Is S5 the appropriate logic tool for examining existential supremacy?

Great Debates
101

I think I now have a bit of a better (and worse) handle on what you’re trying to do. Let me address your OP.

Do you intend them to mean the same as “necessary existence?”

Here you talk about things having views onto worlds. Elsewhere in the thread, I think you talked about worlds having views onto each other. The latter sounds more like things I have heard before–the accessibility relation is sometimes, esp. in pedagogical contexts, referred to as a “seeing” relation. I am not sure, however, what you mean here when you say that things can have views onto worlds. I don’t think you mean to limit the scope of “whatever” in the abovequoted to “whatever possible worlds,” but if that’s not what you mean, I don’t know what you mean.

By “existentially subjective” do you mean “contingently existent?”

Also, why do you say subjective beings can not relate to one another directly? Above you said subjective beings can not have a view onto all possible worlds. But I don’t know how you go from that to something that apparently means “No two subjective beings can have a direct relation with each other.” (What’s a direct relation, and what’s an indirect relation, btw?)

This is why I thought you were interpreting worlds in your model as points of view, by the way. You seem, here, to be relying on either an equivalence or a relation of implication between “having a view onto a world” and “directly relating to a person.” I tried to explain this to myself by supposing you were giving a non-standard interpretation of worlds as viewpoints instead of as possibilities. In this case, the accessibility relation would not have been to be interpreted as representing “possibility relative to a world” but rather “Ability to comprehend one viewpoint from another viewpoint” or something like that. Anyway, turns out that’s not what you were after so never mind.

But is there, in fact, some equivalence or implication relation you are relying on between “having a view onto a world” and “directly relating to a person?” That’s the best way I can make sense of what you’ve said so far, but am I completely off track?

This is more of what makes me think you’re trying to do an interpretation of an S5 model in terms of viewpoints (“views of the world”) intsead of possibilities. But your last post to me in this thread seems to indicate this is not the case. Unfortunately, that leaves me lost. :frowning:

-FrL-

102

Frylock

I deliberately avoided references to NE and such because of posts like the one from Czarcasm that would attempt to derail the discussion, like I said, into yet another argument about the Modal Ontological Proof. I DO NOT WANT to discuss the modal ontological proof or the existence of God, and so far those brush fires have been containable. But the instant that I say that by existential supremacy I mean necessary existence, the thread will be shat upon from multiple directions.

I sense that you can be some help to me, but you’ll need to hurry before the derailment begins in earnest.

Please forget whatever you’ve read already and just listen anew. I want to model a world (that does exist) that is objective or necessary in the sense that it has access to all possible worlds (that do exist), but that do not have COMPLETE access to one another. They can have similarities, but not equalities. I call those subjective because they cannot access the entire system of worlds. But each subjective world has access to the objective world, and through it, some access to each other.

Is there a logic system for examining that?

103

I see!

I wish we had some kind of scratch paper function on these boards. That would be awesome.

The following is far too long. And as it is relatively off the cuff, I may have made a lot of mistakes. I hope someone will help us out in this conversation. (On re-reading this post before submitting, I think a lot of it turned out unnecessary in order for me to make my point, but I’m keeping it in because I kind of like the structural vision it expresses.)

The Euclidean frame doesn’t quite work for what you’re trying to do, for one, possibly two (or more) reasons.

One possible reason turns on the question whether the term “Euclidean” is supposed to already imply, by its use, reflexivity. (I forget.) If that’s the case, then reflexivity plus the euclidean accessibility relation ( “(wRv & wRu) -> vRu” ) taken together imply that every world is related to every other world. Not what you’re looking for.

What about Euclidean without reflexivity? (This is no longer S5.) It seems the picture you then get is of two kinds of worlds. I will call them “Privileged” and “Unprivileged” worlds. (These turn out not to be exclusive classes as we will see.) Unprivileged worlds come in groups which are S5-like in that each world in the group is related to each world in the group. Meanwhile, privileged worlds are like this: They are related to every world in one (and only one) S5-like group. There can be more than one privileged world bearing this relation to a particular S5-like group, but there can not be more than one S-5 group which a privileged world bears this relation to.

There are two kinds of privileged worlds, I’ll call them “selfish” and “generous.” Selfish worlds are related to unprivileged worlds, but not related to by them. Generous worlds are symmetrically related to unprivileged worlds–every unpriviledged world it is related to is also related to it. If an S5-like group has generous privileged worlds associated with it, then that S5-like group together with its geenrous privileged worlds itself forms an S5-like group.

