Introduction
Chapter 1
Previous: Three Remarks
To reason often means to persuade someone of the need for an idea or action.
That someone could be yourself. In the latter case, reasoning may mean following
a line or pattern of thought to arrive at a conclusion, action or decision.
Persuasion or reason can take many forms. There are fair and unfair ways of
persuasion. There are sensible and absurd ways as well. Methods for arriving at
conclusions and judgments in all disciplines are, or should be where possible,
based on the use and recognition of reliable rules and patterns. Where ever
there is a presentation of ideas, there is an element of reason or persuasion.
Reason and persuasion are met in the home, in the print and television media,
in the classroom and in the work place. Rule-based reasoning, that is logic, and
departures from it can be described in and outside of mathematics. The
recognition of rules and patterns, methods with repeatable, reproducible and
thus verifiable results, provides a basis for science, technology and even
accounting.
The first chapters on reason give two logic puzzles to show how rules and
patterns can be used to arrive at conclusions or judgments in all subjects,
mathematical or not. Logos is the Greek word for thought. The puzzles
show the need and so reinforce the ability to precisely read and understand the
statements of rules, patterns, instructions and definitions. The two logic
puzzles in particular show the difference between one- and two-way implication
rules
A one-way implication rule says that when one event occurs, so
should another. A two-way implication rule says that when either of two events
occurs then so must the other. The terminology of one-way and two-way
implication rules may be new to this book. It is a plain language replacement
for the more traditional phrasing which speaks of conditional and
bi-conditional statements.
Not seeing the difference between one- and two-way implications or
suggestions is a source of confusion and false expectations in everyday life,
contracts, instructions and technical areas. Recognizing the difference between
one-way and two-way rules gives an initial step in mastering rule- and
pattern-based thought. Seeing how reliable rules and patterns can be used
one-at-a-time or one after each other to arrive at conclusions gives another
step.
In mathematics courses, logic is often met as the algebraic or symbolic
description and analysis of rule and pattern-based methods used in the
discipline (math) for arriving at conclusions. Some rule and pattern reasoning
methods developed in response to the conclusion reaching needs of mathematics.
The last chapters in this work introduce the algebraic or symbolic
description of logic while leading to an explanation of direct and indirect
methods of reason. The description innovatively employs the simple notions of a
rule being obeyed, disobeyed or not disobeyed, or never disobeyed to
clarify the technical truth-table description of one-way (material)
implications. The very last chapter describes the direct and indirect chains of
reason and persuasion met in mathematical proofs. Indirect methods are also of
service perhaps in the writing and resolution of detective and mystery stories.
In all fields of endeavor and inquiry, the main obstacles to the use of
reliable rules and patterns for arriving at conclusions lie first in
their identification and second in the identification of reliable
information to use with them. To understand or cope with these obstacles, a
knowledge is required of the origins of rules and patterns in daily life as well
as in science and technology. Science, engineering and technology have
empirical, that is experience-based methods, for coping with or circumventing
the two obstacles. Here rules, patterns and procedures which give repeatable and
reproducible results appear to be the most reliable or trustworthy, although not
always optimal. Some rules and patterns appear to be more reliable or secure
than others, but not all is certain.
Next: Chapter 2, Communication of
Ideas and Skills
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