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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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Previous: Chapter 12, Islands
and Divisions of Knowledge
The road and door analogies in the previous chapter describe the division of
human knowledge into sections or islands. The knowledge in each section may be
strongly or weakly linked together by implications. Knowledge in one section may
touch or not touch that of another. All depends on what implication rules are
known. Our minds can explore each section of knowledge as we meet it.
In this chapter, the Euclidean model for organizing reason and knowledge is
discussed. In this Euclidean model for reason and knowledge, each area or
segment of knowledge is derived via chains of reason from a few secure first
principles or assumptions about data and implication rules. This Euclidean model
is an ideal which we would like to attain. Can we?
Deduction From First Principles
The aim of the axiomatic/deductive method is to gather and to organize an
island or body of knowledge so that all parts of it can be reached from a few
basic, clear and self-evident ideas or principles. This is the axiomatic goal.
The simplicity of this goal, an ideal, is appealing.
Where or with what should we begin? The starting points and the rules used
are human selections. If one point can be reached from another, and vice-versa,
then each is as good as the other as a starting point. Changing the starting
place in this manner does not change the destinations or results reachable.
Finally, different starting points for the organization of knowledge have
different advantages. A central starting place may provide faster or easier
access to the various parts and results.
The axiomatic deductive method is used in mathematics, in the physical
sciences and in human laws. The first model of the axiomatic method comes from
Euclidean geometry: the works of Euclid and his school in mathematics some two
thousand years ago. This Euclidean model of reason is deductive. It is based on
supposedly self-evident facts and implication rules. Here the fewest
possible rules are used to avoid conflicts and contradictions. Euclidean models
for reason in all disciplines has been an ideal and goal for some philosophers
and religious thinkers in Europe and possibly elsewhere. The framers of the Bill
of Rights in the United States Constitution were perhaps influenced by the
Euclidean example when they started by declaring certain rights self-evident.
The axiomatic, deductive, chain-of-reason approach to a subject requires a
starting point. We try to build our knowledge and our judgments and conclusions
on a few laws, principles, rules or facts that can be assumed, or viewed as
self-evident. Self-evident rules and principles represent starting points,
sometimes held beyond debate.
The laws, principles or facts we start with and pretend or assume to be true
are called hypotheses, first principles, assumptions, postulates or axioms.
Which word or phrase you use is a matter of choice. For the sake of variety,
we may use all these words and phrases interchangeably.
A set of assumptions together with their consequences, that is, the
conclusions which can be obtained from them, form and define a theory. (We
mortals will only see a finite number of the consequences.) The set of
assumptions on which a theory is built is called a foundation. Again, the
assumptions forming the foundation are supposed to be self-evident, clear and
credible. Identifying the self-evident ones has for mankind been a matter of
trial and error, and perhaps a matter of culture.
Next: 13. Clever Mortals - the challenges of forming
explanations.
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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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