(There can also be “outlier” worlds–worlds related to no world other than, at most, themselves–but I will ignore them for our purposes. There can also be “freaky” groups–groups where no world is related to more than one world–but again I will ignore these for our purposes.)

Take an S5-like group together with its associated privileged worlds (if any). I’ll call this a galaxy.

So you get a few different kinds of galaxies (each completely self-contained–none of the worlds in a galaxy is related to any world from any other galaxy):

Complete galaxies: Every world in the galaxy is related to every world in the galaxy. Note that in this kind of galaxy, the privileged/unprivileged distinction dissolves.

Fractured galaxies: At least one world in the galaxy is selfishly privileged, and at least one is generously privileged. In such a galaxy, the generous world(s) is not related to by any of the selfishly privileged worlds in that galaxy. Also note, the generous worlds in that galaxy form another s5-like group together with the s5-like group out of which that galaxy was composed by the mental act I described when defining “galaxy.” But the generous worlds in this galaxy are not just unprivileged–they are not related to by the selfish worlds associated with this galaxy.

Structured galaxies: There are selfishly privileged worlds in the galaxy, and no generous worlds.

Note that if a galaxy’s privileged worlds are all generous, then that galaxy is a complete galaxy–the kind where the distinction between privilege and unprivileged disappears. Awwww, Isn’t that sweet? :stuck_out_tongue:

There can be any number of galaxies in a model like this.

Anyway, one model satisfying the euclidean frame is the one in which there is a single privileged world (it would have to be selfish, sense if it were generous the model would contain just a single complete galaxy–it would be an S5 model after all, which as I’ve argued does not fit what you’re looking for) and several unprivileged worlds. That’s kind of like the hub with spokes you were talking about–but unfortunately, doesn’t work for you I think, since each of the unprivileged worlds is related to each of the unprivileged worlds in this model.

Any other model satisfying this description would have more than one privileged world, which again doesn’t seem to be what you’re looking for.

What you’re looking for (I think: A single hub, a bunch of spokes, the spokes relating to some other spokes but not to all other spokes), then, does not satisfy a euclidean frame.

I think the kind of model you want to build would be sort of “ad hoc” (for lack of a better term coming to mind) in that there would be no easy way to describe the accessibility relation in a single pithy line, and there would be no nice set of axioms valid on the model. You would just have to draw a diagram, call a substantial number of worlds “related” and a substantial number “unrelate” by fiat, and call that your model.

I’ll do some more thinking about it, though.

104

:stuck_out_tongue:

Thanx for 'splaining, dude. Till I got to your post I thought I was in the SDMB’s greatest collective whoosh, like, evaaahh!

:::shuts door. walks away quietly though still smiling:::

105

Frylock, thank you! I’ll be watching for more! :slight_smile:

106

Sorry, here’s a much better way to explain what I was trying to explain:

First of all, I assume you want one world to be related to all other worlds, and you want all those other worlds not to all be related to each other.

Euclidean + Reflexive (i.e. S5) doesn’t work because in that frame, every world is related to every other world.

Euclidean without reflexive doesn’t work for the following reason.

A model satisfying Euclidean without reflexive turns out to consist of one or more “galaxies” of worlds. A galaxy consists in a set S of worlds which are all related to each other, plus one or more world which are related to at least some of the worlds in S, but which are not related to by any of the worlds in S.* There can be any number of galaxies in a Euclidean-without-reflexive model. Galaxies are closed in the sense that no world from any galaxy is related to any world from any other galaxy. (If one were to be so related, the two galaxies end up “merging”–their two S5-like sets of worlds turning out to be a single S5-like set of worlds.)

If there is only one galaxy in a model, then it either is just an S5 model, or else it is a model in which there is one world related to some or all of the other worlds and every other world all related to each other, or else it is a model in which there is a set of worlds like S above, and another set of worlds each of which is related to one or more of the worlds in S, but none of which are related to by any of the worlds in S.

In none of the three cases I described is there a structure like the one I think you’re looking for.

And if there is more than on galaxy in a model, then of course that doesn’t fit the structure you’re looking for either.

I’ve given statements about what euclidean models would look like. I wish I could do a better job of explaining why they would have to look like that. (In the absence of scratch paper for diagramming, I could do it in symbols, but to be honest I don’t have the mental or temporal resources to work that out right now. Apologies!)

But maybe you already see why Euclidean models have to appear as I’ve described them?

-FrL-

*there can also be other kinds of galaxies–galaxies in which no world is related to more than one world. I’m ignoring these for our purposes–they’re clearly not what you’re looking for.

