Download 16 Lecture 10.11.23

LGIC 310/MATH 570/PHIL 006 Fall, 2010 Scott Weinstein
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Lecture 10.11.23
We proceed to prove the
Abstract Completeness Theorem: The set of valid sentences of an effectively presented first order language is semi-decidable.
Proof: The proof of the theorem will proceed via a number of lemmas. First
observe that a sentence is valid, if and only if, its negation is unsatisfiable. So it
suffices to show that the set of unsatisfiable sentences of an effectively presented
first order language is semi-decidable. We begin with the
Skolem Normal Form for Satisfiability Lemma: We can effectively
construct for every first order sentence ϕ a universal sentence θ (in an expanded
language) such that ϕ is satisfiable, if and only if, θ is satisfiable. (θ is called
the Skolem normal form for satisfiability of ϕ.)
The lemma is proven by introducing symbols for Skolem functions.
Let θ = ∀x1 . . . ∀xn χ, be a universal sentence with quantifier free matrix χ.
Let H be the set of closed terms in the language of θ (we suppose without loss
of generality that H is nonempty). An H-instance of χ is a sentence of the form
χ(x1 |t1 , . . . , xn |tn ) for some t1 , . . . , tn ∈ H. The H-expansion of θ is the set of
all H-instances of θ.
Expansion Lemma: Let θ be an identity free universal sentence. θ is
satisfiable, if and only if, the H-expansion of θ is (truth functionally) satisfiable.
The lemma is proven by constructing a model whose universe is the set of
closed terms.
Quotient Lemma: Let E be a distinguished binary relation symbol and
let θ be a sentence containing E. There is a universal sentence γ (which depends
on the language of θ) such that θ has a model A in which E A is the identity
relation on A, if and only if, θ ∧ γ is satisfiable.
Construct γ to say that E is an equivalence relation which is a congruence
with respect to the relation and function symbols appearing in θ. Then take A
to be the quotient by E B of a model B for θ ∧ γ.
We are now in a position to describe a semi-decision procedure for unsatisfiability. Let ϕ be a first order sentence. Effectively construct the universal
sentence θ which is the Skolem normal form for satisifiability of ϕ. Next, replace
all occurrences of = in θ with a new distinguished binary relation symbol E
and construct γ to satisfy the conditions of the Quotient Lemma. Let ζ be an
effectively constructed universal sentence equivalent to θ ∧ γ. Observe that ϕ is
satisfiable, if and only if, ζ is satisfiable. By the Expansion Lemma, ζ is satisfiable, if and only if, the H-expansion of ζ is truth-functionally satisfiable. Now,
construct an effective enumeration of the H-expansion of ζ and test the conjunction of ever larger initial segments of this enumeration for truth functional
satisfiablity. It follows at once from the compactness theorem for sentential logic
that this procedure terminates with a “no,” if and only if, the H-expansion of
ζ is truth functionally unsatisfiable.