Chapter3, Chains of Reason
Introduction
This chapter shows how reliable rules and patterns can be directly employed
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.
Rules used to get or suggest conclusions are called implications. Just as
there are methods for adding and multiplying numbers carefully, there are also
methods for using implication rules by themselves to get conclusions. There are
also methods for linking, threading and chaining implication rules together to
get more implication rules. This chapter uses examples to explain two basic
ideas:
- how to directly use a single implication rule to get conclusions, and
- how to link, chain or thread implication rules together to obtain or
derive more rules and more conclusions.
The examples are not important (and are perhaps ridiculous) but they
illustrate some rule-based methods in reason. Examples which involved real-life
situations might distract from mastering these methods. That is, in real-life
situations, each of us may have opinions or prejudices about what should occur.
That could spoil an explanation of the use and linkage of implication rules.
There is a need for neutral examples to illustrate the use of implication rules
one at a time or one after another.
Arithmetic, algebra and geometry give many neutral examples for this. The
examples below involve no mathematics. Bon Appetite.
Conclusions From a Single Rule
Direct and Indirect Usage
Pretend the following implication rule is never disobeyed.
Each time Suzy the cat is on the ground and Suzy sees a dog, Suzy climbs a
tree and stays in it for at least five minutes.
Direct Usage
What can we say for sure when Suzy the cat sees a dog? One possible answer is
that Suzy the cat stays in a tree for at least five minutes. Another possible
answer is that Suzy the cat climbs a tree. A more complete answer is that Suzy
the cat climbs a tree and stays there for at least five minutes. Each of these
answers or conclusions is correct. The last conclusion or result is fuller and
more complete than the others. It gives more information. Which answer or
conclusion is wanted here depends on who is interested in what. When many
conclusions are possible, we state only those conclusions of interest to us. We
do not have to state the most complete conclusion. The choice is ours.
Indirect Usage
What can you say for sure if Suzy the cat has not climbed nor stayed in any
tree for at least five minutes? To check your answer, you might have to remember
or revisit the questions in the chapter Implication Rules. But you should
do this after you have read the following words.
Linking and Chaining Two Rules Together
The examples below and in the next page show how to chain, link or connect
implication rules to get information and conclusions. The examples in themselves
are not important. The information in them is silly. But these examples just
show how to put implication rules together. So read on, with patience.
- Every time Suzy the cat climbs a tree, it gets stuck in the tree
- Every time Fred the dog visits the park in which Suzy the cat lives, Suzy
climbs a tree.
By linking or chaining these implication rules, we can make three
conclusions:
- Whenever Fred the dog visits the park where Suzy the cat lives, Suzy
climbs a tree.
- Whenever Fred the dog visits the park where Suzy the cat lives, Suzy gets
stuck in a tree
- Whenever Fred the dog visits the park where Suzy the cat lives, Suzy
climbs a tree and gets stuck.
Each of these conclusions is correct. Each conclusion gives a new implication
rule which we could use in our reasoning process. The third implication rule is
the most informative. It contains the most information. When we view each
correct conclusion as a possible destination for our reasoning process, we may
sometimes select our destination.
Putting Several Rules Togethe
We can chain or link not only two but also several implication rules together.
This sometimes yields useful, new information. As an exercise, we ask the
question: What happens whenever Fred the dog visits the one-tree park?
Several answers are possible. Some have more details than others. All are
correct. To answer the question, assume or pretend the next five implication
rules are never disobeyed. Further, assume that Suzy the cat lives in the
one-tree park.
- When Suzy the cat climbs the tree in the one-tree park, Suzy gets stuck
in the tree.
- Each time Fred the dog visits the one-tree park, Suzy the cat climbs
the tree.
- Every time Charles the human visits the park, Charles sits on a bench
for one hour.
- Whenever a cat climbs the tree in the one-tree park, the five birds
living in the tree fly around in the park.
- Each time birds fly around in the park, sensible worms go
underground.
