Math 430: Problem Set II
due: Friday, 2/3
Proposition 0.1. Fix narbitrary propositional formulas ϕ1, . . . , ϕn. If a formula ψ=
ψ(A0, . . . , An−1)in propositional symbols {A0, . . . , An−1}is a tautology, then so is ψ∗=
ψ(ϕ1, . . . , ϕn), the formula obtained by substituting the entire formula ϕifor the propositional
symbol, Ai−1.
1. By Prop 0.1 above, to verify that (ϕ1→ϕ1) is a tautology, for an arbitrary formula
ϕ1, it suffices to verify that (A0→A0) is a tautology. Using this fact, verify that the
following groups of axioms are tautologies:
(a) Group I(1) Axioms
(b) Group IV(1) Axioms
Definition 0.2. Say that two propositional formulas ϕ0, ϕ1are logically equivalent if for any
truth assignment ν,ν(ϕ0) = ν(ϕ1).
2. Show that (A0→A1) and ((¬A0)∨A1) are logically equivalent. (Hint: by Theorem
1.16 you only need to worry about the propositional symbols occurring in both for-
mulas, which is only a finite number. Thus, you may generate all relevant possibilities
using truth-tables.)
3. Let Γ be any set of propositional formulas. Prove the following claim or refute it with
a counterexample: If s=hs1, . . . , sniis a Γ-proof and t=ht1, . . . , tniis a Γ-proof,
then s+tis a Γ-proof.
4. Consider the proof system that mirrors the one given in Definition 1.25 except that
it admits one more group of axioms: (ϕ1→ϕ2). Is this new proof system sound?
Prove or refute with a counterexample.
5. Prove using the Soundness Theorem (Lemma 1.27): If a set of formulas Γ is satisfiable,
then it is consistent. [Hint: if Γ is inconsistent, it proves any formula ψ, in particular,
it proves ψ:= ((A0)∧(¬A0)). Now show that Γ is not satisfiable (see Definition
1.17.)]
6. Lemma 1.32 suggests that for a consistent set Γ ⊆ L0and a formula ϕ∈ L0, it is
possible that Γ ∪ {ϕ}and Γ ∪ {(¬ϕ)}are both consistent.
Consider Γ := {(¬A0)}. Find a formula ϕthat makes both Γ∪ {ϕ}and Γ ∪ {(¬ϕ)}
consistent (there are many options.) Use Problem 5. to verify that your example
works.
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