Re: tautology

From: Marchal <marchal.domain.name.hidden>
Date: Thu Dec 9 08:04:55 1999

Russell Standish:

>> Bruno: I'm not sure I understand what you mean
>> by ``\models_\alpha^W"
>
>I'm using LaTeX symbols. In English, the above phrase reads "is a
>theorem in world alpha of model W".

I guess you mean is true in word alpha. A theorem, semantically, is
something true in all worlds in a (suitable) model.


>Consider the following diagram:
>
> /------ [False]
> |
> [\Box False]{--- [False]
> |
> \------[False]
>
>I was reading this as saying "False is a true statement", therefore
>\Box False is true is the predecsessor world. Does one rule out
>statements like "False is a true statement" from the picture utterly?

Yes. Unless you did *have* met a white rabbit !
Or unless you are doing non standard modal logic. But please don't
do that. One of my motivation of the use of classical
modal logic is the hope of getting a classical picture of the
non standard (non modal) logic like intuitionistic logic or
quantum logic.

So it's better to remember that FALSE is never a true statement. In fact
classical propositional proposition (tautologies) are true in ALL the
world in all models based on any frame.
You can still get BOX FALSE true in a world. But that makes that world
terminal.

>> The interesting formula to find here is a formula with one
>> variable "p" which would caracterize Kripke frames with no
>> terminal worlds. Solution : (BOX p -> DIAMOND p). This formula
>> is verified, for all the valuation of p, iff the frame is
>> ideal. There are no cul-de-sac at all.
>
>True, but idealism is a sufficient, but not necessary condition for
>absence of cul-de-sacs. One can have a model ...

OK, but the sufficient and necessary condition are defined with
respect to frames. In that sense BOX P -> DIAMOND P is suff & nec.

>... in which every terminal
>world is preceded by a world that acesses non-terminal states. Such a
>model is required for QTI to hold.

OK. We can distinguish the frames where you are immortal but where
you still can die. In that case your immortality is given by luck.
But the measure we are searching is defined only on the world where
we survive. In that case we abstract away the terminal worlds.
This is natural for the modelisation of the first person point of view.
In that case we need the "ideal" formula BOX p -> DIAMOND p.

Bruno
Received on Thu Dec 09 1999 - 08:04:55 PST