For any $\alpha$, does a set $\gamma$ always exist so $\gamma\vdash\alpha$ or $\gamma\vdash\lnot\alpha$? In propositional Calculus, for any proposition $\alpha$ does there always exist a set of propositions $\gamma$ such that $\gamma$ $\vdash$ $\alpha$ or $\gamma$ $\vdash$ $\neg\alpha$?
 A: There is a natural interpretation of your question that makes it important mathematically.
We ask:  Is it true that there is a consistent set $\Gamma$ of sentences such that for any $\alpha$ in the language, we have $\Gamma\vdash \alpha$ or $\Gamma\vdash \lnot\alpha\;$? 
The answer is yes, both for the predicate calculus and  propositional calculi.
The following is the full statement of a result that applies equally well to the predicate calculus and to propositional calculi.
Theorem Let $\Sigma$ be any consistent set of sentences over a language $L$, propositional or predicate.   Then there is a consistent set $\Gamma$ of sentences such that $\Sigma \subseteq \Gamma$, and for any sentence $\alpha$ of $L$, $\Gamma\vdash \alpha$ or $\Gamma\vdash \lnot\alpha\;$.
The usual proof for propositional calculi uses essentially the same Zorn's Lemma argument as the usual proof for the predicate calculus.  Indeed the predicate calculus result can be derived from the one for propositional calculi by using a couple of tricks. 
