# Is S5 the appropriate logic tool for examining existential supremacy?

The title doesn’t really express exactly what I want to cover, but you get only so many characters. I do want to discuss S5 and whether it’s the appropriate logic tool as posed, but I also want to discuss how to describe and define the relations between subjective agents and the objective agent as well as among the subjective agents themselves and see whether others agree that these relations are best described by a Euclidean frame.

Consider the base axiom for S5:

<>A -> <>A

In other words, if A is possible, then it is necessary that A is possible. Or if A is possible in any possible world, then A is possible in every possible world.

And now have a look at the frame conditions imposed:

(wRv & wRu) -> vRu

where w, v, and u are simple variables that range over some world in which all are related. This is a Euclidean relation.

Now that the context is out of the way, let me define a couple of things I’m talking about and state a couple of informal premises. (I want this to be a non-technical discussion.)

I consider existential objectivity and existential supremacy to be synonyms. Whatever is objective must have a view onto all possible worlds, otherwise it is subjective. By the same token, whatever is subjective must NOT have a view onto all possible worlds, otherwise it is objective.

So if you and I are both subjective beings (i.e., existentially subjective), then we cannot relate to one another directly. I cannot look onto your view of the world, and you cannot look onto mine. This makes sense to me since we cannot possibly have ever experienced exactly the same event in exactly the same way.

On the other hand, both you and I MUST be able to relate to the objective being. (There can be only one for obvious reasons.) That’s because the objective being by definition (or premise, however you care to skew it) looks onto both our views of the world.

There arises, it seems to me, the notion of a hub with spokes. The hub is the objective being and the spokes are subjective beings like you and me. And so there is a way for you and I to relate to one another through the hub. Suppose, for example, that the hub could communicate to each of us. It would be possible to receive its communication in exactly the same way at exactly the same time since a circle has no endpoint (and therefore no way to discern the ordinality of spokes.)

With that model in mind, the S5 axiom and the Euclidean frame seem perfectly suited to examining existential objectivity and subjectivity. Does anyone disagree? And if so, why? I have heard some people argue before that the selection of S5 is arbitrary for this purpose, but I’m not so sure. What do you think?

First, I’m no expert on modal logics, and especially not on system S5, but could you explain your reasoning in the following excerpt:

I get the notion of hub and spoke, but where does the circle come into this?

The circumference of the hub, from which the spokes originate.

ETA:

If you prefer to think three dimensionally, it could be conceived as a sphere as well. I was just using the simple idea of a wheel.

I’m not so sure this follows through. You seem to be assuming that we each have exactly the same view and relationship to the objective being. While it’s possible there’s the hub and “standard” spokes, surely it’s also possible that those spokes differ in quality (and even quantity) with each person? Thus we’d have a problem akin to cryptology in which we each have half of the key but can’t access the other’s piece.

The hub may send out the same signal at the same time, but the spokes regulate that signal as much as the hub does.

I see one fallacy right away, which is that not all relationship properties necessarily are transitive. Euclidean ones are, but it is not clear that your definition of relationship to an objective being is Euclidean. As an example of a relationship that isn’t, consider the relationship defined by the sharing of genes. (We’ll ignore that we all have common ancestors for the moment.) I’m related to my maternal grandfather under this relationship, and also to my paternal one, but they are not related to each other. Notice that I’m the hub of this relationship, just like your objective being is.

Second, you haven’t shown that an objective being exists, or, to be more specific, that it is necessarily possible for one to exist in any possible world. If one does exist, I don’t see any notion of supremacy from being objective. An objective being might be able to see into the views of all subjective beings as well as the views of other objective beings. I don’t see any particular problem with this, maybe you can point one out.

What if A isn’t possible?

Keeping in mind that there about half a million ways of applying the ‘possible worlds’ model to discussions about reality, and that thusly I may be totally misunderstanding you, I ask: why can’t two subjective things share the same world? They can’t be in all possible worlds, but why can’t they be in some possible worlds, with the possibility of some of those possible worlds being shared by both? I was fairly sure this was definitely allowed under S5.

To clarify: Example world set: (Ws are worlds, A, B, C etc are beings in the worlds)
W1: A B C
W2: A D
W3: A B C D
W4: A

A is of course objective, (assuming any other unlisted worlds also contain A); B, C, and D are of course subjective. In this scenario, B C and D Share some worlds; there are actually no examples of B and C not co-existing.

To my understanding, this is all very much allowed under S5. What other conditions are you applying such that it must be otherwise?

Bolding mine, and my other question regards that secton: why? Why can’t there be two objective beings?

Even if you’re using some form of supremacy argument (a type of argument that I find to be quite flawed, but that’s another discussion), under which you’re assuming that the supreme being must also be necessary, how do you conclude from that that the penultimate being isn’t also necessary? Necessariness is just the state of existing in all possible worlds, after all. Maybe lots of things do that. Heck, maybe everything does that. As far as I know, that’s all possible under S5.

