|
YOU are better than YOU think. Show
yourself how:
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
|
Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study.
Learn to read notes and textbooks like
a lawyer, so that no nuance, no subtlety and no clause escapes your
attention. |
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
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;
|
// _ _ \\
/\ /\
<| (o) (o) |>
| |
| |
\
/
\ = /
|
Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
-/[]\-
||
_ / \
||||||||||||||||||||||||||||
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.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
| |
Note: This work is self-published. On my return home
in June 2007, paperback copies will again be for sale.
Foreword: This work Pattern
Based Reason surveys rule and pattern based thought in daily life, society,
science and technology. There are simple ideas which should be more widely
known.
Volume 1A, Pattern Based Reason, describes
logic, critical thinking and problem solving skills for many arts and
disciplines. Read it to learn about the benefits, origins,
limits and risks of rule- and pattern-based activities and
explanations; to develop a critical command and understanding of science
and technology before defending or attacking any part; to learn how patterns
are suspected or recognized, and learn what patterns can be tested before
jumping to conclusions or alternatives. . This work provides base for work and
studies, decision-making, in many arts and disciplines at work and
school.
Chapter 1 Introduction:
To reason often means to persuade someone of the need for an idea or action.
That someone could be yourself. In the latter case, reasoning may mean following
a line or pattern of thought to arrive at a conclusion, action or decision.
Chapter 2 Communication
: No area of knowledge is properly mastered until it can be
readily explained to others. Each subject needs paths (or curricula) passing
through easily described and easily repeated ideas and skills. Each such path
permits those who have traveled along it to tell others what to expect and
hopefully why. The existence of such paths may show that an area is
well-understood.
Concludes with Inductive Principles for Instruction
Chapter 3 Elements of Reason
: Chapters four to eight describe the basic elements of rule- and
pattern-based thought and hint at their benefits and limitations. In particular,
the next three chapters, Implication
Rules, Deception and Chains
of Reason describe basic ideas in reason and logic which everyone should
master.
Chapter 4 Implication Rules:
Are you a careful thinker? Can you understand exactly the meaning
of a rule or pattern? Instructions for building or creating provide rules and
patterns which say and suggest that when this is done, that should happen.
Every cook and dressmaker knows the importance of following instructions
carefully. Instructions and suggestions which are not repeatable and results
which are not reproducible are not of interest to a cook or dressmaker.
[ Chapter
Entrance ] [First Puzzle] [Second
Puzzle ] [ One-
Versus Two-Way ] [ Talking About Logic
] [ Implications versus or as Suggestions
] [ Implications Versus Suggestions ]
[ Repeatable & Reproducible ] [
Limits and Benefits ] [ Accidental
Rules ] [ Steps for Better Reason ]
Chapter 5 Deception:
People try to persuade us in many ways. We need to recognize the fair and unfair
ways, or the sensible and nonsensical ways. In persuading ourselves and others,
we need to recognize and appreciate or reward careful logic. Efforts to persuade
and lead us are met in advertising, public relations, political campaigns,
religion, law, business, mathematics courses (yes), and even your family
Chapter 6 Chains of Reason:
This chapter shows how reliable rules and patterns can be directly
employed repeatedly, one at a time, or one after another, to get conclusions or
further reliable rules and patterns. The question of what rules are reliable is
considered in the following chapters.
[Chapter
Entrance] [From a Single Rule] [Linking
and Chaining - Two Rules] [Putting
Several Rules Together] [Deductive,
Inductive and Empirical Reason]
Chapter 7 Longer Chains:
This chapter explains one version of
inductive reason: the recursive or repetitive approach to putting one-way
implication rules together, one after another. This chapter ends with a
description of the principle of mathematical induction – another method for
obtaining conclusions used only in mathematical arguments or computations. There
is more to mathematics than just doing arithmetic.
Chapter 8. A Language Change
(or two). The foregoing development of logic coins the terms one and two-way
implication. The latter can be identified with conditional and
biconditional statements. That being said, if we write B IF A for
the implication A implies B or IF A THEN B, the phrase B IF AND ONLY IF A means
there is no other situation or condition C with B IF C (unless C implies A
as well).
