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BOOLEAN AND HEYTING ALGEBRAS: THE ESSENTIALS

John L. Bell

in Set Theory: Boolean-Valued Models and Independence Proofs

This chapter provides a brief account of the theory of Boolean and Heyting algebras, including the basic representation theorems and their connections with logic.

Introduction to Supervaluational Approaches to Paradox

Hartry Field

in Saving Truth From Paradox

This chapter examines the simplest supervaluationist fixed points, and the ‘internal’ fixed-point theories based on them. These are not gap theories; they are ‘weakly classical’ theories in that they ... More

This chapter examines the simplest supervaluationist fixed points, and the ‘internal’ fixed-point theories based on them. These are not gap theories; they are ‘weakly classical’ theories in that they preserve the implications of classical logic but not certain meta-rules. Most dramatically, they do not allow for reasoning by cases. Two notions of validity are distinguished (strong and weak), and the framework of deMorgan semantics and Boolean semantics is introduced and discussed.Less

Boolean-valued structures

Tim Button and Sean Walsh

in Philosophy and Model Theory

Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin ... More

Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin down the semantic frameworks of language. It transpires that there are not just non-standard models, but non-standard ways of doing model theory itself. In more detail: whilst we normally outline a two-valued semantics which makes sentences True or False in a model, the inference rules for first-order logic are compatible with a four-valued semantics; or a semantics with countably many values; or what-have-you. The appropriate level of generality here is that of a Boolean-valued model, which we introduce. And the plurality of possible semantic values gives rise to perhaps the ‘deepest’ level of indeterminacy questions: How can humans pin down the semantic framework for their languages? We consider three different ways for inferentialists to respond to this question.Less

Logic: a new beginning

Pieter A. M. Seuren

in The Logic of Language: Language From Within Volume II

Entailment, contrariety, and contradiction stand in a triangular relation. Given negation of the predicate and given the duality of all and some, two isomorphic triangles arise, together forming an ... More

Entailment, contrariety, and contradiction stand in a triangular relation. Given negation of the predicate and given the duality of all and some, two isomorphic triangles arise, together forming an improved notation for the traditional Square of Opposition. Logical operators are treated as (abstract) predicates, definable in terms of satisfaction conditions and shaping the logic in which they take part. Carnapian meaning postulates are discussed. Boolean algebra and corresponding standard set theory are shown to underlie standard propositional and predicate logic. The method of valuation‐space modelling is introduced as a means of providing succinct and complete representations of logical systems in such a way that their properties are open to immediate inspection. A survey is given of Russellian and generalized quantification, of internal negation, and De Morgan's laws.Less

Classical Mereology

A. J. Cotnoir and Achille C. Varzi

in Mereology

The aim of this chapter is to provide a comprehensive introduction to classical mereology. It examines this theory by providing a clear and perspicuousaxiom system that isolates several important ... More

The aim of this chapter is to provide a comprehensive introduction to classical mereology. It examines this theory by providing a clear and perspicuousaxiom system that isolates several important elements of any mereological theory. The chapter examines, algebraic, and set-theoretic models of classical mereology, sketching proofs of their equivalence. The new axiom system facilitates algebraic comparisons, showing that models of these axioms are complete Boolean algebras without a bottom element. Then set-theoretic models are presented, and are shown to satisfy the axioms. The chapter explains the important relationship between models and powersets, and the role of Stone’s Representation Theorem in this connection. Finally, a number of significant rival axiom systems using different mereological primitives are introduced.Less

Boolean Algebra

Paul J. Nahin

in The Logician and the Engineer: How George Boole and Claude Shannon Created the Information Age

This chapter discusses how Boole cast logic into algebraic form. Boole was interested in symbolic analysis years before he wrote his Laws of Thought. In fact, others before him—in particular, the ... More

This chapter discusses how Boole cast logic into algebraic form. Boole was interested in symbolic analysis years before he wrote his Laws of Thought. In fact, others before him—in particular, the German mathematician Gottfried Wilhelm Leibniz (1646–1716)—had pursued a similar goal of reducing logic to algebra, but it was Boole who finally succeeded. What Boole described in his books is not exactly what modern users call Boolean algebra, but nevertheless it is from Boole that the modern presentation springs. The chapter first describes the essence of what Boole did using the language of sets. It then discusses Boole's algebra of sets, examples of Boolean analysis, and visualizing Boolean functions.Less

