Logic: Introduction

Chapter 1
Elements of Reason

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.

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.

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.

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.

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.

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.

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.

To Learn More About Logic, see chapters 26 to 31 in this Volume, or the identical (or supposedly identical) 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, section C in the webpage Mathematics Education in General includes a theory of knowledge based on our ability to collect, invent and tell stories with words and symbols, written, spoken or drawn. That theory may move to its own page at some future point in time.

A. More Arithmetic a must for algebra etc

D. Logic In Mathematics

G. Algebra with Take Home Value

I. Vectors & Functions

Decimal Lesson - Reference
Counting & Addition   (8 lessons)
Comparison to Subtraction (9 lessons)
Multiplication ( 11 lessons)
Long Division (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites
-Primes Factorization
-Greatest Common Divisors & Multiples.
-Prime Factorization Aids (Learn how to find factors quickly)
-Prime Factorization Examples
-Counting & Generating. Factors.
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts
Ratios & Fractions (3 lessons): Similarities & Differences
Units in calculations Fractions with Units

B. Basic Algebra

Solving Linear Equations
- in one unknown. Intro with stick diagrams?
the normal way & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of word problems.
- still in more unknowns: Gaussian Elimination via substitution, by equality or comparison, by operations on equations.

C. More Algebra

Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra
What is a Variable. The answer here is a simple prequel to the modern mathematics viewpoint.

First, every rule & pattern U meet in math, logic & science will be used forwards and backwards. Get a head start with this theme by reading Chapter 14 in Three Skills for Algebra. Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial backward use of the proportionality formula.

Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.

If you can not read or write precisely, you will have difficulty in following instructions. One wordy remedy is given by chapters 2 to 5 in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in Euclidean-Geometry In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.

E. More Geometry

The Pythagorean Theorem. Chapter 17 from in Three Skills for Algebra uses algebra and geometry   to show why the Pythagorean equation for right triangles holds. Its forward and backward use is common exercise.. At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be fractional multiples of a unit length. That geometrically implies a need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with Maps and Plans drawn to scale give coordinate-dependent base for senior high school development of similarity, trig, vectors and straight lines.
Complex Numbers: This lesson on Complex Numbers draws on Euclidean and Analytic geometry. Sbortcuts simplifiy trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products.

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry, personal finance, and calculus. To learn more, if you study calculus, see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters

identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry: - Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one. I may combine them later. -In my school days, slopes appeared year after year.   This Why Slopes calculus preview on graphs of functions y = f(x) explains why. Enjoy.
Quadratics and Polynomials: - Operations on Polynomials: Meet a light and ultraquick geometric introduction to multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials. Compare Fractions with Units. Enrichment: A Plus: The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic.
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials high school mathematics has technical value as part of the senior high school mathematics preparation for calculus. This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials, their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.

Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms. Vectors and their properties appear in physics, its mathematical description and formulation.
- Functions - Forwards & Backwards. Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics. comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic.

J. Pre-Calculus Skill Check

- Arithmetic Skill Check. In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.
- All the skills and concepts in Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with your other duties. Good luck.

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster.
The Rest � 1995 onward by site author, Alan Selby, All Rights Reserved.

Chapter 1 Elements of Reason

About the Next Chapters

Chapter 1
Elements of Reason