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The definition for complete theory in Enderton's books: "A Mathematical Introduction to Logic" - "A theory $T$ is said to be complete iff for every sentence $\sigma$ either $\sigma \in T$ or $\neg \sigma \in T$ "

My question is - does the "or" in the definition , means exclusive or? $\sigma \in T$ xor $\neg\sigma\in T$ ?

I ask it because I want to understand whether a theory which includes $\alpha , \neg\alpha$ (and therefore $consequences(\{\alpha ,\neg \alpha\}$) contains all formulas), is considered complete?

user5721565
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It's really a matter of convention. I have seen treatments where a complete theory is required to be consistent by definition and ones where it isn't so that any inconsistent theory is trivially complete. I would guess if Enderton just says 'or' he doesn't mean exclusive (so is taking the second option).

On the semantic side, completeness means any two models are elementarily equivalent and this question becomes whether that 'any two' should be interpreted as vacuously true when there are no models. (Looking it up, I see Enderton hasn't qualified this one either, further suggesting he intends inconsistent theories to be considered complete.)

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