predicate-logic

# Predicate Logic ## When to Use Use this skill when working on predicate-logic problems in mathematical logic. ## Decision Tree 1. **Quantifier Analysis** - Identify: ForAll (universal), Exists (existential) - Scope of quantifiers and free/bound variables - `z3_solve.py prove "ForAll([x], P(x)) implies P(a)"` 2. **Prenex Normal Form** - Move all quantifiers to front - Standardize variables to avoid capture - `sympy_compute.py simplify "prenex(formula)"` 3. **Skolemization (for Exists)** - Replace existential quantifiers with Skolem functions - Exists x. P(x) -> P(c) or P(f(y)) depending on scope - Needed for resolution-based proofs 4. **Resolution Proof** - Convert to CNF, negate conclusion - Apply resolution rule until empty clause or saturation - `z3_solve.py prove "resolution_valid"` 5. **Model Theory** - Construct countermodel to refute invalid argument - Finite model for finite domain - `z3_solve.py model "Exists([x], P(x) & Not(Q(x)))"` ## Tool Commands ### Z3_Forall ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "ForAll([x], Implies(P(x), Q(x)))" ``` ### Z3_Exists ```bash uv run python -m runtime.harness scripts/z3_solve.py sat "Exists([x], And(P(x), Not(Q(x))))" ``` ### Z3_Universal_Instantiation ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "Implies(ForAll([x], P(x)), P(a))" ``` ### Z3_Model ```bash uv run python ...