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From: George Levy <GLevy.domain.name.hidden>

Date: Fri, 23 Mar 2001 10:50:11 -0800

Marchal wrote:

*> Hi George,
*

*>
*

*> I make the foolish promise to give you my
*

*> proof.
*

We're even. I made the foolish promise to ask for your proof. :-)

With my background in electronic engineering, I am moderately versed in

logic, in particular Boolean logic. I am sorry for my long hmmmm. The

going got rough when you started talking about knowability and

believability. But I realize that if we are to investigate consciousness

these are ideas that have to be talked about.

So let's go with it.. I promise I'll give it a shot. It will be very

instructive.... However it would help if together with the string of

symbols there was an English translation.

*> Here is Leibniz semantics for modal
*

*> logic. It is a preamble.
*

*> Don't hesitate to tell me if it is too difficult
*

*> or too easy, or too technical ...
*

*> I suppose you know a little bit of classical logic.
*

*> If you don't, just tell me.
*

I'll tell you!

*>
*

*>
*

*> ====================
*

*>
*

*> We have some atomic propositions p, q, r, ...
*

*> And connectives &, v, ->, - (intended for "and", "or",
*

*> "if...then", "not").
*

*>
*

*> We know what is a semantics for Classical Propositional
*

*> Logic (CL). Basicaly it an assignement of truth values,
*

*> among {FALSE, TRUE} for the atomic propositions.
*

*>
*

*> The semantics of well formed formula like p & (q v r)
*

*> will follow by the usual use of truth table. For exemple:
*

*>
*

*> A v B A -> B
*

*> 1 1 1 1 1 1
*

*> 1 1 0 1 0 0
*

*> 0 1 1 0 1 1
*

*> 0 0 0 0 1 0
*

hmmm.... do you mean?

A v B A -> B

1 1 1 1 1 1

1 0 1 1 0 0

0 1 1 0 1 x

0 0 0 0 1 x

*> Note that the truth value of a formula is completely
*

*> determined by the truth value of its compounds.
*

*>
*

*> The problem is to provide a clean semantics (meaning) for
*

*> sentence like []p -> p or <>p -> []q, ... with the intuitive
*

*> reading "if necessary p then p","if possible p then
*

*> necessary q"...
*

*>
*

OK

*>
*

*> Like the not "-", the box "[]" and the diamond "<>" are unary
*

*> connective and it is obvious that we cannot define them
*

*> truth functionaly (unless we define the box by the
*

*> identity and the diamond by not, but that would be rather
*

*> trivial). The truth value of a formula will no more completely
*

*> be determined by the truth value of its compounds!
*

*>
*

*> In "Leibniz semantics" there is a collection W of worlds.
*

*> That collection of world is called a frame.
*

*>
*

*> The frame W becomes a model (W,V) when there is given
*

*> a valuation V, assigning truth values to the atomic
*

*> propositions in each world.
*

*> Each world is supposed to "obey" classical logic. This means
*

*> that if p is true in world w and if q is true in world w, then
*

*> p & q is true in world w, etc.
*

*>
*

*> I will say that the model (W,V) is based on the frame W.
*

*>
*

*> Here is Leibniz semantics for modal propositions.
*

*> In any world, []p will be considered true if p is true
*

*> in all worlds of W. And in any world, <>p will be considered true,
*

*> if there is (at least one) world in W in which p is true.
*

*>
*

*> This captures the intuitive idea that "p is necessary" means
*

*> p is true in all possible conceivable situations, worlds, states,
*

*> etc. and "p is possible" if there is at least one world (states ...)
*

*> where p is true. That's the idea often attributed to Leibniz.
*

*>
*

OK

*>
*

*> Validity is the key notion (generalising the notion of tautology
*

*> in the non modal case).
*

*> I will say that a formula is valid in a frame, if the formula
*

*> is true in all the worlds of all models based on that frame.
*

*>
*

*> A simple (non modal) tautology is of course valid. For exemple
*

*> "p v -p" is true in all world independently of the valuations.
*

*> []p v -[]p is also valid.
*

*>
*

*> Exercices: 1) is []p -> p valid? 2) is p -> []p valid?
*

*>
*

*> Solution: Yes []p -> p is valid. let take an arbitrary world w
*

*> in an arbitrary model (W,V) if []p is true in w, it means p
*

*> is true in all world of the model (W,V), then it is true
*

*> in particular in w, so []p -> p is true in w. This works
*

*> for all w (note that if []p is false in w, then []p -> p is
*

*> automatically true in w. So []p -> p is valid.
*

*> 2) no p -> []p is not valid. Take a frame with two worlds
*

*> w1 and w2, and take a valuation which makes p true in w1 and
*

*> which make p false in w2. Clearly in w1 you have p and -[]p,
*

*> so in w1 p -> []p is false.
*

*>
*

OK

*>
*

*> Exercices. Show that the following sentences are valid:
*

*>
*

*> p -> <>p
*

*> []p -> [][]p
*

*> p -> []<>p
*

*> <>p -> []<>p
*

*> [](p->q) -> ([]p -> []q)
*

*>
*

*> Of course if <>p -> []<>p is valid, <>TRUE -> []<>TRUE is
*

*> certainly valid to, and so our "godel second theorem"
*

*> <>TRUE -> -[]<>TRUE is certainly NOT valid with Leibniz
*

*> semantics. This just means that formal provability cannot
*

*> play the role of the leibnizian "necessity".
*

*> Kripke generalisation of Leibniz semantics will provide
*

*> the necessary tools.
*

*>
*

OK

*>
*

*> The non logician should note that with a semantics we can
*

*> reason on the validity of sentences without having a formal
*

*> system in which we could *prove* those formula.
*

*> A "difficult" exercice would consist in finding a formal
*

*> system which would axiomatize the Leibnizian formula.
*

*>
*

*> (In fact it is axiomatized by the system known as S5 which
*

*> has the axioms:
*

*>
*

*> [](p->q) -> ([]p -> []q)
*

*> []p->p
*

*> <>p -> []<>p
*

*>
*

*> + the classical tautologies.
*

*>
*

*> with the inferences rules:
*

*>
*

*> p p->q p
*

*> ------- --- + a substitution rule
*

*> q []p
*

*>
*

This string of symbols does not mean anything to me... Is there a real

life model to which it applies, a story, a game, anything to give it

meaning?

*>
*

*> But let us go slowly).
*

*>
*

*> Bruno
*

So far so good

Thanks for making this effort.

George

Received on Fri Mar 23 2001 - 11:48:41 PST

Date: Fri, 23 Mar 2001 10:50:11 -0800

Marchal wrote:

We're even. I made the foolish promise to ask for your proof. :-)

With my background in electronic engineering, I am moderately versed in

logic, in particular Boolean logic. I am sorry for my long hmmmm. The

going got rough when you started talking about knowability and

believability. But I realize that if we are to investigate consciousness

these are ideas that have to be talked about.

So let's go with it.. I promise I'll give it a shot. It will be very

instructive.... However it would help if together with the string of

symbols there was an English translation.

I'll tell you!

hmmm.... do you mean?

A v B A -> B

1 1 1 1 1 1

1 0 1 1 0 0

0 1 1 0 1 x

0 0 0 0 1 x

OK

OK

OK

OK

This string of symbols does not mean anything to me... Is there a real

life model to which it applies, a story, a game, anything to give it

meaning?

So far so good

Thanks for making this effort.

George

Received on Fri Mar 23 2001 - 11:48:41 PST

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