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1. Suppose φ1 , . . . , φn is a deduction of Ax from Φ where A is a unary predicate symbol
and the variable x does not occur free in Φ. Prove that Φ ` ∀xAx by induction on n.
In your proof you may use all tautologies plus the axioms
(a) ∀x(α → β) → (∀xα → ∀xβ)
(b) α → ∀xα, for x not free in α
(c) ∀x1 ∀x2 . . . ∀xn α for any variables x1 . . . xn and any axiom α of the Predicate
Calculus.
2. Show, by using the BSD (Basic Semantic Definition):
a. α → ∃xβ and ∃x(α → β) are logically equivalent if x does not occur free in α.
b. ∃x(Ax → ∀xA) is true in all L-structures.
c. ∃x(∃xAx → Ax) is true in all L-structures.
3. Show that
(a) ` ∀x∀yP xy→∀y∀xP yx
(b) ` Qy ↔ ∀x(x ≈ y → Qx)
(c) ` ∀x(α → β) → (∃xα → ∃xβ)
You may not use completeness of the first-order calculus, but you may use the Deduction Theorem, Generalization over a variable, Reductio, etc., to show that there
is a deduction.
4. The axiom groups of First Order Predicate Logic with Identity, F OL =, are all
universal generalisations of formulas of the following form:
1.
2.
3.
4.
5.
6.
All tautologies
∀xα → αtx where t is substitutable for x in α
∀x(α → β) → (∀xα → ∀xβ)
α → ∀xα where x does not occur free in α
x≈x
x ≈ y → (αxz → αyz )
Let ` be the logical consequence relation of F OL = obtained from these axiom groups
together with the inference rule Modus Ponens.
(a) What does it mean that t is ‘substitutable’ for x in α?
(b) Show that if ` α → β then ` ∀xα → ∀xβ
(c) Show (by induction on the length of a proof) that if Γ ` α and x does not occur
free in any formula of Γ then Γ ` ∀xα
TURN OVER
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5. Let s1 and s2 be (valuation) functions from the set of variables V of a first order
language into |A|, the domain of structure A. Let s1 and s2 agree on all the free
variables of the formula φ.
(a) Show, by induction on the complexity of t, that if t is a term of φ then s1 (t) =
s2 (t).
(b) Show, by induction on the complexity of φ, that A φ[s1 ] iff A φ[s2 ].
6. Let Γ0 be a consistent set of sentences in a language containing only sentential connectives and atomic proposition symbols (i.e. the language of sentential logic). Show
that Γ0 can be extended to a maximal consistent set of sentences Γ0 . [Hint: consider
an enumeration of all sentences of the language, then construct a set Γ0 in stages
Γ0 , Γ1 , . . . such that Γ0 is the union of all the Γi . Then verify that Γ0 is indeed
maximal and consistent.]
7. Let Γ be a set of sentences of a countable first order language L and let ` be the
familiar logical consequence relation of first order logic. Let Lc be obtained from L by
the addition of an ordered countable infinity of new constant symbols {c1 . . . cn . . .}.
Suppose also that we have constructed for every formula φ of Lc , a formula θk of the
form
∃xφ → φxcn
where n > k for every (new) constant ck occurring in φ.
(a) Show that if Γ ∪ {θ1 , . . . , θm+1 } is inconsistent then so is Γ ∪ {θ1 , . . . , θm }. [You
may use the facts that if Γ ∪ {α} is inconsistent then Γ ` ¬α , and that if Γ ` αcx
and c does not occur inΓ or α, then Γ ` ∀xα.]
(b) Let Θ be the set of all θn (for all n). Given that Γ is consistent in the new
language Lc conclude that Γ ∪ Θ is consistent. [Hint: remember that if a set ∆
is inconsistent then there is a deduction of a contradiction from finitely many
premises from ∆.]
8. (a) Use the Compactness Theorem for Sentential Logic to prove that “if Σ |= φ,
then there is some finite Σ0 ⊆ Σ such that Σ0 |= φ”.
(b) Assume that every finite subset of Σ is satisfiable. Show that, for any propositional sentence α, the same holds for at least one of Σ ∪ {α} and Σ ∪ {¬α}.
(c) Let ∆ be a set of sentences of Sentential Logic (using only negation ‘¬’ and
implication ‘→’) such that every finite subset of ∆ is satisfiable and for every
sentence φ, φ ∈ ∆ or ¬φ ∈ ∆. Define the truth valuation v by v(P ) = > if
P ∈ ∆ and v(P ) = ⊥ if P 6∈ ∆ for each propositional variable P . Show that for
every sentence φ we have: v(φ) = > iff φ ∈ ∆.
TURN OVER
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9. (a) Show that a consistent theory Σ is complete iff it is maximal among consistent
theories, that is, it is not included in any other consistent theory.
(b) Show that, for any L structure A, Th(A) is a complete theory.
(c) Show that any consistent theory is included in a complete theory.
(d) Show that any complete theory is of the form Th(A) for some A.
END OF PAPER
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