R just represents an accessibility relation. So (wRv & vRu) -> wRu reads “if w is accessible to v and v is accessible to u then w is accessible to u”. A necessary being would be objective because it is accessible to every possible world — all of your experiences, all of my experiences, etc, and all exactly in the same time and the same way that we each have experienced them (something we subjective beings can’t say for one another).
For the sake of argument, let’s assume that S5 is appropriate for examining existential supremacy. Can we have an example of this tool in action?
Sure. Let’s prove that existential supremacy is true in S5.
Definitions:
Let A be “for every”
Let E be “there exists”
Let != be “not equal”
Let ~ be logical negation
Prove:
AxEy(y=x)
Premises:
- x = x … Identity
- Ay(y!=x) -> x!=x … Quantifier
Proof:
- x=x -> ~Ay(y!=x) … Contraposition on 2
- x=x -> Ey(y=x) … Definition of E on 3
- Ey(y=x) … Modus Ponens on 1 and 4
- Ey(y=x) … Rule of Necessitation on 5
QED:
AxEy(y=x) … Rule of Generalization on 6
Direct experience is possible but then you get a feedback loop ala “Being John Malcovich”
Exactly. The great irony of empirical epistemologies is that they are A observing A. One’s senses are themselves a part of what they are observing.
I believe this is quite a common position, i.e., that anything proved necessary by this particular formulation may have no traits whatsoever, else those traits invalidate the argument. Possession of will is one of the traits.
As for the wheel analogy, it appears to me that the spokes have two “hubs”: one is the hub proper, the other is the rim of the wheel, which each spoke knows equally. Is this a flaw in the analogy, or am I misunderstanding it, or does it point to the possibility of two equally supreme entities?
Daniel
That’s not at all what I understood him to say. He wrote, “Analytic concepts have no mind or will.” I explained why being an analytic concept has nothing to do with having or not having a mind or will. A bachelor is a bachelor: that is an analytic judgment, but does not preclude the bachelor from having a mind or will. As for your secondary point, that anything proved necessary by S5 must be traitless, it just doesn’t make sense. It may be that with a measuring stick, the only thing you can show about something is how long it is. But that doesn’t mean the thing you measured can’t be brown or musical or what-have-you. The nature of a thing’s existence is only one aspect of the thing.
Every analogy breaks down at some point. I would think of a wheel without the rim. When someone described a sphere, the mental image that came to my mind was Sputnik. As to the crux of your question, analogics are for describing, not analyzing. An analogy should be helpful in giving you some idea of what is being discussed. But it shouldn’t be used as a deductive tool.
Are these two statements not contradictory? In other words, if we never experience things the same, can we be said to have received the communication in exactly the same way? And if we did, then that’s at least one event we’ve both experienced the same.
The first statement came before the second one, and was in the context of existential subjectivity. Note that it follows a statement making that very clear: “So if you and I are both subjective beings…”. The second statement is in the context of objectivity, the context of which is similarly clarified. And so the two are reconciled by the fact that no two subjective experiences can possibly be the same, while all objective experiences are necessarily always the same.
That doesn’t clarify things in the slightest - could you restate it some other way? It seems to me that the objective nature of our experience in the second case means that the first case is contradicted i.e. we are not subjective beings. I mean, we are still the same beings from case 1 to case 2, no? That is to say, if we can receive simultaneous, identical experience from your “hub” being, then we are no longer truly subjective beings. And since your hub being communicates with all on the rim, then no being is subjective?
I don’t know, I’m not seeing it.
Not that I accept S5 anyway (I think it leading to <>A -> A is an indication of aesthetic wrongness], but even if I do, I still see the above as contradictory.
There is no rim, as I explained above. Just a hub and spokes. There is an inherent implication that the spokes have a dual nature — one objective and one subjective. Without the hub, there would be the sujbective nature only.
That’s okay. It can be daunting, I’m sure. In modal terms, each spoke is a possible world and the hub is a necessary world. (Each spoke is considered an extension of the hub.) In any case, the analogy is not necessary at all to the discussion. Some other analogy or no analogy at all is just fine. We can still discuss what is the best tool for examining existential supremacy.
<>A -> A is not a sound axiom in S5.
whoops, sorry, I meant <>A -> A…
- to me, (I’m and admitting to my rudimentary reading in modal logics, and my last course in symbolic logic was 1994 AND i failed it through nonattendance), saying if something is possibly necessary, then it is necessary seems not just counterintuitive, but counter-reality.
It’s <>A -> <>A. One way of reading it is that if A is possible in actuality, then it is necessarily possible.
MrDibble’s statement of <>A -> A is also valid in S5.
(Sorry, I don’t yet have anything substantial to contribute to the thread (I’m drowning in end-of-semester work), but I can do these small nitpicks. Speaking of which, I’ll make the same remark about the proof of AxEy(y=x) in post #43 that I’ve made before: it doesn’t go through in all S5 systems of first-order logic, just in some. (Not only that, but in some formulations, it’s not even a well-formed statement! The interplay between S5, the quantifiers, and the rules regarding free/bound variables can be subtle and tricky, and there’s more than one way to do it))
No, that’s Axiom E itself, I’m talking about what you can *derive *in the S5 framework, mostly as a consequence of Axiom 4 (on preview, like Indistinguishable said)
I didn’t say it wasn’t valid. I said it wasn’t an axiom. I would accept a convergent frame axiom: <>A -> <>A.
I never said it was an axiom, just that that was the kind of thing you can get under S5.
That’s great and all, but as far as I can tell, you’ve just proven that everything has a double. I’m assuming you have something bigger in mind than just reiterating a premise of an episode of The Flintstones, so can you give us an example of something tangible in place of the generic label x?
And I think your second premise may have a typo.
No, no typo. Another way to look at it is to negate the whole expression, something like (y) ~(y=x) –> ~(x=x).
As to what I have in mind, I made that clear in the OP. I have in mind finding a tool that is suitable for examining existential supremacy.
Which leads to…?