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If  is a logical language for some logic
 , a set  of formulas with no free variables is called a theory (of
 ). If
 is a first-order logic, then  is called a first-order theory.
We write
 for any formula  if every model
 of
 such that  ,
 .
We write
 is for there is a proof of  from  .
Remark. Let  be an  -structure for some signature  . The theory of  is the set of formulas satisfied by  :
and is denoted by
 .
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