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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
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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Occurrence Table for Two Way Implications, Chapter 21, part
III
Previous: Chapter 21, Part II, Occurrence
Table for Material Implications
3 Two-Way Implication Rules
A rule which can be stated, or restated, in the form
The first situation A occurs when and only when the second
situation B occurs
or in the form
The first situation A occurs if and only if the second situation B occurs
is called a two-way implication rule. For each two-way implication rule
note that:
-
The rule is obeyed when both situations occur.
-
The rule is disobeyed when the first situation A occurs
without the second situation B occurring.
-
The rule is disobeyed when the second situation B occurs
without the first situation A.
-
The rule is not disobeyed when both situations do not occur.
In brief, the two situations in a two-way implication rule must
both occur or both must not occur, for the rule to be not disobeyed.
The next table summarizes the above remarks for any two-way
implication rule A if and only if B.
| row |
situation A |
situation B |
A if and only if B |
| 1 |
occurs |
occurs |
obeyed |
| 2 |
occurs |
occurs not |
disobeyed |
| 3 |
occurs not |
occurs |
disobeyed |
| 4 |
occurs not |
occurs not |
not disobeyed |
As said before, a two-way implication rule is said to be always true when it is
never disobeyed. This requires that the situations in rows 2 and 3 of the above
table do not occur. That is, the above two-way implication rule A iff B is
true (never disobeyed) provided neither of the situations A and B occurs without
the other.
Links to Chapter Sections: [Special
Use of Three Words] [ 21. Material Implication Occurrence Table ] [ Occurence Table for IFF ] [ 21. Converses for 1-Way Implications ]
Next: Chapter 21, Part IV, Converses
for One Implications
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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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