Logical Reasoning Systems derive sound conclusions from formal declarative knowledge. Such systems are usually deened by abstract rules of inference. For example, the rule of modus ponens states that given P , and \P implies Q" (usually written P ! Q), we can infer Q. Logical systems have a rich history starting with Greek syllogisms and continuing through the work many prominent mathematicians...

The word 'logic' in the title of this chapter is ambiguous. In its first meaning, a logic is a collection of closely related artificial languages. There are certain languages called first-order languages, and together they form first-order logic. In the same spirit, there are several closely related languages called modal languages, and together they form modal logic. Likewise second-order logi...

[1] E. BOROS AND A. PRÉKOPA, Closed Form Two-Sided Bounds for Probabilities That Exactly r and at Least r out of n Events Occur, Mathematics of Operations Research, 14 (1989), 317–342. [2] D. DAWSON AND A. SANKOFF, An Inequality for Probabilities, Proceedings of the American Mathematical Society, 18 (1967), 504–507. [3] H.P. EDMUNDSON, Bounds on the Expectation of a Convex Function of a Random ...

ii Contents Introduction v 1 The rationality problem in invariant theory 1 1. iii iv CONTENTS Introduction Invariant theory as a mathematical discipline on its own originated in Eng-land around the middle of the nineteenth century with Cayley's papers on hyperdeterminants and his famous Memoirs on Quantics, followed by Salmon, Sylvester and Boole, and Aronhold, Clebsch and Gordan in Germany. Th...

In 1860, Augustus de Morgan published DeM60], thereby launching an investigation into the algebra of relations. This developed into the subject now called Algebraic Logic, though in the 19th century it was simply thought of as mathematical logic. This work, along with Frege's quantiier logic, became the foundation of modern logic and model theory. In De Morgan's writing there is no sharp separa...

Citation: King R (2016) I Can't Get No (Boolean) Satisfaction: A Reply to Barrett et al. Sometimes history can be philosophically interesting. Barrett (2011) and colleagues (e.g., Barrett et al., 2014, 2015) are to be congratulated on widening the scope of our understanding of animal cognition to include its ecological elements. However, in their eagerness to overturn a narrow model of computat...

‎From Hilbert's theorem of zeroes‎, ‎and from Noether's ideal theory‎, ‎Birkhoff derived certain algebraic concepts (as explained by Tholen) that have a dual significance in general toposes‎, ‎similar to their role in the original examples of algebraic geometry‎. ‎I will describe a simple example that illustrates some of the aspects of this relationship‎. The dualization from algebra to geometr...

We use the level 12 theory of theta functions to derive hypergeometric transformation formulas and show that forty-two functions are equal. We discuss applications to Ramanujan’s series for 1/π and show how to obtain iterations that converge rapidly to 1/π. CUBIC THETA FUNCTIONS A GLIMPSE S.Bhargava [email protected] Department of Studies in Mathematics University of Mysore Manasagango...

algebraic logic (AAL) deals with the algebraic study of logics in a way that views bothterms "algebraic" and "logic" under a general perspective. It studies different abstract processes ofassigning a class of algebras to a given logic, and the general properties of these processes, theirlimitations and their different variants or enhancements, instead of focusing on the resulting cl...

Several universal approximation and universal representation results are known for non Boolean multi valued logics such as fuzzy logics In this paper we show that similar results can be proven for multi valued Boolean logics as well Introduction The study of Boolean algebras started when G Boole con sidered the simplest Boolean algebra the two valued algebra B f g ffalse trueg In this algebra t...