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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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The previous chapters show how reliable rules and
patterns can be used to arrive at conclusions or judgments.
This raises the question: how can we recognize reliable
rules and patterns? This second group of chapters on reason
describes the origins, limitations and organization of rule-
and pattern-based knowledge in all arts and disciplines.
The problem of identifying reliable implication rules and
reliable information is described but not solved, except for
the description of empirical methods of coping in science
and technology. This identification problem touches many
subjects. Students of critical thinking, persuasion,
philosophy, mathematics, science and technology should find
its discussion in these chapters helpful.
About the Next Chapters
Previous chapters also show how rules and patterns can be
used one at a time or one after another to get conclusions.
The use and chaining together of implication rules is called
deductive thinking. The next group of chapters describe more
elements of reason. The chapters in question, namely
give views of reason, persuasion and knowledge in our
schools and communities and in our technical areas of
knowledge. These chapters speak about how and where rules
and patterns are needed, written, discovered or extracted
from experience. This information itself is needed to
recognize when and where the rules and patterns can be
applied, and to be aware of their reliability or
limitations.
The chapter Responsibility tries
to clarify the meaning of liability and responsibility for
actions and accidents. Then it talks about how liability and
responsibility for our actions impose limits on our freedom.
Principles for responsibility are suggested.
The chapter Accidental Patterns
describes how human behavior which we have seen in the past
may be accidental or coincidental. So we can have no
expectation of this behavioral pattern continuing. This
raises the problem of identifying which patterns of behavior
are reliable and not just accidental or coincidental. How
can we be (almost) sure that one event causes another? For
technical areas of knowledge but not for human behavior, an
answer is given in the chapter Origin of Rules. This
chapter echoes and reinforces the discussion of Accidental
Rules in the chapter Implication Rules.
The chapter Islands and Divisions of
Knowledge describes how one- and two-way implication
rules may link together rules and patterns of our
communities and in our technical knowledge. Some parts of
knowledge may be connected to others by chains of reason
while other parts may be separate.
The chapter Objective Processes explains the
notion of objectivity. The following of rules and laws in
a fashion which gives repeatable and reproducible results
independent of whom or what applies them leads to the
concept of objective thought and actions in technological,
in community or bureaucratic processes. Processes and
results which are repeatable and reproducible are not
necessarily optimal.
The chapter Origin of Rules and
Patterns first describes how rules and laws are written
or agreed to in society. Then it describes how reliable
rules and patterns are recognized in our technical areas of
knowledge (except for mathematics). The inference problem in
empirical thought is to identify, draw, induct or extract
from experience or tests, those rules and patterns which are
reliable and not accidental. The word induct means to draw
or extract. Inductive reasoning in empirical thought (not
mathematics) refers to the identification of rules and
patterns from experience.
Inductive reasoning in
mathematics refers to the longer chains of reason associated
with the principle of mathematical induction. The meaning of
the phrase Inductive reasoning in mathematics refers to the
longer chains of reason associated with the principle of
mathematical induction.
The chapter Discovery of Objective
Ways speaks about the non-objective, subjective and
sometimes creative approaches in which objective ways are
found. Problems for which solutions are not dictated by
others leave room for thought and experimentation. Of
course, hard thought and experimentation can be avoided if
we can use a previously found solution known to work well.
Sometimes that is preferable.
The chapter Euclidean Model of Reason
describes a two thousand year old model (and method) for
organizing technical knowledge. This method was first seen
in the geometric works of Euclid and his followers. These
works suggest how clear definitions and clear assumptions
together with chains of reason can be used to firmly derive
conclusions. Law-makers, theologians, scientists and
mathematicians have often tried to follow this model in
their areas of reason besides geometry. One reason for
studying mathematics was to meet this model. The works of
Euclid suggest an ideal we would like to achieve or
approximate as well as possible.
The chapter Views of Mathematics
describes how the first principles of arithmetic-based
mathematics came from experience, and how by trial and
error, calculations that work have been found in many
disciplines. Then this chapter mentions the effort to follow
the Euclidean Model of reason in mathematics and the
associated disappointment. Not all is certain. Finally, the
classroom view of mathematics after elementary school is
described. The latter could skipped on first reading.
The last chapter in this group, Sense
and Knowledge, speculates about the origin of
self-consciousness and of our ability to tell and remember
stories, theories or ideas linked together. As usual, more
can be said or suggested in speculation (or rumor mongering)
than proven. Serious students of reason and mathematics will
not take the speculation seriously.
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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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