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Logic Symbols with Examples

April 14, 2025•692 words

Logic Symbols with Examples

1. Propositional Logic

Symbol Name Meaning Example
¬ Negation Not ¬P: Not P
∧ Conjunction And P ∧ Q: P and Q
∨ Disjunction Or P ∨ Q: P or Q
⊕ Exclusive Or Either but not both P ⊕ Q: P or Q, but not both
→ Implication If..., then... P → Q: If P, then Q
⇐ Converse Q if P Q ⇐ P: Q if P
⇔ Biconditional If and only if P ⇔ Q: P if and only if Q
⊤ Tautology Always true P ∨ ¬P: tautology
⊥ Contradiction Always false P ∧ ¬P: contradiction

2. Predicate Logic (First-Order Logic)

Symbol Name Meaning Example
∀ Universal quantifier For all ∀x (Human(x) → Mortal(x))
∃ Existential quantifier There exists ∃x (Prime(x) ∧ Even(x))
∄ Non-existence There does not exist ∄x (x < 0 ∧ Natural(x))
∃! Unique existence Exactly one ∃!x (x² = 4): x = 2 or x = –2
: Such that Condition on variable {x ∈ ℕ : x is even}
= Equality Same value 3 = 3
≠ Inequality Different values 5 ≠ 4

3. Set-Theoretic Logic Symbols (in FOL)

Symbol Name Meaning Example
∈ Element of x is in a set 2 ∈ {1, 2, 3}
∉ Not in x is not in a set 5 ∉ {1, 2, 3}
⊆ Subset A is subset of B {1,2} ⊆ {1,2,3}
⊂ Proper subset A is subset and not equal to B {1,2} ⊂ {1,2,3}
𝒫(A) Power set All subsets of A 𝒫({1,2}) = {∅, {1}, {2}, {1,2}}

4. Proof Theory and Sequent Calculus

Symbol Name Meaning Example
⊢ Syntactic entailment Provable from P ⊢ Q: Q follows from P
⊨ Semantic entailment True in all models P ⊨ Q: Q is logically implied
⊢ φ Theorem φ is provable ⊢ P ∨ ¬P: Law of excluded middle
⟹ Derives Derivation (in some systems) Γ ⟹ φ: φ derived from Γ

5. Modal Logic

Symbol Name Meaning Example
□ Necessity (Box) Necessarily true □P: P is always true
◇ Possibility Possibly true ◇P: P might be true

6. Temporal Logic (used in CS, model checking)

Symbol Name Meaning Example
G Globally Always in the future G(P): P holds at all future times
F Finally Eventually F(P): P will hold at some point
X Next In the next state X(P): P holds next
U Until P holds until Q P U Q: P until Q becomes true

7. Miscellaneous

Symbol Name Meaning or Use Example
:= Definition Define left side using right side A := {x ∈ ℕ : x is prime}
( ) Parentheses Grouping ¬(P ∨ Q)
{ } Set brackets Set definition {1, 2, 3}
⟨ ⟩ Angle brackets Tuples or sequences ⟨x, y⟩ ∈ ℝ²
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