Computability and Logic

14.2. SOUNDNESS AND COMPLETENESS 175
Consider (R2a). We suppose Ɣ ∪{A}⇒ is secure, and consider any interpretation
that makes all the sentences in Ɣ true. What (R2a) requires is that it should
make some sentence in {∼A}∪ true, and we show it does as follows. On the one
hand, if the given interpretation also makes A true, then it makes all the sentences in
Ɣ ∪{A} true, and therefore by the security of Ɣ ∪{A}⇒ makes some sentence in
true, and therefore makes some sentence in {∼A}∪ true. On the other hand, if
the interpretation does not make A true, then it makes ∼A true, and therefore it again
makes some sentence in {∼A}∪ true.
Consider (R2b). We suppose Ɣ ⇒{A}∪ is secure, and consider any interpretation
making all sentences in Ɣ ∪{∼A} true. What (R2b) requires is that it should
make some sentence in true, and we show it does as follows. The given interpretation
makes all sentences in Ɣ true, and so by the security of Ɣ ⇒{A}∪ makes
some sentence in {A}∪ true. But since the interpretation makes ∼A true, it does
not make A true, so it must be that it makes some sentence in true.
For (R3), we suppose that Ɣ ⇒{A, B}∪ is secure, and consider any interpretation
making all sentences in Ɣ true. By the security of Ɣ ⇒{A, B}∪ the interpretation
makes some sentence in {A, B}∪ true. This sentence must be either A or B or
some sentence in . If the sentence is A or B, then the interpretation makes (A ∨ B)
true, and so makes a sentence in {(A ∨ B)}∪ true. If the sentence is one of those
in , then clearly the interpretation makes a sentence in {(A ∨ B)}∪ true. So in
any case, some sentence in {(A ∨ B)}∪ is made true, which is what (R3) requires.
For (R4), we suppose that Ɣ ∪{A}⇒ and Ɣ ∪{B}⇒ are secure, and consider
any interpretation that makes all sentences in Ɣ ∪{(A ∨ B)} true. The interpretation
in particular makes (A ∨ B) true, and so it must either make A true or make B true.
In the former case it makes all sentences in Ɣ ∪{A} true, and by the security of
Ɣ ∪{A}⇒ it makes some sentence in true. Similarly in the latter case. So in
either case it makes some sentence in true, which is what (R4) requires.
For (R5), we suppose that Ɣ ⇒{A(s)}∪ is secure and consider any interpretation
that makes all sentences in Ɣ true. By the security of Ɣ ⇒{A(s)}∪ it makes
some sentence in {A(s)}∪ true. If the sentence is one in , then clearly the interpretation
makes some sentence in {∃xA(x)}∪ true. If the sentence is A(s), then the
interpretation makes ∃xA(x) true, and so again the interpretation makes some sentence
in {∃xA(x)}∪ true. This suffices to show that Ɣ ⇒{∃xA(x)}∪ is secure,
which is what (R5) requires.
For (R6), we suppose that Ɣ ∪{A(c)}⇒ is secure and consider any interpretation
making all sentences in Ɣ ∪{∃xA(x)} true. Since the interpretation makes ∃xA(x)
true, there is some element i in the domain of the interpretation that satisfies A(x). If
c does not occur in Ɣ or or A(x), then while leaving the denotations of all symbols
that occur in Ɣ and and A(x) unaltered, we can alter the interpretation so that the
denotation of c becomes i. By extensionality, in the new interpretation every sentence
in Ɣ will still be true, i will still satisfy A(x) in the new interpretation, and every
sentence in will have the same truth value as in the old interpretation. But since i
is now the denotation of c, and i satisfies A(x), it follows that A(c) will be true in
the new interpretation. And since the sentences in Ɣ are still true and A(c) isnow
true, by the security of Ɣ ∪{A(c)}⇒, some sentence in true will be true in the