Chapter 5.
Islands and Divisions of Knowledge
Previous Chapter: Longer Chains of Reason
(and Mathematical Induction)
Chapter sections: [Chapter Intro] [ Islands Without Roads Between ] [ Rooms Without Doors Between ]
Recall the difference between one- and two-way implication rules:
A one-way implication rule says that when a first situation occurs, so must
a second. It does not say that when the second occur, 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 implication 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.
The examples in the chapter Chains of Reason involved
one-way implication rules. They showed that one-way implication rules can
sometimes be put together to get further implication rules. You may remember we
had one implication rule about Charles that was not used to get any conclusion.
One and two-way implications can also be joined. The ways in which this can
be done are described below by analogies with one- and two-way streets, and one-
and two-way doors. These analogies indirectly describe how rule-based knowledge
is put together. In particular, rule-based knowledge is divided into separate
segments. Each segment cannot be reached from another by chains of reason. The
two analogies describing this situation further are presented next.
Two analogies (Chapter Sections)
follow.
Chapter sections: [Chapter Intro] [ Islands Without Roads Between ] [ Rooms Without Doors Between ]
Next Chapter: 6 Condition and Biconditional
Statements
| |
Three Skills
For
Algebra
understanding & explaining
Reason and Math
Volume 2
Printed in Canada
ISBN 0-9697564-2-9
|
[ Back ] [ Home ] [ Next ]
Chapters and Appendices
Home
Postscript: The 4-th Skill For Algebra
Foreword
1. Introduction
2. Implication Rules
3. Chains of Reason
4. Romeo and Juliet
4. Induction Mathematical
5 Knowledge Islands
6 Old Language
7 Arith Skill Check
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable
9. Algebra Talk
10 Two More Skills
11 Why Shorthand
12 Shorthand Usage
13 What's Next
14 Compound Interest
15 Linear Equations
PS I. Distributive Law
PS II. Polynomials
16 Painless Proofs
17 Pythagoras
18 Rules of Algebra
19 Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums
23 Summation Notation
24 Your Money
25 Induction & Recursion
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
A. Advice For Learning
Words Before Symbols:
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
|