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

Accepted Answer

An inconsistent statement in logic is one that cannot be true under any circumstances, meaning it is false no matter how the truth values are assigned to its components.

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

Accepted Answer

A simple statement is a grammatically correct construction in a language used to assert that an idea is true. It is a declarative sentence that provides information, makes a claim, or expresses a reality.

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)

Accepted Answer

The statement is a conditional statement in the form "If p, then q." The correct translation must reflect this structure. The statement "If Lauren had not borrowed my car, then she would not have made it back to campus in time" can be represented as "If ~p, then ~q" where p represents Lauren borrowing the car and q represents Lauren making it back to campus in time. This is best represented by option D, which states "~p F1S1S1W1F1F1F10 ~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

Accepted Answer

A conditional statement is a logical statement that asserts that if a certain condition is met, then a certain outcome will occur. It consists of an antecedent (the "if" part) and a consequent (the "then" part). The conditional statement asserts that if the antecedent is true, then the consequent must also be true. Therefore, choice E is the correct answer.

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

Accepted Answer

The bi-conditional (A ≡ B) is a statement that is true if and only if A and B have the same truth value. This means that A and B are logically equivalent, hence the answer is D.

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

Accepted Answer

The answer of Consider this argument: "If we buy tickets...

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

Accepted Answer

The argument involves two statements: "If we bought the movie tickets online, we saved a dollar per ticket" and "We did buy the tickets online." These can be translated into two separate propositions, P and Q. P represents the statement "If we bought the movie tickets online, we saved a dollar per ticket," and Q represents the statement "We did buy the tickets online." Using a truth table with columns for P, Q, and "P implies Q," there will be four rows, all of which are T, in the final "P implies Q" column. Therefore, the argument is valid, and the best choice is D, which states that the truth table will have four rows all 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

Accepted Answer

The last column of the truth table shows that the argument is true in all possible scenarios, indicating that the argument is valid.

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.

Accepted Answer

The negation of a disjunction is the statement that neither of the disjuncts is true. Choice D says "It is not the case that either Joseph is late for work or that he owes me money for gas," which is the same as saying that Joseph is not late for work and he does not owe me money for gas. This is the negation of the disjunction "Either Joseph is late for work or he owes me money for gas."

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.

Accepted Answer

The answer of We can interpret the truth table below...

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

Accepted Answer

The answer of How should the formula in the right...

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

Accepted Answer

A contingent statement is a statement that is neither always true nor always false, but depends on the truth values of its components. Option C states that a contingent statement can be true under one assignment of truth values and false under another, which is the best description of a contingent statement. Option A describes a conditional statement, option B describes a contradiction, and option D describes a tautology. None of these options accurately describe a contingent statement.

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)

Accepted Answer

The correct translation captures the conditional relationship and the conjunction of the outcomes if Lauren does not borrow the car: she doesn't make it back on time and doesn't pass Chemistry.

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)

Accepted Answer

The answer of Which of the following is a grammatically...

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

Accepted Answer

The formula is contingent because its truth value varies depending on the truth values of the variables $p$ and $q$. It is not always true (not a tautology) nor always false (not a contradiction), and it does not describe a relationship of equivalence (not bi-conditional) or a disjunction.

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

Accepted Answer

The argument structure involves a conditional statement and its conclusion based on the premise. A truth table evaluating this argument will include combinations of truth values for the components (buying tickets online, asking for a student discount, and saving a dollar per ticket), leading to a mix of true and false outcomes in the final column, depending on whether the conditions align with the argument's logic.

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.

Accepted Answer

The truth table shows that for all combinations of truth values of p and q, (p & q) and (q & p) have the same truth value, which means they imply each other.

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)

Accepted Answer

Choice B uses the logical operator "and" to express the relationship between Lauren borrowing the speaker's car (p) and the fact that she could have taken her own car (q). This accurately conveys the meaning of the sentence. Choice C and D use complex logical notation that is unnecessary for this sentence, while choice E uses "or" which does not accurately convey the relationship between the two clauses.

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

Accepted Answer

In the Logic of Statements, A implies B if and only if the conditional (A → B) is a tautology. Therefore, the correct choice is C, as it states that the conditional is a tautology. A being a tautology and B being inconsistent do not necessarily imply that A implies B, and the bi-conditional being contingent means that A and B can be true or false independently of each other, so it does not necessarily imply that A implies B.

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.

Accepted Answer

A conjunction is a compound statement formed by combining two statements with the word "and." In option C, "and" is used to combine two statements.

Quiz 17: The Logic of Declarative Statements

Which of the following is the best description of an inconsistent statement in the Logic of Statements? "A grammatically correct expression ___________________.

Which of the following correctly describes a simple statement? "A grammatically correct construction in a given language used to assert ______________.

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."

Which of the following correctly describes a conditional statement? "A grammatically correct construction in a given language used to assert ______________.

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

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 ___________________.

Which of the following is the negation of a disjunction?

Which of the following is the best description of contingent statement in the Logic of Statements? "A grammatically correct expression ___________________.

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."

Which of the following is a grammatically correct formula of declarative logic?

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

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

Which of the following is a conjunction?