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YOU are better than YOU think. Show
yourself how:
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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
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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 3, Elements of
Reason
In this chapter, you will meet two puzzles. They show the difference between
one- and two-way implication rules. Mastering the difference is a simple, first
step, in rule and pattern-based thought. This first step is needed to precisely
read rules, definitions and statements in all disciplines, including
mathematics.
Are you a careful thinker? Can you understand exactly the meaning of a rule
or pattern? Instructions for building or creating provide rules and patterns
which say and suggest that when this is done, that should happen. Every cook
and dressmaker knows the importance of following instructions carefully.
Instructions and suggestions which are not repeatable and results which are not
reproducible are not of interest to a cook or dressmaker.
To read rules carefully, do not imagine too much. To decide or choose among
opinions and actions, you must understand the exact meaning of written and
spoken words. You need this skill to understand, to follow, to write and to
change rules, guidelines, instructions and laws, etc.
Use your imagination in language courses. Use your imagination when you are
reading novels (and newspaper opinion columns). When reading newspapers or
listening to radio and television ask: Is the story presented in a one-sided
way? Headlines may suggest conclusions which are not in the stories or the
text. Look at the details. Here imagination allows you to guess what the full
story might be. But imagination provides only suggestion, not proof. Confidence
in suggestions must come after proof is given, not before.
Also use your imagination for poorly written rules to guess their meanings.
Guesses and speculations give possible meanings. These may or may not be
correct. Proof and evidence, or tests, may decide which among various
possibilities, if any, are correct.
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.
Next Subsection: First Puzzle
Chapter Subsections: [ First Puzzle ] [ Second Puzzle ] [ One- Versus Two-Way ] [ Talking About Logic ] [ Implications vs Suggestions ] [ One Versus Two Way Committments ] [ Repeatable & Reproducible ] [ Limits and Benefits ] [ Accidental Rules ] [ Steps for Better Reason ]
Teachers: Read the following before, besides or
after the two logic puzzles in this chapter, as you like. (Material not in
printed version follows)
Explaining the difference between the meaning of If
A then B and B if and only A is the purpose of
the following two logic puzzles. The questions in the puzzle below
are intended to introduce and emphasize the difference.
Suppose the following:
- the local store sells a newspaper if John enters.
- the same store sells a newspaper if Jeremy enters.
Then we cannot state
the local store sells a newspaper if and only if John
enters it (the local store). .
since the local store also sells a newspaper if Jeremy
enters.
So there is a difference in meaning between the two
suggestions or statements
- the local store sells a newspaper if John enters.
- the local store sells a newspaper if and only if John
enters it (the local store). .
Seeing the difference in meaning in this simple
example is the key to precision reading and writing.
More generally, there is a difference in meaning
between the two suggestions or statements
- situation B arises if and only if situation A
arises
- situation B arises if situation A arises.
Here we may say occurs or happens instead of arise, or
omit the word arise altogether. That be said and done, we make the
convention that the two statements
- Situation B if situation A (B if A
form)
- If situation A then situation B (if A then B)
have the same meaning.
Here we assume that the two following two
statements have the same meaning:
- B if and only if A,
- A if and only B
mean the same.
Understanding there is a difference in
meanings is the key to greater precision and exactness in work and
study. If you see the difference, you will we hope make an effort
to respect the difference and use the difference in following and
writing rules or instruction.
Postscript: October 2006. We may know the following:
Situation A occurs if situation B occurs
We may not know if there are furthers situations C such that
Situation A occurs if situation C occurs.
When further such situations C are possible. we cannot use the two-way
implication rule
Situation A occurs if and only if situation B occurs.
The foregoing observation provides an alternative starting point
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Next Subsection: First Puzzle
Chapter Subsections: [ First Puzzle ] [ Second Puzzle ] [ One- Versus Two-Way ] [ Talking About Logic ] [ Implications vs Suggestions ] [ One Versus Two Way Committments ] [ Repeatable & Reproducible ] [ Limits and Benefits ] [ Accidental Rules ] [ Steps for Better Reason ]
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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.
Vol 1A Postscripts
- online only
+Proof by
Absurdity alias proof by contradiction
+How the demand
for consistency supports the law of the excluded middle
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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