Speculation: Mathematics and logic education, and its
choice of words in North America, or outside of the UK, was influence by
Europeans, expert in subject matter, without a poetic command of English, that
needed to make technical concepts more accessible to students and teachers.
Chapter 9 The Next Chapters:
The problem of identifying reliable implication rules and reliable
information is described but not solved, except for the description of empirical
methods of coping in science and technology. This identification problem touches
many subjects. Students of critical thinking, persuasion, philosophy,
mathematics, science and technology should find its discussion in these chapters
helpful.
Chapter 10 Responsibility:
In this chapter, we give a short story: a conflict between the owners of
a cat and a dog about who or what is responsible for an accident. The murky
situation leads into a discussion of cause and effect, and then responsibility
versus freedom (the limits of freedom) and the absence of liability. Finally,
first principles or patterns for the assignment of responsibility and liability
are stated or suggested last.
[Chapter
Entrance - Felix versus Suzy] [Limits to Freedom ]
[Where does Responsibility
begin or end? who is to blame? Principle to Consider? ]
Chapter 11 Accidental Patterns:
What do we mean, when we say you have caused something to occur? In life
we may see a pattern that whenever a first situation occurs, so does a second.
The pattern could hold true accidentally. There may be no relationship between
the two situations or events. Alternatively, there might be some relationship.
We need in a sense to measure this relationship. We need to measure how much one
event forces, pushes or contributes to the occurrence of another event. This
measurement signals to what extent the first event is a cause or is the
cause of the second. Observation by itself is suggestive but not conclusive.
Examples to support this view follow.
Chapter 12 Knowledge Islands:
Whenever the building we are exploring has sections closed off or
unreachable, we can ignore all maps of those sections. Making a map of the
unreachable sections is not possible, except by guessing. Guessing is
suggestive, yet not reliable.
[ Chapter Entrance
] [12. Two Analogies or Metaphor for the division and
organization of know-how and even know-why ]
Chapter 13 Euclidean Logic:
Knowledge in one section may touch or not touch that of another. All
depends on what implication rules are known. Our minds can explore each section
of knowledge as we meet it. ... In this chapter, the Euclidean model for
organizing reason and knowledge is discussed. In this Euclidean model for reason
and knowledge, each area or segment of knowledge is derived via chains of reason
from a few secure first principles or assumptions about data and implication
rules. This Euclidean model is an ideal which we would like to attain. Can we?
Chapter 14 Views of Math:
This chapter provides several perspectives on mathematics.
[ Chapter
Entrance - Set Theory ] [ Before
& After Set Theory in Pure Mathematics ] [ Euclidean
Model for Physics ] [ Applied
Maths and Electricity Apart from Sets ] [ Decimals
Absent From Pure Mathematics ] [Modern
Mathematics Education ]
Some are slightly at odds. Some are slightly technical. The next chapter Objective
Processes returns to some simpler material.
Volume 2, Chapter 19, Functions and Sets, and Volume 1B,
Mathematics Curriculum Notes, and the rest of this site, material written
later, give further views on mathematics education, what was, what is and what
could be]
Chapter 15: Objectivity:
Recipes and rule-based processes, when carefully done, give results independent
of who obtains them. In this situation, the results cease to be subjective —
that is dependent on the person getting them – and they depend only on the
context. In this situation, the results are said to be objective. ... The main
advantage of objective (rule-based) reason and processes is as follows. Once we
have agreed upon the rules and recipes and on the evidence or ingredients to
use, the results obtained are independent of who or what obtains them.
[Chapter
Entrance] [ The search for Repeatable and Reproducible
Results
Chapter 16 Origin of Patterns:
A rule, law or agreement may say that when one event happens, another event
should also happen or may also happen. Most physical and legal theories, if not
all, use rules which are approximately correct. The rules are like all human
discoveries and creations; some are more reliable than others. The formulation
of laws and rules and agreements by people leads to the chance of error and
incompleteness. Even with uncertainty, once rules or laws or agreements have
been stated, we can use them tentatively, to reach conclusions or judgments.