Synthesis and the Principle of Addition

Kurt Smith

in Matter Matters: Metaphysics and Methodology in the Early Modern Period

This chapter studies the mathematical concept of a group, early ancestors of the concept found in the works of Descartes and Leibniz. Here, it is shown exactly how the four conditions expressed by a ... More

This chapter studies the mathematical concept of a group, early ancestors of the concept found in the works of Descartes and Leibniz. Here, it is shown exactly how the four conditions expressed by a group underwrite all of mathematics, in particular the operations of arithmetic.Less

Introduction

Paul J. Nahin

in The Logician and the Engineer: How George Boole and Claude Shannon Created the Information Age

This introductory chapter considers the work of mathematician George Boole (1815–1864), whose book An Investigation of the Laws of Thought (1854) would have a huge impact on humanity. Boole's ... More

This introductory chapter considers the work of mathematician George Boole (1815–1864), whose book An Investigation of the Laws of Thought (1854) would have a huge impact on humanity. Boole's mathematics, the basis for what is now called Boolean algebra, is the subject of this book. It is also called mathematical logic, and today it is a routine analytical tool of the logic-design engineers who create the electronic circuitry that we now cannot live without, from computers to automobiles to home appliances. Boolean algebra is not traditional or classical Aristotelian logic, a subject generally taught in college by the philosophy department. Boolean algebra, by contrast, is generally in the hands of electrical engineering professors and/or the mathematics faculty.Less

 Mid-plural logic

Alex Oliver and Timothy Smiley

in Plural Logic

Plural formal logic comes in two phases. This chapter develops mid-plural logic, got by adding three logical items to the singular base of the previous chapter: plural variables, a predicate ... More

Plural formal logic comes in two phases. This chapter develops mid-plural logic, got by adding three logical items to the singular base of the previous chapter: plural variables, a predicate expressing inclusion, and an operator symbolizing exhaustive description. Like singular logic, mid-plural logic is axiomatizable, but the price of axiomatizability is that plural variables can only occur free. As to non-logical vocabulary, the system accommodates plural constants, and predicates and function signs which take plural as well as singular arguments. It is given an appropriately topic neutral semantics. Mid-plural logic’s expressive power is illustrated by highlighting a fragment of the system—the algebra of plurals—which does for logic what Boolean algebra does for sets. The appendix to the chapter proves the metatheorems listed in the text, and proves the soundness and completeness of the axioms presented for mid-plural logic.Less

Mid-plural logic

Alex Oliver and Timothy Smiley

in Plural Logic: Second Edition, Revised and Enlarged

Plural formal logic comes in two phases. This chapter develops mid-plural logic, got by adding three logical items to the singular base of the previous chapter: plural variables, a predicate ... More

Plural formal logic comes in two phases. This chapter develops mid-plural logic, got by adding three logical items to the singular base of the previous chapter: plural variables, a predicate expressing inclusion, and an operator symbolizing exhaustive description. Like singular logic, mid-plural logic is axiomatizable, but the price of axiomatizability is that plural variables can only occur free. As to non-logical vocabulary, the system accommodates plural constants, and predicates and function signs which take plural as well as singular arguments. It is given an appropriately topic neutral semantics. Mid-plural logic's expressive power is illustrated by highlighting a fragment of the system—the algebra of plurals—which does for logic what Boolean algebra is supposed to do for sets. The appendix to the chapter proves the metatheorems listed in the text, and proves the soundness and completeness of the axioms presented for mid-plural logic.Less

Logic Switching Circuits

Paul J. Nahin

in The Logician and the Engineer: How George Boole and Claude Shannon Created the Information Age

Today's digital circuitry is built with electronic technology that the telephone engineers of the 1930s and the pioneer computer designers of the 1940s would have thought to be magic. The first real ... More

