Re: Nonfirstorderizability

Michael De wrote:
> George Dance wrote:
> > In that light, I was wondering: when you have time (ie, when the real
> > discussion is over), would you mind explaining to a simple lay person
> > why the Geach-Kaplan sentence cannot be 'firstorderized' as:
> >
> > ExEyAz((Cx & Cy) & (Axy & Ayx) & (Axz -> z=y) & (Ayz -> z=x))
> > (Ca =df. a is a critic; Aab =df. a admires b)?
> >
> > Or refer me to a weblink, if that's easier.
>
> I take that sentence to imply the Geach-Kaplan sentence but not the
> converse. That there are at most two critics who admire only each other
> (and not themselves) implies that there are some critics (provided
> those critics are not the same since we using the plural--'critics')
> who admire only each other. But that there are some critics who admire
> only each other does not imply that there are at most two critics who
> admire only each other--there could be three, four, etc.

I certainly did't intend to translate 'some critics' as 'just two
critics.' Since that has to have entered in with the equality
predicate, let me see what happens when that predicate is removed:

ExEy((Cx & Cy) & (Axy & Ayx) & (Axy <-> Ayx)) (?)

.

Relevant Pages

  • Re: Nonfirstorderizability
    ... > The Geach-Kaplan sentence refers implicitly to a domain D ... > including critics and other individuals they may admire, ... > as far as first-order sentences are concerned, ...
    (sci.logic)
  • Re: Nonfirstorderizability
    ... The Geach-Kaplan sentence refers implicitly to a domain D ... including critics and other individuals they may admire, ... as far as first-order sentences are concerned, ...
    (sci.logic)
  • Re: Nonfirstorderizability
    ... would you mind explaining to a simple lay person ... I take that sentence to imply the Geach-Kaplan sentence but not the ... That there are at most two critics who admire only each other ...
    (sci.logic)
  • Re: Nonfirstorderizability
    ... > Not a simplification, I'm afraid, but a better translation of "Some ... > critics admire only each other". ...
    (sci.logic)
  • Re: Nonfirstorderizability
    ... This formalizes "There are two critics who ... reasonable reading of the odd statement "Some critics admire only ...
    (sci.logic)