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: 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 ${\displaystyle T_{2))$ of a theory ${\displaystyle T_{1))$ is model-theoretically conservative if ${\displaystyle T_{1}\subseteq T_{2))$ and every model of ${\displaystyle T_{1))$ can be expanded to a model of ${\displaystyle T_{2))$. Each model-theoretic conservative extension also is a (proof-theoretic) conservative extension in the above sense.[3] 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.

## References

1. ^ a b S. G. Simpson, R. L. Smith, "Factorization of polynomials and ${\displaystyle \Sigma _{1}^{0))$-induction" (1986). Annals of Pure and Applied Logic, vol. 31 (p.305)
2. ^ Fernando Ferreira, A Simple Proof of Parsons' Theorem. Notre Dame Journal of Formal Logic, Vol.46, No.1, 2005.
3. ^ Hodges, Wilfrid (1997). A shorter model theory. Cambridge: Cambridge University Press. p. 58 exercise 8. ISBN 978-0-521-58713-6.