Elsewhere:
DKL: “There exist tables”
PVI: “There do not exist tables”
The deflationist: “something is wrong with the debate”
— p.3
DKL includes tables in his domain. PVI does not. Both make use of the very same rule for quantifier introduction.
No case has been presented for a variance in the quantifier introduction rules, as opposed to a variance in the domain. — Banno
DKL includes tables in his domain. PVI does not. Both make use of the very same rule for quantifier introduction.DKL: “There exist tables”
PVI: “There do not exist tables”
The deflationist: “something is wrong with the debate” — p.3
After thinking on it, it seems not to be a possibility.Never seen one. — fdrake
Indeed, since Universal Generalisation is taken as granted in first order logic. The formalisation is trivial.But you do so in natural language. So you don't care about the underlying formal logic. — fdrake
Formal logic can serve to clarify usage in natural languages. The primary case being how first order logic sets out and separates the three uses of "is" - predication, quantification and equivalence.Since if even mathematical reasoning has both ambiguity and commonality regarding the underlying logic and its quantifier introduction rules, why would we expect logic to behave as more than a prop, crutch or model of quantification in natural language? Never mind ontology! — fdrake
that is, "...for arbitrary a..." just is "for any a you might pick", or "for any a" - it's introducing a quantifier. — Banno
How would you account for people's differences in use? — fdrake
You haven't interacted with any of this. — Leontiskos
I think this is the first time that paper has been quoted in this thread. — Leontiskos
and logic...In what salient way is logic like chess? Why would we assume such a thing? Chess is just a made-up game we created to have some fun and amusement. — Leontiskos
fdrake has been consistently talking about intensional differences in quantifiers, namely by way of introduction and elimination rules for quantifiers. — Leontiskos
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". — Banno
Well, that's what I'm asking, by way of answering this:How does translation play into it here? — fdrake
I'm still wanting an example of where quantifier variation is not also domain variation. I don't think it can be done - quantifier variation just is domain variation.The challenge I see that presenting is that people can believe statements that block applications of inference rules, or otherwise not believe in appropriate inference rules. Even when two people share the theorems of the underlying logic... You just have to hope that every person innately believes every theorem. If they don't use the quantifiers in the same way, they don't have the same intensions over people. — fdrake
And my answer remains, perhaps they are talking about different domains.How would you account for people's differences in use? — fdrake
Hang on.You can introduce the quantifier onto (Pa->Qa) to get (for all x P(x)->Q(x) ) if you made no assumptions about a anywhere in your reasoning. — fdrake
I'm not sure I follow what you are suggesting here. Yes, sometimes folk make invalid inference, or fail make valid inference. But in order to recognise this, we must understand the distinction between valid and invalid inference.The challenge I see that presenting are that people can believe statements that block applications of inference rules, or otherwise not believe in appropriate inference rules. Even when two people share the theorems of the underlying logic... You just have to hope that every person innately believes every theorem. — fdrake
It might but I don't see that it does - those unbound variables make it hard to see what is going on here. But (Pa→Qa) does not allow (∀x (Px → Qx))... You've lost me.(P(x)=>Q(x) where x is arbitrary lets you derive (for all x P( x ) => Q( x ) ), and they can'd do that). — fdrake
Can you set out why or how the analogy does not work? In what salient way is logic not a game of stipulation?Logic is not just a stipulative game, like chess. The analogy doesn't work. — Leontiskos
Why doesn't it matter how you quantify or which logic you use? Isn't that of the utmost import? That there are multiple logics does not imply that they are all of equal utility or applicability. Propositional logic will be of little help with modal issues, and modal logic might be overkill for propositional problems. Some art is involved in the selection of a logic to use.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
What? For example, how could a "qualitative" difference in domain in a first-order logic lead to a difference in quantification? The quantification rules are defined extensionally....within a single logic qualitative differences of domain reflect qualitatively different understandings of quantification — Leontiskos
Seems to me that such equivocation is still about the domain. I think I showed that , above. Can you show otherwise?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
The paradigmatic example is existential generalisation, f(a)⊢∃(x)f(x). The claim is that Universal Instantiation, Universal generalisation, Existential Instantiation and Existential generalisation have differing uses in different logics. And indeed, these do vary in form from one logic to another.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'd be surprised if there were a substantive difference.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 bit of history only partially answers the question. It remains that we might move bishops anywhere we like on the board, but to do so would be to cease playing chess, or at the least to play it differently. There is a way in which the answer to "Why do Bishops move diagonally?" is, that is just how the game is played, that its what we do. Seeking further explanation is redundant.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.
What? I can't make anything of this, nor much of what follows. Talk of nominalists and universalists seems oddly anachronistic.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
Sure. I don't believe that what I have said implies otherwise. language games are embedded in the world. What was novel in their introduction is the idea that we do things to the world by using words.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's my point?What is your point? — 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
Pretending 8/2 = 5 won't get you very far. You will not be able to divide the berries between two people fairly. It will be functionally inadequate. It won't work.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
There is a bit more going on.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
If you want to say "nouns 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.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
And the conclusion to that section,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
Good questions. The property analogy will only go as far as "counts as..." or "as if...". And as I've said, we do treat numbers to quantification, equivalence and predication - all nice neat uses of "is". Numbers are in many ways not like property.Why this huge difference? — Count Timothy von Icarus