Tableau formulations are given for the relevance logics E (Entailment), R (Relevant implication) and RM (Mingle). Proofs of equivalence to modus-ponens-based formulations are vialeft-handed Gentzen sequenzen-kalküle. The tableau formulations depend on a detailed analysis of the structure of tableau rules, leading to certain global requirements. Relevance is caught by the requirement that each node must be used; modality is caught by the requirement that only certain rules can cross a barrier. Open problems are discussed.

Carnap's early system of inductive logic make degrees of confirmation depend on the languages in which they are expressed. They are sensitive to which predicates are, in the language, taken as primitive. Hence they fail to be ‘linguistically invariant’. His later systems, in which prior probabilities are assigned to elements of a model rather than sentences of a language, are sensitive to which properties in the model are called primitive. Critics have often protested against these features of his work. This (...) paper shows how to make his systems independent of any choice of primitive predicates or primitive properties.The solution is related to another criticism of inductive logic. It has been noticed that Carnap's systems are too all-embracing. Hisc(h, e) is defined for all sentencesh ande. Yet for manyh ande, the evidencee does not warrant any assessment of the probability ofh. We need an inductive logic in whichc(h, e) is defined only whene really does bear onh. This paper sketches the measure theory of such a logic, and, within this measure theory, provides ‘relativized’ versions of Carnap's systems which are linguistically invariant. (shrink)

From the point of view of non-classical logics, Heyting's implication is the smallest implication for which the deduction theorem holds. This book studies properties of logical systems having some of the classical connectives and implication in the neighbourhood of Heyt ing's implication. I have not included anything on entailment, al though it belongs to this neighbourhood, mainly because of the appearance of the Anderson-Belnap book on entailment. In the later chapters of this book, I have included material that might be (...) of interest to the intuitionist mathematician. Originally, I intended to include more material in that spirit but I decided against it. There is no coherent body of material to include that builds naturally on the present book. There are some serious results on topological models, second order Beth and Kripke models, theories of types, etc., but it would require further research to be able to present a general theory, possibly using sheaves. That would have postponed pUblication for too long. I would like to dedicate this book to my colleagues, Professors G. Kreisel, M.O. Rabin and D. Scott. I have benefited greatly from Professor Kreisel's criticism and suggestions. Professor Rabin's fun damental results on decidability and undecidability provided the powerful tools used in obtaining the majority of the results reported in this book. Professor Scott's approach to non-classical logics and especially his analysis of the Scott consequence relation makes it possible to present Heyting's logic as a beautiful, integral part of non-classical logics. (shrink)