Converting Trees to Proofs in K

Not only is the tree method useful for checking validity in modal logics, but it may also be used to help construct proofs. Trees provide a mechanical method for finding proofs that might otherwise require a lot of ingenuity. If an argument has a proof at all in a system S, the tree method can be used to provide one. The process is easiest to understand for system K, so we will explain that first, leaving the stronger systems for Sections 7.3–7.9. The fundamental idea is to show that every step in the construction of a closed tree corresponds to a derivable rule of K.

It is easiest to explain how this is done with an example, where we work out the steps of the tree and the corresponding steps of the proof in parallel. We will begin by constructing a proof of □(p→q) / □p→□q using the steps of the tree as our guidepost. The tree begins with □(p→q) and the negation of the conclusion: ∼(□p→□q). The first step in the construction of the proof is to enter □(p→q) as a hypothesis. In order to prove □p→□q, enter ∼(□p→□q) as a new hypotheses for Indirect Proof. If we can derive ⊥ in that subproof, the proof will be finished.