Mathematical Logic

Truth tables, proofs, and predicates.

Controls

Formula (p, q, r, &&, ||, !, ->, <->)

Browse the propositional logic laws on the right.

Premises (one per line, p, q, r) Conclusion

Domain (comma-sep integers) Predicate P(x) (JS expression)

Inference Rules

MP: p→q, p ⊢ q

MT: p→q, ¬q ⊢ ¬p

HS: p→q, q→r ⊢ p→r

DS: p∨q, ¬p ⊢ q

Quantifier Laws

Results

∀x P(x)?—

∃x P(x)?—

Mathematical Logic & Truth Tables Handbook

Mathematical Principles & Theorems

Investigates propositional and Boolean logic. Operators: Negation (\(\neg\)), Conjunction (\(\land\)), Disjunction (\(\lor\)), Implication (\(P \implies Q \equiv \neg P \lor Q\)), and Biconditional (\(P \iff Q\)). Formulas are classified as Tautologies (valid in all valuations), Contradictions (unsatisfiable), or Contingencies. Transformation to Conjunctive Normal Form (CNF) and Disjunctive Normal Form (DNF) via distributive laws and De Morgan's identities.

Operating Instructions

Enter a logical proposition using propositional variables (P, Q, R) and operator buttons.
Click Generate Truth Table to evaluate all \(2^n\) truth assignments.
Inspect truth table column valuations, main connective evaluation, and tautology checks.
View simplified equivalent expressions and CNF/DNF canonical expansions.