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YOU are better than YOU think. Show yourself how:
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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
Do not leave here without it - Logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
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Caution: Site advice is approximately
correct, for some circumstances, not all. Site How-TOs are logically
developed, but not tried and tested. That leaves room for thought and
refinement.. |
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After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
linear2007 Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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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Explore collaborative whiteboards from
groupboard, twiddla or
scriblink.
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Previous: Chapter 21, Occurrence
Tables for Implications
- PS: Use of the contrapositive form of an implication B IF A
provides one form of indirect reason.
- PS: The occurrence
table in the earlier chapter 21
for B IF A will be used to explain or provide a justification for truth
tables for material implications B IF A (or equivalently, IF A THEN B).
1 Introduction
In the chapter Implication Rules,
we asked the following question: What can you say for sure about Aunt Jane
when Tom does not go out to play and the following rule is never-disobeyed:
Each time Aunt Jane visits her nephew Tom's house, Tom goes out to play. The
answer was: NOT Aunt Jane visit. That is, when the previous rule holds,
the following rule also holds
Each time her nephew Tom does not go out to play,
Aunt Jane does not visits Tom's house.
This is a contrapositive way or form of writing the original
rule.
With the foregoing in mind, we can define the contrapositive way of writing
other implication rules. The contrapositive form of writing the
implication (or conditional statement) if A then B is if NOT B then
NOT A. For example, the contrapositive way of writing
if Aunt Jane visits her nephew Tom's house
then Tom goes out to play is
if NOT (Tom go out to play) then
NOT (Aunt Jane visits her nephew Tom's house).
Language (or grammar) courses would prefer us to write
if (Tom does not go out to play) then
(Aunt Jane does not visit her nephew Tom's house).
2 Equivalence of a one-way implication
with its contrapositive
Note that a hint or preview of the contrapositive was
provided by the discussion of the first logic puzzle (questions 4 and 5) in
the chapter Implication Rules. (You might wish to revisit that puzzle.)
The occurrence table below is intended to show you that if an implication
rule if A then B is true (never disobeyed) then the contrapositive rule if
NOT B then NOT A is true (never disobeyed), and vice versa.
| row |
A |
B |
if A
then B |
NOT B |
NOT A |
if NOT B
then
NOT A |
| 1 |
occurs |
occurs |
obeyed |
occurs
not
|
occurs
not
|
not
disobeyed
|
| 2 |
occurs |
occurs
not |
disobeyed |
occurs
not |
occurs |
disobeyed |
| 3 |
occurs
not |
occurs |
not
disobeyed |
occurs
not |
occurs |
not
disobeyed |
| 4 |
occurs
not |
occurs
not |
not
disobeyed |
occurs |
occurs |
obeyed |
Table for the contrapositive assertion:
(A implies B)
if and only if
(NOT B implies NOT A).
Filling The Table
First we look at the four combinations of the occurrences of the situations A
and B. When A occurs we have two possibilities for B. When A does not occur, we
have two possibilities for B as well. This gives a total of four cases or rows
and fills in the first three columns.
In the fourth column, headed by the rule if A then B for each
combination of occurrences of A and B, we note if the rule is obeyed, disobeyed
or not disobeyed.
Next, in the fifth and sixth columns headed by situations NOT B and NOT A,
for each of the four combinations we note if these situations occur or not.
In the last column, we finally note if the rule if NOT B then NOT A is
obeyed, disobeyed or not disobeyed. The entries in the last column depend on
those in the fifth and sixth columns. The entries in the latter two in turn
depend on those in the previous columns.
Answers to Two Questions
Now we can answer the questions: when are the two one-way implication rules (if
A then B) and (if NOT B then NOT A) true? Remember we say these
implication rules are true if each is never disobeyed. Both implications are
true, that is, never disobeyed, when the situation row 2, A and NOT B, never
occurs. Both implications are false when the situation in row 2, namely (A
and NOT B), occurs. So we conclude from the table that the two rules are
equivalent: each implies the other.1
1The
rule if NOT B then NOT A is disobeyed if the NOT B occurs but NOT A
does not. That is, it is disobeyed precisely when B does not occur, while A
does. But the rule if A then B is disobeyed precisely in this situation
where A occurs and B does not. This tells us that both rules are not disobeyed
provided the situation where A occurs and B does not never occurs. So
if one rule is true (never disobeyed), then so is the other.
Question
Recall that the rule if NOT B then NOT A is called the contrapositive way
of saying if A then B. What is the contrapositive of the contrapositive? The
answer is essentially the original implication: why? Hint: Replace NOT (NOT A)
by A in the statement of the contrapositive of the contrapositive.
Next: Chapter 22, Part II, Vacuously
True Implication Rules
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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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