Stephen Cole Kleene was one of the greatest logicians of the twentieth century and this book is the influential textbook he wrote to teach the subject to the next generation. It was first published in 1952, some twenty years after the publication of Godel's paper on the incompleteness of arithmetic, which marked, if not the beginning of modern logic. The 1930s was a time of creativity and ferment in the subject, when the notion of computable moved from the realm of (...) philosophical speculation to the realm of science. This was accomplished by the work of Kurt Gode1, Alan Turing, and Alonzo Church, who gave three apparently different precise definitions of computable. When they all turned out to be equivalent, there was a collective realization that this was indeed the right notion. Kleene played a key role in this process. One could say that he was there at the beginning of modern logic. He showed the equivalence of lambda calculus with Turing machines and with Godel's recursion equations, and developed the modern machinery of partial recursive functions. This textbook played an invaluable part in educating the logicians of the present. It played an important role in their own logical education.". (shrink)

This completely self-contained study, widely considered the best book in the field, is intended to serve both as an introduction to quantification theory and as ...

A proof method for automation of reasoning in a paraconsistent logic, the calculus C1* of da Costa, is presented. The method is analytical, using a specially designed tableau system. Actually two tableau systems were created. A first one, with a small number of rules in order to be mathematically convenient, is used to prove the soundness and the completeness of the method. The other one, which is equivalent to the former, is a system of derived rules designed to enhance computational (...) efficiency. A prototype based on this second system was effectively implemented. (shrink)

In this paper, by using the method of natural deduction, via the method of subordinate proofs, we develop a hierarchy of natural deduction logical systems NDCn containing just deduction rules with no axiom schema. We prove that these systems NDCn , 1 n , are logically equivalent to the systems of Da Costa’s hierarchy of paraconsistent logics Cn , 1 n . Some of the deduction rules used to introduce these systems are new and do not correspond to Da Costa’s (...) axioms rewritten, permitting the definition of a new paraconsistent semantics, such that soundness and completeness of the systems NDCn , 1 n< , may be directly obtained. Other natural deduction systems logically equivalent to Da Costa’s systems Cn, 1 n , are also introduced. (shrink)