FOM: Re: SOL confusion

[Roger Bishop Jones]

In response to: Harvey Friedman Thursday, September 07, 2000 3:32 AM


> SOL is a semantic system, not a deductive system. It is an interesting
> semantic system for some purposes, and is a lot weaker than set theory in
> its role as a semantic system. Both SOL and set theory can be made into
> deductive systems, the latter being much stronger and far more suitable
for
> the formalization of mathematics.

Thankyou for this clarification of your use of the term SOL, which greatly
helps in understanding your previous message.

Since "confusion" is in our thread title I shall mention my reasons for
considering this usage ill-advised:

1. The word "semantics" does not appear in the phrase for which SOL is an
acronym.
2. The acronyms FOL and HOL have established usage in computer science
referring respectively to first order and w-order logic, not specifically to
the semantics of these logics.
3. If SOL is understood to have the broader reference which seems more
natural, specific references such as "the semantics of SOL" or "the
deductive system of SOL" remain reasonably concise and convenient, however,
if "SOL" refers exclusively to the semantics, mention of the whole is less
easy to contrive.
4. Your usage does not seem to be well established, since at least half a
dozen people have been engaged in a discussion in which its use has been
broader without anyone raising objection until now.

Even if the narrow usage were firmly established, it would still be correct
use of English to say that "SOL has a deductive system" if there ever had
been devised, or even if there existed in some more abstract sense, a
deductive system suitable for use with SOL.
Consequently the statement "SOL does not have any axioms or rules of
inference", even under the usage of SOL which you have explained, is simply
false.
"SOL is not a deductive system" or "SOL does not encompass a deductive
system" would do the trick.

> >In relation to the recent discussion comparing SOL and FOL the crucial
> >differences are semantic, and the alleged advantage of SOL over FOL (with
or
> >without set theory) is that the notion of standard model used in the
> >semantics gives greater expressiveness to the language.
>
> Comparing SOL and FOL is inappropriate since they have virtually no common
> purposes.

Whether you think it appropriate or not, comparing SOL and FOL is what we
have been doing recently in some of the threads on fom.
The point has been made that it is possible to express in SOL propositions
which cannot be expressed in FOL (under its usual semantics).
Though I am surprised that this point should cause much debate, I am under
the impression that it has not yet been conceded by all parties.

Though I am curious as to whether you consider this comparison
"appropriate", I am much more interested to know whether you think it true.

Roger Jones