Deck 17: The Logic of Declarative Statements

Question 1

Which of the following is the best description of an inconsistent statement in the Logic of Statements? "A grammatically correct expression ___________________.
(a) used to assert that if an antecedent statement is true, then a consequent statement is true
(b) that turns out to be false under every possible assignment of truth values to its component simple statements
(c) that turns out to be true under at least one possible assignment of truth values to its component simple statements and false under another possible assignment of truth values to its component simple statements
(d) that turns out to be true under every possible assignment of truth values to its component simple statements

Answer 1

B

Question 2

Which of the following correctly describes a simple statement? "A grammatically correct construction in a given language used to assert ______________.
(a) that an idea is true
(b) that a statement is false
(c) that two statements are both true
(d) that one or both of two statements are true
(e) that if an antecedent statement is true, then a consequent statement is true

Answer 2

A

Question 3

Which of the following is the best translation of "If Lauren had not borrowed my car, then she would not have made it back to campus in time."

A) p & q
B) (p & q)
C) (p $\rightarrow$ q)
D) (~p $\rightarrow$ ~q)
E) (p v ~q)

Answer 3

(~p $\rightarrow$ ~q)

Question 4

Which of the following correctly describes a conditional statement? "A grammatically correct construction in a given language used to assert ______________.
(a) that an idea is true
(b) that a statement is false
(c) that two statements are both true
(d) that one or both of two statements are true
(e) that if an antecedent statement is true, then a consequent statement is true

Question 5

Given two statements, A and B, we can say that A is equivalent to B in the Logic of Statements provided that ___________________.

A) A is a tautology and B is contingent
B) A is insistent and B is a tautology
C) the conditional (A $\rightarrow$ B) is a tautology
D) the bi-conditional (A ≡ B) is a tautology

Question 6

Consider this argument: "If we buy tickets online, then we save money. If we ask for the student discount, then we save money. We did not ask for the student discount, but we did buy tickets online. So we saved money." Using a truth table to evaluate that argument for validity will produce a final column that ___________________.
(a) has eight rows all of which are T
(b) has eight rows some of which are T and some F
(c) has four rows all of which are F
(d) has four rows all of which are T
(e) has two rows both of which are T

Question 7

Consider this argument: "If we bought the movie tickets online, we saved a dollar per ticket. We did buy the tickets online. So we must have saved a dollar per ticket." Using a truth table to evaluate that argument for validity will produce a final column that ___________________.
(a) has eight rows all of which are T
(b) has eight rows some of which are T and some F
(c) has four rows all of which are F
(d) has four rows all of which are T
(e) has two rows both of which are T

Question 8

The truth table below demonstrates that the argument "(q v r) and ~r, so therefore q" is ___________________. $\begin{array}{|c|c|c|c|c|c|}\hline q&r&(qvr)&-r&((qvr)\&-r)&(((qvr)\&-r)\rightarrow q)\\\hline T & T & T & F & F & T \\\hline T & F & T & T & T & T \\\hline F&T&F&F&F&T\\\hline F&F&T&T&F&T\\\hline\end{array}$

A) valid
B) not valid
C) contingent
D) inconsistent

Question 9

Which of the following is the negation of a disjunction?
(a) It is not the case that Joseph is late for work.
(b) Joseph was on time for work every day last month.
(c) It is not true that both Joseph is late for work and that Joseph owes me money for gas.
(d) It is not the case that either Joseph is late for work or that he owes me money for gas.
(e) If Joseph is not late for work one more time, then he will keep his job.

Question 10

We can interpret the truth table below to be a demonstration that (q & p) and (p & q) are related to each other in what way(s)? $\begin{array}{|c|c|c|c|c|c|}\hline p&q&(p\&q)&(q\&p)&((p\&q)\rightarrow(q\&p))&((q\&p)\rightarrow(p\&q))\\\hline T & T & T & T & T & T \\\hline T & F & F & F & T & T \\\hline F & T & F & F & T & T \\\hline F & F & F & F & T & T \\\hline\end{array}$

A) They imply each other.
B) The first implies the second.
C) They are both tautologies.
D) They are both inconsistent.
E) The second implies the first.