107

If it makes a difference, I would amend this part slightly: “A single hub, a bunch of spokes, the spokes relating to some other spokes but not to all other spokes”. I would say instead that what I’m looking for is “A single hub, a bunch of spokes, the spokes relating to all other spokes but only through the hub”. In other words, they are accessible, but only in a sort of transitve sense. (But even then, some accessibility does not obtain. You could say, maybe, that in spoke A, {xyz} is the set of true statements while in A’, {abx} is the set of true statements. In the hub, though, {abcxyz} obtains (with c being common to neither spoke).

108

It’s unclear how to translate the notion of “accessible, but only in a sort of transitive sense” into the usual framework of modal logic. But with “You could say, maybe, that in spoke A, {xyz} is the set of true statements while in A’, {abx} is the set of true statements. In the hub, though, {abcxyz} obtains (with c being common to neither spoke).”, it sounds as though you are conflating the notion of accessibility with the notion of agreeing on the truth of certain statements. (I may be misinterpreting you, however.) Just because world A can access world B doesn’t mean that world A accepts as true everything accepted in world B, or even that there are any particularly significant points of commonality between the two truth sets. Rather, it essentially just means that whenever a statement of the form X holds in A, then X holds in B, and whenever a statement of the form Y holds in B, <>Y holds in A. One could motivate it as the world A being able to “see” world B, but this isn’t the sort of “sight” that indicates direct agreement, just the sort of “sight” that means A is aware that there are worlds like B, even if those worlds drastically differ from A itself.

109

I’m surprised to hear that. I thought modal logic was adaptable to modals other than possibility and necessity. Aren’t there doxastic modals, and deontic modals, and temporal modals? Why couldn’t there be modals for Full Understanding and Partial Understanding? Or Full View and Partial View?

110

It may well be possible; it would just probably be somewhat out there, so to speak. It seems like you don’t want an accessibility relation that’s just “Yes, this world can access that one” or “No, this world can’t access that one”, but one with degrees in between, and that would certainly be a very unusual system, though probably accomodatable in some fashion. (Even though there are modals other than possibility and necessity, as you say, all those ones you mention are usually modelled with pretty plain-vanilla frames of the Yes/No type).

But to really find the best modal logic for your uses, it would be necessary to understand exactly what it is that you intend by Full View and Partial View, what exactly this View relation on the possible worlds of your model is supposed to capture.

I’m afraid I’m not being very eloquent right now. I’ll hopefully be able to better phrase the questions in my head tomorrow, when I will finally be done with the schoolwork currently pressing on me in very annoying fashion, and thus will have some guilt-free free time at last.

111

Something that models, basically, the real world, where you and I can have similar, but never exactly the same, experiences. We can never experience the exact same event the exact same way at the exact same time. Even standing side-by-side, say looking at a museum sculpture, we having different viewing angles. Even if we mashed our heads together, the angles would not be the same. In fact, owing to the nature of electromagnetism, we cannot merge our bodies together, and even if we could, your sense of sight, smell, touch or whatever might be better or worse than mine.

We can have a partial understanding of one another, but never a full understanding. For you to understand me fully, you would have to be me, complete with all my experiences and the loss of yourself. That’s why I say that we have a subjective relation, and I would like to model it logically so that it can be examined. Maybe modal logic of any kind is the wrong tool.

112

(I’m about to point out a bunch of problems with what you’re proposing. I’m not trying to be discouraging though–I’m genuinely interested in seeing if something interestingly like what you’re proposing can be pulled off. I list the problems with a view toward making them clear in order to see them solved.)

The problem is this. It appears you are hoping that Full Understanding will be the interpretation of the necessity operator, and Partial Understanding will be the interpretation of the possibility operator. Yet it also appears you want the interpretation of a world to be “The set of all of a single agent’s experiences.” This is a problem because, in the real world, what you’re calling “Understanding” is a relation between agents, but in the model theoretic world, the modal operators do not denote relations between worlds. Rather, the accessibility relation denotes a relation between worlds. But a single accessibility relation can not stand for both Full and Partial understanding–because those are two different relations.

There are logics, I think, with two accessibility relations. You would use subscripted modal operators: [1] and [2], and <1> and <2>. I’ll give that some thought. The interesting kind of case will be one in which there are axia which contain modal operators of both varieties. And indeed, that seems to be what you’re going to need–because you want the subjective relation to hold between agents only when the objective relation holds between each of these agents and a particular third agent.

