Re: Biology, Buddha and the irreflexive Multiverse (was Re: Modal Logic (Part 3: summary + 1 exercise)
On 07 Feb 2014, at 23:21, meekerdb wrote: On 2/7/2014 10:40 AM, Bruno Marchal wrote: On 06 Feb 2014, at 21:29, meekerdb wrote: On 2/6/2014 12:14 PM, Bruno Marchal wrote: In Kripke semantic all statements are relativized to the world you are in. []A can be true in some world and false in another. The meaning of [] is restricted, for each world, to the world they can access (through the accessibility relation available in the Kripke multiverse). []A still keep a meaning, but only in each world. So everything is said when we define the new meaning of [] by the rule []A is true in alpha, by definition, means that A is true in all world beta *accessible* from alpha. And A is true in alpha iff there is a world beta; where A is true, accessible from alpha. Suppose A is true in alpha, OK. Nice. but alpha is not accessible from alpha OK. and A is not true in any other world accessible from alpha. OK. Does it follow that A is not true in alpha? Yes. That does follow. How frustrating! A is true, but not possible. How could that makes sense? Well, this does not make sense ... in the Leibnizian multiverse. For sure. I don't see the point allowing that worlds may not be accesible from themselves? Does that have some application? Yes. First you prove to everybody that I can see in the future, as I announced yesterday the discovery of a Kripke multiverse violating the law []A - A. You just did. Well, in alpha, to be sure, []A - A is true (OK?), but []~A - ~A is falsified, as []~A is true (~A is true in all accessible world from alpha), and ~A is false in alpha, as A is true is true in alpha, and worlds obeys CPL). That amounts to the same, as the laws do not depend on the valuation. If []A - A is a law, []~A - ~A should follow. Note that []~A - ~A, is equivalent with (contraposition, double negation): ~~A - ~[]~A = A - A A - A is the dual formulation of []A - A. As law, they are equivalent. But as formula in one world, they can oppose to each other. So you did find a Kripke multiverse violating the *law* []A - A. And you did find the culprit: those bizarre world which does not access to themselves. Does that have some application? Yes. 1) An easy one, which plays some role in what I like to call the simplest buddhist theory of life ever! And that theory is a subtheory of G, and so will stay with us. That theory models life by worlds accessibility. To be alive at alpha means that t is true in alpha. It means that there is, at least, one world accessible from alpha. To die at alpha means that t is false in alpha. But t is true in alpha, as t is true in all worlds, so the only way to have t false, is that there are no accessible worlds from alpha, at all, including itself. That makes alpha into a cul-de-sac world. So in Kripke semantics, ~t, or equivalently []f, characterizes the cul-de-sac world. Then the simplest buddhist theory of life ever is just the statement, If you are alive, then you can die. It means that for all worlds alpha where you are alive (t is true), you can access to a cul-de- sac world. It means that everywhere, in all worlds we t - []f, or equivalently t - ~[]t. 2) If you interpret t by intelligent, and []f by stupid, you get with the same multiverse, my general theory of intelligence and stupidity. 3) if you interpret [] by provability (in PA, or in ZF), again, t - ~[]t is a law. Read: if I am consistent, then I can't prove that I am consistent. It is easy to see that the law t - ~[]t is a direct consequence of the formula of Löb []([]A - A) - []A. Just put t in place of A, and keep in mind that A - f is just ~A, and then contra-pose: []([]A - A) - []A []([]f - f) - []f [](~[]f) - []f ~[]f - ~[](~[]f) t - ~[]t The worlds in the Kripke mutiverse characterizing G are like that, they don't access to themselves. []A- A is not an arithmetical law from the 3p self-referential view of the machine, but that is why the Theaetetus idea is applicable and will give the non trivial S4Grz for the knower, or first person, fro which []A - A is indispensable. Some might be astonished that []f is true in a cul-de-sac world. But kripe semantics say that []f is true in alpha then f is true in all accessible worlds from alpha. This really means (for all beta): (alpha R beta) - (beta satisfy f). But (alpha R beta) is always false, and (beta satisfy f) is always false, so (alpha R beta) - (beta satisfy f). OK? Dunno. I'll have to think about it. Normally, we will discuss this a lot. One thing I find puzzling is that accessible seems ill defined. Of course, it means just binary relation, on some non empty set called multiverse (here). I have an intuitive grasp of what possible and necessary mean. But that is only the alethic modalities. In PA the modal box [] will represent provability, and the worlds will be non
