Conservative extension

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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 is a (proof theoretic) conservative extension of a theory if every theorem of is a theorem of , and any theorem of in the language of is already a theorem of .

More generally, if is a set of formulas in the common language of and , then is -conservative over if every formula from provable in is also provable in .

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 would be a theorem of , so every formula in the language of would be a theorem of , so 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, , that is known (or assumed) to be consistent, and successively build conservative extensions , , ... 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[edit]

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


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