FOM: Book recommendation

[Roger Bishop Jones]

responding to Steve Simpson:

> Roger Bishop Jones (FOM, 21 Aug 2000) writes:
>
>  > The purpose of my posting was to resist the suggestion that any
>  > possible reduction would settle the case against "alternative
>  > logics", which I am comfortable doing without having seen
>  > Manzano's.
>
> Manzano does not present any "case against alternative logics".

The suggestion I refer to above was yours, not Manzano's, in the posting in
which you mentioned both Manzano's book and the previous thread on
"second-order logic is a myth".
I hope I have not misunderstood your intention.

> Of course it is true that the interest of Manzano's reductions depends
> on what properties are preserved.  But actually, her reduction of
> second-order logic to first-order logic is truth-preserving and
> meaning-preserving, in an appropriate sense of these terms.  Since you
> have not seen Manzano's book, why do you assume otherwise?

I make no assumptions about Manzano's book.

My own observation in relation to truth preservation should have included
the word "effective".
I know that Manzano's book contains no effective truth preserving reduction
of standard second order logic to first order logic because the true
sentences of first order logic are recursively enumerable, whereas those of
standard second order logic are not.

Clearly, the use of the word "standard" here is essential in making clear
which language I am talking about, given that semantics is of the essence in
this context.
I guess from your comment that (you or) Manzano considers non-standard
models "appropriate", but this is a significant variation in the semantics
of second order logic.

Roger Jones
(retired software engineer, unaffiliated)