Extensions and Limitations to Logic

Classical (Aristotelean or Boolean) logics provide a solid foundation for mathematical reasoning, but are limited in expressivity and necessarily incomplete. Effective understanding of logic in the modern world entails for the instructor and advanced students an understanding of the wider context. This chapter surveys standard extensions used in mathematical reasoning, artificial intelligence and cognitive science, and natural language reasoning and understanding, as well as inherent limitations on reasoning and computing. Initial technical extensions include equality of terms, integer arithmetic and quantification over sets and relations. To deal with natural reasoning, the chapter explores temporal and modal logics, fuzzy logic and probabilistic models, and relevance logic. Finally, the chapter considers limitations to logic and knowledge, via an overview of the fundamental results of Turing, Gödel, and others, and their connection to the state of mathematics, computing and science in the modern world.

The Philosophy of African Logic

The chapter aims to do two things: 1) a rigorous presentation of philosophy of African logic and 2) to do this from the perspective of Ezumezu (an African) logic. The chapter will proceed by defining the three aspects of Ezumezu logic namely: 1) as a formal system, 2) as methodology, and 3) as a philosophy of African logic. My inquiry in this work primarily is with the philosophy of African logic but it will also cut across formal logic and methodology in addition. In the first section, I will attempt to show how the cultural influence behind the formulation of the principles of African logic justifies such a system as relative on the one hand, and how the cross-cultural applications justify it as universal on the other. I believe that this is where African philosophical assessment of African logic ought to begin because most critics of the idea of African logic agitate that an African system of logic, if it is ever possible, must necessarily lack the tincture of universal applicability. Afterwards, I will narrow my inquiry down to the African philosophy appraisal of African logic with an example of Ezumezu system. This focus is especially critical because it purveys a demonstration of a prototype system of an African logic. In the section on some principles of Ezumezu logic, I will attempt to accomplish the set goal of this chapter by presenting and discussing some principles of Ezumezu logic which I had formulated in earlier works in addition to formulating a few additional ones. The interesting thing to note here is that these principles are/will all (be) articulated from the African background ontology. I will conclude by throwing further light on the merits, nature and promises of an African logic tradition.

Logic and Proof in Computer Science

Computer software pervades our lives today. Nevertheless, software is one of the few products for which producers generally provide no express or implied warranties, a truly striking fact since peoples' lives depend in such fundamental ways on these products. This article addresses why such an unintuitive (and undesirable) situation might exist. It will catalog a range of computer science proof techniques and their historical antecedents, the purposes they serve, and several foundational concerns that elude proof techniques of any kind. Along the way, the concept of intractability and its role in computing will be explored as it pertains to algorithmic complexity and to proofs of the meanings of computer programs.

Logic and Order

Traditional human representation is unable to conserve complete information. Therefore ignorance, uncertainty, ambiguity to mankind's best conceivable worldview are even more amplified. To minimize this problem, we need to develop a reliable and effective ontological uncertainty management (OUM) approach. To reach this goal requires starting from traditional mankind worldview to arrive at a convenient OUM framework. Learning from neuroscience helps to develop neuromorphic systems able to overcome previous representation limitations by appropriate OUM solution. Furthermore, according to CICT (computational information conservation theory), the information content of any symbolic representation emerges from the capturing of two fundamental coupled components, i.e. the linear one (unfolded) and the nonlinear one (folded), interacting with their environment. Thanks to its intrinsic self-scaling properties, this system approach can be applied at any system scale, from single quantum system application to full system governance strategic assessment policies and beyond. A detailed OUM application example, taking advantage of the well-known EPM (elementary pragmatic model) by De Giacomo & Silvestri, to achieve full information extraction and conservation, is presented. This chapter is a relevant contribution to effective OUM solution development framework for learning and creativity, emerging from a Post-Bertalanffy General Theory of Systems.

The Impact of Using Logic Patterns on Achievements in Mathematics Through Application-Games

The chapter presents research into the impact of logic patterns, based on logical reasoning, focusing on order and sequence in series, on achievements in mathematics using an application-game which was developed neither specifically for the purpose of the current research nor to address Attention Deficit Hyperactivity Disorder (ADD). Both experimental and control groups were used for checking the central hypothesis on subjects of the same age – first and fourth graders at a similar learning level. The experimenters were Bachelor's Degree (BA) students majoring in special education. The method employed an application-game providing virtual simulation in real time offering the unique opportunity to observe and manipulate normally inaccessible objects, variables and processes. The focus was on qualitative research comparing subjects' achievements in mathematics in pre- and post-intervention. The findings showed that using logic patterns through the application games had an impact on the subjects' mathematical skills, especially verbal problem-solving. Their mathematical achievements increased quickly to the surprise of the experimenters who reported improvement in subjects' logic, mathematical and concentration skills, sometimes even the total stoppage of involuntary tics among those who received the intervention program as opposed to a lack of improvement or even a significant regression among the controls. Moreover, the motivation of both experimenters and subjects was enhanced, and their self-confidence improved. All the findings led to the conclusion that using application-games, although not developed for improving mathematics, can serve as a bridge between using logic patterns and improving or increasing mathematical achievements involving especially verbal problem-solving based on order, sequence and probability, among others.