Today's digital circuitry is built with electronic technology that the telephone engineers of the 1930s and the pioneer computer designers of the 1940s would have thought to be magic. The first real digital technology took the form of electromagnetic relays in telephone switching exchanges. Then came vacuum tube digital circuitry, discrete transistors, integrated transistor circuits, and so on. But the one thing that remains the same is the math, the Boolean algebra that is the central star of this book. This chapter describes the technology that Shannon himself used in his switching analyses. It covers Switches and the logical connectives, a classic switching design problem, electromagnetic relay, the ideal diode and the relay logical AND and OR, and the bi-stable relay latch.Less

George Boole and Claude Shannon: Two Mini-Biographies

Paul J. Nahin

in The Logician and the Engineer: How George Boole and Claude Shannon Created the Information Age

This chapter presents brief biographical sketches of George Boole and Claude Shannon. George was born in Lincoln, a town in the north of England, on November 2, 1815. His father John, while simple ... More

This chapter presents brief biographical sketches of George Boole and Claude Shannon. George was born in Lincoln, a town in the north of England, on November 2, 1815. His father John, while simple tradesman (a cobbler), taught George geometry and trigonometry, subjects John had found of great aid in his optical studies. Boole was essentially self-taught, with a formal education that stopped at what today would be a junior in high school. Eventually he became a master mathematician (who succeeded in merging algebra with logic), one held in the highest esteem by talented, highly educated men who had graduated from Cambridge and Oxford. Claude was born on April 30, 1916, in Petoskey, Michigan. He enrolled at the University of Michigan, from which he graduated in 1936 with double bachelor's degrees in mathematics and electrical engineering. It was in a class there that he was introduced to Boole's algebra of logic.Less

What You Need to Know to Read This Book

Paul J. Nahin

in The Logician and the Engineer: How George Boole and Claude Shannon Created the Information Age

This chapter details the background knowledge needed to read this book. Specifically, it assumes some knowledge of mathematics and electrical physics and an appreciation for the value of analytical ... More

This chapter details the background knowledge needed to read this book. Specifically, it assumes some knowledge of mathematics and electrical physics and an appreciation for the value of analytical reasoning—but no more than a technically minded college-prep high school junior or senior would have. In particular, the math level is that of algebra, including knowing how matrices multiply. The electrical background is simple: knowing (1) that electricity comes in two polarities (positive and negative) and that electrical charges of like polarity repel and of opposite polarity attract; and (2) understanding Ohm's law for resistors (that the voltage drop across a resistor in volts is the current through the resistor in amperes times the resistance in ohms) and the circuit laws of Kirchhoff (that the sum of the voltage drops around any closed loop is zero, which is an expression of the conservation of energy; the sum of all the currents into any node is zero).Less

Beyond Boole and Shannon

Paul J. Nahin

in The Logician and the Engineer: How George Boole and Claude Shannon Created the Information Age

Boole and Shannon never studied the physics of computation. Obviously Boole simply could not have, as none of the required physics was even known in his day, and Shannon was nearing the end of his ... More

Boole and Shannon never studied the physics of computation. Obviously Boole simply could not have, as none of the required physics was even known in his day, and Shannon was nearing the end of his career when such considerations were just beginning. And yet, both Boole's algebra and Shannon's information concepts to make many of our calculations. This chapter touches on how fundamental physics—the uncertainty principle from quantum mechanics, and thermodynamics, for example—constrain what is possible, in principle, for the computers of the far future. It argues that while there are indeed finite limitations, present-day technology falls so far short of those limits that there will be good employment for computer technologists for a very long time to come.Less

Epilogue: For the Future: The Anti-Amphibological Machine

Paul J. Nahin

in The Logician and the Engineer: How George Boole and Claude Shannon Created the Information Age

This chapter present the author's vision of the sort of logical problem that may soon be one that even a quantum computer would find a struggle to deal with—the decipherment of entangled legalese, ... More

This chapter present the author's vision of the sort of logical problem that may soon be one that even a quantum computer would find a struggle to deal with—the decipherment of entangled legalese, the sort of monstrous gobbledegook one finds, for example, in the increasingly convoluted IRS tax code. In the form of a short story, that vision is “The Language Clarifier,” which first appeared in the May 1979 issue of Omni Magazine. The original title was “The Anti-Amphibological Machine,” but Omni's fiction editor thought that a tad too mystifying for readers and “suggested” the change.Less

