Deck 8: Statement Logic: Proofs

Question 1

p → q, p \ q is the implicational rule

A) modus tollens.
B) modus ponens.
C) constructive dilemma.
D) simplification.

Answer 1

modus ponens.

Question 2

Which of the following states the implicational rule hypothetical syllogism?

A) p → q, q → r \ p → r
B) p ⋁ q, p → r, q → s \ r ⋁ s
C) p ⋁ q, ~p \q
D) p → q, ~q \ ~p

Answer 2

p → q, q → r \ p → r

Question 3

Which of the following states the implicational rule constructive dilemma?

A) p → q, q → r \ p → r
B) p ν q, p → r, q → s \ r ν s
C) p ν q, ~p \q
D) p → q, ~q \ ~p

Answer 3

p ν q, p → r, q → s \ r ν s

Question 4

Which of the following states the implicational rule conjunction?

A) p → q, ~p \q
B) p • q \ p
C) p, q \ p • q
D) p \ p ν q

Question 5

Which of the following arguments is an instance of modus ponens?

A) A,A→~(B•C)\~A
B) C•B,(C•B)→(DνE)\DνE
C) ~B,C→B\~C
D) A•B,(C•D)→(A•B)\C•D

Question 6

Which of the following arguments is an instance of simplification?

A) ~A→~(B•C)\~A
B) C•B,(C•B)→(D↔E)\D↔E
C) ~B,C→B\~C
D) (C•D)•(A•B)\C•D

Question 7

Which rule of inference is exemplified by the argument (R → S) \ (R → S) ν T?

A) conjunction
B) modus ponens
C) addition
D) simplification

Question 8

Which of the following is a correct application of the equivalence rule commutation?

A) A•(BνC) \(A•B)ν(A•C)
B) Aν(B•C)\(AνB)•(AνC)
C) (AνB)•C\C•(AνB)
D) (AνB)•C\C•(BνA)

Question 9

Which of the following is a correct application of the equivalence rule exportation?

A) A→(B•C) \(A→B) • (A→C)
B) A→(B•C)\(A→B)•(A→C).
C) A→(B•C)\(A9B)→C
D) (A→B)•C\C•(B→A)

Question 10

Which of the following is a correct application of the equivalence rule distribution

A) A→(BνC) \(A→B)ν(A→C)
B) Aν(B•C)\(AνB)•(AνC)
C) Aν(B•C)\(A•B)ν(C•B)
D) (A ν B) • C \ (A • B) v (B • C)

Question 11

Which of the following is a statement of one of the two De Morgan's laws?

A) ~(p ν q) :: (~p ν ~q)
B) ~(p • q) :: (~p • q)
C) ~(p • q) :: (~p ν ~q)
D) ~(p ν q) :: (p ν ~q)

Question 12

Which of the following is an instance of the rule of disjunctive syllogism?

A) (~p ν ~q), ~p \ ~q
B) (~p v ~q), ~~p \ ~q
C) ~(p ν ~q), p \ ~~q
D) ~(p ν q), ~p \ ~q)

Question 13

Which of the following is not a statement of one of the two De Morgan's laws?

A) ~(p ν q) :: (~p • ~q)
B) (~p • ~q) :: ~(p ν q)
C) (~p ν ~q) :: (~p • ~q)
D) ~(p ν q) :: (~p ν ~q)

Question 14

p ν ~p states which of the following laws?

A) law of excluded middle
B) De Morgan's laws
C) law of noncontradiction
D) law of exclusive dilemma

Question 15

Which of the following is an expression of the law of noncontradiction?

A) p ν ~p
B) p • ~p
C) ~(p → ~p)
D) ~(p • ~p)

Question 16

Which of the following is an expression of the rule of distribution?

A) (p ν (q • r)):: ((p ν q) • (p ν r))
B) p:: p • p
C) ~(p • q):: (~p ν ~q)
D) ((p • q) → r):: (p → (q → r))

Question 17

Which of the following is an expression of the rule of material equivalence?

A) (p • (q ν r)) :: ((p • q) ν (p • r))
B) ((p • q) → r) :: (p → (q → r))
C) (p ↔ q) :: ((p → q) • (q → p))
D) (p ν (q • r)) :: ((p • q) ν r)

Question 18

p :: (p • p) expresses which rule of equivalence?