Biology, Buddha and the irreflexive Multiverse (was Re: Modal Logic (Part 3: summary + 1 exercise)
On 06 Feb 2014, at 21:29, meekerdb wrote: On 2/6/2014 12:14 PM, Bruno Marchal wrote: In Kripke semantic all statements are relativized to the world you are in. []A can be true in some world and false in another. The meaning of [] is restricted, for each world, to the world they can access (through the accessibility relation available in the Kripke multiverse). []A still keep a meaning, but only in each world. So everything is said when we define the new meaning of [] by the rule []A is true in alpha, by definition, means that A is true in all world beta *accessible* from alpha. And A is true in alpha iff there is a world beta; where A is true, accessible from alpha. Suppose A is true in alpha, OK. Nice. but alpha is not accessible from alpha OK. and A is not true in any other world accessible from alpha. OK. Does it follow that A is not true in alpha? Yes. That does follow. How frustrating! A is true, but not possible. How could that makes sense? Well, this does not make sense ... in the Leibnizian multiverse. For sure. I don't see the point allowing that worlds may not be accesible from themselves? Does that have some application? Yes. First you prove to everybody that I can see in the future, as I announced yesterday the discovery of a Kripke multiverse violating the law []A - A. You just did. Well, in alpha, to be sure, []A - A is true (OK?), but []~A - ~A is falsified, as []~A is true (~A is true in all accessible world from alpha), and ~A is false in alpha, as A is true is true in alpha, and worlds obeys CPL). That amounts to the same, as the laws do not depend on the valuation. If []A - A is a law, []~A - ~A should follow. Note that []~A - ~A, is equivalent with (contraposition, double negation): ~~A - ~[]~A = A - A A - A is the dual formulation of []A - A. As law, they are equivalent. But as formula in one world, they can oppose to each other. So you did find a Kripke multiverse violating the *law* []A - A. And you did find the culprit: those bizarre world which does not access to themselves. Does that have some application? Yes. 1) An easy one, which plays some role in what I like to call the simplest buddhist theory of life ever! And that theory is a subtheory of G, and so will stay with us. That theory models life by worlds accessibility. To be alive at alpha means that t is true in alpha. It means that there is, at least, one world accessible from alpha. To die at alpha means that t is false in alpha. But t is true in alpha, as t is true in all worlds, so the only way to have t false, is that there are no accessible worlds from alpha, at all, including itself. That makes alpha into a cul-de-sac world. So in Kripke semantics, ~t, or equivalently []f, characterizes the cul-de-sac world. Then the simplest buddhist theory of life ever is just the statement, If you are alive, then you can die. It means that for all worlds alpha where you are alive (t is true), you can access to a cul-de-sac world. It means that everywhere, in all worlds we t - []f, or equivalently t - ~[]t. 2) If you interpret t by intelligent, and []f by stupid, you get with the same multiverse, my general theory of intelligence and stupidity. 3) if you interpret [] by provability (in PA, or in ZF), again, t - ~[]t is a law. Read: if I am consistent, then I can't prove that I am consistent. It is easy to see that the law t - ~[]t is a direct consequence of the formula of Löb []([]A - A) - []A. Just put t in place of A, and keep in mind that A - f is just ~A, and then contra-pose: []([]A - A) - []A []([]f - f) - []f [](~[]f) - []f ~[]f - ~[](~[]f) t - ~[]t The worlds in the Kripke mutiverse characterizing G are like that, they don't access to themselves. []A- A is not an arithmetical law from the 3p self-referential view of the machine, but that is why the Theaetetus idea is applicable and will give the non trivial S4Grz for the knower, or first person, fro which []A - A is indispensable. Some might be astonished that []f is true in a cul-de-sac world. But kripe semantics say that []f is true in alpha then f is true in all accessible worlds from alpha. This really means (for all beta): (alpha R beta) - (beta satisfy f). But (alpha R beta) is always false, and (beta satisfy f) is always false, so (alpha R beta) - (beta satisfy f). OK? Bruno Brent -- You received this message because you are subscribed to the Google Groups Everything List group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. http://iridia.ulb.ac.be/~marchal/ -- You received this
Re: Biology, Buddha and the irreflexive Multiverse (was Re: Modal Logic (Part 3: summary + 1 exercise)
