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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 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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-/[]\- 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. |
Chapter 23
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| row | A | not(A) |
| 1 | T | F |
| 2 | F | T |
The statement A is always true when statement NOT A is never true.
The statement NOT A is always true when statement A is never true. Here instead of saying never true, we may say always false.
The truth (T) or falseness (F) of the statement A and B depends on the respective truth or falseness of the statements A and B. This situation is summarized in the following table.
| row | statement A | statement B | A and B |
| 1 | T | T | T |
| 2 | T | F | F |
| 3 | F | T | F |
| 4 | F | F | F |
The statement A and B is said to be always true (to always hold) if the situations in rows 2, 3 and 4 of the above table never occur.
The statement A or B is said to be (mathematical usage) when and only when at least one of the statements A and B is true. The following table summarizes this situation. It shows when the statement A or B is true and when it is false.
| row | statement A | statement B | A or B |
| 1 | T | T | T |
| 2 | T | F | T |
| 3 | F | T | T |
| 4 | F | F | F |
With this usage, the statement A or B is guaranteed to be true provided the situation in row 4 of the above table never occurs.
| row | statement A | statement B | if A
then B |
| 1 | T | T | T |
| 2 | T | F | F |
| 3 | F | T | T |
| 4 | F | F | T |
The implication if A then B is said to be vacuously true when statement A is always false.
The following truth table if for the two-way implication A if and only if B. We observe the two-way implication is always true if the situations in rows 2 and 3 never occur.
| row | statement A | statement B | A if and
only if B |
| 1 | T | T | T |
| 2 | T | F | F |
| 3 | F | T | F |
| 4 | F | F | T |
Remember the letter F signals false, and corresponds to the idea of rule
being disobeyed. Also remember that the letter T signals true and corresponds to
the ideas of a rule being obeyed, or not disobeyed.
Next: Chapter 24: Direct and Indirect Reason or Proof Methods
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 middleThere 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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