Propositional Logic

Abstract

In this chapter we analyse reasoning patterns of which the validity only depends on the meaning of the propositional connectives ‘if . . ., then . . .’, ‘and’, ‘or’ and ‘not’. By giving a precise description of the meaning of these propositional connectives one is able to give a precise definition of the notion of logical or valid consequence. Two such definitions are given: a semantic one, in terms of truth values and hence in terms of the meaning of the formulas involved, and a syntactic one in terms of logical axioms and rules of which only the form is important. The semantic and the syntactic definition of logical consequence turn out be equivalent, giving us confidence that we gave a proper characterization of the intuitive notion of logical consequence. We prove or disprove all kinds of statements about the notion of logical or valid consequence, which is useful in order to get a good grasp of this notion. The last section treats a number of paradoxes which have been important for the progress in science and philosophy; it also contains a number of historical and philosophical remarks.