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III. CHAPTER THREE: TRUTH TABLES
A. General Theory
1. Determine the sentence forms of which the following are substitution instances:
a. ~ (A B)C
b. A (B~C)
2. If a sentence form contains five variables, how many lines or rows must its
complete truth table analysis have?
3. Assume you know of an argument only that its premises are not consistent. What,
if anything, can you tell about the argument’s validity? (Defend your answer.)
4. Why is it true that any argument with a conclusion with the form p~pis valid?
5. a. We can define a tautologous sentence as one that is a substitution instance of
some tautologous sentence form, and a contradictory sentence as one that is a
substitution instance of some contradictory sentence form. Why can’t we
analogously define a contingent sentence as one that is a substitution instance of
some contingent sentence form? (Defend your answer, including examples.)
b. We cannot define a contingent sentence as one that is a substitution instance
of some contingent sentence form. But then how can we define that term?
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6. Suppose one of the premises of an argument is logic equivalent to the conclusion.
What, if anything, can you conclude about the argument’s validity?
A. Answers
3. If an argument’s premises are not consistent then it will be impossible for all of
4. Any argument with a conclusion of the form p~pwill have a conclusion that is
6. If one of the premises of an argument is logically equivalent to the conclusion,
B. Tautologies, Contradictions, and Contingent Sentences
Determine by truth table analysis which of the following sentence forms are tautologous,
which are contradictory, and which are contingent:
1. (p q) [(p~q)q] 4. (p~q) ~ (q~p)
2. (p q) (q r) 5. ~ [(~ p q) (~ q~p)]
3. p(q p)
B. Answers
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C. Logical Equivalences
Use a truth table to determine which of the following pairs of sentence forms are
logically equivalent.
1. ~ (p q), p~q3. p(q r), (p q) (p r)
2. ~ (p q), (~ p q) (p~q) 4. ~ [(p q)p], ~ [(p q)q]
C. Answers
D. Proving Validity of Argument Forms
Determine by truth table analysis which of the following argument forms are valid and
which are invalid:
(1) 1. (~ p q) (~ p q)
2. q p / ~q
(2) 1. p~ (~ q r)
2. ~ r q / ~p q
(3) 1. (p q) (q r)
2. ~ (r~q)/~r p
(4) 1. ~ (p~q) ~ p
/ (~ q p) (~ q~p)
D. Answers
E. Short Truth Table Test for Invalidity
Use the short truth table method to show that the following arguments are invalid and
provide the truth-value assignments that show invalidity for each:
(1) 1. A B
2. C~B
3. ~ C/A
(2) 1. A(B C)
2. B(C D)
3. ~ D/ ~ A
(3) 1. A(B~C)
(3) 1. A(B~C)
2. ~ [A~ (C B)]
3. B C
4. ~ B/ ~ A
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(4) 1. ~ (A B) ~ C
2. ~ (~ A ~ C) ~ B
3. ~ (B D)
4. B (E ~ A)
/ ~ A C
E. Answers
F. Short Truth Table Test for Consistency
Use the short truth table method to show that the following sets of premises are consistent
and provide the truth-value assignments that show consistency for each:
(1) 1. (A B) ~ C (3) 1. A (B E)
2. C D 2. (C D) ~ A
3. D (A B) / B 3. B (C D) / E
(2) 1. D (A C) (4) 1. ~ B C
2. ~ [B (D E)] 2. (D C) (A~C)
3. ~ B~C / ~ D 3. (C~C)B
4. B (D A) / B
F. Answers