Question 1

Given the statements: A ⊃ (B • C) / B ⊃ (D ∨ ∼ E) / C ⊃ (E ∨ ∼ D) / A • E.
These statements are:
A) Consistent.
B) Tautologous.
C) Invalid.
D) Logically equivalent.
E) Inconsistent.

Accepted Answer

The statements are consistent because there is no inherent contradiction among them. Given A, it follows that both B and C must be true. From B and C, it follows that either D or ~E, and E or ~D must be true, respectively. Since A and E are true, the conditions can be satisfied without contradiction, making the set of statements consistent.

Question 2

Given the argument: (A • B) ≡ (C ∨ D) / A ⊃ E / B ⊃ F // C ⊃ (E • F).
This argument is:
A) Cogent.
B) Invalid.
C) Sound.
D) Valid.
E) Uncogent.

Accepted Answer

The argument is valid. 
To prove this, we can use the method of logical equivalences:
(A • B) ≡ (C ∨ D)    (given)
(A ⊃ E)               (given)
(B ⊃ F)               (given)
Therefore:
(A ⊃ E) ≡ (~A ∨ E)         (material implication)
(B ⊃ F) ≡ (~B ∨ F)         (material implication)
(A • B) ≡ (~( ~A ∨ E) ∨ ~( ~B ∨ F))    (substituting equivalent forms)
(A • B) ≡ (~A ∧ ~B) ∨ (E ∧ F)          (De Morgan's law)

Now, we want to show that C ⊃ (E • F) is a logical consequence of the three premises:
(A • B) ≡ (~A ∧ ~B) ∨ (E ∧ F)
(A ⊃ E)
(B ⊃ F)

Assume C for conditional proof:
We need to prove E • F
From (A • B) ≡ (~A ∧ ~B) ∨ (E ∧ F) and (A ⊃ E), we can infer ~A ∨ E.
Similarly, from (A • B) ≡ (~A ∧ ~B) ∨ (E ∧ F) and (B ⊃ F), we can infer ~B ∨ F.
Since C ≡ (A • B) ∨ D by the first premise, we know one of two things:
1. A • B is true, in which case either ~A ∨ E or ~B ∨ F must be true. Consequently, E • F must be true because E and F are the only two remaining options.
2. D is true, in which case the conclusion still holds since the conditional is vacuously true when the antecedent is false.

In either case, we have shown that C ⊃ (E • F) is a logical consequence of the premises. Therefore, the argument is valid.

Question 3

If dinosaur DNA has deteriorated, then it is impossible to recreate dinosaurs. If dinosaurs perished long ago, then dinosaur DNA has deteriorated. Hence, if dinosaurs perished long ago, then it is impossible to recreate them.
A) Constructive dilemma.
B) Affirming the consequent.
C) Pure hypothetical syllogism.
D) Modus tollens.
E) Invalid.

Accepted Answer

This argument is a pure hypothetical syllogism because it follows the form: If P then Q, if Q then R, therefore if P then R. It connects two conditional statements to conclude a third conditional statement directly related to the first two.

Question 4

If our company reduces wages, then it will lose its labor force; but if it does not reduce wages, then it will go bankrupt. Thus, our company will either lose its labor force or go bankrupt, since it will either reduce wages or not reduce them.
A) Constructive dilemma.
B) Invalid.
C) Denying the antecedent.
D) Pure hypothetical syllogism.
E) Destructive dilemma.

Accepted Answer

This is a constructive dilemma, which involves two conditional statements and a disjunction, leading to a conclusion that is a disjunction.

Question 5

If computers can play chess, then they can think. But computers cannot think. Thus, computers cannot play chess.
A) Disjunctive syllogism.
B) Invalid.
C) Denying the antecedent.
D) Affirming the consequent.
E) Modus tollens.

Accepted Answer

This argument follows the structure of modus tollens, which is a valid form of argument. It starts with a conditional statement ("If computers can play chess, then they can think"), and then denies the consequent ("But computers cannot think"), leading to the conclusion that denies the antecedent ("Thus, computers cannot play chess").

Question 6

If deficit spending is halted, then interest rates will drop; but if inflation heats up, then interest rates will not drop. Accordingly, either deficit spending will not be halted or inflation will not heat up, since interest rates will either drop or not drop.
A) Destructive dilemma.
B) Pure hypothetical syllogism.
C) Constructive dilemma.
D) Denying the antecedent.
E) Invalid.

Accepted Answer

This argument presents a destructive dilemma, which is a type of argument where two options are presented, both of which lead to negative consequences. In this case, either deficit spending will continue or inflation will heat up, both of which result in interest rates not dropping. Therefore, the argument concludes that interest rates will not drop regardless of whether deficit spending is halted or inflation heats up.

Question 7

If rainfall is normal, then grain reserves will not be tapped, because if the harvest is not abundant, then grain reserves will be tapped, and if rainfall is normal, then the harvest will be abundant.
A) Pure hypothetical syllogism.
B) Invalid.
C) Denying the antecedent.
D) Modus tollens.
E) Affirming the consequent.

Accepted Answer

The answer of If rainfall is normal, then grain reserves...

