FOM: first order and second order logic: once more

Roger Bishop Jones rbjones at
Mon Sep 4 15:48:14 EDT 2000

In response to Martin Davis Monday, September 04, 2000 5:20 AM

> I meant the term "his understanding of second order logic" simply to refer
> to his use of the term as involving the entire set of VALID sentences of
> SOL, as opposed to the "textbook" use of the term that singles out a
> particular r.e. subset of that set.

Well I'm pleased that you didn't mean what I thought you meant, but I don't
recognise your description of "my understanding" of SOL.
It is not my intention to talk of anything other than the standard textbook
second order logic with semantics defined using "standard models".
This does of course "involve" all the valid sentences in some sense, and all
the rest, but the set of theorems remains r.e.

> What I was after is this. Bishop claims that the notion of validity in SOL
> has the virtue of enabling one to "say what you mean." Now I hold, and
> expect that Bishop will agree, that saying what you mean should be in a
> context in which you are saying it to someone, and I take it that what he
> is maintaining is the he can communicate such a meaning that would be
> difficult or impossible to convey in other ways.

No, I don't agree.
People, either a speaker or a hearer are not relevant to the issue on which
I spoke.
The subject matter is entirely abstract and the point at issue is whether
propositions can be expressed in second order logic which cannot be
expressed in first order logic.
Of course, there is some difficulty in attaching meaning to such words as
"proposition" and "expressed".
I have made an attempt to treat the matter slightly more formally in my last
posting, which also includes an extremely unwitty example.

Roger Jones

NB for the record, since it seems to be causing confusion, my full name is
Roger Bishop Jones, though Roger Jones is sufficient for most purposes.
(both Roger and Bishop are christian names, the latter is also a family
I use my full name in mail headers and in some other contexts, because
"Roger Jones" is by itself very ambiguous.

RBJones at

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