Problem 1. Determine whether φ ! ( ! χ); : φ. If
this is an entailment, show that it is. If it is not, construct a
countermodel to show that it is not. Show all relevant work.
Problem 2. Let’s introduce $ to our language as an abbreviation
φ $ =df ((φ ! ) ^ ( ! φ))
Show that, for any valuation V, V (φ $ ) iff: V φ iff
V . (That is, V (φ $ ) iff φ and have the same truth
value in V.) You may use the derived truth-conditions for ! in
Problem 3. What truth function does ^ express?
Problem 4. Show that, for any A and φ, if A [ :φ is unsatisfi
able then A φ.
Problem 5. Suppose B0; B1; : : : ; Bk; : : : is the sequence of finitely
satisfiable extensions of A in the construction used in the
direct proof of compactness. We showed in class that each Bk
in this chain is finitely satisfiable. Show that B = B0[· · ·[Bk[: : :
must also be finitely satisfiable.2
Problem 6. Let VB be the valuation we constructed in the direct
proof of compactness based on the opinionated finitely satisfi
able extension B of A. Show that VB (φ ^ ) iff (φ ^ ) 2 B.
Problem 7. Construct a formal derivation showing that
p ! :q; q :p
Problem 8. Construct a formal derivation showing that
:(p _ q) :p ^ :q
Problem 9. Show that for any A and any φ; : if A; φ then
A φ ! .
Problem 10. Show that the proof rule _E is entailment preserv
ing. That is, show that if: (i) A φ _ , (ii) B; φ χ, and (iii)
C; χ, then (iv) A; B; C χ.
Problem 11 (Extra credit). Show that Cn(Cn(A)) ⊆ Cn(A).