Another problem is this. You want two to have the subjective relation only when (I call the following condition C:) each holds the objective relation to some particular third. But that third you want to have the objective relation to all others. This makes the obtaining of C trivial–it is always true of any two worlds. This makes it unnecessary to even formulate C in one’s condition on the subjective relation. The way I think this should be handled is by constructing a frame broader than what you’re looking for, and then specify the way in which your particular model turns out to have an interestingly unique way of satisfying that frame. Sort of like the way in which the model on which there is only one privileged world and a host of other worlds all forming an S5-like group is a model which satisfies Euclidean in a uniquely interesting way. “Interesting” is subjective, or course, and “unique” is almost an empty term since every model differs from every other model in some way. Ideally, to highlight the ‘uniqueness’ of a model we would need to specify in what interesting way it differs from every other model satisfying the frame.

Here’s another problem. (This is basically what Indistinguishable just said, I think.) Modal operators generally denote a relation (not between worlds but) between a world, a set of worlds, and a proposition. The set of worlds is determined by the accessibility relation associated with the modal operator. A modal statement is true in a world (here I call it “the world in question”) when some single proposition fulfills a particular relation (generally: that of being true in a set of worlds for necessity, or being true in at least one of a set of worlds for possibility) with a set of worlds determined by the world in question and by the model’s accessibility relation. The problem is that it appears you want your modal operators to hold, not based on a relation between a single proposition and a set of worlds, but between a set of propositions and a set of worlds. (By the way, it appears you want propositions to be interpreted not as propositions strictly speaking but as experiences or something along those lines.*) You want the subjective relation to hold between two “worlds” only when they share some propositions in common. (Btw it appears you want the objective relation to hold from w onto v for any w and v just when w is a particular, previously designated world in the model, call it W.) But it also appears you want this to be reflected by the possibility operator in the following way: You want a world to turn out <>P for some P only when there is some set of propositions in the world which is shared by some world having the relevant accessibility relation to that world. Or something like that. The upshot of all this is, I think you are trying to get your accessibility relation and your modal operators to reflect, roughly, the same thing–and that doesn’t really work. (I’m having trouble formulating what I want to say about the problem here. Sorry about that.) Modal operators should be interpreted as standing for a relation between a proposition (whatever that means in the model) a world (whatever that means) and a set of worlds. An accessibility relation should be interpreted as standing for a relation between worlds, and should relate to the modal operators in that the accessibility relation determines the set of worlds relevant to any modal proposition found in any particular world.

The first two problems I’ve mentioned above I think can be solved in interesting ways without damaging the spirit of what you’re trying to propose. (I’ve gestured in the direction of such solutions above, in some cases.) The fourth problem I think indicates that something fundamentally different needs to be done than whatever you’re hoping to do. But it still might be interesting.

I’m still thinking about it!

-FrL-

*It looks to me like you want “a”'s being a member of a world to mean, not that some sentence “a” is true in that world, but rather, that some experience “a” is had by the person that world stands for. Another thing you might mean, though, is that what is supposed to be contained in worlds are indeed sentences like “two plus two equals four,” “the earth revolves around the sun,” and “Melody is my sister.” The first two of these, presumably, would be contained in every world, while the last would be contained in only two worlds–those standing for Melody’s two siblings.

So what is contained in a world, on this last proposal, are sentences (not propositions!) which are true for a person. This is not a relativistic notion–to acknowledge that the sentence “I am typing” is true for me but not for you is not ipso facto to be a relativist. (It would be relativism if I said the proposition represented by the sentence “I am typing” is true for me but not for you. Because that would mean it is true that I am typing for me, and false that I am typing for you.) This latter proposal is probably more likely to be amenable to model theoretic treatment. I am not sure how much trouble it would cause putting sentences instead of propositions inside worlds. I think none, but I suspect some.

113

“Fourth problem” should have read “third problem” of course.

I’ve also just realized that putting sentences in worlds instead of propositions causes problems if the model is supposed to be of a predicate (i.e. “quantified”) logic rather than of a propositional logic. This is because in a predicate logic, the meaning of a proposition (or, as in my proposal, a “sentence”) is supposed to be determined by the meanings of its constituents–but “I” has no determinate meaning according to the way I treated it in my proposal. (It means something different when used in different worlds.) “I” and other terms like “this,” “our,” and so on, would have to recieve special treatment in a logic like this. (The treatment would have to allow them to appear as constituents of sentences, and do the same work toward determining the truth of a sentence no matter who utters (or would affirm) the sentence–yet without requiring that they always denote the same thing.)

