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 ${\displaystyle T_{2}}$ is a (proof theoretic) conservative extension of a theory ${\displaystyle T_{1}}$ if every theorem of ${\displaystyle T_{1}}$ is a theorem of ${\displaystyle T_{2}}$, and any theorem of ${\displaystyle T_{2}}$ in the language of ${\displaystyle T_{1}}$ is already a theorem of ${\displaystyle T_{1}}$.

More generally, if ${\displaystyle \Gamma }$ is a set of formulas in the common language of ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$, then ${\displaystyle T_{2}}$ is ${\displaystyle \Gamma }$-conservative over ${\displaystyle T_{1}}$ if every formula from ${\displaystyle \Gamma }$ provable in ${\displaystyle T_{2}}$ is also provable in ${\displaystyle T_{1}}$.

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 ${\displaystyle T_{2}}$ would be a theorem of ${\displaystyle T_{2}}$, so every formula in the language of ${\displaystyle T_{1}}$ would be a theorem of ${\displaystyle T_{1}}$, so ${\displaystyle T_{1}}$ 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, ${\displaystyle T_{0}}$, that is known (or assumed) to be consistent, and successively build conservative extensions ${\displaystyle T_{1}}$, ${\displaystyle T_{2}}$, ... 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.