ABSTRACT
This chapter presents a procedure for finding counterexamples to arguments that have counterexamples. The procedure works by looking at the shape of sentences rather than their semantics. Despite the focus on ‘mere shape’ the procedure gives a reliable and useful method for figuring out whether an argument is valid. The procedure involves tableaux, to which this chapter is devoted.1
You know what it is for an argument from premises A1,A2, . . . ,An to conclusion B to be valid: namely, absence of counterexample. A counterexample to the argument is a case in which the As are all true and B is not. But for any such case there may be many ways of making all of the As true while making the given B untrue. (For example, if the argument is from premise p ∨ q to conclusion p ∧ q, a case might make p true but q untrue to be a counterexample; but a case might instead make q true and p untrue to be a counterexample.) Tableaux may be seen as a
