Question 1

The rule of exportation applies to disjunctions.

Accepted Answer

The rule of exportation applies to implications, not disjunctions. It allows for the transformation of a conditional statement into a conjunction within the antecedent.

Question 2

According to De Morgan's laws, ~(p ⋁ q) :: (~p • ~q).

Accepted Answer

De Morgan's laws state that the negation of a disjunction is the conjunction of the negations, and vice versa.

Question 3

According to contraposition (the equivalence rule), ~A → ~B is logically equivalent to B →
A. True False

Accepted Answer

Contraposition states that if A implies B, then not B implies not A. In other words, ~A → ~B is logically equivalent to B → A.

Question 4

The law of excluded middle states that for any given syllogism, the middle term drops out.

Accepted Answer

The law of excluded middle is a principle of classical logic stating that for any proposition, either that proposition is true or its negation is true. It is not related to syllogisms or the middle term in them.

Question 5

According to the law of noncontradiction, ~(p • ~p).

Accepted Answer

The law of noncontradiction states that a proposition cannot be both true and false at the same time, which is expressed as ~(p • ~p), meaning it is not the case that both p and not p are true simultaneously.

Question 6

The rule of material equivalence is: (p → q) :: (~p ⋁ q)

Accepted Answer

The rule of material equivalence states that a conditional statement (p → q) is logically equivalent to the disjunction of the negation of the antecedent and the consequent (~p ⋁ q).

Question 7

According to the rule of redundancy, we should never repeat any line of a proof.

Accepted Answer

The rule of redundancy does not prohibit repeating lines in a proof; it emphasizes avoiding unnecessary repetition that doesn't contribute to clarity or understanding.

Question 8

The rule of distribution tells us that (C ⋁ (B • A)) is logically equivalent to ((C ⋁ B) • (C ⋁ A))

Accepted Answer

The rule of distribution states that the disjunction of a conjunction and a proposition is logically equivalent to the conjunction of the disjunction of each proposition with the first proposition. In this case, (C ⋁ (B • A)) is logically equivalent to ((C ⋁ B) • (C ⋁ A)) because the disjunction of the conjunction of B and A is logically equivalent to the conjunction of the disjunction of C and B and the disjunction of C and A.

Question 9

The basic idea behind conditional proof is that we can prove a conditional

Accepted Answer

Conditional proof is a method in logic where one assumes the antecedent of a conditional statement to prove its consequent, thereby establishing the truth of the entire conditional statement.

Question 10

To discharge an assumption is to quit using it in our proof.

Accepted Answer

Discharging an assumption means that it is no longer needed as a separate assumption because it has been incorporated into the proof, often by showing that the conclusion follows from it.

Question 11

If an assumption for a conditional proof has been discharged, it cannot be used in subsequent lines of the proof.

Accepted Answer

In a conditional proof, once an assumption is discharged, it is no longer available for use in subsequent lines because it has served its purpose in deriving the conditional statement.

Question 12

An indirect proof is a proof that makes use of assumptions.

Accepted Answer

An indirect proof, also known as proof by contradiction, involves assuming the opposite of what you want to prove and showing that this assumption leads to a contradiction.

Question 13

The basic principle underlying reductio ad absurdum is that whatever implies a contradiction is false.

Accepted Answer

Reductio ad absurdum is a logical argument technique where one assumes a statement to be true and then shows that this assumption leads to a contradiction, thereby proving the original statement to be false.

Question 14

A proof that employs reductio ad absurdum is a direct proof.

Accepted Answer

Reductio ad absurdum is an indirect proof method where the opposite of the statement to be proven is assumed, and this assumption leads to a contradiction, thereby proving the original statement.

Question 15

All theorems are tautologies.

Accepted Answer

A tautology is a statement that is always true, regardless of the truth values of its components. A theorem is a statement that has been proven to be true. Therefore, all theorems are tautologies.

Question 16

A contingency is the negation of a tautology.

Accepted Answer

A contingency is a statement that is neither a tautology nor a contradiction; it can be true or false depending on the truth values of its components.

Question 17

A theorem is a statement that can be proved independently of any premises.

Accepted Answer

The answer of A theorem is a statement that can...

Question 18

Any argument that has a theorem as a premise is valid, regardless of the conclusion.

Accepted Answer

An argument's validity depends on its logical structure, not the truth of its premises. A theorem as a premise does not guarantee a valid argument if the conclusion does not logically follow from the premises.

Question 19

(A • (A → B)) → B is the corresponding conditional for the argument A, A → B \ B.

Accepted Answer

The formula (A • (A → B)) → B is a tautology, meaning it is always true, which corresponds to the argument form A, A → B \ B being valid.

Question 20

A symbolic argument is valid if and only if its corresponding conditional is a theorem.

Accepted Answer

A symbolic argument is valid if the conclusion logically follows from the premises, which means the conditional statement formed by the premises leading to the conclusion is a theorem in a logical system.

Quiz 8: Statement Logic: Proofs