## 502 – Propositional logic (2)

September 10, 2009

I now proceed to present a series of results whose overall goal is to make the handling of formal proofs more efficient and natural. These results formalize different methods of reasoning commonly used in practice. A second goal in presenting these results is in preparing the way for the proof of the completeness theorem.

Theorem 7 (The deduction theorem) If ${S}$ is a set of formulas, and ${A}$ and ${B}$ are formulas, then ${S\cup\{A\}\vdash B}$ if and only if ${S\vdash A\rightarrow B.}$