# FOM: more on conservative extension results

Neil Tennant neilt at mercutio.cohums.ohio-state.edu
Fri Oct 9 09:19:40 EDT 1998

```In reply to Robert Tragesser's earlier question about conservative
extension results, Joe Shipman instructively replied:

> A conservative extension result takes the form "axiomatic system X+Y
> proves the same S-sentences as system X", where S is a syntactic
> criterion.  If S is perfectly general (all sentences are S-sentences)
> then this just amounts to saying that Y is already derivable from X.
> On the other hand, if S is the simplest possible type of sentence,
> namely a logical term ("true" or "false") this is simply the relative
> consistency result "if X is consistent so is X+Y".

Joe then talked about Hilbert's attempt to make ideal elements
acceptable via conservative extension results concerning theories that
introduced them.

It may also be worth adding, for Robert's benefit in the light of his
original inquiry, that quite a lot has been made of certain other
kinds of conservative extension result in the literature. Two other
applications come to mind.

1) One can look at conservative extension results concerning logical
theoremhood and logical consequence. Language L1 is extended to
language L2 by the introduction of a new logical operator. Rules of
inference are given for this new operator. One shows that these rules do
not extend the field of the deducibility relation within L1.

Results of this kind feature in the arguments of anti-realists such
as Dummett for the superiority of intuitionistic logic over classical
logic. Classical logic is not conservative in this sense, since upon
expanding the language of disjunction and implication by the
introduction of (classical) negation, one can prove (A->B)v(B->A),
which one could not prove using just the old rules for -> and v. (I am
assuming here that one is dealing with a natural deduction system.)

2) Hartry Field once gave an influential argument for the irreality of
numbers by giving a conservative extension result of (roughly) the
following kind (see his book "Science without Numbers"):

The theory (S+M) that results from a synthetic physical theory
S by grafting mathematics M onto it conservatively extends S.
That is, the addition of the math doesn't enable one to derive
any new strictly physical consequences (predictions) from the
theory S.

Other philosophers (e.g. Shapiro and Burgess) have since challenged
whether this is actually true. Even if the jury is still out on that
score, it was a very ingenious strategy that Field tried to
employ. (If memory serves me correctly, I once came across a paper by
Philip Frank, the logical positivist, in an old collection of writings
by members of the Vienna Circle, in which there was a remarkable
anticipation of Field's methodological idea. Not surprising, given the
recency of Hilbert's influence on the thinking of the Vienna Circle.)

Neil Tennant

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