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AN INTRODUCTION TO OUR UNITS ON MATHEMATICAL LOGIC Developed by The Center For Conceptual Studies Presented by: Richard Singer ccsfrs@worldnet.att.net (314) 931-5862 and (314) 721-2779 Edition Date: 7/99 Conceptual study focuses on clarifying and refining concepts. In most areas conceptual study is dominated by concerns about the application of concepts to matters which are not primarily conceptual. Pure conceptual study is guided by the desire to understand or shape some portion of a conceptual net, where a conceptual net is a set of concepts and conceptual relations. Pure conceptual study first emerged in mathematics, but it is also beginning to emerge elsewhere. It has even emerged in areas not closely related to mathematics, such as in the work of by Peter Ossorio in the area of Descriptive Psychology. The center for conceptual studies is a non-profit organization devoted to the enhancement of conceptual studies in a wide variety of areas. Since we regard mathematical logic as the paradigm case of pure conceptual study, we have developed a collection of unit which emphasis logic from this perspective. For a more general perspective on conceptual study see our paper entitled “An ?? Introduction To Conceptual Studies”. Perspective This paper presupposes that you have a copy of our paper entitled “An Initial Perspective On Mathematical Logic” and that you want to examine some portion of logic from this perspective. Our main purpose in this present paper is to introduce you to a collection of units that takes a manifest approach to the to some of the main concepts of contemporary mathematical logic. We discuss the organization of these units and how they unit can be used in the study of mathematical logic. We also discuss the kind of manifest approach we are using and why we have taken this approach. If you decide to use any of these units, keep this paper and use it as a reminder of how we recommend using them. A Manifest Approach to Mathematical Logic The main purpose of our logic units is to provide a wide variety of materials for relating the fundamental concepts of contemporary mathematical logic to concepts that are more manifest. These units are written for anyone who wants to think about logic from a manifest perspective. They are designed as a resource for self-directed personal study by a student who has access to a mentor. Since this type of study is rare in traditional schooling, we give some fairly detailed advice on how to use these units. You do not need any previous knowledge of logic in order to use these units. If have already studied logic, or if you are currently taking a logic course, these units can provide a more intuitive grasp of the remote concepts you have encountered and an intuitive basis for those you are about to encounter. Regardless of your background, we recommend that you pay attention to the prerequisite structure given at the end of this paper. Even units that focus on concepts that you already understand can provide additional perspective. For the student, who is also a teacher of logic, these units can provide materials or ideas which they can easily adapt and use in their own teaching. Organization All of these units are all available as Microsoft word files. Each unit begins with a section giving an overview which may be read without focusing on details. The other sections expand on this thru a closely related combination of exposition and activities. Important ideas are then reinforced by exercises and extended by problems and projects. A mere reading of the unit can acquaint you with a perspective on the ideas but to obtain a functional mastery of the ideas the majority of your time should be spent in active participation in the activities and the exercises. Ac Activities are an integral part of the core units, allowing an immediate check on your understanding of the ideas just covered or providing a basis for understanding the next ideas to be introduced. Respond to all activities, or at least read them carefully and imagine a response. In order to provide obtain immediate feedback, we usually give our response to an activity on the page which suggests the activity. Try to make a response before reading ours. On some occasions you may want to read our response first. When this happens return to the activity at a later time. Ex Exercises are usually given at the end of each section. They are designed to reinforce ideas covered in the main text. Read all the exercises and work enough of them to feel comfortable with the ideas. If you understand the materials then the exercises should be fairly easy, although some may be tedious, and you may make careless errors. If you have any other kind of difficulty with an exercise, this probably indicates an inadequate understand something in the text, so either study the text again or obtain help. Pr Problems are supplementary. Investigate as many or as few of them as you like, either as or after you study the core materials. You may have a good understanding of the ideas, but find some of the problems difficult. Many problems involve using the ideas in ways that require imagination or insight. They can be used in place of exercises to reinforce ideas, but they are also designed to enrich your understanding. Some problems also introduce ideas not covered in the text. If you find most problems too difficult focus on the exercises. These should be sufficient to help you obtain a functional mastery of the core concepts and their use. Pj Projects are like problems, but more extensive. They are intended for anyone who likes to investigate some single topic in depth. A single project, fully developed, can serve to reinforce ideas as well or better than a large number of problems or exercises. However projects may also involve effort not directly related to these ideas, so much of the learning from projects may not relate directly to the concepts of mathematical logic. The best projects are usually those that you put creative effort into. Thus students should feel free to radically modify the project ideas we have suggested. 