Appetizers and Lessons for Mathematics and Reason  ( Français)  
www.whyslopes.com  
 Logic mastery is key to easing or avoiding learning difficulties in work & studies. 
What does it mean to use a formula forwards and backwards? 

Online Volumes (Book Orders)
1,  Elements of Reason. 1996
1A. Pattern Based Reason  1995
1B. Math Curriculum Notes 1996
2. Three Skills for Algebra  1995
3.
Why Slopes & More Math 1995

Links To Tutoring Services

Site  Folders for Instructors & Adults
A.  Public Policy Matters -  Essays
B.  Mathematics  Education Essays  2006-7
C -Logic & Applied Math Program  
    for education,  June 22, 2008 
D. Quebec English Math Ed  1997-2005
E. Help your child or teen
How TOs/ Ref.-08- 2008
1. Arithmetic Reference
2. Algebra 
3. More Algebra 
4. Geometry  
5. More Geometry
6. Calculus
7. Logics in Maths
4. First Puzzle

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Pattern
Based
Reason

Printed in Canada
ISBN 0-9697564-5-3

This folder has a tree like structure. The child, same level and parent level webpages for this webpage follow. 

4. First Puzzle
4. Second Puzzle
4. IF versus IFF
4. Joking About Logic
4. Imply or Suggest
4. 1 vs 2 way Committents
4. Repeat- & Reproduc-ible?
4. Rules Limits & Benefits
4. Accidental Rules
4. Steps for Better Reason

Home
Foreword
PS. Three Remark
1. What is reason
2. Inductive Ed Principles
2. Communication
3. Elements of Reason
4. Implication Rules
5. Hype & Deception
5. Hype & Ethics
6. Chains of Reason
7.  Longer Chains of Reason
7. Mathematical Induction
8. Language Change
9. Next Chapters, About.
10. Limits to Freedom
11. Accidental Patterns
12. Two Analogies
12.  Knowledge Islands
13. Euclidean Model
13. Euclidean Reason
14 Math: Deductive/Empirical
15. Objectivity
15. Objectivity, More
16 Rules-Patterns Origins
17. Objective Ways
17. Trial & Error Discovery
18. Conciousness
19. Symbols & Logic
20. Pronouns & Symbols
21. Truth Tables I.
22. Contrapositive
22. Vacuously True
24. Indirect Reason More
24PS. Excluded Middle Law
24PS.  Proof by Absurdity
PS. Reality vs Imagination
PS. Ahistorical Logic
Links Elsewhere - Go GoGo

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
+History Lost or Missing

 

Chapter 4, Implication Rules

Previous Section: Introduction to Chapter 4.

The First Puzzle

A One-Way Implication Rule

To help you think and possibly cook more carefully, we look at a very simple puzzle. The puzzle consists of a rule and five questions. The questions test your ability to think carefully and to read exactly what is written. Once you have understood the answers and why they are true, your ability to think carefully and clearly will have advanced. The rule for the puzzle is as follows:

When Aunt Jane visits her nephew Tom's home,
Tom goes out to play.

Try to answer the five questions below. Be careful. The questions may trip you. Answers follow. See if you agree with them.

Five Questions

Answers are given twice
  • in popup boxes, and
  • in text below (as in the printed version)

See if you agree with them.2

  1. When the rule is obeyed, what can you say happens for sure when Aunt Jane visits her nephew's home? This is easy.  [Answer
  2. When the rule is not disobeyed, what can you say happens for sure about Aunt Jane when Tom is out playing? Be careful. [Answer]
  3. When the rule is not disobeyed, what can you say happens for sure about Tom when Aunt Jane is not visiting? Be careful, again. [Answer]
  4. What must happen for the given rule to be disobeyed? This is another easy question. [Answer]
  5. When the rule is not disobeyed, what can you say for sure about Aunt Jane when Tom does not go out to play? See the answer to the fourth question. [Answer]

Hint: The rule provides no information and no reason explaining why Tom goes out to play whenever his Aunt Jane visits. The rule only describes what happens when Aunt Jane visits. We cannot say if he goes out to play to avoid Aunt Jane. We cannot say if he looks forward to her visits. The answers to the above questions only depend on the wording of the question and the given information or rule(s). So control your imagination. Do not assume or imagine too much.

