Three Skills
For
Algebra
Volume 2
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Chapters and Appendices
Book Entrance
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
Would you like to show yourself or others how to be algebra
power users?
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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Chapter 2. Implication Rules
Previous Section: Second
Logic Puzzle
One Versus Two Way Implications
The two puzzles give examples of implication rules. The first puzzle gives a
one-way implication rule, while the second gives a two-way implication rule. The
following words should further help you to see the difference between one- and
two-way implication rules. Seeing this difference may help you understand better
the answers to the above questions. They may also help you answer the five
questions again using the two-way implication rule.
- A one-way implication rule says that when a first situation occurs, so
must a second. It does not say that when the second occurs so must the
first. The second situation may occur without the first.
- A two-way implication rule says that
- when a first situation occurs, so must a second, and
- when the second situation occurs, so must the first.
A two-way rule says that when each situation occurs, so must the other.
Therefore if the two-way rule is to be obeyed, when one situation does not
occur, neither can the other.
Seeing or recognizing the difference between one- and two-way implication
rules makes you a more careful thinker.
One- and two-way rules, recognized or not, are what we use to reach
conclusions or make judgments. One and two-way rules can be used to suggest or
persuade us of what needs to be done or avoided.
Talking About Logic
As suggested above, you can give people the above rules or similar ones
before asking five questions. Before you do this, you should wait for a
receptive mood, especially if you are not in a classroom. For the sake of an
argument and some fun, you may ask after getting an answer, are you sure?
Or you may pretend a correct answer is wrong. Of course, you will admit this
ruse later, and explain why you really agree (or disagree) with the answers. The
aim is to see how people reason and more importantly to strengthen their
thinking skills.
Logic is supposed to give rules for thought, that is rules for arriving at
conclusions. Yet the only rule needed in the reasoning shown above is as
follows: Read exactly what is written and don't assume nor imagine too
much.
Implications Versus Suggestions
In a dictionary you may find that the verb to imply also means to
suggest. Words which say when one event occurs so does or will a second are
called suggestions or implications. Suggestions and implications can be true.
True here means obeyed or at least not disobeyed. Suggestions and implications
can be false. False here means disobeyed. In our reasoning process, we want to
say with certainty that when this occurs so will that. In
practice, we may have to be content with saying when this occurs, so may that.
Knowing which of our rules are sure or which are uncertain identifies the
weaknesses in our reasoning processes. The implication rules that are never
disobeyed provide the most certain suggestions in reason.
In logic, when we speak of implication rules, we speak of rules
which we hope are never disobeyed. Rules which might be disobeyed are called
conjectures, suggestions or guesses. Evidence (persuasion) should be required
to convince us that a conjecture or suggestion is a reliable implication. We
can imagine or suggest more than we can prove. Caution is advised on hearing a
rule. Before applying a rule, you need to know how certain it is. Is it a
reliable implication or merely an uncertain suggestion?
Implications Versus Suggestions
In a dictionary you may find that the verb to imply also means to
suggest. Words which say when one event occurs so does or will a second
are called suggestions or implications. Suggestions and implications can be
true. True here means obeyed or at least not disobeyed. Suggestions and
implications can be false. False here means disobeyed. In our reasoning process,
we want to say with certainty that when this occurs so will that.
In practice, we may have to be content with saying when this occurs, so
may that. Knowing which of our rules are sure or which are uncertain
identifies the weaknesses in our reasoning processes. The implication rules that
are never disobeyed provide the most certain suggestions in reason.
In logic, when we speak of implication rules, we speak of rules which
we hope are never disobeyed. Rules which might be disobeyed are called
conjectures, suggestions or guesses. Evidence (persuasion) should be required to
convince us that a conjecture or suggestion is a reliable implication. We can
imagine or suggest more than we can prove. Caution is advised on hearing a rule.
Before applying a rule, you need to know how certain it is. Is it a reliable
implication or merely an uncertain suggestion?
Chapter Sections: [ Up ] [ First Logic Puzzle ] [ Second Logic Puzzle ] [ One-versus Two-Way Implications ] [ Implications versus Suggestions ]
Next Section: Implications
versus Suggestions
Next Chapter: Chains of Reason - Euclidean Model for
Reason
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1996
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Reason 1995
1B. Math
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3 .Why.Slopes.&.More.Math.1995
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