A NonconservativeExtension differs from a ConservativeExtension in the following ways.
First, we need some terminology. Let Ti refer to FormalOntology. Thus Ti consists of a set of axioms employing some language (of non-logical symbols) L(Ti). Additionally, let Mod(Ti) refers to the set of models that satisfy the axioms that comprise Ti. Given two ontologies T1 and T2, we say that T2 nonconservatively extends T1 if the following are true:
- L(T2) is contained in L(T1)
- T1 contained in T2 if and only if Mod(T2) contained in Mod(T1)
Or in plain English, T2 is a nonconservative extension of T1 if:
every non-logical term in ontology T2 is describable using the language of T1 and every model that satisfies ontology T2, satisfies the models of T1.
in plain English:
Imagine we have an ontology for the lineage of living creatures T1. We have a relation called "biological parent", with axioms that states than any creature living today has at least one parent. Now we want to extend our ontology to speak only about humans. To be more precise, we alter the axioms for "biological parent" including the additional constraint that any living human has exactly two biological parents. We can now say that our additional axiom, resulting in T1' non-conservatively extends T1.
(note: I'm not sure how to include mathematical logic symbols in this wiki, it would be useful to recast the above definition in such notation)