On 24 January 2014 01:06, Bruno Marchal <[email protected]> wrote: > > On 23 Jan 2014, at 08:57, LizR wrote: > Everybody loves my baby. Therefore my baby loves my baby. But my baby > loves nobody but me. Therefore - the only way this can be true - is if > Alicia *is* her baby. So the answer is yes! >
> Excellent. > > And that was "predicate" logic! So you are in advance! > I don't know what predicate logic is (but I watch a lot of TV detective shows. Perhaps that helps!) > > To give a taste of first order logic, it is: > > Alicia theory: (with "Ax" = "for all x"). > > Ax (x loves MyBaby) (everybody loves my baby) > Ax ((MyBaby loves x) -> (x = Me)) (my baby loves nobody but me) > > You deduce correctl, in that theory, that MyBaby = Me, and that everybody > loves Me. Nice! > It seemed to make more sense as a puzzle in English than with symbols! > > And now, given that we talk first order logic (the logic with quantifier > like "A" and "E" (it exists)), I suggest a little meditation on *duality*. > Ah, Stephen will be happy :) > > Do you agree that the "Ex" in "ExP(x") (it exists some x such that it is > the case that P(x)) is a dual of "Ax", in a similar sense that <> is a dual > of [] in propositional modal logic? > ...and you lost me completely. OK, I will take a deep breath and try and break down the problem... Ex means "some x exists", which is like saying <>x perhaps (in some world, x is true) Ax means "for all x" I think which is like saying []x (in all worlds, x is true) These seem kind of parallel, except I guess Ex and Ax operate within a single world, not a "logical multiverse" ? Or do they? (Or does it matter?) > > We have defined <>A by ~[]~A. Can we define ExP(x) by ~Ax ~P(x) ? > ExP(x) means that for at least one x, P(x) is true - P(x) is some proposition regarding x, so if P is "loves" and x is "my baby" ExP(x) would be "Someone loves my baby" So ~P(x) is "doesn't love my baby" So Ax ~P(x) is "Nobody loves my baby" So ~Ax ~P(x) is "Somebody loves my baby" - which is the same!! :) So the answer is yes. > > Do you agree with the following: > > ~[]p <-> <> ~p > p isn't true in all words <-> there is a world in which p is false - yes > ~<>p <-> []~p > there is not a world in which p is true <-> in all worlds p is false - yes > []p <-> ~<>~p > p is true in all worlds <-> it isn't true that there is any world in which p is false - yes (the inverse of the one I did above) > > Can you write those equivalence for A and E in predicate logic? Are they > intuitively valid? > ~AxP(x) <-> Ex~P(x) It isn't the case that for all x, P(x) is true ... hence there exists an x for which P(x) is false ~Ex P(x) <-> Ax ~P(x) There doesn't exist an x for which P(x) is true ... hence for all x, P(x) is false Ax P(x) <-> ~Ex ~P(x) For all x, P(x) is true ... hence there doesn't exist an x for which P(x) is false. > > Let us come back on modal logic. > > The idea of the modal box "[]" is an idea of necessity. The dual (<>) is > read "possible". > Can you find the most common english term for the following possible > modalities: > > [] = necessary, <> = possible > [] = obligatory, <> = ? > preferable? > [] = everywhere, <> = ? > somewhere > [] = always, <> = ? > sometime > > And what about the most important modality which plays the key role in our > comp context (and which is the reason why we do all this): > > [] = provable, <> = ? > possible? > > 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.
