Re: tautology

From: Russell Standish <>
Date: Fri, 10 Dec 1999 10:24:28 +1100 (EST)

> 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.

I guess so. I'm not familiar with the terminology used in modal logic
- I interpreted a theorem as meaning a provably true statement (in a
world). I guess that's not how the term is used. In anycase, I was
just following the notation used in Chellas, in which the symbol
\models_\alpha^W is used.

> >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.

I shall take that in board. That makes a few more of your theorems in
the apendix of your thesis easier to prove.

Can you please explain the concept of your 1st person white rabbit
problem - in English (or French if you prefer) rather than modal
logically. I'm afraid the paper you sent me didn't elighten me much -
in fact I was guessing what part of it corresponded - the UDA paradox

> >> 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.

What do you mean by frames? And why does this make \Box p \rightarrow
\Diamond p sufficient?

> >... 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

Since we know that we can die in the 3rd person picture, it is clear
we do not live in an "ideal" model. Why then is it natural to abstract
away the terminal worlds?

Dr. Russell Standish Director
High Performance Computing Support Unit,
University of NSW Phone 9385 6967
Sydney 2052 Fax 9385 6965
Room 2075, Red Centre
Received on Thu Dec 09 1999 - 15:23:33 PST

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