Proof Systems

Consider the following exercise. You are given a sound and complete proof system. You are tasked to define a new proof system, which is based on the first proof system. The new proof system deals with statements about sets of statements from the original proof system and considers a set to be true if at least one statement in the set is true.

Let

be a proof system which is sound and complete (consider $\mathcal{S}$ to be finite).

Now define $\phi_2$ and $\mathcal{P}_2$ such that the following proof system is sound and complete (and prove these properties):

with

where $\sigma \in P(\mathcal{S})$.