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AN INTRODUCTION TO OUR UNITS ON MATHEMATICAL LOGIC

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AN INTRODUCTION TO OUR UNITS ON MATHEMATICAL LOGIC Powered By Docstoc
					       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.




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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.
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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



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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.




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