Hi Liz, Others,
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?
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.
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).
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).
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).
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.
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.
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/
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