Abstract

This thesis is concerned with the problem of entailment. Entailment is a form of implication, perhaps the strongest found in logical calculi. The development of entailment logics is rather new in the history of logic. The entailment concept has however, been employed for the designation of a strong implication relation between antecedent and consequent. It has been used by some to designate deducibility. There are, however, alternative systems of implication which have been fanned and equated by some to the concept of entailment. A central question then, is what conditions shall be given to an acceptable statement of entailment, and how does entailment differ from alternative implication systems? The properties which are inherent in logical implication are shown by a historical sketch in this thesis. The first form of implication discussed is ‘material implication.’ Material implication as found in Russell and Whitehead's Principia Mathematica is discussed. Following upon this the evolution of strict implication and the ‘S’ systems of C.I. Lewis and E.H. Langford are analyzed. In the context of these discussions both the common and diverse properties are discussed. It is found that both material and strict implication have internal weakness in so far as both produce implicational paradoxes. The question then becomes, can an entailment logic overcome these difficulties, or rather must an entailment logic avoid their paradoxes peculiar to the systems cited? With the development of system E by Alan Anderson and Noel Belnap, we find a paradox-free system of implication. The system is designated ‘E’ for entailment by the authors. The system E is formed by combining the semantical character “relevance” with that of necessity. It is shown that the system E does not contain the paradoxical theorems nor any analogues of the theorems which give the paradoxes. In the context of this discussion a proper focus is given for several logical terms used in describing the relations between antecedent and consequent. The research of this thesis deals with certain definite problems. Is strict implication the same as entailment? According to the research of thesis this identification is doubtful. Of further concern is whether or rather what logically follows from contradiction. This research rejects the claims that anything whatsoever “follows from” a contradiction. A final concern of this research is the relation of entailment in modal semantics. This problem which is more general is concerned with such developments as modal models on the one hand and the nature of possibility on the other. It is found that an impossible proposition need not have the essential form of a contradiction.