Sunday, January 29, 2006

Of Unicorns and Possible Worlds

A few posts back I suggested that there might be two senses of "exists": (1) a "there is" sense that functions linguistically as a second-order predicate (i.e., it says of a concept or predicate expression that its extension is non-empty), and (2) an "is actual" sense that functions linguistically as a first-order predicate of individuals.

I tried to support this distinction by appealing to what seem to be noncontroversial cases of possible but non-actual things (like dinosaurs, unicorns, the WTC, etc.). Consequently, it seems right to say that "There are possible things that do not exist." But for that to make sense, what is affirmed by "there are" cannot be what's denied by "do not exist". With two senses of "exists", however, this can be read without contradiction as saying "There are possible things that are not actual."

Commentator Ocham disputes my logic. He thinks I'm guilty of fallaciously reasoning from

(1) Possibly, there are unicorns.
(2) There are possible unicorns.

Perhaps I am guilty of some simple mistake like this, but I don't think so. In the interests of furthering discussion I'm going to try and restate my argument in more precise logical terms using possible worlds semantics.

Let A be the actual world, let U be a possible world accessible from A, and let U contain unicorns. Then, the following are true:

(3) It is not the case that unicorns are actual. (b/c A does not contain unicorns)
(4) It is possible that unicorns are actual. (b/c U does contain unicorns and is accessible from A)

I take (4) to say the same thing as (1).

Notice, we have here two distinct possible worlds, A and U. But if we can have two of a thing, then we can quantify over it. So let's quantify over possible worlds. The usefulness of this is that it allows us to restate (4) without using any modal qualifiers.

Let w be a variable that ranges over possible worlds, and let x be a variable that ranges over all individuals within a possible world. Also, let W_ stand for "... is a world", let U_ stand for "... is a unicorn", and let _E_ stand for "... exists (is actual) in ...". In these terms, (4) states:

(4') (∃w)(Ww & (∃x)(xEw & Ux))

which is just a generalization of the claim that U is a possible world that contains unicorns:

(5) (∃w)(Ww & w=U & (∃x)(xEU & Ux))

Ex hypothesi, the quantifiers (∃w) and (∃x) cannot mean "is actual" because we're quantifying over possible worlds and their contents without implying anything about whether they are actual or not. The predicate _E_, however, denotes world-relative actuality, which in the actual world A reduces to actuality simpliciter (i.e., xEA ↔ x is actual).

In light of the foregoing, my argument for two senses of "exists" can be restated as follows: We can quantify over non-actuals. Hence, if, in Fregean terms, existence is simply the denial of the number nought, there must be a sense of "exists" that does not imply actuality. On the other hand, we need to distinguish between actuals and non-actuals. Hence, we need a sense of "exists" that does imply actuality.


At 1/30/2006 9:10 AM, Blogger Tom said...

Alan: I take (4) to say the same thing as (1).

Tom: I see your point. Totally agree. I just can’t get away from ‘tense’. Maybe that’s my problem. When we say U is ‘accessible’ from A, I understand A can ‘get to’ U or A can ‘become’ U, and to me that seems to introduce tense into the equation; it just means that (1) and (4) mean:

(5) Unicorns might become actual,

which is just to recognize the following are conjointly true,

(6) Unicorns are not now actual
(7) The existence of unicorns is logically possible.

Combining (6) non-actuality with (7) logical possibility (viz., ‘possible-but-not-actual’) gets you “might become actual.”

Once the inescapability of tense is recognized and let go in possibible worlds semantics, it’s easy to see the distinction you make between two senses of ‘exist’. The distinction is made everyday by people everywhere who talk about possible-but-non-actual and possible-and-actual entities.

An interesting question might be, What grounds the intelligibility of ‘there are’ in the first sense? Or rather, what grounds the truth of (1)? It has to be some actual entity, some way the world now is that defines its openness to becoming a world in which unicorns become actual. And THAT actual reality is what is behind the intelligibility of your first sense of ‘exists’ as a second-order predicate. This might be what Ocham misses, namely, that there is actuality behind every ‘possible-but-non-actual S’ we discuss. It’s just that this grounding actuality is not S but some other actuality G that makes it true that S might becomem actual.

