From semantic tableaux to Smullyan trees: a history of the development of the falsifiability tree method

Abstract

The tree method is one of several decision procedures available for classical propositionai logic and first-order functional calculus and for nonclassical logics, inducing intuitionistic propositional logic and intuitionistic first-order logic, modal logics, and multiple-valued logics The tree method provides exceptionally elegant proofs of the consistency and satisfiability of formulae. Falsifiability trees allow easy testing of the validity of proofs and are a canonization of proof by contradiction for natural deduction systems, while truth trees allow easy derivation of theorems in these systems. Tree proofs permit graphical-geometric representations of logical relations, and appear to be of greater intuitive accessibility than either the axiomatic method or the method of natural deduction. The proofs of the completeness and soundness of the tree method and its variants are also straightforward, and the method combines insights and results of model theory and proof theory in a fashion that clearly identifies the most basic concepts of proof involving such model-theoretic results as Craig's Interpolation Lemma, Beth's Definability Theorem, and Robinson's Consistency Theorem.

I trace the contemporary history of the tree method, from its its tentative origins in the Gentzen sequent-calculus and the method of natural deduction, through its evolution and comprehensive development as the Smullyan tree (1964-1968) from Beth tableaux (1955) and Hintikka's theory of model sets (1953-1955). Also considered is van Heijenoort's development of the falsifiability tree as a special case of the truth tree (1966-1974) and work in the 1970s on proving the soundless and completeness of the tree method.