Rules of Inference and Replacement

Rules of Inference and Replacement

100

 P ⊃ Q 

P 

∴ Q

Modus Ponens

200

P ⊃Q

Q ⊃ R

∴ P ⊃ R

Hypothetical Syllogism

300

P

Q

∴ P • Q

Conjunction

400

(P ⊃ Q) • (R ⊃ S) 

P v R

∴ Q ∨ S

Constructive Dilemma

500

(P ⊃ R) • (Q ⊃ S) 

P v Q

∴ R ∨ S

Constructive Dilemma

Replacement Rules

100

~ (P • Q) becomes ~P ∨ ~Q

De Morgan’s Theorems

200

~ ~ P becomes P

Double Negation

300

(P ⊃ Q) becomes (~P ∨ Q)

Material Implication

400

~ (P ∨ Q) becomes (~P • ~Q)

De Morgan’s Theorems

500

P ∨ (Q • R) becomes (P ∨ Q) • (P ∨ R)

Distribution

Recognize the rule

100

P • Q 

∴ P

Simplification

200

P ∨ Q 

~ P

∴ Q  

Disjunctive Syllogism

300

P ⊃ Q 

~ Q 

∴ ~ P

Modus Tollens

400

(P ≡ Q) ≡ [(P ⊃ Q) • (Q ⊃ P)]

Material Equivalence

500

P • (Q ⊃ R) 

∴ P

Simplification

Name the Rule

100

From (P) derive P ∨ Q

Addition

200

P • Q becomes Q • P

Commutation

300

(P • Q) • R becomes P • (Q • R)

Association

400

P ⊃ Q becomes ~Q ⊃ ~P

Transposition

500

P • P becomes P

Tautology

Determine the rule

100

P ⊃ Q 

Q

∴ P

Affirming the Consequent

200

P ⊃ Q 

~P 

∴ ~Q

Denying the Antecedent

300

P ∨ Q 

P 

∴ Q

Affirming a Disjunct

400

P ∨ Q 

Q

∴ P

Affirming a Disjunct

500

P ⊃ Q 

Q ⊃ R

R

 ∴ P

Affirming the Consequent (chained reasoning)