# Conservative extension

## From Wikipedia, the free encyclopedia

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^{[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**.