On 2/7/2014 10:40 AM, Bruno Marchal wrote: On 06 Feb 2014, at 21:29, meekerdb wrote: On 2/6/2014 12:14 PM, Bruno Marchal wrote: In Kripke semantic all statements are relativized to the world you are in. []A can be true in some world and false in another. The meaning of [] is restricted, for each world, to the world they can access (through the accessibility relation available in the Kripke multiverse). []A still keep a meaning, but only in each world. So everything is said when we define the new meaning of [] by the rule []A is true in alpha, by definition, means that A is true in all world beta *accessible* from alpha. And A is true in alpha iff there is a world beta; where A is true, accessible from alpha. Suppose A is true in alpha, OK. Nice. but alpha is not accessible from alpha OK. and A is not true in any other world accessible from alpha. OK. Does it follow that A is not true in alpha? Yes. That does follow. How frustrating! A is true, but not possible. How could that makes sense? Well, this does not make sense ... in the Leibnizian multiverse. For sure. I don't see the point allowing that worlds may not be accesible from themselves? Does that have some application? Yes. First you prove to everybody that I can see in the future, as I announced yesterday the discovery of a Kripke multiverse violating the law []A - A. You just did. Well, in alpha, to be sure, []A - A is true (OK?), but []~A - ~A is falsified, as []~A is true (~A is true in all accessible world from alpha), and ~A is false in alpha, as A is true is true in alpha, and worlds obeys CPL). That amounts to the same, as the laws do not depend on the valuation. If []A - A is a law, []~A - ~A should follow. Note that []~A - ~A, is equivalent with (contraposition, double negation): ~~A - ~[]~A = A - A A - A is the dual formulation of []A - A. As law, they are equivalent. But as formula in one world, they can oppose to each other. So you did find a Kripke multiverse violating the *law* []A - A. And you did find the culprit: those bizarre world which does not access to themselves. Does that have some application? Yes. 1) An easy one, which plays some role in what I like to call the /simplest buddhist theory of life ever/! And that theory is a subtheory of G, and so will stay with us. That theory models life by worlds accessibility. To be alive at alpha means that t is true in alpha. It means that there is, at least, one world accessible from alpha. To die at alpha means that t is false in alpha. But t is true in alpha, as t is true in all worlds, so the only way to have t false, is that there are no accessible worlds from alpha, at all, including itself. That makes alpha into a cul-de-sac world. So in Kripke semantics, ~t, or equivalently []f, characterizes the cul-de-sac world. Then the /simplest buddhist theory of life ever/ is just the statement, If you are alive, then you can die. It means that for all worlds alpha where you are alive (t is true), you can access to a cul-de-sac world. It means that everywhere, in all worlds we t - []f, or equivalently t - ~[]t. 2) If you interpret t by intelligent, and []f by stupid, you get with the same multiverse, my general theory of intelligence and stupidity. 3) if you interpret [] by provability (in PA, or in ZF), again, t - ~[]t is a law. Read: if I am consistent, then I can't prove that I am consistent. It is easy to see that the law t - ~[]t is a direct consequence of the formula of Löb []([]A - A) - []A. Just put t in place of A, and keep in mind that A - f is just ~A, and then contra-pose: []([]A - A) - []A []([]f - f) - []f [](~[]f) - []f ~[]f - ~[](~[]f) t - ~[]t The worlds in the Kripke mutiverse characterizing G are like that, they don't access to themselves. []A- A is not an arithmetical law from the 3p self-referential view of the machine, but that is why the Theaetetus idea is applicable and will give the non trivial S4Grz for the knower, or first person, fro which []A - A is indispensable. Some might be astonished that []f is true in a cul-de-sac world. But kripe semantics say that []f is true in alpha then f is true in all accessible worlds from alpha. This really means (for all beta): (alpha R beta) - (beta satisfy f). But (alpha R beta) is always false, and (beta satisfy f) is always false, so (alpha R beta) - (beta satisfy f). OK? Dunno. I'll have to think about it. One thing I find puzzling is that accessible seems ill defined. I have an intuitive grasp of what possible and necessary mean. And I know what provable means. But my intuitive idea of accessible say every world should be accessible from itself. Logic is about formal relations of sentences so I understand that accessible will have different applications, but what are some examples? Is Robinson arithmetic accessible from Peano? Is ZFC accessible from arithmetic? Brent -- You
