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??