# Conservative extension

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 ("everything follows from a contradiction"), every theorem in the original theory as well as its negation would belong to the new theory, which then would not be a conservative extension.] 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.

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.

## 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 every model of $T_{1}$ can be expanded to a model of $T_{2}$ . 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.