FOM: first order and second order logic: once more

Martin Davis martin at
Mon Sep 4 00:20:56 EDT 2000

At 05:17 PM 9/2/00 +0100, replying to my statement

> > This is one of a series of comments in which Roger Jones extols the
> > of his understanding of second order logic.

Roger Bishop Jones wrote:

>This is entirely false.
>I have made no comments whatsoever on my understanding of second order
>logic, which I am confident is limited by comparison with the average
>contributor to the fom list.

I apologize for my lack of clarity: I was in no way intending to impugn 
Roger Bishop's degree of comprehension. Indeed, as fom referee, I'd have 
bounced such a suggestion from anyone else as being rude and inappropriate. 
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.

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. That's why I asked for a 
sample dialog to show concretely how that could happen.

I'm grateful to Robert Black for answering my request (and so amusingly as 
well), and I found his reply intelligent and interesting. I will only say 
that personally, I care for neither the position of Argle nor that of 
Bargle. I have no problem understanding arithmetic truth and  agree that 
every arithmetic sentence has a determinate truth value. I even believe 
that CH has a determinate truth value, and would even bet that in twenty 
years that truth value will be known and generally accepted. And if 
challenged, I believe that I could explain why I believe these things, 
proceeding botttom-up, without resorting to Argle's top-down ontology. I 
have not managed to understand the notion of "arbitrary collection" where 
there is no knowledge of what sort of things are being collected. Perhaps 
some day there will be such an understanding, but I do not believe that day 
has arrived.


                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at
                          (Add 1 and get 0)

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