Chapter 22, Contrapositive and
Vacuously True Implications
1. Introduction
Use of the contrapositive form of an implication B IF A provides one
form of indirect reason - the first?
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).
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).
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 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]
[1] The 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.
4. Vacuously True Implications
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.
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