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From: oholden@idirect.com (Owen Holden)
Subject: Re: Errors in (Hughes and Crestwell)'s Intro to Modal Logic
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ilias kastanas 08-14-90 wrote:
> 
> In article <331812B3.6DE4@idirect.com>,
> Owen Holden <oholden@idirect.com> wrote:
> @My claim is that the following formulas, given as theorems in
> @INTRODUCTION TO MODAL LOGIC by Hughes and Crestwell, are invalid.   (A
> @means all, E means some, [] means necessarily, <> means possibly, '
> @means 'the' in the singular, E! means uniqueness e.g.
> @E!('xFx)<->EyAx(x=y<->Fx)).
> @
> @Ax(([]Fx v [](~Fx)) v Ax((<>Fx & <>(~Fx)) page 185  invalid
> @
> @EyAx(x=y<->[]Fx)->F('xFx)                 page 204  invalid
> @
> @[]E!('xFx)->Ex[]Fx                        page 205  invalid
> @
> @Each of these formulas are shown to be invalid by granting a model,
> @universe of discourse, of two individuals a,b and S5.
> 
>         The first one looks like a typo  ( ... v Ax   should be  ... v Ex ).
> 
>         The other two are variations on the Barcan formula   Ex []P(x)  <=>
>    [] ExP(x)  where either E is taken to be E!  I don't have the book to
>    check this; presumably it discusses rigidity.
> 
>                                                         Ilias

Hi Llias,

There is no typo in the first formula. Could you expand your claim that
the other two are variations of the Barcan formula, I don't see it. The
barcan formula is not 
Ex[]P(x)<=>[]ExP(x), this formula is invalid. We do have
Ex[]P(x)=>[]ExP(x) as a valid formula, usually as an axiom. The Barcan
formula is Ax[]P(x)<=>[]AxP(x).

Owen