1 Unit Objectives Most units provide a suggested list of specific objectives. Examine these as you proceed through the unit. We also encourage you to formulate your own objectives, as you find this useful. Emphasis on objectives that can be clearly formulated can divert energy which could be better channeled by simply following your intuition about what seems most interesting. Approach In these logic units explanation and discussion of remote ideas is usually preceded by less remote activities and illustrations that allow you to start with special cases of these ideas. This strategy is used because most persons acquire remote concepts more effectively when they arise from practice with closely related manifest concepts. Most of the manifest materials in the core text involve puzzles like those about the set BAS introduced the initial perspectives paper. Such puzzles can be used to illustrate purely analytic reasoning in a context which almost anyone can easily grasp. Furthermore such puzzles can be designed at various levels of complexity, depending on the logical principles being introduced at the time. The puzzles in the core text are given in the units as needed, but they are also listed in reference files so you can play with them without reference to any of the units. There is also a file of puzzles that refer to an expanded set XAS consisting of 24 items. While these puzzles are seldom used in the text of any core unit, they appear in some of the problems and in some of the supplementary materials. The main limitation of puzzles is that while many people enjoy them, they may not find them very significant. In the initial perspective paper we showed that the same form of reasoning applied to clues about attribute items could be applied to the ordinary matter of buying a car. However it takes considerable effort to imagine and present ordinary situations in which only analytic reasoning is used. Analysis of most ordinary situations usually assumes some context. Unless this context is described in detail or directly experienced, misunderstandings are almost certain to arise. Furthermore, ordinary reasoning is seldom purely analytic. It is usually a mixture of analysis, synthesis, implicit beliefs, extrapolation, emotional appeals, etc. For these reasons it can be tedious to use ordinary reasoning as a source of initial illustrations for mathematical logic. How to communicate the significance of analytic reasoning early in an introduction to mathematical logic is a problem which we do not know how to approach in any direct fashion. We hope the use of puzzles will at least make the study of such reasoning interesting and manifest. Mathematical Reasoning Mathematical reasoning provides the richest source for illustrating the concepts of mathematical logic. We did not choose this as our main source, because although mathematical reasoning can be formulated in terms of the logical principles in these units, most such reasoning uses principles not introduced in early units. Furthermore, unless you are comfortable with mathematical concepts, such illustrations are difficult to follow. If you are comfortable with mathematical proofs may want to go rapidly through some of the earlier units and focus on more mathematical units. You may also want to skip ahead to more remote concepts in a unit and then return to see how they are illustrated by more manifest activities and examples. Ordinary Reasoning While the same form of reasoning applies attribute puzzles and situations involving ordinary matters, it takes considerable effort to imagine and present ordinary situations in which only analytic reasoning is used. Analysis of most ordinary situations usually assumes some context. Unless this context is described in detail or directly experienced, misunderstandings are almost certain to arise. The car buying situation from the Initial Perspective paper the premise that they will either buy a Dodge or a Chevy, when all we really know is that these are the only cars they are considering. We interpreted this situation as if they were limited to the alternatives mentioned. Clearly other interpretations are possible. Perhaps they might change their minds before making a final decision, or find a good compromise. Such possibilities can be either included or excluded by giving more information about a situation, however this means spending more time on the situation than on the logic. Furthermore, ordinary reasoning is seldom purely analytic. It is usually a mixture of analysis, synthesis, implicit beliefs, extrapolation, emotional appeals, etc. For these reasons it can be tedious to use ordinary reasoning as a source of initial illustrations for mathematical logic. 2 3 The Core Units The 12 core units are named below. A indicates that the structure to which propositional reasoning is being applied only involves attributes. R indicates that structure to which such reasoning is being applied also involves relations. M indicates that propositional reasoning is being applied to mathematical structures, and hence functions are involved. Q indicates that reasoning with quantifier is used. The number indicate the level of the unit in relation to other units of that type. A0: Primary Concepts For Attribute Structures A1: Informal Propositional Reasoning For Attribute Structures A2: Formal Propositional Theories For Attribute Structures R0: Primary Concepts For Relational Structures R1: Informal Propositional Reasoning For Relational Structures R2: Formal Propositional Theories For Relational Structures O0: Primary Concepts For Mathematical Structures O1: Informal Propositional Reasoning For Operational Structures O2: Formal Propositional Theories For Operational Structures Q0: Primary Quantification Concepts Q1: Informal Quantification Reasoning For Various Structures Q2: Formal Quantified Theories For Various Structures The main prerequisite outline for the core units is given below. However it is possible to do the Q units without doing the O units as long as you are comfortable with the concept of a mathematical structure as used in contemporary mathematics. A0 R0 O0 Q0 A1 R1 O1 Q1 A2 R2 O2 Q2 Most units have a main test along with several appendixes. Appendixes are labeled according type. EP: gives extra practice on the core materials, BP: provides a broader perspective, AP: provides a more advanced perspective. There are also a number of reference files which aren’t directly related to any particular unit. Binary Attribute Puzzles Extra Attribute Puzzles Miscellaneous Logic Puzzles Logic Terms Glossary Inference Rule List Logic For Secondary Math 4 To Begin Before using any of these units we suggest that you first solve at least 10 attribute puzzles. Puzzles involving the 8 item set can be found in “Binary Attribute Puzzles”. Puzzles involving the larger 24 item set can be found in “Extra Attribute Puzzles”. Course Use Suggestions While we are developing materials for student to use for personal study, either with a mentor or to supplement their understanding in more traditional courses, these materials have also been used as the main text in various courses. The core materials from Units A0,A1,R0,R1,Q0,Q1 have been used for a logic course for students whose background in mathematics is extremely limited. The only mathematics competence these units presuppose is a functional mastery of the basic concepts from ordinary algebra. These units, along with M0 and M1 have been used as the main core of a course on algebraic structures, a course which focused on developing skill in the discovery and presentation of mathematical proofs and on gaining an understanding of the nature of mathematical proofs by using mathematical logic. We have also used a number of these units in a the study of discrete algebra focusing on boolean algebra and other concepts for computer science majors. We also recommend the total collection of these material to student in the usual graduate level courses on mathematical logic, since they provide a number of more elementary illustration of the concepts used in such courses, or the total collection could be used as a basis for a 2 semester course on mathematical logic which is less remote than the usual graduate level courses. 5