The logic of indexicals is notoriously difficult to deal with (I certainly don’t know more than a tiny bit about how this goes) so on my “sentence” proposal your model will have a bit of baggage to lug around.

(Though come to think of it, maybe you don’t need a predicate logic. Maybe a propositional logic (where “P” can, all by itself, stand for an entire sentence) would do everything you need sentences to do. In that case, I don’t presently think there’s a problem, at least not one like the one I outlined (poorly) above.)

-FrL-

114

I’m a bit lost at this part: “you want the subjective relation to hold between agents only when the objective relation holds between each of these agents and a particular third agent”. That might have come from my poor communication, but there is no third agent.

Let’s call the Hub H and the Spokes S1 through Sn. Then S1 can relate to S2, but only through H. (H is the third party in that sense, I suppose.) That’s why I called it transitive. S1 relates to H and H relates S2, so in that manner (and only that manner) S1 relates to S2. Does that make sense?

Meanwhile, H relates to every S, from S1 to Sn. And in general, any S can relate to any other S through H. But only partially because H may contain information from S2 that S1 knows nothing about and vice-versa. It would therefore be only gibberish (or at best a bad translation) for the “target” S because there is no common frame of reference with the “source” S.

Does that help at all, or only muddy things further?

115

Yes, that’s pretty much what I thought you meant. (By “third agent” I did indeed mean H)

For starters (and only for starters) I can offer this pair of accessibility relations.

R1 goes like this:

R2 goes like this:

A different version of R1 which I like better (but don’t know why) would go like this:

This condition (this version of it) does not, all by itself, make W relate to every other world. It just makes it the case that W is the only thing that relates (relates-subscript-1) to any other world.* But you can specify your model as one that satisfies this frame in the interestingly unique fashion such that W doesn’t just relate to some worlds, but to all worlds.

So I’ve got two accessibility relations (two versions of one, one version of the other). What do they stand for? Maybe one can be thought of as the “objective relation” and the other as the “subjective relation.” I’m not sure. As the model is more fully specified, if these turn out to be the right relations to capture what you’re looking for, the right way to interpret them for your purposes will become more clear.

-FrL-

*To be clear, plenty of other worlds relate to worlds, in the relates-subscript-2 sense

116

Is “W” the objective world and “w”, “v” the subjective worlds?

117

Almost: W is the objective world, and “w” and “v” are variables ranging over all worlds.

-FrL-

118

Does anyone know if there are complications that necessarily arise from having two kinds of modality in a single model like this? Esp. where the two kinds are related together in the specification of the frame? (And so, also, I suppose, in the axiom schemata that will be valid in the model?) This strikes me as just the kind of thing that ends up getting you in trouble with completeness/consistency and things like that.

Also, is there a relatively easy way to figure out modal axioms from their concommitant frames, or do you just have to noodle around til you figure it out?

Also, is it kosher to mention a particular world in the specification of an accessibility relation? I ask this because it strikes me that if I do this, then my modal axioms are going to have to explicitly mention this world as well–and I think you definitely get problems when you have propositions in one world mentioning other worlds by name. Or do you? (It seems like you would, since you now have a part of a model talking about the model itself. Self-referentiality is scary, woooooo…)

-FrL-

119

Then, how are subjective worlds differentiated, if w and v can just as well be W? Maybe it would help if you would write out the relations. That might also help me grasp the mixture of implication and equivalence.

120

You’re right. For example, an S5 model would satisfy the frame I’ve outlined, and in an S5 model there is no distinction makeable-outable between any particular “objective” world and the other “subjective” worlds.

But there are, I think, always an infinite number of models that can satisfy a given frame. You can’t capture everything in the frame. What you need is a theoretically productive frame, and then to further specify your model as some theoretically interesting way to satisfy that frame.

Notice that the kind of model you want to build does satisfy the frame I’ve outlined, but does not satisfy, for example, a Euclidean frame. So in that sense, the frame I’ve outlined does seem to have an important relationship with the model you’re after.

Anyway, the frame as I’ve outlined it so far goes as follows:

There are two accessibility relations, R1 and R2, and there is a particular designated world, W.

R1: wR1v -> w == W

R2: wR2v -> (WR1w & WR1v)

Is that what you mean by “writing out the relations?”

Like I said, this is just for starters. It seems to me to basically capture the spirit of what you’re after, but from your last post it looks like you want the relations to do more of the work in making the objective world and the subjective worlds necessarily (i.e., on any model satisfying the frame) distinguishable. If that can be done, I think it will make the frame specification more complicated, but that doesn’t have to be bad.

-FrL-