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More primitive: restricted or unrestricted quantification?

More primitive: restricted or unrestricted quantification?

muxol
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Posted 01/03/06 - 05:36 PM: Subject: More primitive: restricted or unrestricted quantification?	#1
When we quantify over some class of things we usually implicitly restrict that class to a subclass of all things. E.g. when I say "There's nothing in this fridge" I mean e.g. that there is nothing I wish to eat in the fridge. Formally we have 1. ~(Ex:x is wished to be eaten by me)InFridge(x). The unrestricted meaning would be that there really is nothing in my fridge---no atoms, molecules, etc. which would be paraphrased 2. ~(Ex)InFridge(x). Of course if we were to really translate (1) into unrestricted notation we would use 3. ~(Ex)[Infridge(x) & Desirable(x,me)]. But the second conjunct occurring in (3) is implicit when we speak, so why make it explicit in the sentence (outside of restricting the quantifier) when we symbolize it? Are we always implicitly restricting quantification whenever we speak of something or everything? Is the case different for quantification involving 'anything'? If so, why? (For those formally inclined, which is more formally primitive---restricted or unrestricted quantification? See Barwise and Cooper.)

Owen
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Posted 01/04/06 - 05:31 AM:	#2
muxol wrote: When we quantify over some class of things we usually implicitly restrict that class to a subclass of all things. E.g. when I say "There's nothing in this fridge" I mean e.g. that there is nothing I wish to eat in the fridge. Formally we have 1. ~(Ex:x is wished to be eaten by me)InFridge(x). The unrestricted meaning would be that there really is nothing in my fridge---no atoms, molecules, etc. which would be paraphrased 2. ~(Ex)InFridge(x). Of course if we were to really translate (1) into unrestricted notation we would use 3. ~(Ex)[Infridge(x) & Desirable(x,me)]. But the second conjunct occurring in (3) is implicit when we speak, so why make it explicit in the sentence (outside of restricting the quantifier) when we symbolize it? Are we always implicitly restricting quantification whenever we speak of something or everything? Is the case different for quantification involving 'anything'? If so, why? (For those formally inclined, which is more formally primitive---restricted or unrestricted quantification? See Barwise and Cooper.) imho, I think we usually assume a restricted universe of discourse, such as: physical things, humans, real numbers, edible foods, etc. When we do not restrict the universe then we are talking about any existent thing. In Ex(x>2), x is restricted to numbers, implicity. In Ex(x is a tree), x is restricted to physical things, implicitly. The predicates used govern the (implied) universe of discourse that makes sense. For example: (muxol > 2) has no sense, (5 is a tree) has no sense, but, muxol=muxol and 5=5, both have sense. x=x, is sensible for all existent x's in all universes of discourse, hence unrestricted. What is restricted from 'all universes of discourse' is described objects which do not refer. Non-existent things are not values of any variable.

muxol
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Posted 01/04/06 - 06:06 PM:	#3
Hi Owen, I think your restriction to existent things is too strong--i.e. regarding x=x. I can assert that redness=redness without accepting the existence of a universal, redness. You say "What is restricted from 'all universes of discourse' is described objects which do not refer", but I think you mean that descriptions that do not refer. Described objects always exist, if they are objects at all, and generally objects do not refer, unless they are linguistic. In any case, this point about irreferential terms is not the issue. What I am asking is if it is more appropriate to use restricted quantifiers as basic rather than following the standard of using unrestricted quantifiers and modifying the formulas appropriately. It seems more faithful to at least Aristotelian logic that we take restricted quantification, not unrestricted quantification, as basic.