A) conjunction
B) distribution
C) commutation
D) redundancy

Question 19

(p → q) :: (~p ν q) expresses which rule of equivalence?

A) material implication
B) commutation
C) material equivalence
D) contraposition

Question 20

(p → q) :: (~q → ~p) expresses which rule of equivalence?

A) material implication
B) commutation
C) material equivalence
D) contraposition

Question 21

Which of the following is a correct replacement for (D • C) → ~(A ⋁ B) using material equivalence?

A) ((D • C) → ~(A ⋁ B)) • ((~(A ⋁ B) → (D • C))
B) (D•C)↔~(A⋁B)
C) ~(D•C)ν~(A⋁B)
D) (A⋁B)→~(D•C)

Question 22

The basic idea behind conditional proof is that

A) we can prove anything true if given the appropriate assumptions.
B) we can prove a conditional true by assuming that its antecedent is true and showing that its consequent . can be derived from that assumption.
C) a proof is always conditioned on acceptance of the law of excluded middle.
D) conditional statements can only be proved indirectly-that is, by making an assumption.

Question 23

The guiding principle of reductio ad absurdum is that

A) whatever implies a contradiction is false.
B) systems of natural deduction often lead to absurd conclusions.
C) it is not always possible to reduce a set of statements to absurdity.
D) often the most straightforward method for deriving a contradiction is by assuming an absurdity.

Question 24

A theorem is a statement that

A) can be false only under circumstances specified by the appropriate theory.
B) is true only within the context of a specific theory.
C) can be proven independently of any premises.
D) guarantees the validity of an argument of which it is a premise.

Question 25

Which of the following is a correct application (or are correct applications) of the rule Contraposition to ~L → ~M?

A) L → M
B) M → L
C) ~~M → ~~L
D) both M → L and ~~M → ~~L

Question 26

Which of the following is a correct application (or are correct applications) of the rule Material Implication to ~L → ~M?

A) ~~LνM
B) M ν L
C) ~~M ν ~~L
D) both M ν L and ~~M ν ~~L

Question 27

Using which rule can one infer (A ↔ B) using the following statements: (B → A) and (A → B)?

A) material implication
B) exportation
C) material equivalence
D) simplification

Question 28

Using which rule can one infer (A ν (B • C)) using the following statement: (A ν B) • (A ν C)?

A) simplification
B) distribution
C) association
D) redundancy

Question 29

Using which rules can one infer B using the following statements: (A → B), (C → B), and (C ν A)?

A) hypothetical syllogism and simplification
B) material equivalence and addition
C) constructive dilemma and redundancy
D) disjunctive syllogism and material implication

Question 30

A proof is a series of steps that show how the premises lead, by way of valid rules of inference, to the conclusion.

Question 31

Implicational rules of inference may, within truth-functional logic, be validly applied to parts of compound statements or lines in a proof.

Question 32

The rule of conjunction indicates that if we have a conjunction, you may validly infer either conjunct.

Question 33

The rule of modus ponens indicates that if we have a conditional statement and the affirmation of its consequent, we may validly infer its antecedent.

Question 34

The rule of addition indicates that if we have a disjunction, we may validly infer the denial of one of its disjuncts.

Question 35

The rule of modus tollens indicates that if we have a conditional and then denial of its antecedent, we may validly infer the denial of its consequent.

Question 36

A single line in a proof may constitute more than one application of a rule or rules of inference.

Question 37

Hypothetical syllogism is an implicational rule.

Question 38

Conjunction is an equivalence rule.

Question 39

We can apply equivalence rules to parts of lines in a proof as well as to entire lines.

Question 40

The rule of commutation applies to conditionals.

Question 41

The rule of exportation applies to disjunctions.

Question 42

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

Question 43

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

Question 44

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

Question 45

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

Question 46

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

Question 47

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

Question 48

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

Question 49

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

Question 50

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

Question 51

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

Question 52

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

Question 53

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

Question 54

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

Question 55

All theorems are tautologies.

Question 56

A contingency is the negation of a tautology.

Question 57

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

Question 58

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

Question 59

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

Question 60

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