Of course, it's supposed to be confusing, one is supposed to ponder it in its aporia.
I didn't consider a naming operator. Objectual quantification, which I've taken Quine at his word that it's the standard kind, does not require that everything in the variable's range be indicable, able to be singled out, by names, definite descriptions, or, for that matter, trails of physical evidence, much less that the singling out need to be registrable on a common central list.
I just noticed that by the simple expedient of replacing the existential particular's "or" with "and" in the expansion, one gets a quantification that seems paradoxical when treated as objectual, such that one wonders what it could mean outside the context of a regimented system of names, descriptions, & objects, and what sort of limits, short of such a regimented system, one would have to impose in order to get something meaningful out of it. Those happen to be like some of the questions that arose in my mind when I first encountered the claim that "everything exists." Does it mean that everything that is "legitimately" indicable, by any sign or evidence such as would be grasped by an observer, exists? If the indicated thing seems to contingently or necessarily not to exist in "our" universe, for instance, where an indication of it nevertheless appears, then must it exist in some other universe? Can something impossible in our universe be "legitimately" indicated by something in our universe? Penrose talks about how not only counterfactuals but impossibles could be factored into particle behavior.
Then there's an added consideration that, if this objectual "omniversal" quantification is confusing, why doesn't its close relative, the objectual existential particular, seem anywhere near as confusing? Are there similar issues there, only in concealed form? Probably such "similar" issues are held legitimately at bay, but it seems strange, anyway.
Ben Udell
----- Original Message -----
From: "Brent Meeker" <meekerdb.domain.name.hidden>
To: <everything-list.domain.name.hidden>
Sent: Monday, March 06, 2006 4:45 PM
Subject: Re: Why is there something rather than nothing?
Benjamin Udell wrote:
> In standard first-order logic, the phrase "everything exists" would be taken to trivially mean "“that, that is, is," or the like. Is there a way to say it in a non-trivial sense in first-order logic at all? Is it an idea that can be logically expressed at that basic level? What would it mean if it can't? I'm not a logician, but there does appear to be a way to say it in a specially restricted kind of first-order logic, by use of a special kind of quantificational functor. As for whether this leads to a coherent logical idea in less restricted logic, you be the judge. The result is, at least, a kind of statement which seems to lead to an area of logical issues raised by the "Everything Exists" picture, in any case, with regard to saying that every "potential" particular definite individual is actualized somewhere and somewhen, or the negative, that the world in all times and places lacks some particular definite individual. Now, in defining the existential particular quantification, one may start with a finite universe of objects named by constants "a" through "h", and say “There is a such that...Ja...or there is b such that...Jb...or... [etc.] ...or there is h such that...Jh....” and agree to write this as "Ex ...Jx...." Then one drops the substitutionalist requirement that x shall range over only named objects a, b, c, etc. Then the variable x is no longer _substitutional_ but instead is _objectual_. To get to our new special functor will be a matter of replacing the repeated "or" with a repeated "and". Let’s define a functor "Æ" such that "Æx ...x...." is equivalent to "There is a such that...a...AND there is b such that...b...AND... [etc.] ...AND there is h such that...h...." In effect one is saying that every name names something. Now, what happens when the substitutionalist requirement is dropped? In considering just what it is that x now ranges over, and whether the objectual statement "Æx ...x..." is contingently or formally true or contingently or formally false or formally or contingently undecidable or (despite its fraternal-twin relationship with the existential particular) just plain ill-defined,
To me this seems confusing because you're not distinguishing between names and objects. Let yNx be the relation "y names x". Then your functor is equivalent to Ex("x"Nx). But that is really no different that Ex(yNx) unless you postulate that there is an operator " " that produces a canonical name of an object, i.e. given x then "x" is (by construction) a name for x. But if you use such a naming operator then you've already assumed that every name so generated names something. The question is the converse; whether every name names something. The usual way of addressing this is to reinterpret the name as a definite description: "Did Moses exist?" -> "Did a person who took the ten commandments from God exist?"
>one is led to consider some of the logical problems which arise in any case in entertaining the general idea that “everything exists.” In other words, we seem to arrive at some of the right problematics. Then if you negate it, you're saying that there lacks a something, some particular thing is failing to exist. If you say "~Æx Jx," you're saying that something's missing or it exists but isn't J (e.g., but isn't jumping). So you could say "[AxJx] & ~[ÆxJx]" (Note: "Æx" should NOT be called the "existential universal" which would instead be properly applied to whatever is equivalent to the conjunction or predicative combination of the existential particular and the hypothetical universal, where you say, e.g., "there's some food that’s good, and any food is good" or "there's some food that's good such that any food is good" or “there’s food and any food is good” I suppose that "Æx” could be called the "omniexistential."
There are also logics in which existence is treated as a predicate instead a quatifier, e.g.
http://en.wikipedia.org/wiki/Free_logic
Brent Meeker
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Received on Mon Mar 06 2006 - 17:42:42 PST