Am I close?

Alan: In light of the foregoing, my argument for two senses of "exists" can be restated as follows: We can quantify over non-actuals. Hence, if, in Fregean terms, existence is simply the denial of the number nought, there must be a sense of "exists" that does not imply actuality.

Tom: Maybe it’s just that some prefer one way over another to express the same thing, but saying we can “quantify over non-actuals” looks very much like a denial of presentism, which I understant entails the claim (which I actually ran across in reading some presentist but can’t now remember) that among all the entities over which we are able to quantify, there is not a single non-present, non-actual entity.

Here’s how I’d try to express it. When we quantify of possible-but-non-actual worlds what we are really doing is explicating our own actual world in its totality, since what our actual world A is is what grounds the possible-but-not-actual worlds we describe. They are all ‘accessible’ from A (either in the sense that they might yet become actual or in the sense that they might have become actual but cannot now become actual--thus the inescapable reality of tense). What I mean to say is that quantifying over non-actual worlds is in truth nothing more than describing what our actual world ‘might have’ or ‘might yet’ become. But saying we “quantify over non-actual worlds” is confusing to me as a presentist. We quantify over actualites, but the actualites we quantify over are themselves either causally open or causally closed with regard to future becoming. They possess attributes (actual attributes) or dispositions that are indetermiate with respect to how they will, will not, or might/might not change in the future. It’s this we quantify over when we posit future possible worlds. And it’s also this we quantify over when we posit what might have but cannot now become (to which your paper about how past-tense truths are grounded speaks).

In short—we can only quantify over present actualities (that much is presentism). But when we do so we are quantifying over all they actual are, which in some cases means quantifying over their determinate tendencies and in others over their tendencies toward indeterminacies—thus possible worlds. And thus we see the sense in which ‘actuality precedes possibility’.


At 1/30/2006 10:39 AM, Blogger Ocham said...

Alan, I have replied to your posting with another complete posting here.

At 1/30/2006 5:01 PM, Blogger Alan Rhoda said...

Hi Tom, Ocham,
I'll try to get back with you both tommorow. I promised my wife I'd take her dancing tonight.


At 1/31/2006 9:33 AM, Blogger Tom said...

Found that quote I referred to. It’s Markosian (in his contribution to Oxford Studies in Metaphysics). He writes:

According to Presentism, if we were to make an accurate list of all the things that exist—i.e., a list of all the things that our most unrestricted quantifiers range over—there would be not a single non-present object on the list. Thus, you and I and the Taj Mahal would be on the list, but neither Socrates nor any future grandchildren of mine woujld be included.

It has been an objection to presentism that it’s unable to provide an adequate explanation for the truth conditions of past- and future-tense propositions. Since past and future objects are not present realities, and since only present realities can function as grounds for propositions, the objection is that presentism can’t adequately account for such truths.

How presentists account for past- and future-tense truths is another issue, but it seems clear to me that we don’t do so by positing the past and future objects described in such propositions. On presentism, they don’t exist, aren’t real/actual, and so can’t be quantified over. How does a presentist DO possible world semantics then? But grounding the truth of past- and future-tense propositions in present reality. But that has to mean, Alan, that present reality is what we quantifying over, and all that we can quantify over. But in my view, that’s perfectly fine.

A word about ‘accessibility’. One has to explain this a bit. I’m actual in this world, world A. But there’s a possible world U, let’s say, in which I don’t exist because I was never born. How is U accessible from A? Certainly not in the sense that A can become U. Is it just ‘logically’ accessible? What’s that even mean? This again is where tense can’t be avoided (and why I don’t like saying “Possibly, unicorns exist”). When we say there’s a possible world U in which I don’t exist, what we mean is that my existing in world A is contingent, that Feb. 25, 1960 might not have seen my birth, and that had I not been born the world would have gone on without me and without any of the consequences my existing is necessary to. This world we can call possible world U. It’s accessible to us in the sense that we can imagine the world’s going on after 1960 had I not been born. But the truth conditions for “U is a possible world,” aren’t U, they’re A. And it seems to me to be A that we quantify over when positing “U is a possible world." If we’re talking merely logical possibility, then “U is a possible world.” But if we’re talking concrete possibilities, worlds that can yet become actual, then “U was a possible world.” I love tense, and I don’t like doing possible world semantics without it.

