For example, you have done in the above mail an excellent work of narrative pedagogy, unlike in other cases. But sorry I don´t want to deviate you from the subject of modal logic, that is very interesting. Please forget my responses for now.
2014/1/21, Alberto G. Corona <[email protected]>: > Thanks for the info. It is very interesting and It helps in many ways. > > The problem with mathematical notation is that it is good to store and > systematize knowledge, not to make it understandable. The transmission > of knowledge can only be done by replaying the historical process that > produces the discovery of that knowledge, as Feyerabend said. And this > historical process of discovery-learning-transmission can never have > the form of some formalism, but the form of concrete problems and > partial steps to a solution in a narrative in which the formalism is > nothing but the conclussion of the history, not the starting point. > > Doing it in the reverse order is one of the greatest mistake of > education at all levels that the positivist rationalsim has > perpetrated and it is a product of a complete misunderstanding that > the modern rationalism has about the human mind since it rejected the > greek philosophy. > > Another problem of mathematical notation, like any other language, is > that it tries to be formal, but part of the definitions necessary for > his understanding are necessarily outside of itself. Mathematics may > be a context-free language, but philosophy is not, as well as > mathematics when it is applied to something outside of itself. but > that is only an intuition that I have not entirely formalized. > > 2014/1/21, Bruno Marchal <[email protected]>: >> On 20 Jan 2014, at 23:47, LizR wrote: >> >>> On 21 January 2014 08:38, Bruno Marchal <[email protected]> wrote: >>> >>> If you remember Cantor, you see that if we take all variables into >>> account, the multiverse is already a continuum. OK? A world is >>> defined by a infinite sequence like "true, false, false, true, true, >>> true, ..." corresponding to p, q, r, p1, q1, r1, p2, q2, ... >>> >>> I assume it's a continuum, rather than a countable infinity because >>> if it was countable we could list all the worlds, but of course we >>> can diagonalise the list by changing each truth value. >> >> >> Very good. >> >> (Those who does not get this can ask for more explanations). >> >> >> >> >> On 21 Jan 2014, at 01:32, LizR wrote: >> >>> On 21 January 2014 08:38, Bruno Marchal <[email protected]> wrote: >>> >>> Are the following laws? I don't put the last outer parenthesis for >>> reason of readability. >>> >>> p -> p >>> >>> This is a law because p -> q is equivalent to (~p V q) and (p V ~p) >>> must be (true OR false), or (false OR true) which are both true >>> >>> (p & q) -> p >>> >>> using (~p V q) gives (~(p & q) V q) ... using 0 and 1 for false and >>> true ... (0,0), (0,1) and (1,0) give 1, (1,1) gives 1 ... so this is >>> true. So it is a law. I think. >>> >>> (p & q) -> q >>> >>> Hmm. (~(p & q) V q) is ... the same as above. >>> >>> p -> (p V q) >>> >>> (~p V (p V q)) must be true because of the p V ~p that's in there >>> (as per the first one) >>> >>> q -> (p V q) >>> >>> Is the same...hm, these are all laws (apparently). I feel as though >>> I'm probably missing something and getting this all wrong. Have I >>> misunderstood something ? >> >> No, it is all good, Liz! >> >> What about: >> >> (p V q) -> p >> >> and >> >> p -> (p & q) >> >> What about (still in CPL) the question: >> >> is (p & q) -> r equivalent with p -> (q -> r) >> >> Oh! You did not answer: >> >> ((COLD & WET) -> ICE) -> ((COLD -> ICE) V (WET -> ICE)) >> >> So what? Afraid of the logician's trick? Or of the logician's madness? >> Try this one if you are afraid to be influenced by your intuition >> aboutCOLD, WET and ICE: >> >> ((p & q) -> r) -> ((p -> r) V (q -> r)) >> >> Is that a law? >> >> And what about the modal []p -> p ? What about the []p -> [][]p, and >> <>p -> []<>p ? Is that true in all worlds? >> >> Let me an answer the first one: []p -> p. The difficulty is that we >> can't use the truth table, (can you see why) but we have the meaning >> of "[]p". Indeed it means that p is true in all world. >> Now, p itself is either true in all worlds, or it is not true in all >> worlds. Note that p -> p is true in all world (as you have shown >> above, it is (~p V p), so in each world each p is either true or false. >> >> If p is true in all worlds, then p is a law. But if p is true in all >> world, any "A -> p" will be true too, given that for making "A -> p" >> false, you need p false (truth is implied by anything, in CPL). So if >> p itself is a law, []p -> is a law. For example (p->p) is a law, so [] >> (p->p) -> (p->p) for example. >> But what if p is not a law? then ~[]p is true, and has to be true in >> all worlds. With this simple semantic of Leibniz, []p really simply >> means that p is true in all world, that is automatically true in all >> world. If p is not a law, ~[]p is true, and, as I said, this has to be >> true in all world (in all world we have that p is not a law). >> So []p is false in all worlds. But false -> anything in CPL. OK? So >> []p -> p is always satisfied in that case too. >> So, no matter what, p being a law or not, in that Leibnizian universe: >> []p -> p *is* a law. >> >> In Leibniz semantic, you have just a collection of worlds. If []p is >> true, it entails that p is true in all worlds, so []p is true in all >> world too. >> >> Can you try to reason for []p -> [][]p, and <>p -> []<>p ? What about >> p -> []p ? What about this one: >> ([]p & [](p -> q)) -> []q ? >> >> Ask any question, up to be able to find the solution. tell me where >> you are stuck, in case you are stuck. I might go too much quickly, you >> have to speed me down, by questioning! >> It may seem astonishing, but with the simple Leibnizian semantic, we >> can answer all those questions. >> >> With Kripke semantics, the multiverse will get more structure. But in >> Leibniz, all worlds are completely independent, so if p is a law, the >> fact that it is a law is itself a law, and []p will be true in all >> worlds, and be a law itself. Indeed if []p was not a law, there would >> be a world where ~p is true, and p would not be a law, OK? >> >> Take those questions as puzzle, or delicious torture :) >> >> 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. >> > > > -- > Alberto. > -- Alberto. -- 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]. 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