Question 8

Given the statements: A ≡ (B ∨C) / B ⊃ D / D ⊃ E / E ⊃ ∼ B / A ≡ ∼ C.
These statements are:
A) Valid.
B) Tautologous.
C) Consistent.
D) Contradictory.
E) Inconsistent.

Accepted Answer

The answer of Given the statements: A ≡ (B ∨C)...

Question 9

Either taxes will be imposed on sugary drinks or public health will deteriorate. Therefore, public health will deteriorate, since taxes will not be imposed on sugary drinks.
A) Modus tollens.
B) Modus ponens.
C) Disjunctive syllogism.
D) Invalid.
E) Denying the antecedent.

Accepted Answer

This argument is a disjunctive syllogism because it presents two options (either taxes will be imposed on sugary drinks or public health will deteriorate), rejects one (taxes will not be imposed on sugary drinks), and concludes the other must be true (therefore, public health will deteriorate).

Question 10

Either we recycle our trash or we will be inundated in garbage. Therefore, we will not be inundated in garbage, since we recycle our trash.
A) Disjunctive syllogism.
B) Destructive dilemma.
C) Pure hypothetical syllogism.
D) Modus ponens.
E) Invalid.

Accepted Answer

The argument is invalid because it assumes that recycling alone will prevent being inundated in garbage without considering other factors that could contribute to the problem.

Question 11

Given the argument: A ⊃ (B • ∼ C) / B ⊃ (C ∨ A) / C ⊃ ∼ A // C ⊃ ∼ B.
This argument is:
A) Invalid; fails in 6th line.
B) Valid.
C) Invalid; fails in 5th line.
D) Invalid; fails in 2nd line.
E) Invalid; fails in 3rd line.

Accepted Answer

The answer of Given the argument: A ⊃ (B •...

Question 12

If the nuts are stale, then they should not be put into the salad. Thus, the nuts are stale, since they should not be put into the salad.
A) Modus ponens.
B) Denying the antecedent.
C) Invalid.
D) Affirming the consequent.
E) Disjunctive syllogism.

Accepted Answer

This is an example of affirming the consequent, a logical fallacy. The argument incorrectly assumes that because the consequence (should not be put into the salad) is true, the antecedent (the nuts are stale) must also be true.

Question 13

Given the argument: (A • B) ∨ (C • D) / E ⊃ ∼ B / F ⊃ ∼ D / E ∨ F // A • C.
This argument is:
A) Valid.
B) Invalid.
C) Uncogent.
D) Cogent.
E) Sound.

Accepted Answer

The premises are (A • B) ∨ (C • D), E ⊃ ∼ B, and F ⊃ ∼ D. 
Using the first premise, we can separate it into two cases: A • B or C • D. 
Assume E is true. From E ⊃ ∼ B, we can conclude that ∼ B is true. Similarly, assume F is true. From F ⊃ ∼ D, we can conclude that ∼ D is true. 
If A • B is true, then from ∼ B, we can conclude that ∼ A is true. But we cannot make any conclusion about C. 
If C • D is true, then from ∼ D, we can conclude that ∼ C is true. But we cannot make any conclusion about A. 
Therefore, we cannot conclude that A • C necessarily follows from the premises and conclusion (E ∨ F) given, making the argument invalid.

Question 14

Given the pair of statements: A ∨ (B • C) and (B • ∼ A) ∨ (A • ∼ C).
These statements are:
A) Logically equivalent.
B) Valid.
C) Contradictory.
D) Consistent.
E) Inconsistent.

Accepted Answer

The two statements can both be true under some interpretations (e.g., when A is true, B is false, and C is false), which means they are consistent. They are not logically equivalent, contradictory, or necessarily valid on their own, and they are not inconsistent because there is at least one interpretation under which they can both be true.

Question 15

Statement 3G
Given the following statement:
[A • (B ∨ C)] ∨ [∼ (A • B) ∨ ∼ (A • C)]
-The truth table for Statement 3G has how many lines?
A) Four.
B) Nine.
C) Twelve.
D) Six.
E) Eight.

Accepted Answer

The truth table for a statement involving three variables (A, B, C) has 2^3 = 8 lines, corresponding to all possible combinations of truth values for these variables.

Question 16

Given the argument: A ⊃ ∼B / B ∨ A // ∼B ≡ A.
This argument is:
A) Invalid; fails in 4th line.
B) Invalid; fails in 2nd line.
C) Valid.
D) Invalid; fails in 1st line.
E) Invalid; fails in 3rd line.

Accepted Answer

This is a valid argument that uses the principle of proof by contradiction.

1. A ⊃ ∼B  (premise)
2. B ∨ A  (premise) 
3. Assume ∼(∼B ≡ A) for the sake of contradiction. 
4. From 3, we derive ∼B ∧ ¬A by applying the definition of the biconditional and DeMorgan's Law. 
5. From 4, we derive ∼B by simplification. 
6. From 5 and 1, we derive ¬A by modus tollens. 
7. From 2 and 6, we derive B by disjunctive syllogism. 
8. From 5 and 7, we obtain a contradiction. 
9. Therefore, our assumption in 3 is false, and ∼B ≡ A (equivalence).

