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Implication Rules
Previous: Chapter Introduction
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.
| Teachers: Read the following before, besides or after
the two logic puzzles in this chapter, as you like.
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.
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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 |
- When the rule is obeyed, what can you say happens for sure when Aunt Jane
visits her nephew's home? This is easy. [Answer]
- When the rule is not disobeyed, what can you say happens for sure about
Aunt Jane when Tom is out playing? Be careful. [Answer]
- 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]
- What must happen for the given rule to be disobeyed? This is another
easy question. [Answer]
- 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.
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Some Vocabulary
The above rule is called a 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.
Chapter Sections: [ Up ] [ First Logic Puzzle ] [ Second Logic Puzzle ] [ One-versus Two-Way Implications ] [ Implications versus Suggestions ]
Next Section: Second Logic
Puzzle
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Three Skills
For
Algebra
Volume 2
Printed in Canada
ISBN 0-9697564-2-9
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Read slowly, this work may enrich your
skills & knowledge. Take the risk.
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Chapters and Appendices
First Logic Puzzle Second Logic Puzzle One-versus Two-Way Implications Implications versus Suggestions
Foreword 1. Introduction 2. Implication Rules [4] 3. Chains of Reason [3] 4. Induction Mathematical 4. Romeo and Juliet 6 Old Language 5 Knowledge Islands [2] 7 Arith Skill Check [4 X 2] Arith Webvideos 7. The Next Chapters 8 The Three Skills 8 VNR-Concise-Encyclopedia PS. What is a Variable [8] 9. Algebra Talk [7] 10 Two More Skills[5] 11 Why Shorthand 12 Shorthand Usage [10] 13 What's Next PS: The 4-th Skill For Algebra 14 Compound Interest [6] 15 Linear Equations [5] 16 Painless Proofs 17 Pythagoras PS I. Distributive Law PS II. Polynomials 18 Rules of Algebra [20] 19 Functions & Sets 20 Degrees & Radians 21 What's Next 22. Arith & Geometric Sums [2] 23 Summation Notation 24 Your Money [3] 25 Induction & Recursion [4] 26 What's Next 27 Pronouns in Logic 28 Occurrence Tables 29 Contrapositive 30 Truth Tables 31 Indirect Reason Pathways for Learning
What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent
or Independent
Variable, a Matter of Choice
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For
Senior
High School & Calculus Students
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Words to clearly
introduce algebra and variables
have been missing in course design. For people who cannot do
algebra,
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the missing words may
explain or ease their difficulties. Volume 2 ,Three
Skills for Algebra, in Chapters
8 to 14 & 18 etc, puts words before symbols to
providing the missing words in a way that enrich the
comprehension of all. Those words form the middle part of a algebra
(and logic) lessons aimed at helping or improving all
of high school mathematics and also calculus course
design & delivery.
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For Avid Readers in School & Out -
Online Books
1. Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
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Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3.Why
Slopes & More.Math
1995
Tour their forewords.
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Mostly
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Fraction
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Study Slopes.
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