Locating the weakest links in our reasoning gives us a chance to strengthen or
replace them.
[ Chapter
Entrance - Origin of Patterns ] [ Private
Agreements ] [ Public Laws
] [ Physical Laws ]
[ Accidental Patterns ]
[ Reliable(?) Patterns ]
[ Scientific Method ] [
Reaction to Failed Tests
] [ Chaos ] [ Statistical
Inference ] [ End Notes ]
Chapter 17 Discovery
of Objective Ways: Knowledge of what others have done or tried
to do may help and guide our actions. Without previous know-how and knowledge,
we need to improvise and look for patterns, rules and recipes that work. This is
where the search for objective reason, or simple rules to follow, becomes
subjective. Each may have a different idea of where to look. This is because
each person has a different background and varied preferences. The road to
objectivity is in part subjective and creative.
[Discovery of
Objective Ways - Yours Objectively in Creative and Subjective Manners] [17.
Discovery Process - Trial and Error Discovery)]
Chapter 18
Sense+Knowledge: Consciousness and thought appears in infancy or
childhood. There they may be initially taken for granted or not explicitly
noticed. Only later are they questioned, if they are questioned at all.
Vagueness of memory may hide the days when consciousness and thought began. A
few speculative remarks follow.
More About Logic:
The last five chapters 20 to 24 give a technical view of logic and also enter
the discussion of direct and indirect methods for reason. The latter discussion
is continued in online postscripts - material not in the printed or printable
version of Volume 1A.
Chapter 20, Shorthand or Pronouns in
Logic, introduces the use of letters A and B, and possibly others
first to represent situations that can occur or not, and second to represent
phrases or statements that may be true or false (or neither). Talking about
pronouns, the pronoun metaphor, and talking about shorthand, represent one or
two ways to introduce the the shorthand role of letters in logic and more
generally in mathematics.
The online Volume 2, Three Skills For Algebra, in Chapters 8
and 9, and in the online postscript, What is a Variable, go further in
Euclidating or clarifying the shorthand role of letters and symbols in logic
and algebra, or symbol based, shorthand paths, for arriving at conclusions
with implication rules and formulas (or numbers)
Chapter 21 coins or introduces Occurrence
Tables. for three phrases A AND
B; A OR B; and NOT A; for one
way implications B IF A, and for two-way
implications B IF and ONLY IF A. The last section of Chapter 21 defines Converses
to One Way Implications and so digresses from the earlier content of the
chapter.
The occurrence (or obedience) tables invented and introduced
in Chapter 21, Occurrence Tables,
identify those situations in which implication rules are obeyed, disobeyed or
not disobeyed. The latter notions are intended to simplify or justify the
explanation of truth tables for the implication B IF A, or if you prefer, the
implication, IF A THEN B.
Chapter 22, The Contrapositive
shows the equivalence of an implication rule with its contrapositive
formulation - meaning B IF A holds when and only when NOT A IF NOT B
holds. The analysis is based on the three notions of a rule being (i)
obeyed, (ii) disobeyed or v(iii) not disobeyed. An implication rule B IF A
or IF A THEN B is Vacuously True
when and only when it never applies - that is when situation A never occurs.
In the latter case B or NOT B implies NOT A is a tautology.
Chapter 24, Direct
and Indirect Reason describes and explains direct and indirect methods for
reaching or proving conclusions. Among the indirect methods, this chapter
describes in particular, how an implication rule can be shown to always hold by
(a) showing its contrapositive form always hold, or by (b) looking for
absurdities that would occur if the implication rule did not hold. The second
method (b) is more indirect than the first method (a).
Online Postscripts: While we may not know that a
theory is consistent, we use the requirement for consistency as part of
the reasoning process without loss of generality or harm we
hope. See Proof by
Absurdity alias proof by contradiction and see How
the demand for consistency supports the law of the excluded middle
| |
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.
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
+Three Remarks
+History
Lost or Missing
There 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.
.
|