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From: torkel@sics.se (Torkel Franzen)
Newsgroups: comp.ai.philosophy
Subject: Consistency theorems
Message-ID: <1991Feb28.172555.12897@sics.se>
Date: 28 Feb 91 17:25:55 GMT
References: <16462.9102272325@s4.sys.uea.ac.uk>
Sender: news@sics.se
Organization: Swedish Institute of Computer Science, Kista
Lines: 41
In-Reply-To: jrk@information-systems.east-anglia.ac.uk's message of 27 Feb 91 23:25:44 GMT

In article <16462.9102272325@s4.sys.uea.ac.uk> jrk@information-systems.east-
anglia.ac.uk (Richard Kennaway CMP RA) writes:

>I'm not saying it's no evidence at all, but that it is only evidence
>relative to the consistency of the system in which the proof is conducted.
>If one accepts that consistency, the proof is perfectly good evidence; to
>the extent that one doubts it, so is that evidence weakened.

  What you are saying is that the proof is evidence for the
hypothetical statement "if S is consistent, then T is consistent".
Thus, the proof in itself is no evidence at all for the consistency of T.
Whether we have (empirical?) evidence apart from the proof for the
consistency of S - and thereby, given the consistency proof, for the
consistency of T - is a different matter.

>What is special about consistency theorems is that they are consistency
>theorems, and that they generally cannot be proved in systems weaker than
>those whereof they speak.  As a result, a proof of the consistency of a
>system X conducted in a system Y containing X does not give one any reason
>to believe in that consistency that one did not have already.

  There's an unwarranted step here from "not weaker" to "stronger". A theory T
may well be proved consistent in a theory S that is not stronger than T. For
example, the consistency of elementary arithmetic + "elementary arithmetic
is inconsistent" is provable in elementary analysis. And I think you will
agree that this consistency theorem is far from obvious.

  What you have in mind is a special class of consistency proofs:
proofs by reflection, in which we prove that a theory T is consistent
by noting, in a stronger theory S, that all theorems of T are true.
These proofs are essentially trivial, granted. However, we must be
careful in considering what is meant by saying that "the consistency
proof is worthless as evidence to support one's belief in the
consistency of T". Certainly it is true that if one has no confidence
in the theory S, the proof gives one no reason whatever to believe in
the consistency of T. But this applies to any theorem proved in S! A
proof of a statement A in S is worthless as evidence to support one's
belief in A unless there is evidence for the validity (in a stronger
or weaker sense, depending on the logical form of A) of the axioms and
rules of inference of S. What remains concerning consistency proofs by
reflection is just this, that they're trivial.