Question 17

French has many irregular verbs. But if that is so, then French is harder to learn than English. Thus, French is harder to learn than English.
A) Denying the antecedent.
B) Constructive dilemma.
C) Modus ponens.
D) Affirming the consequent.
E) Modus tollens.

Accepted Answer

The argument follows the structure of modus ponens, which is a form of argument where it is stated that if P, then Q; P is asserted to be true, therefore Q must also be true. In this case, P is "French has many irregular verbs," leading to Q, "French is harder to learn than English." Since the premise that French has many irregular verbs is accepted, the conclusion that French is harder to learn than English is drawn, fitting the modus ponens form.

Question 18

Given the pair of statements: (A ∨ B) • (∼ A ∨ ∼ B) and (B • A) ∨ (∼ A • ∼ B).
These statements are:
A) Inconsistent.
B) Contradictory.
C) Invalid.
D) Logically equivalent.
E) Consistent.

Accepted Answer

The answer of Given the pair of statements: (A ∨...

Question 19

Statement 3G
Given the following statement:
[A • (B ∨ C)] ∨ [∼ (A • B) ∨ ∼ (A • C)]
-Statement 3G is:
A) Valid.
B) Self-contradictory.
C) Tautologous.
D) Inconsistent.
E) Contingent.

Accepted Answer

The statement is tautologous because it is true for all possible truth values of A, B, and C. Regardless of whether A is true or false, if either B or C is true, the first half of the statement is true. If neither B nor C is true, the second half of the statement is still true because either ∼(A • B) or ∼(A • C) must be true. Therefore, the entire statement is always true and is a tautology.

Question 20

Statement 2G
Given the following statement:
∼ (B ∨ A) ≡ (∼ A ⊃ B)
-Statement 2G is:
A) Tautologous.
B) Contingent.
C) Consistent.
D) Self-contradictory.
E) Inconsistent.

Accepted Answer

The answer of Statement 2G
Given the following statement:
∼ (B ∨...

Quiz 6: Propositional Logic

Given the statements: A ⊃ (B • C) / B ⊃ (D ∨ ∼ E) / C ⊃ (E ∨ ∼ D) / A • E. These statements are:

Given the argument: (A • B) ≡ (C ∨ D) / A ⊃ E / B ⊃ F // C ⊃ (E • F) . This argument is:

If dinosaur DNA has deteriorated, then it is impossible to recreate dinosaurs. If dinosaurs perished long ago, then dinosaur DNA has deteriorated. Hence, if dinosaurs perished long ago, then it is impossible to recreate them.

If our company reduces wages, then it will lose its labor force; but if it does not reduce wages, then it will go bankrupt. Thus, our company will either lose its labor force or go bankrupt, since it will either reduce wages or not reduce them.

If computers can play chess, then they can think. But computers cannot think. Thus, computers cannot play chess.

If deficit spending is halted, then interest rates will drop; but if inflation heats up, then interest rates will not drop. Accordingly, either deficit spending will not be halted or inflation will not heat up, since interest rates will either drop or not drop.

If rainfall is normal, then grain reserves will not be tapped, because if the harvest is not abundant, then grain reserves will be tapped, and if rainfall is normal, then the harvest will be abundant.

Given the statements: A ≡ (B ∨C) / B ⊃ D / D ⊃ E / E ⊃ ∼ B / A ≡ ∼ C. These statements are:

Either taxes will be imposed on sugary drinks or public health will deteriorate. Therefore, public health will deteriorate, since taxes will not be imposed on sugary drinks.

Either we recycle our trash or we will be inundated in garbage. Therefore, we will not be inundated in garbage, since we recycle our trash.

Given the argument: A ⊃ (B • ∼ C) / B ⊃ (C ∨ A) / C ⊃ ∼ A // C ⊃ ∼ B. This argument is:

If the nuts are stale, then they should not be put into the salad. Thus, the nuts are stale, since they should not be put into the salad.

Given the argument: (A • B) ∨ (C • D) / E ⊃ ∼ B / F ⊃ ∼ D / E ∨ F // A • C. This argument is:

Given the pair of statements: A ∨ (B • C) and (B • ∼ A) ∨ (A • ∼ C) . These statements are:

Statement 3G Given the following statement: [A • (B ∨ C) ] ∨ [∼ (A • B) ∨ ∼ (A • C) ] -The truth table for Statement 3G has how many lines?

Given the argument: A ⊃ ∼B / B ∨ A // ∼B ≡ A. This argument is:

French has many irregular verbs. But if that is so, then French is harder to learn than English. Thus, French is harder to learn than English.

Given the pair of statements: (A ∨ B) • (∼ A ∨ ∼ B) and (B • A) ∨ (∼ A • ∼ B) . These statements are:

Statement 3G Given the following statement: [A • (B ∨ C) ] ∨ [∼ (A • B) ∨ ∼ (A • C) ] -Statement 3G is:

Statement 2G Given the following statement: ∼ (B ∨ A) ≡ (∼ A ⊃ B) -Statement 2G is: