YOU are better than YOU think. Show yourself  how:  

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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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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.

Chapter 22
The Contrapositive

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:

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
 

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.

Filling The Table

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

Next: Chapter 22, Part II, Vacuously True Implication Rules

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.

Vol 1A Postscripts
- online only

+Proof by Absurdity alias proof by contradiction
+How the demand for consistency supports the law of the excluded middle

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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