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Chapter 10
Deductive Arguments II
Truth–Functional Logic
When students first confront truth-functional logic, some recoil from the symbolizations. They
get the idea that there will be a lot to memorize—an idea that can cause trouble if they’re not
disabused of it right away. In fact, the basic truth tables for the truth-functional symbols are very
easy to learn. But a student who doesn’t learn them immediately, of course, is going to get
absolutely nowhere. The instructor should give a brief quiz on the basic truth tables as soon as
possible, to keep students from getting behind.
In Chapter 9, the translation of informal claims into standard-form categorical claims gets
students to do some hard thinking about their language. And, when they get things wrong, they
realize that things are not as obvious as they had thought. The same holds true for the display of
claims’ truth-functional forms, the result of the type of symbolization discussed in this chapter.
This point can be made early by calling students’ attention to items like those found in Exercise
10-14 given below. This is sometimes unpleasant news since more students miss those questions
than one might think. Things look better after students do their first successful symbolizations.
Most of the symbolizations in Exercises 10-1 and 10-2 in the text are not difficult, and students
learn quickly that if they’re careful, they can learn this stuff.
Chapter Recap
The following topics are covered in this chapter:
• Logical symbols, their truth tables, and their English counterparts: negation, conjunction,
disjunction, conditional (see Figure 1, page 288, for a summary).
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• Symbolizations of truth functions can represent electrical circuits because “true” and
“false” for sentences can be made to correspond to “on” and “off” for circuits. (See the
large box at the end of the chapter.)
• Deductions can be used to prove the validity of propositional arguments; they make use of
the rules on the Figure 2, page 323, and the rule of conditional proof, pages 327–329.
Test Yourself Answer
Since you’ve gone to the trouble to seek answers to some exercises, we’ll throw in an answer to
the test in the box on page 290. Two cards must be turned over, the one with the “e” and the one
with the “3.”
Answers to Text Exercises
Exercise 10-1
1. ▲Q → P
Exercise 10-4
Note: We’ve given the most obvious symbolizations, but we feel obliged to take anything that’s
truth-functionally equivalent as correct.
[Notice that the only difference between (1) and (2) is the location of the comma. But the
symbolizations have two different truth tables, so moving the comma actually changes the
meaning of the claim. And we’ll bet you thought that commas were there only to tell you
when to breathe when you read aloud.]
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Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill
Education.
6. (P → Q) & (R → Q)
7. (P v R) → Q
8. P v (Q → R)
9. (P v Q) → R
10. P → (Q v R)
15. F v C
16. ▲S → ~C [Ordinarily, the word “but” indicates a conjunction, but in this case it is present
only for emphasis—“only if” is the crucial truth-functional phrase.]
17. C → ~S
Exercise 10-5
1. ▲
P
Q
R
(P → Q)
(P → Q) & R
T
T
T
T
T
T
T
F
T
F
T
F
T
F
F
T
F
F
F
F
F
T
T
T
T
F
T
F
T
F
F
F
T
T
T
F
F
F
T
F
2. ▲
P
Q
R
(Q & R)
P → (Q & R)
T
T
T
T
T
T
T
F
F
F
T
F
T
F
F
T
F
F
F
F
F
T
T
T
T
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(R → Q)
T
T
T
T
T
T
T
T
F
T
T
T
T
F
T
F
F
F
T
F
F
F
T
F
F
T
T
T
T
T
F
T
F
T
T
T
F
F
T
T
F
F
F
F
F
T
T
T
7.
P
Q
R
(P v R)
(P v R) → Q
T
T
T
T
T
T
T
F
T
T
T
F
T
T
F
T
F
F
T
F
F
T
T
T
T
F
T
F
F
T
F
F
T
T
F
F
F
F
F
T
8.
P
Q
R
(Q → R)
P v (Q → R)
T
T
T
T
T
T
T
F
F
T
T
F
T
T
T
T
F
F
T
T
F
T
T
T
T
F
T
F
F
F
F
F
T
T
T
F
F
F
T
T
9.
P
Q
R
(P v Q)
(P v Q) → R)
T
T
T
T
T
T
T
F
T
F
T
F
T
T
T
T
F
F
T
F
F
T
T
T
T
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F
S
F v S
~ (F v S)
T
T
T
F
T
F
T
F
F
T
T
F
F
F
F
T
Since ~(F v S) is exactly equivalent to ~F & ~S, the latter can be substituted for the former in the
preceding table and it will still be correct. Columns for ~F and for ~S would need to be added to
make it complete.
1. ▲Invalid:
(Premise) (Premise) (Conclusion)
P
Q
~ Q
P v ~ Q
~ P
T
T
F
T
F
T
F
T
T
F
F
T
F
F
T
F
F
T
T
T
(Row 2)
For invalid examples in the remainder, we’ll just provide the row (or rows, in case there
(Conclusion) (Premise) (Premise)
P
Q
R
(P → Q)
~ (P → Q)
(Q → R)
P → (Q →
R)
T
T
T
T
F
T
T
T
T
F
T
F
F
F
T
F
T
F
T
T
T
T
F
F
F
T
T
T
F
T
T
T
F
T
T
F
T
F
T
F
F
T
F
F
T
T
F
T
T
F
F
F
T
F
T
T
(Row 4)
5. Invalid. It is shown by the 2nd row of a full table.
Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill
Education.
Exercise 10-11
We’ve used the short truth-table method to demonstrate invalidity.
3. ▲Invalid. There are two rows that make the premises T and the conclusion F:
M P R F G
T T T F T
T T F T F
4. ▲Invalid. There are three rows that make the premises true and the conclusion F:
D G H P M
5. ▲Invalid. There are two rows that make the premises T and the conclusion F:
6. Invalid.
7. Invalid.
P D J M L
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3. P v Q 2, ADD
4. R 1, 3, MP
6. 1. ~P (Premise)
2. ~(R & S) v Q (Premise)
3. ~P → ~Q (Premise) /∴ ~(R & S)
4. ~Q 1, 3, MP
5. ~(R & S) 2, 4, DA
10. ▲1. (T v M) → ~Q (Premise)
2. (P → Q) & (R → S) (Premise)
Exercise 10-15
1. ▲4. 1, 3, CA
6. 4, 5, MP
3. 4. 1, CONTR
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Education.
5. 3, 4, CA
6. 2, 5, CA
7. 6, IMPL
8. 7, EXPORT
4. ▲4. 3, CONTR
5. 2, 4, MP
6. 2, 5, CONJ
7. 1, 6, MP
5. 4. 1, DEM
5. 4, SIM
6. 2, 5, MT
7. 4, SIM
8. 5. 2, COM
6. 5, IMPL