A logical approach to competition in industries (original) (raw)
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Tennessee Technological University Mathematics Department Technical Report, No. 1999-1., 1999
Mathematics departments rarely require students to study very much logic before working with proofs. Normally, the most they will offer is contained in a small portion of a "bridge" course designed to help students move from more procedurally-based lower-division courses (e.g., calculus and differential equations) to more proof-based upper division courses (e.g., abstract algebra and real analysis). What accounts for this seeming neglect of an essential ingredient of deductive reasoning? We will suggest a partial answer by comparing the contents of traditional logic courses with the kinds of reasoning used in proof validation, our name for the process by which proofs are read and checked. First, we will discuss the style in which mathematical proofs are traditionally written and its apparentutility for reducing validation errors. We will then examine the relationship between the need for logic invalidating proofs and the contents of traditional logic courses. Some topics emphasized in logic courses donot seem to be called upon very often during proof validation, whereas other kinds of reasoning, not often emphasized in such courses, are frequently used. In addition, the rather automatic way in which logic, such as modus ponens, needs to be used during proof validation does not appear to be improved by traditional teaching, which often emphasizes truth tables, valid arguments, and decontextualized exercises. Finally, we will illustrate these ideas with a proof validation, in which we explicitly point out the uses of logic. We will not discuss proof construction, a much more complex process than validation. However, constructing a proof includes validating it, and hence, during the validation phase, calls on the same kinds of reasoning. Throughout this paper we will refer to a number of ideas from both cognitive psychology and mathematics education research. We will Þnd it useful to discuss short-term, long-term, and working memory, cognitive load, internalized speech and vision, and schemas, as well as reßection, unpacking the meaning of statements, and the distinction between procedural and conceptual knowledge.
Logical Reasoning including Mathematical Reasoning: Practical Problems and Possible Solution
Logical reasoning in any form is an important aspect of life; it is persuading or convincing others with logic through writing or speech, for example, scientists, politicians, businessmen, financiers, solicitors and many others do this. This paper points out the frequent inefficacy of logical presentations, arguments and debates per se in bringing about the correct and wonted outcomes. It describes the scenario of people frequently involved in fruitless arguments and debates, and shows why the application of logic, for example, in logical argument or debate, could not often achieve the desired outcomes, much of the time ending up with frustration, unhappiness, bad feelings and poor relationships. Scenarios from mathematics, which probably represents the most rigorous form of logical reasoning, and science are described as well. The paper also delves into the problems encountered in logical reasoning as well as some modes of reasoning. It would be difficult and might be impossible to reason with and convince someone with a closed mind-set, someone who has made up the mind not to be convinced, or even someone who is not intelligent enough to be convinced. The paper, which is published in an international mathematics journal, presents a resolution to this serious problem, which is important, as that would be conducive to peace and harmony.
Formalization as the Immanent Part of Logical Solving
Logical Investigations, 2018
The work is devoted to the logical analysis of the problem solving by logical means. It starts from general characteristic of the applied logic as a tool: 1. to bound logic with its applications in theory and practice; 2. to import methods and methodologies from other domains into logic; 3. to export methods and methodologies from logic into other domains. The precise solving of a precisely stated logical problem occupies only one third of the whole process of solving real problems by logical means. The formalizing precedes it and the deformalizing follows it. The main topic when considering formalization is a choice of a logic. The classical logic is usually the best one for a draft formalization. The given problem and peculiarities of the draft formalization could sometimes advise us to use some other logic. If axioms of the classical formalization have some restricted form this is often the advice to use temporal, modal or multi-valued logic. More precisely, if all binary predicates occur only in premises of implications then it is possible sometimes to replace a predicate classical formalization by a propositional modal or temporal in the appropriate logic. If all predicates are unary and some of them occur only in premises then the classical logic maybe can replaced by a more adequate multi-valued. This idea is inspired by using Rosser–Turkette operator Ji in the book [22]. If we are interested not in a bare proof but in construction it gives us it is often to transfer to an appropriate constructive logic. Its choice is directed by our main resource (time, real values, money or any other imaginable resource) and by other restrictions. Logics of different by their nature resources are mutually inconsistent (e.g. nilpotent logics of time and linear logics of money). Also it is shown by example how Arnold's principle works in logic: too " precise " formalization often becomes less adequate than more " rough " .
The Role of Logic in the Validation of Mathematical Proofs, July 1996
Technical Report of the Tennessee Technological University Mathematics Department, No. 1, 1999. Also presented at the DIMACS Symposium on Teaching Logic and Reasoning, Rutgers University, 25-26 July 1996.
Mathematics departments rarely require students to study very much logic before working with proofs. Normally, the most they will offer is contained in a small portion of a bridge course designed to help students move from more procedurally-based lower-division courses (e.g., calculus and differential equations) to more proof-based upper division courses (e.g., abstract algebra and real analysis). What accounts for this seeming neglect of an essential ingredient of deductive reasoning?1 We will suggest a partial answer by comparing the contents of traditional logic courses with the kinds of reasoning used in proof validation, our name for the process by which proofs are read and checked. First, we will discuss the style in which mathematical proofs are traditionally written and its apparent utility for reducing validation errors. We will then examine the relationship between the need for logic in validating proofs and the contents of traditional logic courses. Some topics emphasized in logic courses do not seem to be called upon very often during proof validation, whereas other kinds of reasoning, not often emphasized in such courses, are frequently used. In addition, the rather automatic way in which logic, such as modus ponens, needs to be used during proof validation does not appear to be improved by traditional teaching, which often emphasizes truth tables, valid arguments, and decontextualized exercises. Finally, we will illustrate these ideas with a proof validation, in which we explicitly point out the uses of logic. We will not discuss proof construction, a much more complex process than validation. However, constructing a proof includes validating it, and hence, during the validation phase, calls on the same kinds of reasoning. Throughout this paper we will refer to a number of ideas from both cognitive psychology and mathematics education research. We will find it useful to discuss short-term, long-term, and working memory, cognitive load, internalized speech and vision, and schemas, as well as reflection, unpacking the meaning of statements, and the distinction between procedural and conceptual knowledge.