In
mathematical logic
, a
conservative extension
is a
supertheory
of a
theory
which is often convenient for proving
theorems
, but proves no new theorems about the language of the original theory. Similarly, a
non-conservative extension
is a supertheory which is not conservative, and can prove more theorems than the original.
More formally stated, a theory
is a (
proof theoretic
) conservative extension of a theory
if every theorem of
is a theorem of
, and any theorem of
in the language of
is already a theorem of
.
More generally, if
is a set of
formulas
in the common language of
and
, then
is
-conservative
over
if every formula from
provable in
is also provable in
.
Note that a conservative extension of a
consistent
theory is consistent. If it were not, then by the
principle of explosion
, every formula in the language of
would be a theorem of
, so every formula in the language of
would be a theorem of
, so
would not be consistent. Hence, conservative extensions do not bear the risk of introducing new inconsistencies. This can also be seen as a
methodology
for writing and structuring large theories: start with a theory,
, that is known (or assumed) to be consistent, and successively build conservative extensions
,
, ... of it.
Recently, conservative extensions have been used for defining a notion of
module
for
ontologies
[
citation needed
]
: if an ontology is formalized as a logical theory, a subtheory is a module if the whole ontology is a conservative extension of the subtheory.
An extension which is not conservative may be called a
proper extension
.
Examples
[
edit
]
Model-theoretic conservative extension
[
edit
]
With
model-theoretic
means, a stronger notion is obtained: an extension
of a theory
is
model-theoretically conservative
if
and every model of
can be expanded to a model of
. Each model-theoretic conservative extension also is a (proof-theoretic) conservative extension in the above sense.
[3]
The model theoretic notion has the advantage over the proof theoretic one that it does not depend so much on the language at hand; on the other hand, it is usually harder to establish model theoretic conservativity.
See also
[
edit
]
References
[
edit
]
External links
[
edit
]