Examples of Predicate Logic Proofs

Example 1 Argument: ( ∀ x ∈ D, P(x)) ˅ ( ∃ y ∈ D, Q(y)) ∃ z ∈ D, ~P(z) ∃ w ∈ D, Q(w) Proof: 1. ( ∀ x ∈ D, P(x)) ˅ ( ∃ y ∈ D, Q(y)) premise 2. ∃ z ∈ D, ~P(z)premise 3. ~ ∀ z ∈ D, P(z)2, gen. De Morgan or quad. negation 4. ~ ∀ x ∈ D, P(x) 3, var, change (may skip) 5. ∃ y ∈ D, Q(y) 1,4, elimination 6. ∃ w ∈ D, Q(w)5, var. change

Example 2 Argument: ∀ x ∈ D, P(x) → Q(x) ∀ y ∈ D, Q(y) → R(y) ∀ z ∈ D, ~R(z) → ~P(z) Proof: 1. ∀ x ∈ D, P(x) → Q(x)premise 2. ∀ y ∈ D, Q(y) → R(y)premise 3. P(g) → Q(g)1, univ. inst. g –a generic element of D 4. Q(g) → R(g)2, univ. inst. 5. P(g) → R(g)3,4, trans. of → 6. ~R(g) → ~P(g)5, contrapositive 7. ∀ z ∈ D, ~R(z) → ~P(z)6, univ. gen.

Example 3 Argument: ∀ x ∈ D, P(x) → Q(x) ∃ y ∈ D, Q(y) → R(y) ∃ z ∈ D, ~R(z) → ~P(z) Proof: 1. ∀ x ∈ D, P(x) → Q(x)premise 2. ∃ y ∈ D, Q(y) → R(y)premise 3. Q(w) → R(w)2, exist. inst. w – not used before 4. P(w) → Q(w)1, univ. inst. 5. P(w) → R(w)3,4, transitivity 6. ~R(w) → ~P(w)5, contrapositive 7. ∃ z ∈ D, ~R(z) → ~P(z)6, exist. gen.