Question about maximally consistent sets in logic Problem:
Σ is "proofwise stronger" than a set Γ if {$\alpha$: Σ ⊢ $\alpha$} $\supseteq$ {$\alpha$ : Γ ⊢ $\alpha$ }. Show that for every maximally consistent set of propositions Σ, for every set Γ, either Σ is proofwise stronger than Γ or Σ $\cup$Γ is not consistent, or both (I am confused about "or both"). 
Question:
I was able to prove  either Σ is proofwise stronger than Γ or Σ $\cup$Γ is not consistent. However, the "or both" part confuses me. I handed in my proof without showing that they could be "both" and for some reason the marker give me full marks. But then the professor explicitly said we need to prove "or both" in class and on the assignment. Perhaps "or both" is some trivial case? Could someone clarify what is going on here?
My proof (sketch):
If Σ is maximally consistent, then $\forall \alpha,$ Σ ⊢ $\alpha$ or Σ ⊢ $\neg\alpha$. 
Suppose  Σ is not proofwise stronger than Γ then $\exists\beta$ such that  Γ⊢$\beta$, but Σ $\nvdash$ $\beta$. if Σ ⊢ $\neg\beta$ then Σ $\cup$ Γ is not consistent (by monotonicity and the definition of consistency).
but if Σ $\nvdash$ $\neg\beta$ then Σ $\nvdash$ $\neg\beta$ and Σ $\nvdash$ $\beta$. This fails the definition of maximally consistent. Therefore Σ is proofwise stronger.
how to do or both??
 A: The answer hinges on the definition that the course uses for maximally consistent.
On no very good evidence, apart from the fact that you have written down the sentence in your argument, it is possible that "maximally consistent" was defined in your course as follows:
"$\Sigma$ is maximally consistent if  $\forall \alpha$, $\Sigma \vdash \alpha$ or $\Sigma \vdash \neg\alpha$." 
In that case, any inconsistent $\Sigma$ is automatically maximally consistent.  Then the "or both" can certainly hold. If $\Sigma$ is inconsistent, then it is also automatically "proofwise stronger" than $\Gamma$.
It is possible that the definition of "maximally consistent" was different from the guess above, but still allowed for the possibility of inconsistency.
If however, the definition of maximally consistent given in the course is (the more sensible to me) "consistent, but such that no proper extension is consistent" then the "or both" of your question is not possible.
So I am afraid there is no simple answer. You have to check carefully the actual definitions that are used in your course. There is unfortunately no complete uniformity of terminology. 
