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Will_Sawin comments on Notion of Preference in Ambient Control - Less Wrong
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I haven't yet read beyond this point, but this is a kind of confusing thing to write. Definitions can't extend a theory, because they don't give you new theorems. My assumption is that you will add axioms that incorporate the new symbols, and the axioms will extend the theory.
Some axioms are definitions.
Previous theorem: All unmarried men are not married New definition: "Bachelor" means "unmarried man" New theorem: All bachelors are unmarried men.
I'm pretty sure that's what he means. Hopefully clarified, if not made perfectly in accord with standard definitions.
I think that he means something analogous to the way that we can add some axioms involving the symbol "+" to the Peano axioms, and then show in second-order logic that the new axioms define addition uniquely.