Symbol Key, Inferences, and Equivalences

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)

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