FOM: first order and second order logic: once more
Martin Davis
martin at eipye.com
Thu Aug 31 00:33:29 EDT 2000
To add to Joe Shipman's salient remarks:
1. Lost in the discussion is the original notion that logic has to do with
inferences, that a system of logic should provide a means of proceeding
from given premises to an appropriate conclusion. Thus, if a practitioner
of a particular logical system claims that conclusion B follows from
premises A1,....An the system should provide, as the evidence (sufficient
reason) for this claim, a proof or, one may say, a derivation. Moreover, if
the proof is to be convincing there should, at the least, be an algorithm
by which a doubter can verify that the alleged proof really is one
(according, of course, to the rules of the logic). Invoking Church's thesis
this implies that the sets of pairs ((A1,...,An),B) for which this relation
holds is an r.e. set.
The fact that the valid sentences of second order "logic" is not r.e. is
thus not just another property that happens to fail, it is a fatal flaw in
the use of this validity as the basis of a reasoning system. The
second-order logic of the textbooks amounts to first-order reasoning
coupled with a strong second order comprehension principle. And the
derivable sentences of this system do, of course, form an r.e. set.
2. Joe Shipman has indicated some of the set-theoretic cans of worms opened
when one considers full validity for second-order sentences. I want to add
the remark that many open mathematical questions are equivalent to the
validity of corresponding sentences of second-order logic. This includes
Goldbach's conjecture, the twin primes conjecture, and the Riemann
Hypothesis. Likewise the various combinatorial problems that Harvey
Friedman has been showing require large cardinal assumptions for their
proof are each also equivalent to such a sentence. Obviously this fact
casts no particular light on these problems, but rather shows how
intractable second-order validity is.
3. In this last connection, it may be worth observing that the set of
second-order valid sentences does not merely fail to be r.e.: it has a very
high degree of unsolvability, that of true arithmetic. Or stated in terms
of the halting problem, to get its degree you must iterate omega times the
operation ("jump") of forming the halting set for a Turing machine
provided with a given set as "oracle".
None of this is at all novel, but in the light of ongoing discussion, I
though these remarks worth making.
Martin
Martin Davis
Visiting Scholar UC Berkeley
Professor Emeritus, NYU
martin at eipye.com
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http://www.eipye.com
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