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YOU are better than YOU think. Show
yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study.
Learn to read notes and textbooks like
a lawyer, so that no nuance, no subtlety and no clause escapes your
attention. |
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Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
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Chapter 19
Elements of Logic
Logos is a Greek word for thought. Previous algebra and symbol
free chapters on reason showed how implication rules can be directly used or
chained together to arrive at conclusions. In daily life with the exception
perhaps of detective stories, the direct use of rules and patterns is usually
sufficient (enough).
Yet in mathematics, direct and indirect chains of reasoning appear. The study
of logic, that is, methods or laws for rule- and pattern-based thought, has been
motivated by the need in mathematics to reach conclusions. In particular, proofs
based on (1) mathematical induction,
(2) the contrapositive, and (3) proof
by contradiction all stem or originate from the conclusion-reaching needs of
mathematics.
The chapter Direct and Indirect Reason below, will describe methods
(2) and (3). Suggestion: try to read this last chapter to see how much can be
immediately understood.
The subject of logic as it is studied within mathematics courses, is often
presented as an algebraic (or symbolic) perspective of the methods of reason.
The next chapters present the algebraic perspective. They with the earlier
algebra-free discussion of implication rules and chains of reason give some
preparation for the description of the indirect methods (2) and (3) for in the
last chapter Direct and
Indirect Reason
The algebraic description of logic also has a role in the
design and simplification of electrical controls and computing circuits.
The algebraic description of logic further allows algebraic methods for
arriving at conclusions, in particular mathematical induction, to be applied
to the drawing conclusions about rule-based reason and logic. The algebraic
description of logic provides models of mathematical logic. Conclusions drawn
about the models then reflect on the limitations and reach of logical or
rule-based thought in mathematics.
1 About the Next Chapters
The next five chapters
- Shorthand or Pronouns
in Logic
- Occurrence Tables,
- The Contrapositive
- Truth Tables
and
- Direct and Indirect
Reason
continue the description of logic.
The occurrence (or obedience) tables invented and introduced below identify
those situations in which implication rules are obeyed, disobeyed or not
disobeyed. The latter notions are intended to simplify the explanation of truth
tables. An implication rule is said to be true in the case when it is obeyed or
it is at least not disobeyed. An implication rule is said to be false or not
true when it is disobeyed.1
The chapter The Contrapositive
shows the equivalence of an implication rule with its contrapositive
formulation. The analysis is based on the three notions of a rule being obeyed,
disobeyed or not disobeyed.
The chapter Direct and
Indirect Reason describes and explains direct and indirect methods for
reaching or proving conclusions. Among the indirect methods, this chapter
describes in particular, how an implication rule can be shown to always hold by
(a) showing its contrapositive form always hold, or by (b) looking for
absurdities that would occur if the implication rule did not hold. The second
method (b) is more indirect than the first method (a).
1 The
language previously used to explain and justify the entries of truth tables
overuses the word true. The introduction of the three notions of an
implication rule if A then B being obeyed, disobeyed or not
disobeyed aims to avoid this situation. Such implication rule is said to
be false in situations where it is disobeyed, and it is said to hold (or be
true) in those situations where it is obeyed or at least not disobeyed.
Finally, the implication rule is said to be always true in the circumstances
of interest provided it is never disobeyed in those circumstance. See the text
for further explanation.
Next: Chapter 20, Pronouns
or Symbols in Logic
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www.whyslopes.com
Volume 1A, Pattern Based Reason
Chapters 1 to 24
FOREWORD
Three Remarks
1 Introduction
2 Communication
3. Elements of Reason
4 Implication Rules
5. Deception
6 Chains of Reason
7 Longer Chains
For & From Consistency
8. Language Change
9 Next Chapters
10 Responsibility
11 Accidental Patterns
12 Knowledge Islands
13 Euclidean Logic
14 Deductive
& Empirical Views of Mathematics
15 Objectivity
16 Origin of Rules
and Patterns
17 Objective Ways
18. Waking up
19. Symbols & Logic
20. Pronouns or Symbols
21. Truth Tables I.
22. Truth Tables II
22. Biconditional
22. Contrapositive
23. IF-THEN table
24. Indirect Reason Again
To reason often means to persuade someone of
the need for an idea or action. That someone could be yourself. So be
careful.
1A Logic Postscripts
- online only
+Proof by
Absurdity alias proof by contradiction
+How the demand
for consistency supports the law of the excluded middle
+Reality versus or with the aid of Imagination
+Links for reason, logic and crtical thinking
+Three Remarks
+History
Lost or Missing
There is a difference between
knowing how to spend money,
and having money to spend.
There is likewise a difference
between mastering a skill
and having meeting a situation in which it applies.
.
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