Chapter 12, Islands and Divisions of Knowledge
Recall the difference between one- and two-way implication rules:
A one-way implication rule says that when a first situation occurs, so
must a second. It does not say that when the second occur, so must the
first. (The second situation may occur without the first).
A two-way implication rule says that:
- when a first situation occurs, so must a second, and
- when the second situation occurs, so must the first.
A two-way implication says that when each situation occurs, so must the
other. (Therefore if the two-way rule is to be obeyed, when one
situation does not occur, neither can the other.)
The examples in the chapter Chains of Reason involved one-way
implication rules. They showed that one-way implication rules can
sometimes be put together to get further implication rules. You may
remember we had one implication rule about Charles that was not used to
get any conclusion.
Two analogies (Isolated Islands or Ignorable Rooms)
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.
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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
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
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
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
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
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
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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