On 29 January 2014 08:29, Bruno Marchal <[email protected]> wrote: > Hi Liz, Others, >
"Good morning Professor Marchal!" > > In the general semantic of Leibniz, we have a non empty set of worlds W, > and some valuation of the propositional variables (p, q, r, ...) at each > world. > > And we should be convinced than all formula, with A, B, C, put for any > formula, of the type > > [](A->B) -> ([]A -> []B) > []A -> A > []A -> [][]A > <>A -> []<>A > A -> []<>A > > are all laws, in the sense that they are all true in all worlds in all > Leibnizian multiverse. OK? > Yes. > > Most are "obvious" (once familiarized with the idea 'course). Take []A -> > A. Let us prove by contradiction, to change a bit. Imagine there is world > with []A -> A is false. That means that in that world we have []A and ~A. > But []A means that A is true in all world, so in that world we would have > [A and ~A. Contradiction (all worlds obeys classical CPL). > > Test yourself by justifying in different ways the other propositions, > again and again. > > But now, all that was semantic, and logicians are interested in theories. > They want axioms and deduction rules. > > So, the question is: is there a theory capturing all the laws, true in all > worlds in all Leibnizian multiverse? > > Answer: YES. > > Ah? Which one. > > S5. > > S5? > > Yes, S5. The fifth system of Lewis. Who did modal logical purely > deductively, and S5 was his fifth attempt in trying to formalize a notion > of deducibility. > > The axioms of S5 are (added to some axiomatization of CPL, like the one I > gave you sometimes ago): > > [](A->B) -> ([]A -> []B) > []A -> A > []A -> [][]A > <>A -> []<>A > > The rules of S5 are: > > The modus ponens rule, like CPL axiomatization. > The necessitation rule: derive []A from A. > ? derive []A from A ??? > > It can be proved that S5 can prove all the laws satisfied by all worlds in > the Leibnizian multiverse. > > Those axioms are independent. For example you cannot prove <>A -> []<>A > from the other axioms using those rules. But how could we prove that? This > was rather well known by few modal logicians. > > There is a curious article by Herman Weyl, "the ghost of modality", were > Herman Weyl illustrate both that he is a great genius, and a great idiot > (with all my very deep and sincere respect). > He said that our minds "crawl up our worldlines" didn't he? Thereby giving lots of people the wrong idea about how a block universe works. > > Modal logic has been very badly seen by many mathematicians and logicians. > In the field of logic, modal logicians were considered as freak, somehow. > Important philosopher, like Quine were also quite opposed to modal logic. > So it was very gentle from Herman Weyl to attempt to give modal logic some > serious considerations. > He tried to provide a semantic of modal logic with intuitionist logic, but > concluded that it fails, then with quantum logic, idem, then with > provability logic (sic), but it fails. It fails because each time some > axiom of S5 failed! > This shows he was biased by the Aristotelian Leibnizian metaphysics. In > fact he was discovering, before everybody, that there are many modal > logics, and that indeed they provide classical view on many non standard > logics. In fact, somehow, it is the first apparition of the hypostases in > math (to be short). > I really love that little visionary paper (if only I could put my hand on > it). > It comes up a lot if you google - I think you have to belong to various academic groups to read it...maybe you would be able to? > > But if S5 is characterized by the Leibnizian multiverse. What will > characterize the other modal logics? > > Well, there has been many other semantics, but a beautiful and important > step was brought by Kripke. > > It is almost like the passage from the ASSA to the RSSA! The passage from > absolute to relative. The passage from Newton to Einstein. > > Kripke will put some structure on the Leibnizian multiverse. He will > relativize the necessities and possibilities. > How? > By introducing a binary relation on the worlds, called accessibility > relation. Then he require this: > > []A is true in a world alpha = A is true in all worlds *accessible* > from alpha. > > Exercise: what means <>A here? (cf <>A is defined by ~[]~A). > it isn't the case that in all worlds accessible from alpha, A is false. Or in at least one world accessible from alpha, A is true. > > So a Kripke multiverse is just a non empty set, with a binary relation > (called accessibility relation). It is a Leibnizian multiverse, enriched by > that accessibility relation. For []A being true, we don't require it to be > true in all worlds, but only in all worlds accessible from some world (like > the "actual world", for example). > > Again a Kripkean law will be a proposition true in all worlds in all > Kripke multiverse. > > Now I am a bit tired, so I give you the sequel in 2 exercises, or subject > of meditation. > > 1) Try to convince yourself that the formula: > > [](A->B) -> ([]A -> []B) > > is a Kripkean law. It is satisfied in all worlds (meaning also all > valuations of the propositional letters), in all Kripke multiverse. > OK, I think, if the []s refer to the same subset of the multiverse in each case this reduces to Leibniz case. > > 2) try to convince yourself that none of the other formula are laws in all > Kripke multiverse. Try to find little Kripke multiverse having some world > contradicting those laws. > []p -> p I take this to mean that the truth of p isn't available if the world in question isn't accessible from the one under consideration. So []p = "(if) p is true in all worlds accessible from some world" which doesn't imply that p is true in a given world. []A -> [][]A if A is true in all worlds accessible from a given world, then wouldn't that imply that it's true that "A is true in all worlds accessible from a given world" in all worlds accessible from a given world? It seems like because we use [] on either side, we are reducing to a multiverse connected by accessibility, and within that world, "Leibniz still applies". But I must have that wrong. I will have to leave this for now... > Can you find *special* binary relations which would enforce some of those > propositions to be law in those corresponding *special* Kripke multiverse? > > Advise. Draw potatoes for the worlds, with the valuation inside, and draw > big readable arrow between two worlds when one is accessible from the > other. The binary relation is arbitrary. If (alpha beta) is in the relation > (beta is accessible from alpha), we don't have necessarily that (beta > alpha) is in the relation. > > I let you play. (with moderation. No need of brain boiling, and ask any > question if something seems weird, keep in mind I made typo also ...) > > Bruno > > > > > http://iridia.ulb.ac.be/~marchal/ > > > > -- > 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 [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/groups/opt_out. > -- 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 [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
