On 23 Jan 2014, at 08:57, LizR wrote:
On 23 January 2014 08:18, Bruno Marchal <[email protected]> wrote:
OK. A last little exercise in the same vein, for the night. (coming
from a book by Jeffrey):
Alicia was singing this:
"Everybody loves my baby. My baby loves nobody but me".
Can we deduce from this that everybody loves Alicia?
Surely we can't deduce anything about A and her baby, unless we know
that the song is true! :-)
Oh, but I am not asking if everybody loves Alicia. Only if we can
deduce that everybody loves Alicia from Alicia's theory. For this, we
can be agnostic on that theory. We need just to assume it.
But if it is...
Yes, that is what we can assume, if only for the sake of the argument.
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!
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!
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.
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?
We have defined <>A by ~[]~A. Can we define ExP(x) by ~Ax ~P(x) ?
Do you agree with the following:
~[]p <-> <> ~p
~<>p <-> []~p
[]p <-> ~<>~p
Can you write those equivalence for A and E in predicate logic? Are
they intuitively valid?
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, <> = ?
[] = everywhere, <> = ?
[] = always, <> = ?
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, <> = ?
Bruno
http://iridia.ulb.ac.be/~marchal/
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