Whoa! No. That is not the conclusion you should draw. — Srap Tasmaner
This reading is inconsistent with how ∃ is actually used in mathematical texts, at least the ones I am familiar with (which would be math textbooks mostly). — SophistiCat
So by the time we get to asserting all of modern science every time you ask for the salt, you'll still be fine, because holism, right? — Srap Tasmaner
you don't mean the same thing I do by "assert". — Srap Tasmaner
I'd be thrilled if anyone in this thread were prepared to dissolve statements, assertions, beliefs, propositions and truths into one colour — bongo fury
But the configuration of prefixes '~∀x~' figures so prominently in subsequent developments that it is convenient to adopt a condensed notation for it; the customary one is '∃x', which we may read 'there is something that'. — Quine, Mathematical Logic
I’m more going on about how “all rectangles have different length legs” fleshes out to “if something is a rectangle then it has different length legs”, and we can affirm or deny that conditional statement without asserting the existence, in any ordinary sense, of any rectangles at all: a disagreement about that conditional is a disagreement about what would count as a rectangle if any such things existed, not about what kind of things exist. — Pfhorrest
This reading is inconsistent with how ∃ is actually used in mathematical texts, at least the ones I am familiar with (which would be math textbooks mostly). — SophistiCat
Can you elaborate? — Pfhorrest
I'm a little confused now, but it's probably my own fault!
I put on my "speaking for the received view" hat to address a couple of your questions, and if I'm still wearing that hat then absolutely the existential quantifier has existential import, and the universal quantifier doesn't -- it's just a kind of souped-up conditional.
If you want me to put on a "reforming logic" hat, I don't have one of those.
I do have a "logic is swell for math and generally ham-fisted dealing with ordinary language" hat and I'm almost always wearing that one, enough that I forget to take it off even when I meant to, which might have happened in this thread, I'm not sure. — Srap Tasmaner
Yours is, if an expression of math/logic is "logically equivalent" to a natural language proposition about Dodos, then....
And it isn't. — tim wood
for some value of x, [formula involving x] is true — Pfhorrest
I'd be thrilled if anyone in this thread were prepared to dissolve statements, assertions, beliefs, propositions and truths into one colour... — bongo fury
You forget the twin criteria of validity and truth. In short, whichever way you wriggle, you're always a few cards short of a full deck. And your example suffices. Whether or not aliens exist is a matter of speculation that presumably someday will be resolved to a fact. No symbolic or grammatical manipulation will alter that. As with unicorns, in translating to natural language and back, the existential quantifier is always an assumption they exist for the purpose of the argument. — tim wood
that’s what I’m advocating — Pfhorrest
This flexibility, the possibility that what is being said could be hypothetical or fiction, is absent in the logical translations of the above statements. Ex(Ux & Ox) and Ex(Ax & Lx) can only be interpreted in one way - that unicorns and aliens actually exist. — TheMadFool
Please make this explicit. How do you demonstrate that U or A is unicorn or alien?Ex(Ux & Ox) and Ex(Ax & Lx) can only be interpreted in one way - that unicorns and aliens actually exist. — TheMadFool
There's some equivalence of course, but I don't think anyone is going to convince mathematicians to quantify over expressions instead of objects. — Srap Tasmaner
And of course you trade whatever is a pain-in-the-ass about existence for whatever is a pain-in-the-ass about truth. — Srap Tasmaner
I’m not talking about the empty set issue or anything like that. I fully support the standard modern relations between “some”, “all”, and “none”. It is perfect correct in my view to take “some rectangles have equal length legs” as equivalent to “it is not the case that all rectangles have different length legs” or “it is not the case that no rectangles have equal length legs”. — Pfhorrest
I’m more going on about how “all rectangles have different length legs” fleshes out to “if something is a rectangle then it has different length legs”, and we can affirm or deny that conditional statement without asserting the existence, in any ordinary sense, of any rectangles at all: a disagreement about that conditional is a disagreement about what would count as a rectangle if any such things existed, not about what kind of things exist. — Pfhorrest
I'm a little confused now, but it's probably my own fault! — Srap Tasmaner
Ex should also be neutral on the matter of existence like its companion Ax. — TheMadFool
absolutely the existential quantifier has existential import, and the universal quantifier doesn't -- it's just a kind of souped-up conditional. — Srap Tasmaner
That use is not contrary to what I’m saying at all. In fact it’s a great illustration of the alternative reading of the “existential” operator I’m suggesting: instead of “there exists some x such that [formula involving x] is true”, I suggest “for some value of x, [formula involving x] is true”. — Pfhorrest
Are you just concerned about (not) making metaphysical commitments when we write formulas? — SophistiCat
*sigh* All right. Ex(Ax & TMFx). TMF, of course, is you. I guess I just proved you're an A. After all, that's the logic. And any variations n this theme, yes? — tim wood
Please make this explicit. How do you demonstrate that U or A is unicorn or alien? — tim wood
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