Question 11

How should the formula in the right most column of this truth table be characterized? $\begin{array}{|l|l|l|l|l|l|l|l|l|l|l}\hline&p&q&r&-p&(r \rightarrow p) &-(r \rightarrow p) &(-p \rightarrow r) &(-(r \rightarrow p)v(-p \rightarrow r)) &((-(r \rightarrow p)v(-p \rightarrow r))vq)\\\hline 1&T&T&T&F&T&F&T&T&T\\\hline 2&T & T & F & F & T & F & T & T & T \\\hline 3&T & F & T & F & T & F & T & T & T \\\hline 4&T & F & F & F & T & F & T & T & T \\\hline 5&F & T & T & T & F & T & T & T & T \\\hline 6&F & T & F & T & T & F & F & F & T \\\hline 7&F&F&T&T&F&T&T&T&T \\\hline 8&F&F&F&T&T&F&F&F&F\\\hline \end{array}$

A) Tautology
B) Inconsistent
C) Disjunction
D) Conditional
E) Double Negation

Question 12

Which of the following is the best description of contingent statement in the Logic of Statements? "A grammatically correct expression ___________________.
(a) used to assert that if an antecedent statement is true, then a consequent statement is true
(b) that turns out to be false under every possible assignment of truth values to its component simple statements
(c) that turns out to be true under at least one possible assignment of truth values to its component simple statements and false under another possible assignment of truth values to its component simple statements
(d) that turns out to be true under every possible assignment of truth values to its component simple statements

Question 13

Which of the following is the best translation of "If Lauren does not borrow my car, then she does not make it back to the campus on time for her Chemistry final and moreover she does not pass Chemistry."

A) (p $\rightarrow$ q)
B) (p & q)
C) ((p $\rightarrow$ q) v ~q)
D) (~p $\rightarrow$ (~q & ~r))
E) ((p v ~q) $\rightarrow$ r)

Question 14

Which of the following is a grammatically correct formula of declarative logic?
(a) (r v (~p & q)
(b) (A & B)
(c) ~(~p)
(d) (((r & s) = s) v s)
(e) ((~p & q) v ~r)

Question 15

According to the following truth table, how should the formula ((p & q) $\rightarrow$ ~q) be characterized? $$\begin{array} { | c | c | c | c | c | } \hline p & q & ( p \& q ) & - q & ( p \& q ) \rightarrow - q ) \\\hline T & T & T & F & F \\\hline T & F & F & T & T \\\hline F & T & F & F & T \\\hline F & F & F & T & T \\\hline\end{array}$$

A) Contingent
B) Inconsistent
C) Disjunction
D) Tautology
E) Bi-Conditional

Question 16

Consider this argument: "If we bought the movie tickets online or asked for the student discount, we saved a dollar per ticket. We did save a dollar per ticket. So we must have bought the tickets online or asked for the student discount." Using a truth table to evaluate that argument for validity will produce a final column that ___________________.
(a) has eight rows all of which are T
(b) has eight rows some of which are T and some F
(c) has four rows all of which are F
(d) has four rows all of which are T
(e) has two rows both of which are T

Question 17

We can interpret the truth table below to be a demonstration that (q & p) and (p & q) are related to each other in what way(s)? $$\begin{array} { | c | c | c | c | c | } \hline p & q & ( p \& q ) & ( q \& p ) & ( p \& q ) = ( q \& p ) ) \\\hline T & T & T & T & T \\\hline T & F & F & F & T \\\hline F & T & F & F & T \\\hline F & F & F & F & T \\\hline\end{array}$$

A) They imply each other.
B) The second does not imply the first.
C) The first does not imply the second.
D) They are both tautologies.
E) They are both inconsistent.

Question 18

Which of the following is the best translation of "Lauren borrowed my car, although she could have taken her own."

A) p & q
B) (p & q)
C) (p $\rightarrow$ q)
D) (~p $\rightarrow$ ~q)
E) (p v ~q)

Question 19

Given two statements, A and B, we can say that A implies B in the Logic of Statements provided that ___________________.

A) A is a tautology
B) B is inconsistent
C) the conditional (A $\rightarrow$ B) is a tautology
D) the bi-conditional (A ≡ B) is contingent

Question 20

Which of the following is a conjunction?
(a) Joseph is late for work.
(b) Joseph is not late for work.
(c) Joseph is late for work and he owes me money for gas.
(d) Either Joseph is late for work or he owes me money for gas.
(e) If Joseph is late for work one more time, then he will lose his job.

