Logic is not just a stipulative game, like chess. The analogy doesn't work. — Leontiskos

And as I said, if you embrace logical pluralism then it doesn't matter how you quantify or which logic you use, for everything is stipulation and no one stipulation is any better than any other. — Leontiskos

...within a single logic qualitative differences of domain reflect qualitatively different understandings of quantification — Leontiskos

Quantifiers are not subject to second-order equivocation; therefore QV fails — Leontiskos

Those quantifiers are introduced differently, and as the paper "Quantifier Variance Dissolved" notes that provides a strong argument for a form quantifier variance without a reduction of quantifier meaning to underlying entity type it quantifies over, and without committing yourself to the claim that there's a whole bunch of equally correct logics for the purposes of ontology. — fdrake

the sort of assertions you get when you try to squeeze a big set of phenomena into a tiny box of explanation. — Count Timothy von Icarus

No doubt the inferential role of “there is” or “exists” in natural language is more complex than the role of “∃” in formal logical languages, but the formal-syntactic role of “∃” provides a tidy approximation of the informal inferential role of “exists” or “there is” in English. The expression “there is” is an existential quantifier, in English, roughly because for name “a” and predicate “F”, from “a is F”, “there is an F” follows; and if a non-“a” claim follows from “a is F”, with no auxiliary assumptions made about “a”, then that same thing also follows from “there is an F”. Expressions that obey this role unrestrictedly, for all names and predicates that could be introduced into the language, express the language’s unrestricted concept of existence.

We argue that ∃ always has this role, as it invariably has the function of ranging over the domain and signaling that some, rather than none, of its members satisfy the relevant formula. Yet the quantifier-variance theorist requires ∃ to have multiple meanings. — Quantifier Variance

I think we are largely on common ground then. — Count Timothy von Icarus

I dunno OLP heads, "is" sure crops up in a lot of language games with different grammars. Almost as if there are different uses of it! — fdrake

This piece originally began life as a symbol of the elephants in the Indian army. It's original movement was 2 squares diagonally in any direction. It was a piece of only moderate power.

It was only when the game was carried to Europe that it's fortunes began to improve. The Europeans were not as familiar with the elephant as the Indians so they needed to change the piece to something that people in Europe could relate to. The church was very powerful in Europe when these changes were going on. It's influence on political life in the Middle Ages was recognized when the piece became a Bishop.

The Europeans also wanted to speed the game up as they found it laboriously slow. The Bishop was one of a number of pieces to see it's powers increase, gaining unlimited range on the diagonals.

if one attempts to quantify over all mammals but omits unicorns because they were not known to exist (or vice versa) then a quantitative difference of domain results in a merely artificial difference of quantifier-qua-extension. But if one attempts to quantify over all things but omits universals because they are a nominalist (cf. QVD 295) then a qualitative difference of domain results in a substantial difference of quantifier-qua-extension. — Leontiskos

Use itself doesn't float free of the rest of the world. — Count Timothy von Icarus

Now how exactly do we manage that? Attributing a predicate to an identified individual looks straightforward, but in ordinary life we only reach for the existential quantifier in the absence of such an individual. (One of you drank the last beer. Someone left these footprints. There's something really heavy in this box.) — Srap Tasmaner

What is your point? — Srap Tasmaner

Something is B — Srap Tasmaner

Can you think of an edge case where it's not clear whether something counts as a berry? — Srap Tasmaner

As it happens, this is what the thread should be about. — Srap Tasmaner

What I am objecting to is an explanation that seems to say that prior to an act of counting there is nothing that affects how counting is done. — Count Timothy von Icarus

what explains this? — Count Timothy von Icarus

If some society somehow stipulated that 8/2 = 5, we tend to feel we could give them a good demonstration of why this is not the correct way to do division. — Count Timothy von Icarus

The attempt to reduce mathematics to 'speech acts' is inadequate to account for the 'unreasonable effectiveness of mathematics in the natural sciences' — Wayfarer

But just stating the trivial fact that "numbers are something humans use," or "words are things we say," as if this pivot to activity makes the explanation an unanalyzable primitive strikes me as essentially a non-explanation. — Count Timothy von Icarus

It is unclear that there is a coherent way of formulating any such quantification and the resulting maximal domain. If the maximal domain is a set, then unrestricted quantification would require quantifying over everything, and there would have to be a set of everything, including, in particular, a set of all sets, among other inconsistent totalities, since all of these things are in the scope of an unrestricted quantifier: everything is in its scope, after all! But that is clearly inconsistent. — p.294

