Here are the symbols of various logical operators as I use them on this website (there is no one, universal list of symbols, though certain norms are adhered to across the spectrum):

∧ = And

∨ = Or

→ = If…then

¬ = Not (negation)

≡ = Logical Equivalence

Here are the most common inferences in symbolic logic:

*Modus Ponens (MP)*

P → Q

P … Therefore, Q

*Modus Tollens (MT)*

P → Q

¬Q … Therefore, ¬P

*Hypothetical Syllogism (HS)*

P → Q

Q → R … Therefore, P → R

*Disjunctive Syllogism (DS)*

P ∨ Q

¬P… Therefore, Q

*Simpliﬁcation (SIMP)*

P ∧ Q … Therefore, P

*Addition (ADD)*

P … Therefore, P ∨ Q

*Double Negation (DN)*

P … Therefore, ¬¬P

*Conjunction (CONJ)*

P

Q … Therefore, P ∧ Q

*Constructive Dilemma (CD)*

(P → Q) ∧ (R → S)

P ∨ R … Therefore, Q ∨ S

*Destructive Dilemma (DD)*

(P → Q) ∧ (R → S)

¬Q ∨ ¬S … Therefore, ¬P ∨ ¬R

Finally, here are the 9 logical equivalences:

*De Morgan’s Theorem (DeM)*

¬(P ∧ Q) ≡ ¬P ∨ ¬Q

¬(P ∨ Q) ≡ ¬P ∧ ¬Q

*Commutation (Comm)*

(P ∨ Q) ≡ (Q ∨ P)

(P ∧ Q) ≡ (Q ∧ P)

*Association (Assoc)*

[P ∨ (Q ∨ R)] ≡ [(P ∨ Q) ∨ R]

[P ∧ (Q ∧ R)] ≡ [(P ∧ Q) ∧ R]

*Distribution (Dist)*

[P ∧ (Q ∨ R)] ≡ [(P ∧ Q) ∨ (P ∧ R)]

[P ∨ (Q ∧ R)] ≡ [(P ∨ Q) ∧ (P ∨ R)]

*Double Negation (DN)*

¬¬P ≡ P

*Transposition (TRANS)*

P→Q ≡ ¬Q→¬P

*Material Implication (IMP)*

P→Q ≡ ¬P ∨ Q

*Material Equivalence (Equiv)*

P≡Q ≡ [(P→Q) ∧ (Q→P)]

P≡Q ≡ [(P ∧ Q) ∨ (¬P ∧ ¬Q)

*Exportation (Exp)*

[(P ∧ Q)→R] ≡ [P→(Q→R)]

*Tautology (Taut)*

P ≡ (P ∧ P)

P ≡ (P ∨ P)