# Is S5 the appropriate logic tool for examining existential supremacy? **URL:** \ **Category:** Great Debates\ **Created:** [December 14, 2007, 11:29pm UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050 "2007-12-14T23:29:00Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [December 15, 2007, 11:54pm UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/41 "2007-12-15T23:54:00Z")
[QUOTE=Voyager] I guess my problem is that you haven’t defined R (at least as far as I can tell.) I agree that R being necessity is transitive, but in the second part of your post you used the transitivity of R in an argument about objectivity and subjectivity. Why would a necessary being be objective in your sense? [/QUOTE] 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). ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [December 16, 2007, 7:17am UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/42 "2007-12-16T07:17:59Z")
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? ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [December 16, 2007, 10:02am UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/43 "2007-12-16T10:02:23Z")
[QUOTE=Bryan Ekers] 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? [/QUOTE] 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: 1. x = x … Identity 2. Ay(y!=x) -\> x!=x … Quantifier Proof: 1. x=x -\> ~Ay(y!=x) … Contraposition on 2 2. x=x -\> Ey(y=x) … Definition of E on 3 3. Ey(y=x) … Modus Ponens on 1 and 4 4. Ey(y=x) … Rule of Necessitation on 5 QED: AxEy(y=x) … Rule of Generalization on 6 ---
**Author:** ![jackdavinci](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jackdavinci/32/3259_2.png) [@jackdavinci](https://boards.straightdope.com/u/jackdavinci)\ **Post date:** [December 16, 2007, 11:19am UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/44 "2007-12-16T11:19:14Z")
Direct experience is possible but then you get a feedback loop ala “Being John Malcovich” ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [December 16, 2007, 11:24am UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/45 "2007-12-16T11:24:49Z")
[QUOTE=jackdavinci] Direct experience is possible but then you get a feedback loop ala “Being John Malcovich” [/QUOTE] 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. ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [December 16, 2007, 1:16pm UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/46 "2007-12-16T13:16:21Z")
[QUOTE=Liberal] If by mind you mean “brain”, then I agree. But will? By what rationale do you take such a renegade position? [/QUOTE] 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 ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [December 16, 2007, 3:39pm UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/47 "2007-12-16T15:39:39Z")
[QUOTE=Left Hand of Dorkness] 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. [/quote] 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. > [@](#): > > 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? 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. ---
**Author:** ![MrDibble](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mrdibble/32/114_2.png) [@MrDibble](https://boards.straightdope.com/u/MrDibble)\ **Post date:** [December 16, 2007, 5:26pm UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/48 "2007-12-16T17:26:05Z")
[QUOTE=Liberal] This makes sense to me since we cannot possibly have ever experienced exactly the same event in exactly the same way. … It would be possible to receive its communication in exactly the same way at exactly the same time [/QUOTE] 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. ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [December 16, 2007, 6:28pm UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/49 "2007-12-16T18:28:10Z")
[QUOTE=MrDibble] 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. [/QUOTE] 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. ---
**Author:** ![MrDibble](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mrdibble/32/114_2.png) [@MrDibble](https://boards.straightdope.com/u/MrDibble)\ **Post date:** [December 16, 2007, 7:51pm UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/50 "2007-12-16T19:51:52Z")
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. ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [December 16, 2007, 8:04pm UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/51 "2007-12-16T20:04:20Z")
[QUOTE=MrDibble] 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? [/quote] 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. > [@](#): > > I don’t know, I’m not seeing it. 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. > [@](#): > > 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. \<\>A -\> A is not a sound axiom in S5. ---
**Author:** ![MrDibble](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mrdibble/32/114_2.png) [@MrDibble](https://boards.straightdope.com/u/MrDibble)\ **Post date:** [December 16, 2007, 8:15pm UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/52 "2007-12-16T20:15:11Z")
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. ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [December 16, 2007, 8:19pm UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/53 "2007-12-16T20:19:03Z")
It’s \<\>A -\> \<\>A. One way of reading it is that if A is possible in actuality, then it is necessarily possible. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 16, 2007, 8:46pm UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/54 "2007-12-16T20:46:42Z")
**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)) ---
**Author:** ![MrDibble](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mrdibble/32/114_2.png) [@MrDibble](https://boards.straightdope.com/u/MrDibble)\ **Post date:** [December 16, 2007, 9:12pm UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/55 "2007-12-16T21:12:37Z")
[QUOTE=Liberal] It’s \<\>A -\> \<\>A. One way of reading it is that if A is possible in actuality, then it is necessarily possible. [/QUOTE] 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) ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [December 16, 2007, 9:33pm UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/56 "2007-12-16T21:33:42Z")
I didn’t say it wasn’t valid. I said it wasn’t an axiom. I would accept a convergent frame axiom: \<\>A -\> \<\>A. ---
**Author:** ![MrDibble](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mrdibble/32/114_2.png) [@MrDibble](https://boards.straightdope.com/u/MrDibble)\ **Post date:** [December 16, 2007, 9:42pm UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/57 "2007-12-16T21:42:37Z")
I never said it _was_ an axiom, just that that was the kind of thing you can get under S5. ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [December 16, 2007, 10:12pm UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/58 "2007-12-16T22:12:36Z")
[QUOTE=Liberal] Sure. Let’s prove that existential supremacy is true in S5. [/quote] 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. ---
**Author:** ![Liberal](https://avatars.discourse-cdn.com/v4/letter/l/848f3c/32.png) [@Liberal](https://boards.straightdope.com/u/Liberal)\ **Post date:** [December 16, 2007, 10:53pm UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/59 "2007-12-16T22:53:58Z")
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. ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [December 17, 2007, 9:50am UTC](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050/60 "2007-12-17T09:50:22Z")
[QUOTE=Liberal] 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. [/QUOTE] Which leads to…? [Previous page](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050.md?page=2) [Next page](https://boards.straightdope.com/t/is-s5-the-appropriate-logic-tool-for-examining-existential-supremacy/430050.md?page=4)