Propositional Deduction via Sequent Calculus

We have left out negation. We regard ¬φ as a shorthand for φ→ ⊥. A sequent is a pair that packages together some assumptions and a conclusion. We write Γ ` φ for the sequent whose assumptions are the finite set of formulas Γ and whose conclusion is the formula φ. We call Γ the antecedent and φ the consequent of the sequent Γ ` φ. A sequent is satisfied in a model M iff either M |= φ or for some ψ ∈ Γ, M 6|= ψ. Later, we will consider sequents Γ ` ∆ where the consequent is also a finite set, with the semantics that Γ ` ∆ is satisfied in a model M iff either, for some φ ∈ ∆, M |= φ, or else, for some ψ ∈ Γ, M 6|= ψ. Thus, a sequent says that if all of the formulas in its antecedent are true, then some