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

Although of course it does commit you to there being non-unicorns :wink:

O statements — TheMadFool

Par for the course. Obviously I meant A.

I think it's just a matter of two languages and validity and truth. And the valid statement, stripped of any content that could be false, becomes itself true because valid. E.g., Exyz: x+y=z is true, but not for the numbers 1,5,9.

Small point: I myself distinguish between false and not-true. For particular cases it doesn't really matter, but as class statements they're not quite the same thing, per Godel et al.

:up:

:up:

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

No, it's vacuously true. The suggestion you make toward the end of your post:

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

That's already what we do. You can do the proof yourself:

1. (x)(~Vx)
2. ~(x)(Vx → Bx)
3. (∃x)~(Vx → Bx) ...... 2
4. ~(Va → Ba) ............ 3
5. Va ........................... 4
6. ~Va ......................... 1

Are you just concerned about (not) making metaphysical commitments when we write formulas? — SophistiCat

Yes. — Pfhorrest

I don't think that's an issue above and beyond the old realist/non-realist divide. Non-realists simply mean something different than realists when they say "there exists x such that..." - or so they say. I am not even convinced that there is a substantive difference between these positions.

It's alright. There's no problem with the existential quantifier.

Since I've already made such a hash of things, I'll continue! (Apologies to @Pfhorrest.)

What makes me uncomfortable about the predicate calculus is that sortals aren't really like attributes, and sortals are the natural way to talk about what exists. 'To be' is substantive hungry: if you say 'X is', the question is, 'Is a what?'

We now have coyotes where I live, but we didn't when I was a kid. On first hearing them howl at night, 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. A coyote might or might not be howling -- that's a predicate; but could a coyote be a dog, or a block of cheese, or a representative democracy? And we recognize this in our grammar: 'is ...' is not the same as 'is a ...' We'll never say that a coyote is a howling, though it might be a-howlin'.

I think this is why I'm inclined to bring up the old syllogistic and the square of opposition when talk turns to existence. It feels like there's room there to make the distinction I want.

It also occurs to me that what I'm calling "sortals" here might be how we specify the domain for our variables on the fly.

If I say, 'Some dogs over there are howling,' I'm attributing "howling" to some of the individuals in the domain "the dogs over there". But suppose they're coyotes, not dogs. There may be dogs over there and they're not howling, and that would be one kind of error; but the main error seems to be picking the wrong collection of individuals to consider attributing "howling" to. That looks to me like a very different kind of mistake.

But what if there aren't any dogs over there, howling or otherwise? I've implied there are. When corrected, I might say I thought the coyotes were dogs.

It's plain enough what I mean, but the plain language of that sentence is ludicrous. I'll only add that I couldn't have said this before being informed that there were coyotes over there, and if I had known that beforehand I wouldn't have been tempted to say that the coyotes over there are dogs, and they're howling.

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

I.e., that you assigned the right predicate to the wrong things? Apparently so:

the main error seems to be picking the wrong collection of individuals to consider attributing "howling" to. — Srap Tasmaner

In which case I get:

that what I'm calling "sortals" here might be how we specify the domain for our variables on the fly. — Srap Tasmaner

... As per @jamalrob's comment and probably others.

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. Mending ruptures is difficult, and hence (maybe):

the plain language of that sentence ("I thought the coyotes were dogs") is ludicrous. — Srap Tasmaner

But then, isn't mending or patching together and reconciling domains what science is about? In which case I am surprised if,

What makes me uncomfortable about the predicate calculus — Srap Tasmaner

... is the quantifiers requiring us to make our selections in terms of predicates, define our sorts in terms of attributes. Which it does, so that the selection can be from a maximally inclusive domain. Maybe?

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

I'm not seeing what you're seeing, so maybe you can fill me in. I don't know why sortals would be especially problematic. It's still just public language, public conceptual apparatus, picking out individuals in the way a speech community does. "Dogs". "Those dogs over there." "Some of those dogs over there." What struck you as uniquely problematic about this, more problematic than what we do with predicates?

Anyway it seems natural to me that insofar as trouble arises, the parties to a conversation will negotiate through it, as my dogs and coyotes example runs right into. (I think David Lewis talks about this in Scorekeeping, which I ought to reread.) I'm just splitting the negotiation into (a) what are we talking about? and (b) what are we saying about it?

Does that seem terribly unnatural to you?

(a) what are we talking about? — Srap Tasmaner

What things, or what kinds of thing?