All the information has been stated. We start our reasoning process. That is,
we will answer the question: What happens whenever Fred the dog visits the
one-tree park?
To answer the question, suppose or assume Fred the dog visits the park. Then
from the implication rule (2), we see that Suzy the cat climbs a tree. Next,
from the implication rule (1) we see that Suzy the cat gets stuck and from the
implication (4) we see that birds fly around the park. Finally from the
implication (5), we note sensible worms go underground.
We could list all that occurs when Fred the dog visits the park. Or, we could
state only those results of Fred's visit to the park which are of most interest
to us. The choice is ours. For instance, one of our possible conclusions
follows:
| If Fred the dog visits the park then sensible worms
go underground. |
This conclusion is not of interest unless you are a fisherman (or woman)
looking for worms, sensible or not, for use as bait. The conclusion selected and
stated here hides the reasoning process. That is, it hides the chain of
implications leading to it. Our last conclusion does not mention the
intermediate events where a cat climbs a tree and birds fly around the park.
The long path by which we get conclusions shows that implication or
rule-based thinking can lead to surprising results. These surprising results are
true if the initial implications are also true.
In the long path by which we got the conclusions, the information in the
third implication (3) about Charles the human is not used. The conclusion we
reached is independent of implication (3). In fact, without further information,
I see no way of linking the rule about Charles with the other rules. The third
rule is extra information. It can be ignored.
In answering questions, we often have extra information. Indeed, you can
imagine the five rules given above are stated in random positions among a list
of twenty, or hundred and twenty rules. An answer to the question
What happens when Fred the dog visits the one-tree park?
now depends on finding the rules in the list which can be used. This is a
game of hide and seek. So we have to be selective, observant or fussy in
deciding or seeing what information leads to our conclusions.
The scenery or route by which a conclusion is reached may contain as much
useful information as the conclusion itself. A conclusion may contain a fraction
of the information we could have stated or written. Being aware of the route or
proof by which a conclusion is attained will sometimes suggest how more
conclusions can be reached. This awareness is often more important that any
conclusion we state because it allows us to state more conclusions, as needed.
Mathematics students take note. Remembering the route taken in
solving a problem is worth more to the development of skills than
remembering the solution.
Deductive, Inductive or Empirical Reason
Deductive reason uses or chains together supposedly (or preferably)
never-disobeyed implication rules to suggest, to make or to reach conclusions.
See the examples above. The implication rules in question may come from
assumptions. The assumptions may be tentative.
The phrase inductive reason has one role in mathematics and another
outside of mathematics. To induce (or induct) literally means to draw or
extract. When you see a rule or pattern that no one has suggested, you are
extracting or drawing that pattern from your observations. This process of
recognizing rules and patterns that may hold, accidentally or not, is called
inductive reasoning. Inductive reason outside of mathematics refers to the
identification and recognition of rules and patterns from data and observations.
Here rules and patterns may hold accidentally.
Reason which relies on a single or several, experience-found, rules and
patterns to arrive at conclusions is called empirical. The underlying problem of
inductive, empirical reason is to extract (infer, draw, induct or identify) from
experience, in particular, data and observations, rules and patterns not
satisfied merely by accident and which appear to be reliable. Self-deception
needs to be avoided here.
Inductive reason inside mathematics refers to another process, namely, the
extraction or drawing of conclusions from ladder-like chains of reason. See the
next chapter for a more precise image or explanation. The rules or
assumptions here are usually so certain, that we deliberately ignore the
experience-based origins of mathematical reason.
Criteria for the recognition of reliable, non-accidental rules and patterns
are described later in chapter 16, Origin
of Rules and Patterns .
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For home-tutoring or -schooling, or for schools or colleges
with course content control: Secondary
Mathematics for Ages 11+, A Practical Approach.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
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Parent Center: Help your child or teen
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Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
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Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
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Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
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Is your child able to add, subtract and multiply amounts
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work with maps and plans, and measure length, weight-mass and
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Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
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Algebra
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Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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