Hi Liz, and others,
I explain the classical modal logic.
It extends classical propositional logic (CPL), that we have already
encounter.
I will recall it first, and present it in a way which will suit well
the modal extensions of CPL.
One big advantage of CPL on all other propositional logic, is the
extreme simplicity of its semantics, which is truth functional, but I
will explain this later.
And what is logic? Also.
Logicians are interested in "reasoning", and they want the validity of
the reasoning guarantied by its formal structure. He want to present
the reasoning in a way which does not depend on the interpretation of
the formula involved. That will guarantied that the reasoning is
independent of the interpretation, or of the world which might satisfy
the formula, and the formula will be true in all worlds, or in all
situations, or in all interpretations, etc. making a formula true
independently of the interpretation makes it into a law. It makes it
into something necessary, universally true.
But what is a world? For logician, worlds are rather abstract things.
Here, let me give the simple definition which suits well the goal of
classical propositional logic (CPL).
We make this formal, so we have a list of so-called propositional
variable p, q, r, p1, q1, r1, p2, ...
It helps some people, but distracts others, to instantiate the
propositional variable by concrete propositions like "Obama is
president of the US, there is a planet called Earth, Earth has two
natural satellites, 34 is prime, etc."
The idea is that we take *all* propositions (later represented by
sentences, and restricted to more particular language, like predicate
calculus, or arithmetic, or set theory, ...)
Having those propositional variables we can define now a world, or an
interpretation, by an assignment of truth value (true, false) to each
proposition.
To simplify, let us use a simple finite finite set of propositions {p,
q, r}.
With just that set we get 8 worlds (2^3):
- the world where p is true, q is true, and r is true
- the world where p is true, q is true, and r is false
- the world where p is true, q is false, and r is true
- the world where p is true, q is false, and r is false
- the world where p is false, q is true, and r is true
- the world where p is false, q is true, and r is false
- the world where p is false, q is false, and r is true
- the world where p is false, q is false, and r is false
OK?
If you want the set of all possible worlds, having three propositional
variables, can be said to be the mutiverse of that logic.
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, ...
If p is true in a word, I will often express this by saying that the
world satisfies p, or obeys p.
Now, what makes CPL much more easy than any other propositional logic,
is that the truth of the compound non atomic formula, those build with
the usual symbol "&, V, ~, ->, <->, ... is entirely determined by the
worlds, that is by the assignment of truth value to the components of
the formula (eventually the atomic formula, that is the propositional
variables).
For example, if a world satisfies p, and if it satisfies q, then it
satisfy the compound formula (p & q).
And in fact, the truth value (true or false) of a conjunction (p & q)
is entirely determined in all 8 worlds above.
(And also in richer "multiverse" with assignment to all variables)
Is the formula (p & q) a law? is it true in all possible worlds (in
the multiverse, which here contains 8 worlds).
Obviously not. OK? (p & q) is true only in the two first world
described above. All others does not satisfy (p & q), they contradicts
it. They constitute counter-examples.
Do you remember the semantic (the association of truth value) for (p V
q)? for ~p?
I recall that the semantic of (p -> q) is the one for (~p V q), or
similarly ~(p & ~q). OK? (p -> q) is false only in the worlds where p
is true and q is false. If p is false in some world, then p -> q is
true (trivially true, said some logicians).
Can you find laws? It means a formula true in all worlds (in our
little multiverse, or bigger one).
Are the following laws? I don't put the last outer parenthesis for
reason of readability.
p -> p
(p & q) -> p
(p & q) -> q
p -> (p V q)
q -> (p V q)
p -> (q -> p)
(p -> (q -> r)) -> ((p -> q) -> (p -> r))
What about, with the propositional variable WET, COLD, ICE:
((COLD & WET) -> ICE) -> ((COLD -> ICE) V (WET -> ICE))
Is that a law? (hint: beware the logician's trick!)
Anyone asks any question if something is not understood. I use the
fact that we have already done a bit of CPL on FOAR more or less
recently, and on this list too, but much less recently.
Let us begin the modal logic.
The syntax is the same, we have the same propositional variables, p,
q, r, ... and the same logical connectors (&, V, ->, ~, ...). But we
have one more symbol, usually represented by a box [].
"[]" is a unary connector. so (p [] q) is NOT a formula, but ([]p) and
([] q) are. I will abbreviate them by []p and []q, again for reason of
readability.
The bad news is that [] will not be truth functional, and in most
circumstance the truth value of "[]p" will not be derivable from the
truth value of p.
The intended semantics, indeed meant by Aristotle (who invented or
discovered the modal logic), or by Leibniz, who got somehow the
"multiverse" or "possible world" semantic of modal logic, is that
[] p
means that p is a law.
We want be able to express in the language the very idea that some
formula are laws, or that they are *necessary truth*. We want express
in the language that p, or some formula is true in all possible worlds.
According to Leibniz, we inhabit some world (even the best possible
one). We have clue that it is probably the world where "Obama is the
president of the US" is assigned to TRUE, "Earth has to satellites is
assigned to FALSE, "34 is prime" is assigned to false, etc.
But our worlds is just a worlds among all possible worlds, which here
includes worlds where "Obama is the president" is assigned to FALSE,
and where "34 is prime" is assigned to TRUE. That might look absurd,
but the logician, at this stage does not care, because he is
interested only in the laws, that is what is true in *all* worlds,
even, the most insane one.
Now, you will tell me, that such a notion of necessity is perhaps
trivial. Indeed we can find the laws in those "propositional worlds"
just by doing the truth table, and when they got TRUE at each line, we
know that it is a law, so []A will be true in all worlds if and only
if A is a tautology.
Well, that solves the problem when A is a non modal formula. But what
can be said about the following formula, like
[] p -> p
Is that true in all world? (With the present semantic where []x means
that x is true in all worlds, with the definition of world above).
What about []p -> [] [] p ?
Let us define <>p by ~[] ~p. "<>p" can be read as "not necessary not
p", or more simply "possible p". (OK?, mediate on this).
So what about <>p -> []<>p, what about p -> []<>p, ?
I let you think. You can test those formula in the little multiverse
of size 8 that we have above.
In a sense, modal logic is the math of the multiverses. All
multiverses, in a quite larger sense than the quantum multiverse, of
course.
Hope your head is not boiling hot, in which case I encourage a good
refreshing tisane :)
Take it easy, it is not yet finished. I have to sum up more than 2300
years of studies, with capital and subtle development those last
centuries.
Bruno
http://iridia.ulb.ac.be/~marchal/
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