select
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Posted 01/04/06 - 09:32 PM:	#4
I think you have to restrict it, unless you want to leave the expression ambigious. You can restrict either by assuming a domain of discourse or through predication as you just said. I don't agree that the predicate of treeness is so obviously physical, you're assuming that all trees are physical which clearly isn't the case. If we don't restrict the domain of discourse to physical things to begin with, you could easily be talking about imaginary trees, cartoon trees, concepts of trees, and so on. I think Aristotle dealt with this sort of ambiguity with his ontological category scheme, so maybe that's what you need muxol.
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AKG
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Posted 01/05/06 - 12:51 AM:	#5
select wrote: I think you have to restrict it, unless you want to leave the expression ambigious. You can restrict either by assuming a domain of discourse or through predication as you just said. I don't agree that the predicate of treeness is so obviously physical, you're assuming that all trees are physical which clearly isn't the case. Out of curiousity, what was the first thing you thought of? Indeed, you could be talking about imaginary trees, or cartoon trees, but then you'd probably say so. The first thing that comes to mind when someone says "trees" is that we're talking about physical objects called trees. I.e. I think that given it is implicit, it is relatively clear that mention of "trees" restricts discussion to that of physical things. So when Owen said, "Ex(x is a tree), x is restricted to physical things, implicitly," did you immediately think that we might be talking about cartoon trees, or did you implicitly assume we meant physical trees, and then later thought that we might also consider cartoon trees when you consciously looked for possible exceptions to his restriction?
"The only reason we die... is because we accept it as an inevitability." -- Stewie "To enslave nuance to dogma is folly." -- Lord Hillyer

select
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Posted 01/05/06 - 10:30 AM:	#6
AKG, the point is to remove that ambiguity when/if we translate it into formal language, which is what I thought this thread was about. It's true that we usually think of physical trees, but unless you can provide a better argument, that needs to be considered an assumption and dogma that all trees are physical. If you look at different religious or metaphysical worldviews, they would reject just this assumption. Part of what I was getting when I was trying to argue for modal ontology earlier is that there are certain "special" predicates that can't be defined descriptively or functionally or anyway else (that is, in terms of other properties). They include predicates such as physical, concept, imaginary, and fictional. muxol writes: muxol wrote: The unrestricted meaning would be that there really is nothing in my fridge---no atoms, molecules, etc. If you go farther in this direction, you also have to rule out conceptual and imaginary things. If you want to remove ambiguity, then you have to also specify a mode of being (such as physical, conceptual, imaginary, and so on). As to why should we remove ambiguity, simply for the other reasons we desire to remove ambiguity: so different people are indeed talking about the same thing. Yes, in certain modes of discourse the domain is specified, although I doubt it is as clear as people often assume. I've been working on a syllogism lately: In any form of art, nothing created can faithfully mimic or recreate the original experience. Experience here, depending on the work of art, can mean an emotion, a place, an object, or so on. A painting of the grand canyon is always less than the grand canyon. A literative description of a city is always less than the city. For everything that qualifies as art, it is in addition always an expression of something else. Nothing from the artist comes ex nihilo but is always in some way an expression of his own experiences, desires, or feelings. Therefore, all art is a pale imitation of the original experience of the artist. I'm not done with it because I'm stuck on matters of domain. It's a genuine philosophical argument, not unlike one that Plato might offer, but there are technical problems by which I can't properly call it a valid syllogism. For instance, do works of art have to be physical? Can an emotion be a work of art? And it just wouldn't do to say, "Well, do I usually associate works of art as emotional things?" Well, if a work of art brings forth certain emotions, then those emotions could also be considered a part of the work of art. And that would falsify my syllogism. See? I'll let muxol answer whether this is an answer to his question or not, or whether I've brought this discussion onto a tangent, which isn't uncommon for me
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AKG
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Posted 01/05/06 - 02:37 PM:	#7
It's true that we usually think of physical trees, but unless you can provide a better argument, that needs to be considered an assumption and dogma that all trees are physical. It is not so say that all trees are physical, it is to say that a sentence about trees is implicity restricted to physical trees. That is, the restriction to physical trees is primitive, and the inclusion of imaginary trees, conceptual trees, etc. is contrived. So the question can be asked another way: Is EVERYTHING the natural, primitive domain of discourse, where subsets such as "physical things" or "things muxol would like to eat" are meaningful, yet contrived domains? OR Are restricted domains primitive, and when we speak of EVERYTHING, we should properly regard this as a contrived union of primitive domains? We have restricted domains, and the unrestricted domain. Restricted domains are parts of the restricted domain. Which is more primitive, the parts or the whole? It might seem, at least it does at first glance, that natural sentences are implicitly restricted, and symbolic sentences are implicitly unrestricted. This might suggest that the primitive interpretations of natural sentences restricts the domain, but, for some reason, when we go to symbolize those sentences, the primitive interpretations of those symbolic sentences don't restrict to domain, so an additional predication must be made explicit. I think that what really happens is that natural sentences pretty much always come with a context, and it is this context which potentially bears the restriction to the domain, and not the sentence itself. Morever, contexts almost always do restrict the domain. I think we can see easily that restriction is a matter of context, and not a property of the sentence. If I said, "go find yourself something to eat," would suggest a context different from, "can you find all the Easter eggs I've hidden around the house." As such, "there's nothing in the fridge," would have two different "natural" interpretations. Also, I don't think a separate analysis is required for symbolic sentences. If you told me "go get yourself something to eat," and I said to you "~(∃x)In(x,fridge)" you'd give me a (or maybe a :rolleyes, but you'd still know that I meant that my quantification was restricted to things I like, and did not include atoms and imaginary trees, etc. As a better example, I might write something in a math assignment, "(∀X)(forallY)(\|XY\| = \|X\|\|Y\|)". If it were an assignment about complex numbers, quantification would be implicitly restricted to complex numbers, it would go without saying. If it were about groups, X and Y would implictly be restricted to groups by context, it would again go without saying. So far, I would say: 1) Restriction is depends on context, not the sentence, 2) There is no special difference between natural sentences and symbolic sentences with respect to tendencies of quantification restriction But we can still ask 3) Are contexts with restricted domains primitive, or are contexts with unrestricted domains primitive (or neither)? 4) Does every sentence need a context? 5) If not, is a sentence without a context unrestricted "by default"? I think the answer to 3 is neither. If someone says, "everything is self-identical" then he clearly means EVERYTHING, and moreover, it doesn't seem like this type of sentence can't be primitive, i.e. that it would have to be broken down into a conjunction of sentences, each quantifying over a restricted domain. That is, the notion of everything in general is something we can understand on its own, and doesn't need to be understood in terms of other things. On the other hand, if someone says, "everything in the fridge has meat in it," he's not including the racks, or light bulbs, etc. And I think we can treat this interpretation as primitive. That is, the context where everything is included in the domain of discourse isn't more basic than one restricted to food. For 4, I think the answer is 'Yes'. Otherwise, it is just a set of symbols. So there's no need to answer 5. Note that sometimes, we are not given a context, so we cannot exactly tell what was meant. However, we can sometimes guess it. Or, if the statement is something like a logical truth, then it might not matter what the context was supposed to be. Any guess will do. But there still has to be one.
"The only reason we die... is because we accept it as an inevitability." -- Stewie "To enslave nuance to dogma is folly." -- Lord Hillyer

