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YOU are better than YOU think. Show
yourself how:
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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. |
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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;
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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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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.
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Chapter 24
Direct and Indirect Reason
To prove a statement is to show that it must hold. To refute a statement is
to show that it cannot hold. In between proof and refutation1
lies uncertainty or We don`t know.
PS. The concept of proof comes perversely from the idea of
testing. The first question can we test if a statement is false? If a
consequence of a statement or assertion fails to be true, then the statements
or assertion is false or refuted.
1 Proof and Refutation Methods
Proof and Refutation was the title of a work by the
philosopher Latakos, Cambridge University Press, 1976. More recent editions
are available.
In a given situation:
- what evidence or proof do we need to say a rule if A then B is
never disobeyed?
- what evidence or proof do we need to say the rule is false?
False here means sometimes disobeyed. The proof or disproof techniques described
next are usually employed in a context where some implication rules are assumed
to be never disobeyed. The first question (1) then becomes what further
implications rules are never disobeyed. The techniques described next provide
answers in some but not all circumstances.
Refutation: Proving Falseness
To show that a rule if A then B is false is simple, see if we can find
(or show there exists or must be) a situation in which A occurs and B does
not. Then the implication rule A implies B is false. In other words, it
is not always obeyed or it is sometimes disobeyed.
A Direct Approach: Applying Implication Rules
PS: A proof in mathematics consists of a chain of reason, direct or
indirect, but carefully done, which implies that a statement or
assertion is a consequence of an assumption or set of assumptions. Then if the
assumptions and chains of reason are not false, the statement or assertion
must hold (we hope).
You may show that the rule If A then B is always true by showing that
when situation A occurs so must B. Such a proof could employ a
chain of reason (deduction) using trusted implication rules: rules that are not
disobeyed in the situation at hand. Such a proof shows that when A occurs,
so must B. This says and shows that the situation A and NOT B never
occurs. See the chapter Chains of Reason for an illustration of this
approach.
The Contrapositive Method (Indirect)
Alternatively we can show the (contrapositive) rule If NOT B then NOT A is
always true by showing that when situation NOT B occurs then so must NOT
A. Such a proof again shows the situation A and not B never occurs.
That is what we need. (See the chapter Chapter
22, The Contrapositive, or see Chapter
4, Implication Rules
The Contradiction Method (Indirect)
The aim here is to find a situation C with the following properties:
- the situation C does not occur (is obviously false) and
- (A and NOT B) implies situation C.
These properties tell us that the situation (A and NOT B) cannot occur -
why? So that the implication rule A implies B is never disobeyed. Next
are a few words to explain why:
The contrapositive form of the implication rule if (A and NOT B) then C is
what we need. It says if NOT C then NOT (A and NOT B). But the situation NOT
C, by chance or discovery, occurs. Thus the situation (A and not B) can
never occur. And that is what we mean when we say that the rule A implies B always
holds.
Remark (a-looking we will go). This proof by contradiction method
can be applied without knowing in advance what the situation C will be.
We search for it. That is, for the sake of finding such a situation C, we
assume the situation A and not B occurs. After this we follow whatever
chains of reason we can to reach a conclusion that an absurd (or obviously
false) situation C occurs.
Next: The following three postscripts,(material online only and not as
yet in the printed version,
reflect or discuss further methods of direct and indirect reason.
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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.
.
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