Prove the principle of replacement for LPC:

PHIL 322—Modal Logic

Homework 5

Friday, November 30

1. Prove the principle of replacement for LPC:

Theorem 0.1 (Principle of Replacement). Let α be any wff, x and y any variables, 〈D,V 〉 any model, and µ any assignment. Then, where ρ is just like µ except that ρ(x) = µ(y), then Vρ(α) = Vµ(α[y/x]).

2. Prove the principle of agreement for LPC:

Theorem 0.2 (Principle of Agreement). If µ and ρ are two assignments that agree on all the free variables in a formula α, then Vµ(α) = Vρ(α).

3. Assuming constant domain semantics, for each formula give a validity proof if it is valid and a countermodel if it is invalid. Indicate whether and how the formulas status would change with varying domains.

a. ♦∀xFx→ ∃x♦Fx

b. �∀x(Fx→ Gx) → (∀x�Fx→ �∀xGx)

c. ∀x(�Fx ∨�Gx) → �∀x(Fx ∨Gx)

d. ♦∀xFx→ ∃x♦Fx

e. (�∀x(Fx→ Gx) ∧ ♦∃xFx) → ♦∃xGx

f. ∃x♦Rzx→ ♦�∃x∃yRxy