The paper is concerned with propositional calculi having arbitrary modus inference operations that are analogous to the modus ponens operation. For these calculi the question on the existence and ...cardinality of sets of generators is examined. A criterion for propositional calculi with arbitrary modus inference operations to be finitely generated is put forward. For some calculi, the existence of a finite complete system of tautologies is shown to imply the existence of a basis of arbitrarily large finite cardinality. We show the existence of propositional calculi with countable basis, without any basis, and calculi having a complete subsystem without any basis.
We consider iterative propositional calculi that are finite sets of propositional formulas together with modus ponens and an operation of superposition defined by a set of Mal’tsev operations. For ...such formulas, the question is studied whether the derivability problem for formulas is decidable. In the paper, we construct an undecidable iterative propositional calculus whose axioms depend on three variables. A derivation of formulas in the given calculus models the solution process for Post’s correspondence problem. In particular, we prove that the general problem of expressibility for iterative propositional calculi is algorithmically undecidable.
The paper first introduces fuzzy propositional calculus system ℋ, generalized effective inference and integral truth degree of formulas, then proves integral truth degree decreasing theorem of ...generalized effective inference in fuzzy propositional calculus system ℋ. Finally, the paper compares this theorem with the conclusion calculates integral truth degree using Fibonacci sequence and illustrates the superiority of the theorem.
"Satisfiability (SAT) related topics have attracted researchers from various disciplines: logic, applied areas such as planning, scheduling, operations research and combinatorial optimization, but ...also theoretical issues on the theme of complexity and much more, they all are connected through SAT. My personal interest in SAT stems from actual solving: The increase in power of modern SAT solvers over the past 15 years has been phenomenal. It has become the key enabling technology in automated verification of both computer hardware and software. Bounded Model Checking (BMC) of computer hardware is now probably the most widely used model checking technique. The counterexamples that it finds are just satisfying instances of a Boolean formula obtained by unwinding to some fixed depth a sequential circuit and its specification in linear temporal logic. Extending model checking to software verification is a much more difficult problem on the frontier of current research. One promising approach for languages like C with finite word-length integers is to use the same idea as in BMC but with a decision procedure for the theory of bit-vectors instead of SAT. All decision procedures for bit-vectors that I am familiar with ultimately make use of a fast SAT solver to handle complex formulas. Decision procedures for more complicated theories, like linear real and integer arithmetic, are also used in program verification. Most of them use powerful SAT solvers in an essential way. Clearly, efficient SAT solving is a key technology for 21st century computer science. I expect this collection of papers on all theoretical and practical aspects of SAT solving will be extremely useful to both students and researchers and will lead to many further advances in the field."--Edmund Clarke (FORE Systems University Professor of Computer Science and Professor of Electrical
and Computer Engineering at Carnegie Mellon University, winner of the 2007 A.M. Turing Award).
Das Buch vermittelt die Grundlagen der Aussagen- und erweiterten Prädikatenlogik in 12 Lektionen. Neben Techniken zum Überprüfen der Schlüssigkeit von Argumenten bilden die Kunst des Formalisierens ...wissenschaftlicher Argumente und metalogische Fragen den Inhalt des Buches. Das Buch eignet sich in Verbindung mit begleitenden interaktiven Übungseinheiten und Klausuren, die über Internet zugänglich sind, sowohl zum Selbststudium als auch für Einführungskurse in die Logik. Die zweite überarbeitete Auflage erscheint in einem größeren und besser lesbaren Format.
The book is about Gentzen calculi for (the main systems of) modal logic. It is divided into three parts. In the first part we introduce and discuss the main philosophical ideas related to proof ...theory, and we try to identify criteria for distinguishing good sequent calculi. In the second part we present the several attempts made from the 50's until today to provide modal logic with Gentzen calculi. In the third and and final part we analyse new calculi for modal logics, called tree-hypersequent calculi, which were recently introduced by the author. We show in a precise and clear way the main results that can be proved with and about them.
The year's finest mathematical writing from around the world
This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices ...alongside some of the foremost names in the field,The Best Writing on Mathematics 2018makes available to a wide audience many pieces not easily found anywhere else-and you don't need to be a mathematician to enjoy them. These essays delve into the history, philosophy, teaching, and everyday aspects of math, offering surprising insights into its nature, meaning, and practice-and taking readers behind the scenes of today's hottest mathematical debates.
James Grime shows how to build subtly mischievous dice for playing slightly unfair games and Michael Barany traces how our appreciation of the societal importance of mathematics has developed since World War II. In other essays, Francis Su extolls the inherent values of learning, doing, and sharing mathematics, and Margaret Wertheim takes us on a mathematical exploration of the mind and the world-with glimpses at science, philosophy, music, art, and even crocheting. And there's much, much more.
In addition to presenting the year's most memorable math writing, this must-have anthology includes an introduction by the editor and a bibliography of other notable pieces on mathematics.
This is a must-read for anyone interested in where math has taken us-and where it is headed.
The definition of the notion of intuitionistic fuzzy set is the basis for defining intuitionistic fuzzy logics of different kinds. In this paper, we construct two versions of intuitionistic fuzzy ...propositional calculus (IFPC) and a version of intuitionistic fuzzy predicate logic (IFPL).
Among the non-classical logics, the intuitionistic one stands out in many ways. First of all, because of its properties, it is grateful subject of formal analysis. Moreover, there is small, but very ...significant group of mathematicians and philosophers who claim that intuitionistic logic captures the reasoning utilized in mathematics better than classical one. This article reveals the origins of intuitionistic propositional calculus – it was an outcome of formalization of certain ideas about foundations of mathematics. A large part of the article is devoted to Glivenko’s Theorem – somewhat forgotten, but extremely interesting formal result regarding the relationship between the two logical calculi: classical and intuitionistic propositional logic.
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