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Let $A$ be a structure, $T$ a theory (a set of sentences) and $F$ a sentence, where "sentence" means a formula with no free variables.
1) if $A \models T$ and $A \models F$, is it true that $T \models F$?
2) if $T_0 \subseteq T$ and $A \models T$, is it true that $A \models T_0$?
These questions may be trivial for you, but I am a beginner and very confused.

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1 Answer 1

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Statement (1) is false: Let A = { R, +, ., < } (i.e the real numbers with addition, multiplication, and linear ordering), T = <the set of all true sentences over R involving only the '<' operator>, and F = <any true sentence over R involving only the '+' operator>. Then T is valid in A, F is valid in A, but T doesn't establish anything about F.

Statement (2) is true: If all sentences of T are valid in A, certainly the sentences of any subset of T must be valid in A.

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  • $\begingroup$ What if $F$ is defined as the AND of all sentences in $T$? In this case statement 1) would be true? $\endgroup$
    – 7iat
    Sep 15, 2016 at 17:04
  • $\begingroup$ Or better, if $F$ is defined as the AND of all sentences in $T$, does it follow that $A \models F$ and $T \models F$? $\endgroup$
    – 7iat
    Sep 15, 2016 at 17:33

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