Conservative extension

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In mathematical logic, a theory T_2 is a (proof theoretic) conservative extension of a theory T_1 if the language of T_2 extends the language of T_1; every theorem of T_1 is a theorem of T_2; and any theorem of T_2 that is in the language of T_1 is already a theorem of T_1.

More generally, if Γ is a set of formulas in the common language of T_1 and T_2, then T_2 is Γ-conservative over T_1 if every formula from Γ provable in T_2 is also provable in T_1.

To put it informally, the new theory may possibly be more convenient for proving theorems, but it proves no new theorems about the language of the old theory.

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.


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