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 ```bash uv run python -m runtime.harness scripts/z3_solve.py sat "And(p, Implies(p, q), Not(q))" ``` ### Z3_Tautology ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "Implies(And(p, Implies(p, q)), q)" ``` ### Sympy_Truthtable ```bash uv run python -m runtime.harness scripts/sympy_compute.py truthtable "p & (p >> q) >> q" ``` ### Z3_Modus_Ponens ```bash 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.