The Core of Logics

Like mathematics so often logic is taught to introductory students in a very mechanical way, the emphasis being on memorization and working problems. Particularly egregious is that the logic taught in philosophy departments is devoid of philosophy. Students rarely encounter the deep philosophy underpinning the structures. Logic is the theory of innate order in the universe and is the language of that order. More explicitly the foundation of that order is binary, based on the most fundamental law of all: dialectics. Something is apprehended because of what it is not. This chapter summarizes the development of thinking underpinning this idea of the innate binary structure. It is an ordered binary space starting in one dimension and progressing through three, and beyond. The philosophical basis of single, two (Table of Functional Completeness), and three (three-dimensional hypercube) dimension space provides coherency to ideas like deduction, induction, and inference, in general. The ordering in these spaces is founded on the same thinking giving rise to numbers and arithmetic. An exposition of how binary logical space develops sets the stage for discussing foundational ideas like the relationship between arithmetic (and its follow-on, mathematics) and logic, pattern recognition, and even whether we may be a simulation, a conjector made by Nick Bostrom. Research directions are proposed such as questioning the nature of axioms, exploring the insufficiency of Peano's postulates, proof theory, and ordering of operators based on intellectual complexity.

Against Method, Against Science?

This chapter will focus on two questions, first, the question of Feyerabend's use of analogy, in Against Method, in order to give an account of science, scientific research, and/or scientific institutions in terms of fairy tales; second, it will focus on the question of whether the analogy holds up to critical scrutiny. Feyerabend uses “fairy tale” in a number of senses in Against Method; for example, he uses it to capture some misleading indeed erroneous views about methodology; he insists that the truth is at odds with the fairy tale and that the truth is that “all methodologies have their limits” (1980, p. 32); he takes it to mean not just an erroneous narrative but an erroneous or at the very least, questionable, narrative, which is promoted as true; he uses it in at least three other senses in the book. This chapter will then offer a detailed critique of the use of such analogies in Against Method in order to clarify the strengths and weaknesses of Feyerabend's argument.

Metaphors in Science

Usually it is held that metaphors are expressions in which something is said but it is evoked or suggest another thing. It is also said that they are - or should be - almost exclusive patrimony of literary or vulgar languages and are not relevant in scientific discourse. However, there are three arguments that lead one to suspect that there is something wrong in these points of view. First, the ubiquity of metaphor in past and present sciences. Second, in almost all such cases, metaphorical expressions are not substitutes or paraphrases of other literal expressions that scientists would use with their colleagues but instead are the common way they are expressed; there is just no other language, metaphors are part of the technical vocabulary. Third, the theoretical and practical consequences of metaphors are part of the corpus or theoretical system to which they belong, in the same manner that the consequences of theorems of an axiomatic system are part of the theoretical system. The three preceding arguments allow us to sketch the following hypothesis: metaphors used by scientists (at least a lot of them) say something in themselves, but are not mere subsidiaries of other literal expressions; therefore, they have legitimate and irreplaceable cognitive and epistemic functions. This change in approach challenges at least four different problem fields: 1) the concept of metaphor; 2) the standard epistemological tradition and its postmodern heresies such as the social studies of science; 3) the history of science; and finally, 4) the biological and cognitive sciences. These four problems will be addressed in this article.

A Boolean Logic Approach to Issues of Vagueness, Heuristics, Subjectivity, and Data Mining

This chapter introduces several modules that can be used to supplement an introductory logic course. The modules cover advanced topics such as rule heuristics, association rules, polythetic vs. monothetic grouping, and subsective adjectives. These topics are all approached using ordinary propositional logic, Boolean algebra with 0-1 variables. The topics are presented using a computational approach. The computations and concepts are elementary and accessible to an undergraduate without further prerequisites. The modules besides introducing advanced topics also facilitate discussion of other logic topics such as the law of the excluded middle, the concept of vagueness, and privative adjectives. This chapter presents new solutions to these problems. The chapter reviews the place of these modules within the context of introductory logic courses and the history of science. The supplementation of an introductory logic course with these modules is expected to strongly motivate students to pursue advanced topics and to increase interest in topics related to logic.

Aristotle's “Logical Worldview”

The author presents the idea of a “logical worldview” – an approach to understanding logic by examining philosophical positions in metaphysics, epistemology, and theories of cognition and perception, and exploring how philosophical and logical positions combine to form a complete logical system. Aristotle's logical worldview is examined in some detail, and the logical systems of Francis Bacon and George Boole are examined by exploring how a new logic results when certain Aristotelian philosophical positions are abandoned. The logical worldview approach is also shown to help explain certain puzzles with Aristotle's logic, such as existential import, the form of the syllogism, and Aristotle's “missing” moods and figures from the list of valid moods.