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)

The aim of this paper is to explore the peculiar case of infectious logics, a group of systems obtained generalizing the semantic behavior characteristic of the -fragment of the logics of nonsense, such as the ones due to Bochvar and Halldén, among others. Here, we extend these logics with classical negations, and we furthermore show that some of these extended systems can be properly regarded as logics of formal inconsistency and logics of formal undeterminedness.

The paper discusses regularisation of dualities. A given duality between (concrete) categories, e.g. a variety of algebras and a category of representation spaces, is lifted to a duality between the respective categories of semilattice representations in the category of algebras and the category of spaces. In particular, this gives duality for the regularisation of an irregular variety that has a duality. If the type of the variety includes constants, then the regularisation depends critically on the location or absence of constants (...) within the defining identities. The role of schizophrenic objects is discussed, and a number of applications are given. Among these applications are different forms of regularisation of Priestley, Stone and Pontryagin dualities. (shrink)

In this article we will focus our attention on the variety of distributive bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and involutive bisemilattices. After extending Balbes’ representation theorem to bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas–Dunn duality and introduce the categories of 2spaces and 2spaces\. The categories of 2spaces and 2spaces\ will play with respect to the categories of distributive bisemilattices and De Morgan bisemilattices, respectively, a role analogous to the category of Stone spaces (...) with respect to the category of Boolean algebras. Actually, the aim of this work is to show that these categories are, in fact, dually equivalent. (shrink)

The hallmark of the deductive systems known as ‘conceptivist’ or ‘containment’ logics is that for all theorems of the form , all atomic formulae appearing in also appear in . Significantly, as a consequence, the principle of Addition fails. While often billed as a formalisation of Kantian analytic judgements, once semantics were discovered for these systems, the approach was largely discounted as merely the imposition of a syntactic filter on unrelated systems. In this paper, we examine a number of prima (...) facie unrelated deductive contexts in which Addition fails and attempt to harmonise them by developing a computational interpretation of conceptivist logics. (shrink)

Paraconsistent Weak Kleene logic is the 3-valued logic with two designated values defined through the weak Kleene tables. This paper is a first attempt to investigate PWK within the perspective and methods of abstract algebraic logic. We give a Hilbert-style system for PWK and prove a normal form theorem. We examine some algebraic structures for PWK, called involutive bisemilattices, showing that they are distributive as bisemilattices and that they form a variety, \, generated by the 3-element algebra WK; we also (...) prove that every involutive bisemilattice is representable as the Płonka sum over a direct system of Boolean algebras. We then study PWK from the viewpoint of AAL. We show that \ is not the equivalent algebraic semantics of any algebraisable logic and that PWK is neither protoalgebraic nor selfextensional, not assertional, but it is truth-equational. We fully characterise the deductive filters of PWK on members of \ and the reduced matrix models of PWK. Finally, we investigate PWK with the methods of second-order AAL—we describe the class \ of PWK-algebras, algebra reducts of basic full generalised matrix models of PWK, showing that they coincide with the quasivariety generated by WK—which differs from \—and explicitly providing a quasiequational basis for it. (shrink)

The paper aims at studying, in full generality, logics defined by imposing a variable inclusion condition on a given logic \. We prove that the description of the algebraic counterpart of the left variable inclusion companion of a given logic \ is related to the construction of Płonka sums of the matrix models of \. This observation allows to obtain a Hilbert-style axiomatization of the logics of left variable inclusion, to describe the structure of their reduced models, and to locate (...) them in the Leibniz hierarchy. (shrink)

Płonka sums consist of an algebraic construction similar, in some sense, to direct limits, which allows to represent classes of algebras defined by means of regular identities. Recently, Płonka sums have been connected to logic, as they provide algebraic semantics to logics obtained by imposing a syntactic filter to given logics. In this paper, I present a very general topological duality for classes of algebras admitting a Płonka sum representation in terms of dualisable algebras.