In mathematical logic, a logical theory is a (proof theoretic) conservative extension of a theory if the language of extends the language of ; every theorem of is a theorem of ; and any theorem of which is in the language of is already a theorem of .
More generally, if Γ is a set of formulas in the common language of and , then is Γ-conservative over , if every formula from Γ provable in is also provable in .
To put it informally, the new theory may possibly be more convenient for proving theorems, but it proves no new theorems about the language of the old theory.
Note that a conservative extension of a consistent theory is 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.
The theorem provers Isabelle and ACL2 adopt this methodology by providing a language for conservative extensions by definition.
Recently, conservative extensions have been used for defining a notion of module for ontologies: 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.
With model-theoretic means, a stronger notion is obtained: is a model-theoretic conservative extension of if every model of can be expanded to a model of . It is straightforward to see that each model-theoretic conservative extension also is a (proof-theoretic) conservative extension in the above sense. 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.