Boole, Shannon, and Probability

Paul J. Nahin

in The Logician and the Engineer: How George Boole and Claude Shannon Created the Information Age

George Boole and Claude Shannon shared a deep interest in the mathematics of probability. Boole's interest was, of course, not related to the theory of computation—he was a century too early for ... More

George Boole and Claude Shannon shared a deep interest in the mathematics of probability. Boole's interest was, of course, not related to the theory of computation—he was a century too early for that—while Shannon's mathematical theory of communication and information processing is replete with probabilistic analyses. There is, nevertheless, an important intersection between what the two men did, which is shown in this chapter. The aim is to provide a flavor of how they reasoned and of the sort of probabilistic problem that caught their attention. Once we have finished with Boole's problem, the reader will see that it uses mathematics that will play a crucial role in answering Shannon's concern about “crummy” relays.Less

Characteristic Analysis as an Oil Exploration Tool

H. A. F. Chaves

in Computers in Geology - 25 Years of Progress

Characteristic analysis is well known in mineral resources appraisal and has proved useful for petroleum exploration. It also can be used to integrate geological data in sedimentary basin analysis ... More

Characteristic analysis is well known in mineral resources appraisal and has proved useful for petroleum exploration. It also can be used to integrate geological data in sedimentary basin analysis and hydrocarbon assessment, considering geological relationships and uncertainties that result from lack of basic geological knowledge, A generalization of characteristic analysis, using fuzzy—set theory and fuzzy logic, may prove better for quantification of geologic analogues and also for description of reservoir and sedimentary facies. Characteristic analysis is a discrete multivariate procedure for combining and interpreting data; Botbol (1971) originally proposed its application to geology, geochemistry, and geophysics. It has been applied mainly in the search for poorly exposed or concealed mineral deposits by exploring joint occurrences or absences of mineralogical, lithological, and structural attributes (McCammon et al., 1981). It forms part of a systematic approach to resource appraisal and integration of generalized and specific geological knowledge (Chaves, 1988, 1989; Chaves and Lewis, 1989). The technique usually requires some form of discrete sampling to be applicable—generally a spatial discretization of maps into cells or regular grids (Melo, 1988). Characteristic analysis attempts to determine the joint occurrences of various attributes that are favorable for, related to, or indicative of the occurrence of the desired phenomenon or target. In geological applications, the target usually is an economic accumulation of energy or mineral resources. Applying characteristic analysis requires the following steps: 1) the studied area is sampled using a regular square or rectangular grid of cells; 2) in each cell the favorabilities of the variables are expressed in binary or ternary form; 3) a model is chosen that indicates the cells that include the target (Sinding—Larsen et al, 1979); and 4) a combined favorability map of the area is produced that points out possible new targets. The favorability of individual variables is expressed either in binary form— assigning a value of +1 to favorable and a value of 0 to unfavorable or unevaluated variables—or in ternary form if the two states represented by 0 are distinguishable—the value +1 again means favorable, the value—1 means unfavorable, and the value 0 means unevaluated. Less

Turing Machines

Paul J. Nahin

in The Logician and the Engineer: How George Boole and Claude Shannon Created the Information Age

This chapter discusses Turing machines. A Turing machine is the combination of a sequential, finite-state machine plus an external read/write memory storage medium called the tape (think of a ribbon ... More

This chapter discusses Turing machines. A Turing machine is the combination of a sequential, finite-state machine plus an external read/write memory storage medium called the tape (think of a ribbon of magnetic tape). The tape is a linear sequence of squares, with each square holding one of several possible symbols. The Turing machine's power to compute comes from its tape, for two reasons. First, Turing was the first to conceive of the idea of a stored program that could be changed by the operation of the machine itself. The program, and its input data, exist together on the tape as sequences of symbols. Second, because of the arbitrarily long length of the tape, a Turing machine has the ability to “remember” what has happened in the arbitrarily distant past.Less