Is there something I’m missing here? If you’re importing a discussion from other thread, it’d be nice to hear the relevent points of it.

(I apologise if there’s something standard and obvious that I’m missing; I’m not all that familiar with the modal logic stuff. Still though, what little I know doesn’t jive with your OP, so I’m confused…please rectify my ignorance…)

That’s definitely arguable, and in fact I agree. But the frame is still Euclidean either way. So even if circumstances are such that signals from the objective being are subjectively interpreted, they are our only hope for a relation of any kind with each other. It may well be more analogical than logical.

As you say, Euclidean relations are transitive, and so it is no surprise that S5 subsumes S4, which has a transitive condition on frames: (wRv & vRu) -> wRu. Its axiom is A -> A. Or, if A is necessary, then A is necessary in every possible world.

Yes, but since you are not the hub of every possible ancestor (including mine, for example) you cannot be said to be objective. Note that the implication of the relation is not biconditional, nor did I say that it is.

I think that’s putting the cart before the horse. We all know what the elephant in the room is here, and I’m not going to mention it unilaterally. But agreeing that S5 is an appropriate tool for examining existential supremacy is a prerequisite to agreeing on whether the existence of an objective being as I defined it is possible (not to mention necessary or actual). Like I said, this is an objection I’ve seen (rarely) raised before, and the purpose here is to dispense with it outside the context of the other poison.

Doesn’t this just mean any object can be the “hub”, if two people each have their subjective perceptions of it? God, a statue of Apollo, a fire hydrant…

Then ~<>A -> ~<>A. Or, if A is not possible in some world, then A is not possible in any world, which is obviously false. And that’s what we would expect from a denial of the antecedant fallacy. Just because A -> B doesn’t mean that ~A -> ~B.

Change “two” to “all”, and the answer is yes.

Nope, there’s no high falutin’ way to convince athiests to concede that there might be a god, any more than there is any high falutin’ way to convince believers to not believe anymore.

Then doesn’t this tool always yield positive results? How useful is it?

Oh, sure. No problem. Naturally, S5 will subsume all its weaker cousins. We can assume, for example, that S5 will allow for a reflexive frame, where wRw, which happens to be the Modal Axiom itself: A -> A.

Just as you anticipate in your subsequent text, I am using a supremacy argument. But that’s fair because of how I have defined objective. (For argument about definitions, I’d prefer someone open another thread. These are fairly ordinary definitions that I’m using for this context.) But to cut to the chase, the problem with there being two (or more) necessary beings is that the ultimate one would be both supreme and necessary while the penultimate would be necessary only (owing to the definition of supremacy as a superlative). There could not be a being that is both necessary and NOT supreme. Therefore, by modal modus tolens, there MUST be a being that is both necessary AND supreme. If the necessary being is not also supreme, then the supreme being is not itself supreme, because it is supreme only because it is necessary. That would be a contradiction, and so could not be true.

I’m not sure what you mean by “positive”. There can be negations within the system, but not negation of the axiom that underlies the system. Not to state the obvious, but there it is. You could, however, posit that <>~A -> <>~A, and that would not contradict the base premise. (~A is a straight substitution for A).

ETA:

Forgot the utility question. Modal logic, in general, is useful in a number of applications, like computer science, for example.

Just tiny nits till I read things in more detail (at which point, I may discover I have nothing to contribute to the thread).

[for S5]

Well, you’re leaving a little something out here, aren’t you? S5 also imposes reflexivity (wRw), which this doesn’t imply. For that matter, S5 also imposes symmetry (wRv -> vRw) and transitivity, and this doesn’t imply those, either, though Euclidean + reflexive proves both of them. So I think you just forgot to mention reflexivity.

Well, like I said above, not all Euclidean properties are transitive, though all reflexive Euclidean ones are. As an example of the failure without reflexivity, consider xRy = either (x isn’t the Pope and y is someone’s father) or (x is the Pope and y is the Pope’s father). Euclidean, but not reflexive, symmetric, or transitive.

You need to add reflexivity to Euclideanness to get transitivity.

And… on preview

Not only can you assume that S5 allows for reflexive frames, you must. If you don’t impose the reflexivity condition, it’s not S5.

That every conceivable object has existential supremacy. Even something previously labelled existentially subjective becomes supreme as soon as it is perceived by another existential subjective.

Is this any different from saying “God is everywhere and everything and everyone”?

I should say, Euclidean + reflexive is equivalent to reflexive + symmetric + transitive (i.e., the condition of being an equivalence relation), which is what I usually take to be S5. I find symmetry and transitivity easier to think about than Euclideanness.

Alternatively, one can go for something even easier, and take S5’s frame conditions to just be “Every world is related to every other world”. This is, of course, strictly stronger than the equivalence relation condition, but it induces the same modal logic.

Thanks for dropping in, Indistinguishable. I was hoping you would, and it is an honor. (See you fellows in the morning.)