Re: Nonfirstorderizability

Torkel Franzen wrote:
> "George Dance" <georgedance04@xxxxxxxx> writes:
>
> > 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)) (?)
>
> Can you think of a way of simplifying the above?

Hmmm ... dropping out the last conjunct (as it's implied by the
second)? Without that, though, all that the statement says is that
there are some critics who admire each other, with no 'only.' But the
third conjunct doesn't translate 'only' either (as it says nothing
about whether any of the critics admire anyone else).

> What does it say
> when translated into ordinary language?

Just that some critics admire each other; not that they admire *only*
each other; so apparently dropping the reference to others (z) was a
bad move. The trick, then, seems to be to put that back in without
using the equality predicate as previously; which would look like:

ExEyAz((x/=y) & (Cx & Cy) & (Axy & Ayx) & (Axz -> (Cz & Azx)) & (Ayz ->
(Cz & Azy))).

Not a simplification, I'm afraid, but a better translation of "Some
critics admire only each other". ("There are at least two critics who
admire each other, both or all of whom admire only critics who admire
them."

.