Question 21

Explain why an argument may fail to be valid at the level of the Logic of Statements, but might be discovered to be valid a deeper analytical level.
The Logic of Statements is valuable for determining that a certain set of arguments is valid provided that the validity depends on the grammatical relationships between simple statements. But, as we saw in the chapter entitled "Valid Inferences," the validity of some arguments depends on relationships among classes of objects and their members, and it gives examples of valid inferences based on relationships between individuals, such as transitivity and reflexivity. The examples in those sections of that chapter would not be valid in the Logic of Statements.

Question 22

Compose a conjunction in English from these two simple statements: "We took Grandfather's advice." And "We had a great time." Let the first one be "p" and the second by "q" and translate the conjunction into symbolic logic.

Question 23

Statements A and B are __________________ in the Logic of Statements if, and only if, the bi-conditional (A ≡ B) is a tautology.

Question 24

What is a simple declarative statement?

Question 25

________________ language is much richer than the notational language of the Logic of Declarative Statements. We can express only a limited range of the logical power of natural language using the tilde, ampersand, wedge, arrow, and triple bar.

Question 26

Using "only if" compose a conditional in English from these two simple statements: "We took Grandfather's advice." And "We had a great time." Let the first one be "p" and the second by "q" and translate the conjunction into symbolic logic.

Question 27

Figure out "Who ate the Pizza?" and explain the answer.
The Puzzle: Five roommates, Abe, Bob, Carl, Dave, and Ziggy, share an off-campus apartment. On Monday night they had leftover pizza and so they put it in the refrigerator. On Tuesday when Ziggy went to get some of that pizza he found that it had all been eaten. Later Ziggy confronted his four roommates to find out who ate the pizza. Each roommate made one statement. The person who ate the pizza lied; his statement is false. The other three statements are true. Which one ate the pizza? Here are their four statements:
Abe: I was in class all day.
Bob: Carl ate the pizza.
Carl: Bob's statement is false.
David: Carl's statement is true.

Question 28

An argument is valid at the level of the Logic of Statements if the conditional formed by the conjunction of its premises as the antecedent and the conclusion as its consequent is a tautology.

Question 29

Analyze the logic relationship called Disjunction and diagram a "Logic Circuit" with a power source and a light bulb that shows the conditions under which the light will be lit and not lit.

Question 30

Statements A and statement B are __________________ in the Logic of Statements if A and B have the same truth value under every interpretation of their statement letters.

Question 31

Accurately and elegantly translate ((p → s) & (s → t)). Use these interpretations of the statement
letters:
p = "My son has chickenpox."
q = "I have to stay home from work."
r = "I need to take care of my son."
s = "My son is contagious."
t = "I need to call my boss."

Question 32

Two statements are equivalent at the level of the Logic of Statements if the bi-conditional of the two is a tautology.

Question 33

Accurately and elegantly translate (~s → (~p & (~r & ~t))). Use these interpretations of the statement
letters:
p = "My son has chickenpox."
q = "I have to stay home from work."
r = "I need to take care of my son."
s = "My son is contagious."
t = "I need to call my boss."

Question 34

At the level of the Logic of Statements, a tautology is implied by any statement and an inconsistent statement implies any statement. Moreover, any two tautologies or any two self-contradictory statements are equivalent. With regard to implication and equivalence, how should we resolve the differences between the treatment of these concepts in the Logic of Statements and how these same concepts apply in everyday real-world discourse?

Question 35

Construct the truth table that defines negation.

Question 36

A tautology is a grammatically correct expression that turns out to be true under every possible assignment of truth values to its component simple statements.

Question 37

Construct the truth table for ((~(r → p) v (~p → r)) v q). Based on the truth table, correctly characterize the formula as tautology, inconsistent, or contingent. Explain the basis for the characterization.

Question 38

Construct the truth table that defines the conditional.

Question 39

If a natural language statement is translated into symbolic logic and its truth table ends up with a T on at least one row and an F on at least one row, then the statement can be characterized as _____________.

Question 40

In the Logic of Statements means a grammatically correct expression that turns out to be false under every possible assignment of truth values to its component simple statements is called inconsistent or ______________.