Computability and Logic
Computability and Logic
Computability and Logic
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
20.1. CRAIG’S THEOREM AND ITS PROOF 263<br />
(S3) is just slightly different. Suppose Ɣ = Ɣ L ∪ Ɣ R is an unbarred division <strong>and</strong><br />
(D 1 ∨ D 2 )isinƔ. Again there are two cases, <strong>and</strong> we consider only the one where<br />
(D 1 ∨ D 2 )isinƔ L . Each of the D i is of course a left sentence, since their disjunction<br />
is. We claim that Ɣ ∪{D i }=(Ɣ L ∪{D i }) ∪ Ɣ R is an unbarred division for at least<br />
one i. Towards showing this, note that if Ɣ L ∪{D 1 } is unsatisfiable, then Ɣ L implies<br />
both (D 1 ∨ D 2 ) <strong>and</strong> ∼D 1 , <strong>and</strong> hence implies D 2 . In this case, the proof that (Ɣ L ∪<br />
{D 2 }) ∪ Ɣ R is an unbarred division is just like the proof of the preceding paragraph.<br />
Similarly, if Ɣ L ∪{D 2 } is unsatisfiable, then (Ɣ L ∪{D 2 }) ∪ Ɣ R gives an unbarred<br />
division. So we are left to treat the case where Ɣ L ∪{D i } is satisfiable for both i.<br />
In this case, (Ɣ L ∪{D i }) ∪ Ɣ R can fail to give an unbarred division only because<br />
there is a sentence B i that bars the pair Ɣ L ∪{D i },Ɣ R . What we claim is that there<br />
cannot exist such B i both both i = 1 <strong>and</strong> i = 2. For suppose there did. Then since<br />
B i is implied by Ɣ L ∪{D i } for each i, <strong>and</strong> Ɣ L contains (D 1 ∨ D 2 ), it follows that<br />
B = (B 1 ∨ B 2 ) is implied by Ɣ L . Moreover, since each ∼B i is implied by Ɣ R ,so<br />
is ∼B. Finally, since each B i is both a left <strong>and</strong> a right sentence, the same is true<br />
of B. Thus there is a sentence B that bars Ɣ L ,Ɣ R , contrary to hypothesis. This<br />
contradiction completes the proof for the case where identity <strong>and</strong> function symbols<br />
are absent.<br />
Proof, Part II: We next consider the case where identity is present but function<br />
symbols are still absent. Suppose A implies C. As in section 18.4, we introduce<br />
the new two-place predicate ≡. We write E L for the conjunction of the<br />
equality axioms <strong>and</strong> the congruence axioms for predicates in A, <strong>and</strong> E R for the<br />
corresponding sentence for C. We write * to indicate replacing = by ≡. Since A<br />
implies C, A & ∼C is unsatisfiable. What the proof of Proposition 19.13 tells us is<br />
that therefore E L & E R & A*&∼C* is unsatisfiable. It follows that E L & A* implies<br />
E R → C*. By the interpolation theorem for sentences without identity, there is a sentence<br />
B* involving only ≡ <strong>and</strong> nonlogical predicates common to A <strong>and</strong> C, such that<br />
E L & A* implies B* <strong>and</strong> B* implies E R → C*. It follows that E L & A*&∼B* <strong>and</strong><br />
E R & B*&C* are unsatisfiable. Then we claim B, the result of replacing ≡ by =<br />
in B*, is the required interpolant between A <strong>and</strong> C. Certainly its nonlogical predicates<br />
are common to A <strong>and</strong> C. Further, what the proof of Proposition 18.13 tell us<br />
is is that A & ∼B <strong>and</strong> B & ∼C are unsatisfiable, <strong>and</strong> therefore A implies B <strong>and</strong> B<br />
implies C. The treatment of function symbols is much the same, but using the machinery<br />
of Proposition 19.12 rather than of Proposition 19.13. Details are left to the<br />
reader.<br />
In the remaining sections of this chapter we apply the interpolation theorem to<br />
prove two results about theories: one about the conditions under which the union<br />
of two theories is satisfiable, the other about the conditions under which definitions<br />
are consequences of theories. (Throughout ‘theory’ is being used, as elsewhere<br />
in this book, in a very broad way: A theory in a language is just a set of<br />
sentences of the language that contains every sentence of that language that is a<br />
logical consequence of the set. A theorem of a theory is just a sentence in that<br />
theory.)