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29 Contrapositive
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Chapters and Appendices

Book Entrance

Foreword
1. Introduction
2. Implication Rules [4]
3. Chains of Reason [3]
4. Induction Mathematical
4. Romeo and Juliet
6  Old Language
5 Knowledge Islands [2]
7  Arith Skill Check [4 X 2]
Arith Webvideos
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable [8]
9. Algebra Talk [7]
10 Two More Skills[5]
11 Why Shorthand
12 Shorthand Usage [10]
13 What's Next
PS: The 4-th Skill For Algebra
14 Compound Interest [6]
15 Linear Equations [5]
16 Painless Proofs
17 Pythagoras
PS I.  Distributive Law
PS II. Polynomials
18 Rules of Algebra [20]
19  Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums [2]
23 Summation Notation
24 Your Money [3]
25 Induction & Recursion [4]
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
Pathways for Learning

Book Entrance

Would you like to show yourself or others how to be  algebra power users?

What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent or Independent
Variable, a Matter of Choice

Chapter 29
The Contrapositive

Previous Chapter: Proofs and Logic- 28 Occurrence Tables

Note: Online Book Pattern Based Reason includes this chapters and more on logic and reason.

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

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

Question

3  Vacuously True

The one-way implication rule If A then B is said to be vacuously true if and only if the situation A never occurs.

The contrapositive If NOT B then NOT A is vacuously true if and only if the situation NOT B never occurs, that is if and only if the situation B always occurs. Therefore an implication rule and its contrapositive are vacuously true in different circumstances.

Finally, an innovation perhaps, the two-way implication rule A if and only if B is said to be vacuously true in the situation where A and B are both always true or both always false.

An implication rule says that when a first situation A occurs then so must a second situation B. The associated contrapositive implication rule says that when the second situation B does not occur, then the situation A cannot occur. The previous part of this chapter explains why an implication rule is never disobeyed if and only if its contrapositive is never disobeyed. In consequence, a chain of reasoning which shows the contrapositive form of an implication rule is never disobeyed also shows the implication rule is never disobeyed.

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.)

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.

Next: Chapter 30 Truth Tables, Revisited

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