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