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 T_2 is a (proof theoretic) conservative extension of a theory T_1 if every theorem of T_1 is a theorem of T_2, and any theorem of T_2 in the language of T_1 is already a theorem of T_1. More generally, if \Gamma is a set of formulas in the common language of T_1 and T_2, then T_2 is \Gamma**-conservative** over T_1 if every formula from \Gamma provable in T_2 is also provable in T_1. 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 T_2 would be a theorem of T_2, so every formula in the language of T_1 would be a theorem of T_1, so T_1 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, T_0, that is known (or assumed) to be consistent, and successively build conservative extensions T_1, T_2, ... of it. 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.

Examples

Model-theoretic conservative extension

With model-theoretic means, a stronger notion is obtained: an extension T_2 of a theory T_1 is model-theoretically conservative if and every model of T_1 can be expanded to a model of T_2. 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.