Methods of Deduction

Question 1

Rules of inference

Answer 1

The rules that permit valid inferences from statements assumed as premises.

Question 2

Natural deduction

Answer 2

A method of proving the validity of a deductive arguments by using the rules of inference.

Question 3

Formal proof of validity

Answer 3

A sequence of statements each of which is either a premise of a given argument, or follows from the preceding statements of the sequence by one of the rules of inference, or by logical equivalence, where the last statement in the sequence is the conclusion of the argument whose validity is proved.

Question 4

Elementary valid argument

Answer 4

Any one of a set of specified deductive arguments that serve as rules of inference and that may therefore be used in constructing a formal proof of validity.

Question 5

9 Elementary Valid Argument Forms

Answer 5

• Modus Ponens (M.P.)
• Modus Tollens (M.T.)
• Hypothetical Syllogism (H.S.)
• Disjunctive Syllogism (D.S.)
• Constructive Dilemma (C.D.)
• Absorption (Abs.)
• Simplification (Simp.)
• Conjunction (Conj.)
• Addition (Add.)

Question 6

Modus Ponens (M.P.)

Answer 6

p ⊃ q
p
therefore q

Question 7

Modus Tollens (M.T.)

Answer 7

p ⊃ q
~q
therefore ~p

Question 8

Hypothetical Syllogism (H.S.)

Answer 8

p ⊃ q
q ⊃ r
therefore p ⊃ r

Question 9

Disjunctie Syllogism (D.S.)

Answer 9

p v q
~p
therefore q

Question 10

Constructive Dilemma (C.D.)

Answer 10

A rule of inference.

One of nine elementary valid argument forms.

Constructive dilemma permits the inference that

if
(p ⊃ q) · (r ⊃ s)
is true,

and
p v r
is also true,

then
q v s
must be true

Question 11

Dilemma

Answer 11

An argument in which one of two alternatives must be chosen.

Question 12

Absorption

Answer 12

p ⊃ q
therefore p ⊃ (p · q)

Question 13

Simplification

Answer 13

p · q
therefore p

Question 14

Conjunction (as a rule of inference)

Answer 14

p
q
therefore p · q

Question 15

Addition (as a rule of inference)

Answer 15

p
therefore p v q

Question 16

Rule of replacement

Answer 16

A rule that permits us to infer from any statement the result of replacing any component of that statement by any other statement that is logically equivalent to the component replaced.

Question 17

10 Logically Equivalent Expressions

Answer 17

• De Morgan’s theorems (De M.)
• Commutation (Com.)
• Association (Assoc.)
• Distribution (Dist.)
• Double Negation (D.N.)
• Transposition (Trans.)
• Material Implication (Impl.)
• Material Equivalence (Equiv.)
• Exportation (Exp.)
• Tautology

Question 18

De Morgan’s theorems

Answer 18

~ ( p · q ) ≡⁺ (~p ▽ ~q )

~ ( p ▽ q ) ≡⁺ (~p · ~q)

Question 19

Commutation

Answer 19

( p ▽ q ) ≡⁺ ( q ▽ p )

( p · q ) ≡⁺ ( q · p )

Question 20

Association

Answer 20

[ p ▽ (q ▽ r) ] ≡⁺ [ (p ▽ q) ▽ r ]

[ p · (q · r) ] ≡⁺ [ (p · q) · r ]

Question 21

Distribution

Answer 21

[ p · (q ▽ r) ] ≡⁺ [ (p · q) ▽ ( p · r ) ]

[ p ▽ (q · r) ] ≡⁺ [ (p ▽ q) · ( p ▽ r ) ]

Question 22

Double Negation

Answer 22

p ≡⁺ ~~p

Question 23

Transposition

Answer 23

( p ⊃ q ) ≡⁺ (~q ⊃ ~p )

If P then Q.
If not Q then not P

Question 24

Material Implication

Answer 24

( p ⊃ q ) ≡⁺ ( ~p ▽ q )

p ⊃ q means that either p is false, or q is true.

Question 25

Material Equivalence

Answer 25

( p ≡ q ) ≡⁺ [ ( p ⊃ q ) · ( q ⊃ p ) ]

( p ≡ q ) ≡⁺ [ ( p · q ) ▽ ( ~p · ~q ) ]

Question 26

Exportation

Answer 26

[ ( p · q ) ⊃ r ] ≡⁺ [ p ⊃ ( q ⊃ r ) ]

Question 27

Tautology

Answer 27

p ≡⁺ ( p ▽ p )
p ≡⁺ ( p · p )