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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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Previous: Chapter 12, part I, Skills
and knowledge divide
Two analogies and Ignorable Rooms
Duplicated Paragraph
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.
Islands Without Roads Between
Implications are like streets or roads. They may be traveled one-way or both
ways. Streets (or implications) may lead nowhere. Others may lead to interesting
and sometimes unexpected places.
Each road may touch several others. Each of these others may touch several
more. But by foot or car, from one road, there is no guarantee that all roads
can be reached. Moreover, when some one-way roads are present, poor planning may
imply no return route for every possible starting point.
Maps make the exploration of any road system easy. All we have to do is read
the map. Without a map, we have to explore the neighborhood in which we live,
and hope we can find a path back. One-way streets are a danger here, unless
another path back is available. Without a good map, we cannot say in advance,
when we explore the streets, if we will get to an interesting or boring
destination. To find out what is interesting, our only choice is to explore or
to ask whether any one has made a map. We would like to learn from the
experience of others, perhaps.
By road, not all destinations are accessible or reachable. We may for example
have roads on several islands with no boats, ferries, planes, bridges or ships
to take us between them. Without boats, ferries, planes, bridges, or a very
low-tide, we have no route or connection between one island and the next.
Without these extra routes, the roads (or implications) of one island are not
linked to the roads of another. The streets on even a single island need not all
be connected to each other. For example, imagine on one island that a
mischievous or artless road planner has provided one-way roads all leading from
one end of the island to the other. On such a road system, a return to the
starting point is not possible. We can imagine another island in which the
planner, mischievous or not, has placed a mixture of one- and two-way roads.
From some starting points you can leave but not return. From some parts or
destinations, you cannot leave. Between other starting points and destinations,
you can go back and forth. And after going back and forth several times, you may
forget which place was the destination or the starting point.
All the situations just described with one- and two-way streets can happen
similarly in logic with one- and two-way implication rules. In other words,
knowledge is linked by one- and two-way implication roads, spread over several
islands. The map of this area is not complete. As we explore and forget, roads
and routes new to us or our neighbors are uncovered or rediscovered.
Rooms Without Doors Between
Implication rules are also like doors or gates between sections of a building
or estate. (Implication rules like doors join the rooms of a large palace,
castle, house or prison. ) Some allow two-way passage. Others permit only
one-way passage. All this can be a deliberate design or it could be due to a
poor design.
When we restrict our paths to two-way doors, we can always retrace our steps
exactly and get back to where we started. But one-way doors are different. To
get back after going through a one-way door, we need to find another route back
through some other door or doors. Otherwise, we are shut out of our starting
room. That is, we suppose a one-way door can only be opened from one side, and
that after use it snaps shut. When we go through a one-way door, we can get back
to our initial side of the door only if there is a route back. But by passing
through one-way doors, we may find ourselves locked out of the initial room we
were in. We may further find ourselves locked in another room or section of the
building.
Ignored Rooms
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.
Next: Chapter 13, Euclidean
Model for logic and 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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