Are you just concerned about (not) making metaphysical commitments when we write formulas?
— SophistiCat
Yes. — Pfhorrest
But, the million dollar question is, Does the existential quantifier, ∃, need to make a distinction between fact and fiction?
— TheMadFool
No. — bongo fury
tl;dr: 'Everything is a unicorn' and 'Something is a unicorn' both commit you to there being unicorns. 'Nothing is a unicorn' doesn't. — Srap Tasmaner
I could make the statement, X = "all vampires are bloodsuckers" = Ax(Vx -> Bx). Is X true/false? There are no vampires, at least to the extent we're aware, and so the statement X is false. — TheMadFool
universal statements like "all vampires are bloosuckers", because they're translated as hypotheticals (if...then...) can be assigned the truth value TRUE even when the subject term, here vampires, is empty. — TheMadFool
Are you just concerned about (not) making metaphysical commitments when we write formulas? — SophistiCat
Yes. — Pfhorrest
I might remark, 'Listen to those dogs howling.' If someone else tells me, 'Those aren't dogs; they're coyotes,' they're not telling me I assigned the wrong predicate to an object, they're telling me I picked the wrong sortal. — Srap Tasmaner
the main error seems to be picking the wrong collection of individuals to consider attributing "howling" to. — Srap Tasmaner
that what I'm calling "sortals" here might be how we specify the domain for our variables on the fly. — Srap Tasmaner
the plain language of that sentence ("I thought the coyotes were dogs") is ludicrous. — Srap Tasmaner
What makes me uncomfortable about the predicate calculus — Srap Tasmaner
But "specifying a domain" is flagging up a likely rupture of your individual discourse from the wider "web" :wink:, e.g. your specified domain might be fictional, or (per the OP?) hypothetical, or for other reasons resist identification with any more widely recognised domain. — bongo fury
What things, or what kinds of thing? — bongo fury
quantification in the restricted way math does. — Srap Tasmaner
∃x (x∈R, f(x) = 0) — SophistiCat
∃x∈R ( f(x) = 0) — SophistiCat
As you like, it's just that math never uses quantifiers that range over even all mathematical objects, — Srap Tasmaner
The existential proposition can be expressed with bounded quantification as
∃x∈D P(x)
or equivalently
∃x (x∈D & P(x)) — Wikipedia
A [..] natural way to restrict the domain of discourse uses guarded quantification. For example, the guarded quantification
For some natural number n, n is even and n is prime
means
For some even number n, n is prime. — Wikipedia
you want to separate the process of identifying and setting up a target (which is to be the subject for a forthcoming predication) from the actual predication, the 'firing' of the predicate at that target. Anything like that? — bongo fury
The for() function that takes three arguments, the first being a set of values that some variable can take to satisfy some formula, the second being that variable, and the third being that formula. (This would then be read as "for [these values of] [this variable], [this statement involving that variable] (is true)").
This replicates some of functionality of another function frequently used together with the traditional quantification operators, ∈, which properly indicates that whatever is on the left of it is a member of the set on the right of it, but together with the existential operators is often used to write things like
∀x∈S...
meaning "for every x in set S...", meaning that only the members of S satisfy the formula to follow. Expressions like the usual
∃x∈S...
(meaning "for some x in set S...") can also be formed, with this function, by using the equivalent of an "or" function on the set in the first argument of for(), to yield an expression meaning "some of this set". — Pfhorrest
But there is still a difference right? If you're a coyote, you're always a coyote, wherever you go and whatever you do. If you're howling, that's a temporary state. When a coyote dies, there is one less coyote in the world, though others are probably added. When something stops howling, does a howling thing blink out of existence? There is one less howling thing in the world, okay, but wouldn't we rather just say that fewer of the things in the world are howling? And same for the converse: it's not that a thing that is howling springs into existence; one of the things already here begins howling. If that thing is a coyote, it was already a coyote, and doesn't begin being a coyote at the same time it begins howling. — Srap Tasmaner
A place where we see a more trivial side of ontological relativity is in the case of a finite universe of named objects. Here there is no occasion for quantification, except as an inessential abbreviation; for we can expand quantifications into finite conjunctions and alternations. Variables thus disappear, and with them the question of a universe of values of variables. And the very distinction between names and other signs lapses in turn, since the mark of a name is its admissibility in positions of variables. Ontology thus is emphatically meaningless for a finite theory of named objects, considered in and of itself. Yet we are now talking meaningfully of such finite ontologies. We are able to do so precisely because we are talking, however vaguely and implicitly, within a broader containing theory. What the objects of the finite theory are, makes sense only as a statement of the background theory in its own referential idiom. The answer to the question depends on the background theory, the finite foreground theory, and, of course, the particular manner in which we choose to translate or imbed the one in the other. — Quine: Ontological Relativity
The only tenable attitude toward quantifiers and other notations of modern logic is to construe them always, in all contexts, as timeless. — Quine: Mr Strawson
the aforementioned problem (and other ordinary talk) posable as a finite one, and would it help? — bongo fury
For every fixed function f which sends each thing þ [...] to a proposition f(þ), ∃ sends f to the proposition that f(1) OR f(2) OR f(exponential function) OR f(evenness) OR f(f) OR ... . — Tristan L
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