Semantic Tableaux proof method for predicate logic

The Semantic Tableaux method is a formalisation of proof in predicate logic which works by backward chaining: that is, proof by contradiction rather than deduction. In seeking to establish whether a sentence A is a logical consequence of a set of axioms Γ we instead assume that Γ holds yet A is falsethen proceed systematically by a version of the tree method for logical decomposition and hope to find our derivation blocked by contradiction. These contradictions lead us to conclude the original assertion (Γ true but A false) is itself contradictory and hence if Γ is true A must also be (A rigorous illustration of completeness and correctness is given later in this writeup; but an appreciation of the method is required for that). Of course, we may fail to find contradiction in all casesif we are left with an open branch, then that branch describes an interpretation in which Γ can hold but A is false, so A is not a logical consequence (which requires A true under all Interpretations).