I’m repeating myself. And I gotta run!


At 1/31/2006 10:27 AM, Blogger Tom said...

On his site, Ochum replies to Alan with:

I argued that

(1) It is possible that there are unicorns.
does not imply
(2) There are unicorns

Alan disputes this…

I fail to see how Alan disputes this. Your (1) posits the possibility of unicorns. Your (2), unqualified as it is, posits the actual existence of unicorns. But Alan would never argue that whatever is possible is actual. Possible S doesn’t imply actual S. Alan doesn’t dispute this.

Ochum says:
If, on the other hand, we translate it [Alan’s revised argument] as

(3') There is a possible-world, and there are unicorns, such that the unicorns are in that world

then this does logically imply there are unicorns.

Again, it doesn’t imply that “there are unicorns” if by the latter you mean “there are ‘actual’ unicorns.” All it implies is the ‘conceivability’ of unicorns, the meaningfulness of imagining them. It implies only that unicorns don’t represent something that ‘could not possibly’ exist under any conditions. Don’t you ever imagine what the world might be like had you never been born? To attempt to do so is an exercise in possible worlds theory. You do it. We all do it. But the possibilites we contemplate are not ‘actual worlds’ by virtue of being ‘possible ways the world might have been’. We might attempt to imagine a world in which the law of non-contradiction is false, where square wheels roll up hills defying the laws of gravity, where married bachelors live, etc. Is such a world a ‘possible world’? Most agree it is not. But how do we say it's not a meaningful way the world could have or could be if we don't find it meaningful to employ the category of "possible-but-not-actual"? But we can and do employ the category. That is, we can and do in fact imagine ways the world might have been or might yet be, which ARE ‘logically consistent descriptions’ even if they are not ‘actual’ ways the world, our world, the only world that actually exists, is.

So there’s no question that we do in fact think in such ways, imagine such possibilities, and find it meaningful to do so. Alan’s point, I think, is that we can only do this if in fact we do (and I think we DO) use “there is” or “there exists” in different ways. One use we have for these words is obviously to make a claim about what is actually, concretely the case about the world we live in. But that’s not the ONLY way we use these words. If we really do imagine meaningful ways the world might have been or might yet be, then we must be using the words “there is/are” to describe those ways while recognizing their non-existence.

Personally, I don’t care for atemporal possible-world semantics. I understand the need to have a language-game where we can go and talk just about logical necessity and possibility. That’s the standard language for possible worlds’ talk. But it runs into problems and misunderstandings of the sort you’ve brought up, which temporal langauge would solve for the most part. And it makes it very difficult for untrained philosophers (like me!) who might have something meaningful to contribute to engage in the conversation. But put tense back in and you have an easier time (well, slightly easier, depending on whether you’re an A- or B-theoriest on time).


At 1/31/2006 3:10 PM, Blogger Alan Rhoda said...

Hi Tom,

As to the question of 'accessibility', what it means is this: Possible world B is accessible from possible world A iff B is logically possible in A. On one of the most common systems of modal logic (S5), all possible worlds are equally accessible to each other. Consequently, if something's possible in one world, it's possible in all, which means that Possibly(p) entails Necessarily(Possibly(p)).

Now, once we shift from logical possibility to something like causal or historical possibility, things change considerably. Granting indeterminism, there is more than one world that is causally possible in this world, but that doesn't mean that every world is causally possible in this world. Same goes for historical necessity and possibility. The past cannot be changed. At one point it may have been possible to avert WWII, but now that it's happened, none of the logically possible worlds in which WWII does not happen are historically possible from our world.

In short, different types of possibility generate different accessibility relations among possible worlds.

Regarding presentism, while it's true that some presentists (notably Markosian and Crisp) define it in terms of what's included in the domain of 'our most unrestricted quantifiers', I'm not sure that's the right way to do it. As I've said, it seems to me that we can quantify over non-actuals, in which case the Markosian/Crisp definition of presentism has got to go in favor of something else.


At 2/01/2006 2:12 AM, Blogger MKamer459 said...

