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
Model-theoretic conservative extension
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
References
External links