A note on another of those bits of really elementary logic you don’t (re)think about from one year to the next – except when you are (re)writing an introductory text! This time, the question is which is the quantifier, ‘$latex \forall$’ or ‘$latex \forall x$’, ‘$latex \exists$’ or ‘$latex \exists x$’? Really exciting, eh?
Those among textbook writers calling the quantifier-symbols $latex \forall$’, ‘$latex \exists$’ by themselves the quantifiers include Barwise/Etchemendy (I’m not sure how rigorously they stick to this, though). Tennant also calls e.g. ‘$latex \forall$’ a quantifier, and refers to ‘$latex \forall x$’ as a quantifier prefix.
Those calling the quantifier-symbol-plus-variable the quantifier include Bergmann/Moor/Nelson, Chiswell/Hodges, Guttenplan, Jeffrey, Lemmon, Quine, Simpson, N. Smith, P. Smith, Teller, and Thomason. (Lemmon and Quine of course use the old notation ‘$latex (x)$’ for the universal quantifier.) Van Dalen starts by referring to ‘$latex \forall$’ as the quantifier, but slips later into referring to ‘$latex \forall x$’ as the quantifiers.
It’s clear what the majority practice is. Why not just go with it?
Modern practice is to parse ‘$latex \forall x(Fx \to Gx)$’ as ‘$latex \forall x$’ applied to ‘$latex (Fx \to Gx)$’. However Frege (I’m reading through Dummett’s eyes, but I think this is right, mutating the necessary mutanda to allow for differences in notation) parses this as the operator we might temporarily symbolize ‘$latex \forall x \ldots x \ldots x \ldots$’ applied to ‘$latex (F\ldots \to G\ldots)$’.
To explain: Frege discerns in ‘$latex \forall x(Fx \to Gx)$’ the complex predicate ‘$latex (F\ldots \to G\ldots)$’ (what you get by starting from ‘$latex (Fn \to Gn)$’ and removing the name). Generalizing involves applying an operator to this complex predicate (it really is an ‘open sentence’ not in the sense of containing free variables but in containing gaps — it is unsaturated). Another way of putting it: for a Fregean, quantifying in is a single operation of taking something of syntactic category s/n, and forming a sentence by applying a single operation of category s/(s/n). This quantifying operator is expressed by filling-the-gaps-with-a-variable-and-prefixing-by- ‘$latex \forall x$’ in one go, so to speak. The semantically significally widget here is thus ‘$latex \forall x \ldots x \ldots x \ldots$’. Yes, within that, ‘$latex \forall$’ is a semantically significant part (it tells us which kind of quantification is being done). But — the Fregean story will go — ‘$latex \forall x$’ is not a semantically significant unit.
(Suppose that instead of using ‘x’s and ‘y’s to tell us which quantifier gets tied to which slots in the two place predicate ‘$latex L\ldots,–$’ to give ‘$latex \forall x\exists yLxy$’ we had used ‘$latex \forall \exists L\ldots,–$’ with arrows tying each quantifier to its appropriate slot. Typographical convenience apart, that would serve just as well. But then no one would think that ‘$latex \forall$’ and half a broken arrow is a semantically significant unit!)
So, whether you think ‘$latex \forall x$’ is worthy of being called the universal quantifier is actually not such a trivial matter after all. For it should depend on your view as to whether ‘$latex \forall x$’ a semantically significant unit, shouldn’t it? You might think that the true believing Fregean protests about this sort of thing too much. I’d disagree — but at any rate the underlying issue is surely not just to be waved away by unargued terminological fiat.
I think that David Auerbach’s point about the number of quantifiers is (or ought to be) decisive.
(I suppose fans of saying ‘∀x’ and ‘∃x’ are quantifiers might get around that point by saying ‘∀’ and ‘∃’ are ‘quantifier types’ (or some such terminology), so that there are 2 quantifier types, rather than 2 quantifiers, but does anyone feel so strongly that it’s ‘∀x’ and ‘∃x’ that should be called ‘quantifiers’?)
I don’t agree that “it should depend on your view as to whether ‘∀x’ a semantically significant unit”. A verb phrase is a semantically significant unit in English, for example, but that doesn’t mean it’s verb phrases that should be called ‘verbs’.
BTW, I don’t think it works to argue that “no one would think that ‘∀’ and half a broken arrow is a semantically significant unit”. If someone thought ‘∀x’ was a semantically significant unit, they might well think the direct translation of that as ‘∀’ and half a broken arrow was semantically significant too.
I think “semantically significant” may be a red herring as a phrase; it’s really that ∀ and ∃ do one thing and broken arrows/repetition-of-variables do another. (Of course, together they do yet another…)
YES. I always fussed, in my baby logic class, about: a) not using variables in the glossary (I typically used circled numerals) . So K①②: ① kicked ② .
b) in my introducing of quantifiers (starting with how things go awry if one treats ‘someone’ as just another designator) I use those arrows you mention for at least one class session before observing how typographers hate that notation and we can do better.
Which is to say, it is both conceptually and pedagogically apt to treat ∀ and ∃ as the quantifiers. (And, also it makes it be the case that there are two quantifiers not ω-many.
From a functional point of view it was a step backward when we passed from Peirce’s $latex \sum$ and $latex \prod$ to the present $latex \exists$ and $latex \forall.$ There’s a rough indication of what I mean at the following location:
☞ Functional Logic : Higher Order Propositions