Philosophical Logic is a clear and concise critical survey of nonclassical logics of philosophical interest written by one of the world's leading authorities on the subject. After giving an overview of classical logic, John Burgess introduces five central branches of nonclassical logic (temporal, modal, conditional, relevantistic, and intuitionistic), focusing on the sometimes problematic relationship between formal apparatus and intuitive motivation. Requiring minimal background and arranged to make the more technical material optional, the book offers a choice between an overview and in-depth study, and it balances the philosophical and technical aspects of the subject. – The book emphasizes the relationship between models and the traditional goal of logic, the evaluation of arguments, and critically examines apparatus and assumptions that often are taken for granted. Philosophical Logic provides an unusually thorough treatment of conditional logic, unifying probabilistic and model-theoretic approaches. It underscores the variety of approaches that have been taken to relevantistic and related logics, and it stresses the problem of connecting formal systems to the motivating ideas behind intuitionistic mathematics. Each chapter ends with a brief guide to further reading. – Philosophical Logic addresses students new to logic, philosophers working in other areas, and specialists in logic, providing both a sophisticated introduction and a new synthesis. – Chapter One, Classical Logic; – Ch. Two, Temporal Logic; – Ch. Three, Modal Logic; – Ch. Four, Conditional Logic; – Ch. Five, Relevantistic Logic; – Ch. Six, Intuitionistic Logic. M.-M. V.

We describe E. W. Beth’s use of nonclassical valuations (in his own terminology pseudo-valuations) in propositional logics. Three periods are distinguished. In the first period (1954) he develops the idea of pseudo-valuation intending to apply it to obtain a subformula theorem for arbitrary propositional logics. When this fails, he obtains in the second period (1958-1961) some simple but elegant applications of the idea, mainly with regard to proofs of independance of axioms systems. The thirs period (1961-1964) is the application of the idea towards the introduction of a semantics (his second one) for intuitionistic logics. We will show that it is highly likely that Beth discovered this version of “possible worlds semantics” for intuitionistic and some modal logics essentially independently from Kripke. The history of the concept of semantic tableaux is strongly bound to the birth of the concept of semantic tableaux, but we will touch the latter subject only in so far as is necessary for our considerations.

In this paper we analyze the consideration of intuitionistic logic as an extension of classical logic. This – at first sight surprising – point of view has been sustained explicitly by Jan Lukasiewicz on the basis of a mapping of classical propositional logic into intuitionistic propositional logic by Kurt Gödel in 1933. Simultaneously with Gödel, Gerhard Gentzen had proposed another mapping of Peano’s arithmetic into Heyting’s arithmetic. We shall discuss these mappings in conncection with the problem of determining what are the logical symbols that properly express the idiosyncracy of intuitionistic logic. Many philosophers and logicians do not seem to be sufficiently aware of the difficulties that arise when classical logic is considered as a subsystem of intuitionistic logic. As an outcome of the whole discussion these difficulties will be brought out. The notion of logical translation will play an essential role in the argumentation and some consequences related to the meaning of logical constants will be drawn.