What things, or what kinds of thing? — bongo fury

I'm honestly not sure how to answer. I leaned on the word "about" there but I often find analysis of "about" kinda slippery.

What I have in mind is pretty minimal, just approaching quantification in the restricted way math does.

So the analysis of

Some of those dogs over there are barking.

would not be

There is something such that: it is a dog, it is over there, and it is barking.

but

There is some member of "dogs" such that: it is over there and it is barking.

How do I describe that? If I want to say I'm talking only "about" the dogs that are over there and barking -- there's nothing left to say about them! Yuck.

Honestly it feels like I want to push "over there" back into the subject, that what I'm talking about is all those dogs over there and what I'm saying is that some of them are barking.

That might work, and in a sense it's okay if our sortal isn't a natural kind, but just an ad hoc count noun.

Shrug. The ad-hoc sortal thing is appealing, but we lose some of the other stuff we might want to say, even though the analysis of the sentence feels rightish. For instance, if something is a dog, it's necessarily a dog, but if it's over there it's only per accidens over there. So natural kinds.

In my system of logic, the “over there” would be conveyed through a modal operator “at()”: it would basically be analyzed as “at there, for a non-zero number of dogs, dogs are barking”. (Or rather, accounting for the abstraction of assertive force out into mood operators, "there is, at there, for a non-zero number of dogs dogs, dogs being barking").

quantification in the restricted way math does. — Srap Tasmaner

Does it? Do you mean something involving x∈R as alluded to previously? So the drift from

∃x (x∈R, f(x) = 0) — SophistiCat

to

∃x∈R ( f(x) = 0) — SophistiCat

isn't a mere abbreviation, and ∈ a mere binary predicate? (In math?)

As you like, it's just that math never uses quantifiers that range over even all mathematical objects, much less everything in this and all possible worlds.

And just as a side effect, it's clearer that what you're asserting is that one of the reals is such that f(x)=0; you're certainly not asserting that the set of reals is non-empty -- you know it is, or you wouldn't be saying things like f(x)=0 anyway.

I'm not all that concerned about the metaphysics, but I am interested in finding the least misleading way to analyse ordinary language. "There is something that is a dog and is barking " is not it.

As you like, it's just that math never uses quantifiers that range over even all mathematical objects, — Srap Tasmaner

So thanks, because what I needed was to think of searching "range of quantification", and it turns out that @SophistiCat's drift, which I recognised as a (kind of a) thing, was into what's called "bounded quantification". Thing is, though, it is a mere abbreviation:

The existential proposition can be expressed with bounded quantification as

∃x∈D P(x)

or equivalently

∃x (x∈D & P(x)) — Wikipedia

It's an abbreviation that supports your point of view, that the range of quantification is restricted, but it doesn't resist undoing (as shown here), so that ∈ is a binary predicate and ∃ ranges over the whole domain of discourse - which in a mathematical discourse is presumably all mathematical objects, no?



Also supporting your point of view:

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

So yes, you can (informally at least) use "existence" to highlight more and more predicates (sub-domains?) applying to the specific entities you are about to assign a predicate.



And then I think I might get your whole agenda here: 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?

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

Exactly like that!

Actually I think I'm probably just recreating work Strawson did years ago ...

In my related thread on a sole sufficient quantifier, I set forth a way where this whole bounded quantification thing has to be addressed explicitly:

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

I think I've been running together a few different sorts of concerns.

I want to say that what we predicate of is not a generic unknown object but a more or less specific sort of object, an object taken as some kind of thing, or taken to be some kind of thing. In a sense, I am taking back the incredibly useful step of abstracting; that is, when we say that x*x = -1 has no real solution, but does have a complex solution, there is a step of generalizing what numbers are, abstracting away certain properties and leaving a smaller defining set, thus enlarging the set of potential solutions. (Probably not how complex numbers come about, but how they are eventually understood algebraically, I think.)

I've been thinking there are modal claims that we might want to make about sortals and predicates that are obscured by this phantom abstraction, i.e., the presumption that we predicate of a generic object. But I think that turns out to be at least a little wrong, and it shows in my posts. Yes, to be a coyote is necessarily not to be a dog. But also to be over there is necessarily not to be over here, and to be howling is necessarily not to be silent. On the other hand, none exclude the others: you can be a silent coyote over here, or a howling dog over there, and all the other variations.

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.

