Logic Overview - Rules Of Inference

Front 1

Modus Ponens (M.P.)

Back 1

p⊃q
p
∴q

Front 2

Modus Tollens (M.T.)

Back 2

p⊃q
~q
∴p

Front 3

Hypothetical Syllogism (H.S.)

Back 3

p⊃q
q⊃r
∴p⊃r

Front 4

Disjunctive syllogism (D.S.)

Back 4

p∨q
~p
∴q

Front 5

Constructive Dilemma (C.D.)

Back 5

(p⊃q) ・(r ⊃s)
p∨r
∴q∨s

Front 6

Absorption (Abs.)

Back 6

p⊃q
∴p⊃(p・q)

Front 7

Simplification (Simp.)

Back 7

p・q
∴p

Front 8

Conjunction (Conj.)

Back 8

p
q
∴p・q

Front 9

Addition (Add.)

Back 9

p
∴p∨q

Front 10

De Morgan's Theorems (De M)

Back 10

~(p・q) ≡ (~p∨~q)
~(p∨q) ≡ (~p・~q)

Front 11

Commutation (Com.)

Back 11

(p∨q) ≡ (q∨p)
(p・q) ≡ (q・p)

Front 12

Association (Assoc.)

Back 12

[p∨(q∨r)] ≡ [(p∨q)∨r]
[p・(q・r)] ≡ [(p・q)・r]

Front 13

Distribution (Dist.)

Back 13

[p・(q∨r)] ≡ [(p・q)∨(p・r)]
[p∨(q・r)] ≡ [(p∨q) ・(p∨r)]

Front 14

Double Negation (D.N.)

Back 14

p ≡ ~~p

Front 15

Transposition (Travs.)

Back 15

(p⊃q) ≡ (~q⊃~p)

Front 16

Material Implication (Impl.)

Back 16

(p⊃q) ≡ (~p∨q)

Front 17

Material Equivalence (Equiv.)

Back 17

(p ≡ q) ≡ [(p⊃q)・(q⊃p)]
(p ≡ q) ≡ [(p・q)∨(~p・~q)]

Front 18

Exportation (Exp.)

Back 18

[(p・q)⊃r] ≡ [p ⊃(q⊃r)]

Front 19

Tautology(Taut.)

Back 19

p ≡ (p∨p)
p ≡ (p・p)