Propositional Logic

Propositional Logic

When to Use

Use this skill when working on propositional-logic problems in mathematical logic.

Decision Tree

1. Identify Formula Structure

   • Classify: tautology, contradiction, or contingent?
   • Main connective: AND, OR, IMPLIES, NOT, IFF?
   • z3_solve.py sat "formula" to check satisfiability

2. Truth Table Method

   • For small formulas (<=4 variables): enumerate all valuations
   • sympy_compute.py truthtable "p & (p -> q) -> q"
   • Tautology = all T, Contradiction = all F

3. Natural Deduction

   • Apply inference rules: Modus Ponens, Modus Tollens
   • Conditional proof: assume antecedent, derive consequent
   • z3_solve.py prove "Implies(And(p, Implies(p,q)), q)"

4. Semantic Tableaux

   • Build tree by decomposing formula
   • Closed branches = contradictions
   • All branches closed = valid argument

Tool Commands

Z3_Sat

uv run python -m runtime.harness scripts/z3_solve.py sat "And(p, Implies(p, q), Not(q))"

Z3_Tautology

uv run python -m runtime.harness scripts/z3_solve.py prove "Implies(And(p, Implies(p, q)), q)"

Sympy_Truthtable

uv run python -m runtime.harness scripts/sympy_compute.py truthtable "p & (p >> q) >> q"

Z3_Modus_Ponens

uv run python -m runtime.harness scripts/z3_solve.py prove "Implies(And(p, Implies(p,q)), q)"

Cognitive Tools Reference

See .claude/skills/math-mode/SKILL.md for full tool documentation.