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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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Proof by Absurdity
alias proof by contradiction
Example One: A detective in solving a crime may have a
suspect. Then he may found the suspect has an alibi which directly or
indirectly implies she did not committed the crime. So the alibi and suspicion
are inconsistent - that is incompatible. The detective may drop the suspicion
or challenge the alibi. Lawyers for the prosecution and defense may erect
competing chains of reason, and leave it to a jury or judge to decide which
one, if any, appears to be true. A conclusion may follow or not.
In telling a story or developing a theory, we may look at the consequences of
our assumptions - the situations we tend to assume as holding or being
true. If a chain of reason implies that a situation C occurs and does not
occur, then the story or theory is inconsistent - becomes absurd. For the sake
of consistency, the story or theory needs to be revised or abandoned.
That being said in developing a theory of how matters work, we hope that the
theory will be logically consistent. That we hope that there will be no
contradictions as a consequence of our assumptions. The consistency of a
theory is hard to prove or test, and impossible in a mathematical theory large
enough to include counting with whole numbers 1, 2, 3, 4, ...
In telling a story or developing a theory, there is an inconsistency if a
situation A and its negation Not A both occur. While we are developing a
theory from assumptions, we cannot be certain that an inconsistency A
and Not A will not occur. However, the assuming the law of excluded
middle in the development of a theory or is equivalent to the statement that the
situation A and Not A does not happen. It equivalent to the
assumption that the theory under development is consistent. That be said, when
we are developing a theory from logic and underlying assumptions, we may not be
able to prove that the theory is consistent. If the theory is consistent, the
law of excluded middle holds and so we may used in our logical development of
theory without loss of consistency. But if the theory under development is
inconsistent, assuming or using the law of excluded middle in its
development may lead to the discovery of the inconsistency, sooner rather than
later, if at all. Assuming the law of excluded middle, simply adds another
inconsistency. .
- Example Two: Assume any infinite decimal expansion locates a point
or distance on a real number line. Assume further that each ratio of two
whole numbers can be expressed a ratio of two whole numbers with no
common divisors? The Pythagorean theorem then suggests in an isosceles
right triangle, the ratio of the hypotenuse to each of the others sides, the
legs, by length given by the square root of 2. Is that square root
equal to a rational number? The suspicion or assumption that YES, the square
root of 2 equals a rational number implies an inconsistency.
Namely, that in any ratio or fraction that represents the square root of 2,
the denominator and numerator will both be multiples of 2. So the square
root of two cannot be rational.
The Pythagoreans in finding the inconsistency in example 2 had a problem.
They assume lined segments in the plane represented numbers and they
assumed all such lengths were rational multiples of each other. When these
assumptions or their consequences clashed, reconciliation was not obvious.
Their view of numbers collapsed and a replacement was not available. That was a
serious problem for the Pythagorean school in their theory of knowledge was
based on and assumed rational numbers and only rational numbers.
Today, however, we have an advantage or two. One advantage is a our
assumption that infinite decimals expansions represent rational and irrational
numbers. Physically, if you imagine ruler with a unit length and its
division into tenths, hundredths, thousandths and so on, then you can count the
maximum number of units, tenths, hundredths, thousandths and that may fit
in a line segment. The result is a sequence of numbers which provide a
better and better approximation to the line segment length. You can
further imagine that sequence of approximations given by two, three, four, five
and more decimal places in a decimal expansion locate the end of a line
segment. The decimal representation of numbers does not depend on nor
require the "numbers" to be rational.
Aside: The number 1.0000 represent a single unit of
length. It also represents the limit of the sequence 0.9, 0.99, 0.999, 0.9999
and so on. The sequence of decimal approximations 0.99999 (9 recurring) is
shorthand for a sequence of lengths L(j) =1 - 10-j < 1
where the lengths are increasing, and where the differences d(j) = 1 -
L(j) = 10-j is getting smaller and smaller as j increases. So
the sequence approaches 1. Because the difference tends to zero, the
number 1 has several decimal expansions
- 1, 1.0, 1.00 (0 recurring finitely many times)
- 1.000 (0 repeating indefinitely)
- 0.99999 (9 recurring)
where the last one represents 1 as the limit of a
sequence of approximations 0.9, 0.99, 0.999, 0.9999, 0.99999, ...
.
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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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