Owen
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Posted 01/07/06 - 09:50 AM:	#8
muxol wrote: Hi Owen, I think your restriction to existent things is too strong--i.e. regarding x=x. I can assert that redness=redness without accepting the existence of a universal, redness. Hi muxol, I don't agree, non-existent things are not self identical. Non-existent things are not included in quantification, only existent values of variables are included. For example: (AxFx -> Fa) iff, a exists. Ax(x=x) is true, but when descriptions are used A(x=x) -> (ix:Fx)=(ix:Fx), only if E!(ix:Fx). That is: (the x:Fx)=(the x:Fx) <-> Exists(the x:Fx). (the x:Fx)=(the x:Fx) is false if (the x:Fx) does not refer to an existent object. You say "What is restricted from 'all universes of discourse' is described objects which do not refer", but I think you mean that descriptions that do not refer. Of course. Descriptions are described objects.

Owen
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Posted 01/07/06 - 02:48 PM:	#9
AKG wrote: Out of curiousity, what was the first thing you thought of? Indeed, you could be talking about imaginary trees, or cartoon trees, but then you'd probably say so. The first thing that comes to mind when someone says "trees" is that we're talking about physical objects called trees. I.e. I think that given it is implicit, it is relatively clear that mention of "trees" restricts discussion to that of physical things. So when Owen said, "Ex(x is a tree), x is restricted to physical things, implicitly," did you immediately think that we might be talking about cartoon trees, or did you implicitly assume we meant physical trees, and then later thought that we might also consider cartoon trees when you consciously looked for possible exceptions to his restriction? Yes. Commonsense must prevail. When we talk about physical perceptions, we do not include 'illusions'.

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