Prof, I hadn't made any comments yet, and I wanted to at least get started in the habit of getting involved on here, so sorry if this is a bit short, I assure you more involved responses in the future.

That said, I'd like to re-state the argument I made in class on Thursday - In my mind, a unicorn DOES exist the minute I imagine it. Now, it may CEASE to exist once I stop focusing on it, however I can conjure it in and out of existence in my own mind at will (this is actually something of a basic kabbalistic teaching :: that 'God' exists in everything, and were he to stop existing in object A, object A would cease to be, and would cease to 'have been'. That said, the world could be changing around us, but since when things Exist and 'unexist', we do not notice since they ceased to EVER exist, but that's getting a bit off the subject...

So, I think if anything, you could at least clarify the argument to address the idea of 'sustained' existence.

I'm having difficulty being clear, however I think if you knew my background of what I hold as my own hypothetical model of how the universe is sustained I might be making more try to sum it up without a 10 page paper, I guess you could break it down like this. I hypothesize that::

1) Existence as 'we' know it, is a dream/mental function of a greater being.

2) This being sustains the universe by dreaming/thinking about it.


3) We are all just figments of this greater being/consciousness' imagination.
Now, The Old Testament says we are made in 'God's Image'. I believe that to be able to be read hermeneutic as 'we have the ability to create and sustain realities in our own mind. It's less a physical image we are made similar to than an ability/trait'

Using that kind of assumption, we can then say that in our own minds exist realities which we sustain, just not on the omnipotence and Super-consciousness level of this greater being. However, if we imagine unicorns in these personal universes, they are real.

That said, we may be incapable of sustaining them at all times.
So, to get to my point, perhaps it is possible to factor in sustained-existence in some sense to tighten the formula? Or is sustained-existence so immeasurable that it's a problem/factor we can't really address?

Sorry if that was a lot.

At 2/01/2006 10:32 AM, Blogger Ocham said...

Tom, you say 'It is possible that there are unicorns' doesn’t imply that “there are unicorns” if by the latter you mean “there are ‘actual’ unicorns.” But I don't mean that. If it makes it easier, consider 'some mountains are golden'. Then I hold that

(1) It is possible that some mountains are golden

does not imply

(2) Some mountains are golden

But according to Alan, as I read him, it does imply that, because (1) implies, according to him, that in some possible world some mountain is golden. That's what the formal quantifier expressions say.

You could reply that the formal quantified expressions don't imply the ordinary language statement (2), because formal quantifiers don't translate into the informal quantifiers 'some', 'any' &c. Then, fine – that is my point.

At 2/01/2006 3:14 PM, Blogger Tom said...

Ochum: “…consider 'some mountains are golden'. Then I hold that

(1) It is possible that some mountains are golden

does not imply

(2) Some mountains are golden…

And you’re correct. Possibly S doesn’t imply actually S. But possibly S does entail some actual T which is the ontological ground of possibly S. If there exists no actuality whatsoever, nothing is possible.

Ochum: But according to Alan, as I read him, it does imply that, because (1) implies, according to him, that in some possible world some mountain is golden. That's what the formal quantifier expressions say. You could reply that the formal quantified expressions don't imply the ordinary language statement (2), because formal quantifiers don't translate into the informal quantifiers…

But if we’re able to employ language in a meaningful way to posit possibilities, then there's no reason why the idea of possible worlds should not find expression in formal logic. Now, if the existential operator (and I’ve read enough to know there’s no complete agreement on just what it ‘posits’ or even on what ‘exists’ means) won’t allow positing “possible-but-not-actual” events or entities, then there’s every justification for adjusting the operator to include such non-actuals. It’s our language game. We can change it however we need to suit our needs. So long as Alan defines what he's doing, how he's using the operator, it's a legitimate enterprise. The notation is just a tool we use to express ourselves.

But it really does seem to me that presentists are committed to quantifying over actuals. If truth supervenes on being, and the present is the only time at which there is any ‘being’ at all, then what is actual, what exists, now is the “stuff” over which we quantify. I’ve got no one to quote on that. Just my hunch. So “logically possible S” supervenes over some actual T as its grounds, not over non-actual S as its ground. Possible S is just an attribute of T, a ‘way of T’s being’.



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