Maybe the proponent would take each person, sit them in the same room, and ask them to evaluate the sentence < Ǝx(R(x) ^ A(x)) > (“There exists an x such that x is in the room and x is an apple”). In the corner of the room is a painting by Cézanne, and within the painting is depicted a paradigmatic red apple. One person says that the sentence is true and the second person says that it is false. Upon inspection we realize that the disagreement is not over whether the painting depicts an apple, but is instead over whether the quantifier captures it as an apple. Specifically, it is over whether an imaged thing exists through the image. This is an extensional evidence for quantifier equivocation, different from fdrake's intensional evidence. The paper itself admits this possibility. It begins an argument: — Leontiskos

Now, if you want to say "numbers are a human invention," that seems like fair game. But there has to be some sort of explanation of their usefulness and development across disparate, isolated societies. — Count Timothy von Icarus

Another point that seems to need reinforcing is the nature of quantification. If our domain is {a,b,c} then "U(x)fx" is just "fa & fb & fc"; and "∃(x)fx" is just "fa v fb v fc". If the domain changes to {a',b',c'} then "U(x)fx" is just "fa' & fb' & fc'"; and "∃(x)fx" is just "fa' v fb' v fc'". That is, the definition of each quantification doesn't change with the change in domain; but remains a conjunct or disjunct of every item in the domain. — Banno

...quantifier variance is not meant to entail a multiplicity of logical systems, each with its own quantifiers and conception of validity, but rather it requires that, within a single logic, there should be multiple (existential) quantifiers operating differently. And so, logical pluralism should not be equated with quantifier variance, as having a choice between logical systems is not the same as having a choice of quantifier meaning within a system of logic. — Quantifier Variance Dissolved

What all of this illustrates, is that in tying quantification to existence, two distinct roles are ultimately conflated:
(a) The quantificational role specifies whether all objects in the domain of quantification are being quantified over or whether only some objects are.
(b) The ontological role specifies that the objects quantified over exist.
These are fundamentally different roles, which are best kept apart. By distinguishing them and letting quantifiers only implement the quantificational role, one obtains an ontologically neutral quantification. Ontological neutrality applies to both the universal and the particular quantifier (that is, the existential quantifier without any existential, ontological import). — Quantifier Variance Dissolved

However, once again, no variance in any quantifier is involved.

Knowing what mathematics is seems like one of the biggest philosophical questions out there. Not to mention that a number of major breakthroughs in mathematics have been made while focusing on foundations, so it hardly seems like a useless question to answer either. — Count Timothy von Icarus

Why this huge difference? — Count Timothy von Icarus

With all respect to Banno, the formula "Numbers are something we do" could use some clarification. — J

Austin carefully dismantles this argument, and in the process other transcendental arguments. He points out first that universals are not "something we stumble across", and that they are defined by their relation to particulars. He continues by pointing out that, from the observation that we use "grey" and "circular" as if they were the names of things, it simply does not follow that there is something that is named. In the process he dismisses the notion that "words are essentially proper names", asking "...why, if 'one identical' word is used, must there be 'one identical object' present which it denotes". — Wiki article

So on to ontological pluralism? — J

Nothing whatever to do with Cartesian dualism — Wayfarer

All activities have causes, right? — Count Timothy von Icarus

I don't think we've laid to rest, or explained, the doubts that Hale and Wright express. — J

I may say something true when I assert ‛there exists something which is a compound of this pencil and your left ear’, and in another, you may say something true when you assert ‛there is nothing which is composed of that pencil and my left ear’. — Bob Hale and Crispin Wright

I’m not arguing in favor of it. I’m asking why it’s even necessary. I’m questioning the claim that ‘according to our best epistemic theories, mathematical knowledge ought not to be possible.’ It obviously is possible, so what does that say about the shortcomings of ‘our best epistemic theories’? — Wayfarer

My intuition about the matter is simply that numbers are real but that they don't exist. — Wayfarer

Quine’s belief that we should defer all questions about what exists to natural science is really an expression of what he calls, and has come to be known as, naturalism.

Along the same line of thought, a number (and any other mathematical entity) is a set of neurons that form a specific structure in my brain.
— bioByron
There's a real problem with this view. If "seven" is a structure in your brain, then your "seven" is not the same as my "seven", which would be a distinct structure in my brain.

But when we each say seven is one more than six, we both mean the same thing.

Hence we must conclude that "seven" is not just a structure in your brain. Rather, it is in some way common to both you and I.

Plato answered this problem by positing a world of forms in which we both share. I think there are better answers, to do with how we use words. — Banno

Do I get a prize? :halo: — J

Along the same line of thought, a number (and any other mathematical entity) is a set of neurons that form a specific structure in my brain. — bioByron

Those quantifiers are introduced differently, and as the paper "Quantifier Variance Dissolved" notes that provides a strong argument for a form quantifier variance without a reduction of quantifier meaning to underlying entity type it quantifies over, and without committing yourself to the claim that there's a whole bunch of equally correct logics for the purposes of ontology. — fdrake

a mere difference in the domain of quantification is not enough to deliver a difference in the meaning of the quantifiers, rather a difference in the rules that govern the quantifiers would be required.