From 1912, C. I. Lewis attempted to construct a logic of entailment. In doing so, he created his modal logics, S1–S5, of which his chosen logic of entailment was S2. Although his logics avoid the so-called paradoxes of material implication, they still fall prey to the problem of explosion (that every proposition follows from any contradiction) and the problem of implosion (that every tautology follows from every proposition). These problems, and the inadequate treatment of nested entailments, make Lewis’s logics of limited use as logics of entailment. The chapter also discusses the systems devised by Lewis’s students Everett Nelson and William Parry. Nelson’s connexive logic avoids many of the problems with Lewis’s system but is found to have severe difficulties of its own, and Parry’s analytic implication, although it introduces an interesting version of the notion of meaning containment, does not adequately avoid the problems with Lewis’s logics.

A logic of entailment is one in which it represents its own derivability relation as a connective. This allows it to express nested claims about what is derivable from what. It can say, for example, that if B is derivable from A and if C is derivable from B, then C is derivable from A. Mathematicians and logicians think this way when making proof plans. This chapter sets out the problem of constructing proof plans and looks at the other uses to which logics of entailment have been put. One key use, especially in the context of this book, is as a theory of the closure of scientific theories.