Suggestion: Discuss the questions with your family and friends. Some people will get correct answers immediately. Others require persuasion. Still others will not understand. Talking with people about the questions shows how well they think.

The First Answer

The first question is
When the rule is obeyed, what can you say happens for sure when Aunt Jane visits her nephew's home?
Its answer is easy: Tom goes out to play.

The Second Answer

The second question is
When the rule is not disobeyed, what can you say happens for sure about Aunt Jane when Tom is out playing?
The answer is nothing. The rule only tells what happens when Aunt Jane visits. It does not say what must happen when Tom goes out to play. Tom could go out to play without Aunt Jane visiting. The rule does not say, nor does it suggest that Tom may only play outside when Aunt Jane visits. The rule does not say Aunt Jane must visit when Tom goes out to play.

When the rule is not disobeyed, we cannot say much for sure or certain about Aunt Jane when Tom goes out to play. All we can say for sure is that she may be visiting or she may not be visiting. When she is not visiting, the rule cannot be disobeyed. When she is visiting, the rule is obeyed and so not disobeyed. In either case, the rule is not disobeyed.

The above rule is one way. It says what should happen when Aunt Jane visits without saying that she must be visiting when Tom goes out to play. When Tom goes out to play, the rule is not disobeyed when Aunt Jane is not visiting. It gives no information on her whereabouts. An example of a two-way rule is given later. See the second puzzle.

The Third Answer

The answer to the third question

When the rule is not disobeyed, what can you say happens for sure about Tom when Aunt Jane is not visiting?

is like that of the second. When Aunt Jane is not visiting, the rule is not disobeyed if Tom goes out, and the rule is not disobeyed if Tom does not go out. When the rule is not disobeyed we can say nothing for certain about Tom when Aunt Jane is not visiting. The rule does not say that the only time Tom can go out to play is when his Aunt Jane visits. Again, the rule is only one-way. When Aunt Jane is not visiting, no information can be extracted from the rule. It says nothing about Tom.

The Fourth Answer

The fourth question is
What must happen for the given rule to be disobeyed?
The rule is disobeyed if Aunt Jane visits and Tom does not go out to play. That is, the situation where Aunt Jane visits and Tom does not go out to play must happen for the rule to be disobeyed.

The Fifth Answer

The fifth question is
When the rule is not disobeyed, what can you say happens for sure about Aunt Jane when Tom does not go out to play?
The rule will be disobeyed when Aunt Jane visits and Tom does not go out to play. To avoid the rule being disobeyed when Tom does not go out to play, Aunt Jane must not be visiting. The fifth answer is Aunt Jane is not visiting.

The contrapositive way of writing the above rule is When Tom not go out to play, Aunt Jane not visit. For this contrapositive rule to be never disobeyed, what can you say for sure when Aunt Jane visits? Answer: Not (Tom Not go out to play), that is, Tom goes out to play. The contrapositive of the contrapositive is the original rule. Later chapters on logic give more information, just a little more, about the contrapositive.

Some Vocabulary. The above rule is called an one-way implication rule. The first aim of this chapter is to show you the difference between one- and two-way implication rules. The meaning and use of the word implication will be talked about later. The five questions should help you see the difference between a one-way and a two-way implication rule. Seeing this difference signals that you can interpret precisely what a rule means. 

When the rule is not disobeyed, what can you say happens for sure about Aunt Jane when Tom is out playing? Answer: Aunt Jane must be visiting (the answer has changed). 

Next: Second Puzzle

Chapter Subsections: [ 4. First Puzzle ] 4. Second Puzzle ] 4. IF versus IFF ] 4. Joking About Logic ] 4. Imply or Suggest ] 4. 1 vs 2 way Committents ] 4. Repeat- & Reproduc-ible? ] 4. Rules Limits & Benefits ] 4. Accidental Rules ] 4. Steps for Better Reason ]

 

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Caution: Site advice is approximately correct, for some circumstances, not all. Site How-TOs are logically developed, but not tried and tested. That leaves room for thought and refinement..

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