Chapter 1
Elements of Reason

Previous: Foreword

To reason with someone often means to persuade them of the need for an idea or action. Persuasion and reason can take many forms. Methods for arriving at conclusions and judgments in all skills disciplines are, or should be where possible, based on the use and recognition of reliable rules and patterns, and also data to use with them.

Each of us needs to understand fully or as much as is possible, whatever we might be doing or learning. In reasoning, some rules and patterns are reliable. Others are guidelines. Each of us needs to know which is which.

Mathematics, in and besides arithmetic, depends on rules and patterns, which are used one at time or one after another, to obtain conclusions. The aim of the next chapters is to introduce and provide an image of rule and pattern thought in mathematics and other disciplines.

When ideas in mathematics or another discipline are described instead of being drawn from implication rules, the role of rule-based reason or logic may be forgotten. But in every discipline including mathematics, signs of rule- and pattern-based reason are given by the word and phrases such as from this, then, if, therefore, thus, because, since, as, gives, yields etc.

About the Next Chapters

The next chapters describes some basic elements of rule- and pattern-based thought. In particular, four chapters, Implication Rules, Chains of Reason, Longer Chains of Reason and Islands and Divisions of Knowledge describe basic ideas, keys for reason and logic.

1. The chapter Implication Rules presents two logic puzzles. Each consisting of a rule and five questions. Answers are also provided. The puzzles show the difference between one- and two-way implication rules.

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2. The chapter Chains of Reason describes how to directly use rules one at-a-time or chained together, one-after another, for arriving at conclusions and judgments.

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3. The chapter Longer Chains of Reason starts to indicate the special role of reason in mathematics. It describes, in a very non-mathematical fashion, the concept of induction, a method used in mathematics to arrive at conclusions. This concept of induction is an example of a method of reason employed mainly or only in mathematical subjects.

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4. The chapter Islands and Divisions of Knowledge describes how rule and pattern-based bodies of thought may be organized. Here different starting points, first principles or assumptions, may lead to the same body of rule-based knowledge.

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   In philosophy, the discipline that is literally the love of knowledge, perhaps an infatuation, Euclid's logical or rule based arrangement of geometry provided a model for reason. This chapter with words and images apart from geometry describes the model and the variations possibly within it.

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The chapter A Change of Language introduces the conventional if-then and iff forms for writing one- and two-way implication rules. The one- and two-way implication rules in this work have been identified with condition and bi-conditional statements. But the terminology one and two-way employed here draws on the present-day common experience of one and two-way roads.

Next: Chapter 2, Implication Rules

To Learn More About Logic, see chapters 26 to 31 in this Volume

Chapters 19 to 24 in Volume 1A,. Pattern Based Reason . Volume 1A describes the benefits, origins of rule and pattern based thought, deeds and hopes in greater detail, and still leaves room for thought. Online postscripts in the Volume 1A site area discuss further the methods and context for indirect reason in and outside of mathematics. Finally, Appendices Story Telling includes a theory of knowledge based on our ability to collect, invent and tell stories with words and symbols, written, spoken or drawn.

Multi-Level Logic Skill Mastery - What it Means

Postscript: August 28th, 2011.

Before the study of implication rules of the form A IF B and the form A IF and ONLY IF B, older children and young adolescents may be shown how written work can be done and recorded in steps that can be seen as done or later by the doer, a peer or teacher. Steps observed as done or as recorded allows work to checked and in that, the domino effect of mistakes to be seen. In advanced mathematics, the concept of checking goes further: In derivations and proofes, steps are not only done and recorded for later inspection, reasons for them, where not obvious, are recorded as well, again for later checking. However learning to do and record steps, minus reasons for them, provides the first form of proof - empirical and preductive - in which steps are checked. Empirical reason in science and technology goes further by calling for results to obtained by procedures that are recorded for the sake of confirming that the results are repeatable and reproducible. So logic occurs in many different ways.

To repeat, two kinds of logic children and adolsecent may accept or understand consists of the following:

• Learning to do work that be done and recorded in steps or pieces that can be seen and thus checked as done or later.
• Learning to follow methods or processes where the steps are not necessarily recorded, but the results are repeatable and reproducible, and hence verifiable.

Beyond and besides the foregoing mechanics of empirical and deductive methods for arriving at conclusions, the underlying rules and patterns may form smaller and large islands and bodies of rule and pattern based practices and theory. Different entry points into the bodies and islands will make entry and exploration easier or difficult. Some trial and error, or experience, may be needed to find the easiest routes to follow. For example, site logic and mathematics material includes some easier entry point or easiet routes to follow, but simpler or clearer routes may be yet be found. The latter is not surprising. Only time and comparision will tell. The composition of site material was an iterative affair. Another iteration will likely improve it. Bon Appetit.

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