even within a single sentence such as “I’ve packed everything everyone wanted.” But regardless of how well that works for English, it wasn’t what was being proposed for the specialised languages such as the one for arithmetic or the one for chemistry. It also suggests that we have an idea of absolutely everything after all; we just don’t put everything into one domain.

  2. Daniel Nagase says:

    I think this is exactly right. Unfortunately, a lot of Frege scholarship still followa Goldfarb, Ricketts, et al in taking Frege to have been advocating some kind of “universal logic”, in which (i) metalogic notions are disallowed and (ii) the quantifiers range over anything. Of course, aomw of Frege’s remarks may encourage this (e.g. the Caesar problem). On the other hand, some really good scholarship has recently emerged challenging this picture; have you read Patricia Blanchette’s book on Frege? It ia very congenial to your interpretation.

  3. Rosety Moid says:

    Could cases such as Napoleon + 0 be handled by regarding them as syntactically invalid (as would happen in a programming language), rather than by seeing the quantifiers as implicitly restricted?

    (I don’t see this as the same as S. Motta’s suggestion, because I don’t understand what “grammar” could mean if it’s so easy to slide from “grammar” to “language” and “domain”, or how there can be “language/domain” as if they were two names for the same thing.)

  4. Jan von Plato says:

    A bit late in following this discussion, but a couple of remarks anyway:

    1. The terminology used today is “bounded” and “unbounded” as in the typical bounded quantification (x:N)G(x), “for all x in N, G(x)”

    2. Frege’s explanation of the universal quantifier is syntactic, even if he gives the explanation in the wrong order. So, contrary to Frege’s order, first: (x)G(x) can be judged if G(x) can be judged for an arbitrary x. Secondly, if (x)G(x) can be judged, any instance G(a) can be judged. Frege muddled up the conceptual order here.

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