I also want to say that by avoiding premature abstraction, we get to save it for when we need it, as a step we take to solve a problem. Once there's doubt about the source of the howling, we might find it useful to speak in more abstract terms like "the source of the howling".

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

In a sense, one could say that when a coyote dies, that matter stops doing whatever it was that constituted being a coyote. The matter still exists, it just stopped doing something. To be is to do.

Yeah that's worth considering, but it looks like a category mistake to me.

@bongo fury @Pfhorrest

Last post resorted to heavy use of tenses, and maybe it turns out this is the most obvious difficulty in "applying" classical logic to everyday speech, a difficulty not faced in mathematics.

Goodman famously connects natural kinds to tense through projectibility, but I haven't looked at that in a long time.

I'm torn now between wanting to find a way to distinguish two types of "predicates", say by sortals being tenseless (the difference between being a coyote and howling); or just distinguishing the role in a sentence or an assertion. Maybe I get what I want just by saying this part of the sentence determines the subject -- these predicates constitute a sortal, what we're talking about -- and this other part is what we say about the subject.

I'm trying to think how Goodman and Elgin might solve your dogs vs. coyotes problem...

Btw though:

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

Is the aforementioned problem (and other ordinary talk) posable as a finite one, and would it help?

Also:

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

:chin:

the aforementioned problem (and other ordinary talk) posable as a finite one, and would it help? — bongo fury

If the plan is just to cash out talk that relies on "universals", broadly construed, into talk that doesn't, because it just uses names to refer to individuals, then my mistake of thinking it's dogs howling and not coyotes seems to just drop out. If that's not a big enough problem -- I have clearly made a mistake we want to be able to point at -- how am I supposed to name the individuals in order to make this translation? If not in practice, then in principle I should be able to do so. If I were to try -- "let's go see!" -- I can land right in a de dicto/de re swamp, of a sort Quine talks about somewhere with propositional attitudes: I am looking for a dog that is howling.

(( Right now reading Sellars's "Grammar and Existence: A Preface to Ontology" -- if I understand it, I may report back. ))

I think that the existential quantifier is basically just an infinite OR operator, just as the all-quantifier is basically an infinite AND operator. For every fixed function f which sends each thing þ (all things are actual and abstract and thus eternal btw.) 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 ... . So when we say that odd even rimetales (numbers) don’t exist, we mean that it’s not the case that (1 is odd and even) OR (2 is odd and even) OR (3 is odd and even) OR ... . We don’t mean that there are things called “odd even rimetales” which don’t exist. The sentence “there are things called ‘odd even rimetales’ which don’t exist” already contains a contradiction. That which doesn’t exist wouldn’t have any properties on one hand (since it doesn’t exist), but would have the property of not-existence on the other. At least that’s when we use ∃ when talking about abstract things.

When we talk about concrete stuff (be it mindly or physical), we often use ∃ to mean the existence of a piece of information, which is equivalent to the truth of a proposition. See that comment of mine for more info.

I think you’re on the right track there, and my own thoughts on that same track show up in my names for my proposed versions of “existential” quantification and disjunction: for-some(x,F(x)) and some-of(a,b,c,...).

(Where “some-of(a,b,c,...)” just means “a or b or c or ...”).

I actually construct “for-some()” explicitly from a more general “for()” function with an embedded “some-of()” function; and I similarly construct “for-all()” out of that same “for()” function with an embedded “all-of()”, which in turn is my version of conjunction.

(Where “all-of(a,b,c,...)” just means “a and b and c and ...”).

That does sound similar to what I think.

There’s just one problem with regarding all-quantification and there-is-quantification as mere infinitary AND and OR operators, respectively. I’ll quote myself but make certain important words italic:

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

Also, note that for every property E, “∃x:E(x)” means the disjunction of all propositions of the shape E(x), and “∀x:E(x)” means the conjunction of all propositions of the shape E(x), where “x” is a variable that varies over all things.

We see that the very definitions of the two quantifiers need allness, and the latter cannot be reduced to AND alone. Moreover, there has to be at least one existing thing, for otherwise, the variable couldn’t vary over anything, and there wouldn’t even be a variable to vary (for it is a thing, too). So, we can’t reduce allness and existence to anything else, but we only need to invoke them once (in the definition of the quantifiers or the definition of families indexed by all things), after which we need only deal with infinite AND (or infinite OR; the two can be defined in terms of each other and negation).

That’s at least my take on this thing.