Yes, and this is an important way that the logic reflects the commitments or intentions of its creators. It is not logic qua logic; it is logic qua mathematics. — Leontiskos
Well, I brought up the issue, so I'm bound to say there's something to this.
On the other hand, I'm hesitant to endorse what you say here because mathematics is special, and there's a sense in which mathematics is the goal of logic, the goal of thinking as such. (I think there are hints of the excitement of this discovery almost everywhere in Plato where he rattles off the list -- argument, mathematics, astronomy, and so on.) --- And that means "qua mathematics" is not generally a restriction of anything, a limiting of it to this one domain, but an idealization of it.
And it's historically backwards -- but maybe that was deliberate? Frege was trying to reduce mathematics to logic, not the other way around, and that turns out not quite to work, but in trying to do so, he came up with a formalization of logic which could be extremely useful to mathematics rather than providing its foundation. A sort of logic "adapted to" mathematics, or to the needs of mathematics, which is what I was suggesting --- although this time around I've already suggested this isn't necessarily a deformation of logic by focusing on a limited domain, so much as an idealization of logic by focusing on the domain that most cleanly, we might say, represents human thought. And as it happens, I think Frege thought so as well. I think he was mostly of the opinion that natural languages are too much of a mess to do sound work in.
Do all those steps amount to "logic qua mathematics"? Maybe kinda, in a dyer's hand sort of way. There's a lot that makes it look like a branch of mathematics, and the advanced stuff tends to be called "mathematical logic" and get taught in math departments. But that's a deeply tricky business because basic logic is the fundamental tool of everything done in mathematics, absolutely everything -- it's just taken as given at lower levels of learning, without any suggestion that you're actually borrowing from some rarefied advanced field of mathematics.
So I think advanced "mathematical logic" is something like "mathematized logic" -- that's qua-ish maybe in the sense you meant -- but what that means is applying the tools and techniques of mathematics to the
given material that is logic, which mathematics can treat
of, because mathematics is good at treating
of anything. (That's the whole point.) And one of the techniques mathematics brings to bear in treating of logic is, well, logic, because mathematics was just borrowing it for free in the first place.
Still agree?
I actually think ↪fdrake's post may be most instructive and fruitful. — Leontiskos
Wouldn't be the first time, but he was addressing the topic, and I have yet to develop an interest in doing that.
Now if quantifier variance is occurring—superable or insuperable—then the existential quantifier is doing more than presupposing a univocal notion of existence. Or, if you like, the two secretly competing meanings of existential quantification are each “presupposing” a different notion of existence, and this is the cause of the disagreement. Thus arises the very difficult question of how to adjudicate two different notions of existence, and this is the point of mine to which you initially objected. — Leontiskos
Do as you like, I just don't see the point. We can talk about existence all we like without dragging quantifiers into it, and people -- they're always wandering around the forum -- who get worked up about the meaning of the "existential quantifier" are generally just confused by the name (a name I note Finn and Bueno would like to retire).
It's a funny thing. This is all Quine's fault, as I noted. "To be is to be the value of a bound variable" comes out as a deflationary slogan, but what we was really arguing for was a particular version of univocity: the idea was that if you quantify over it, you're committed to it existing, and he meant "existing" with the ordinary everyday meaning; what he was arguing against was giving some special twilight status to "theoretical entities". If your model quantifies over quarks, say, then your model says quarks are real things, and it's no good saying they're just artifacts of the model or something. --- The reason this is amusing is that all these decades later the consensus of neuroscientists and cognitive psychologists, so far as I can tell, is that absolutely everything we attribute existence to in the ordinary everyday sense -- medium-sized dry goods included -- is an "artifact of the model" or a "theoretical entity", so the threat to univocity Quine was addressing never actually existed, if only because the everyday meaning of "exist", the one Quine wanted to stick with, is in fact the "twilight" meaning he wanted to tamp down. And so it goes.
What do you think of the claim that discrete entities only exist as a product of minds? That is, "physics shows us a world that is just a single continuous process, with no truly isolated systems, where everything interacts with everything else, and so discrete things like apples, cars, etc. would exist solely as 'products of the mind/social practices.'" — Count Timothy von Icarus
I don't think we are any more justified in saying this than we are in saying the world is full of distinct objects. All we have is signal processing. Is the source one signal? Two? Two trillion? How can you tell when you're receiving and analysing them all at once? It makes a difference in your metaphysics, but in nothing else at all that I can see.
We don't see individual objects in isolation, but as embedded in and different from their surroundings, so difference if not a property of some putative completely isolated object, but a property it displays in its situatedness. — Janus
And you don't see any circularity here?
Remember the issue was whether number could be a property of an object, and it just obviously can't unless sets count as objects. It's really straightforward and it pissed Quine off considerably.
What's more ...(It is curious that we don't adjectivize numbers much at all, so even sets aren't said to be two-ish but to have cardinality of two. ((We have "once" "twice" and "thrice" for adverbs, but then it's on to "repeatedly" or "continually" or something.)) Maybe it's an Indo-European thing.)
But then you brought in this other stuff about "diversity, sameness, and difference being real" which just begs another pile of questions. I'm at a loss.