# Relations of language/theory/signature

Say that the language of the first order logic is the collection of symbols that can be used in the formulas + the grammar (the rules that specify how they can be combined)?

1) However, the signature of the system can include additional symbols to be used in the formula. So, it is wrong to say that a formula can be constructed only from the language of predicate logic, because it needs additional symbols from the signature.

2) When specifying a theory, we need to include a signature that models the domain of the theory? So that the theory will be defined as $\mathfrak{T} = (\Sigma, \Gamma)$, where $\Sigma$ is it's signature and $\Gamma$ the collection of formulas of the signature.

Example. The language of predicate logic includes all the standard symbols: $\forall$, $\land$, .... Say that our signature is the signature for Peano Arithmetic, composed of $\Sigma = \{ \times, +, 0, \dots \}$. A formula $\forall x(x + 0 = x)$, can't be composed only from the language of predicate logic, it need's the signature $\Sigma$ too. And the theory of Peano Airthmetic too will need this signature.

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Specifying a language as part of a theory is also mostly a matter of general tidyness -- it's good form to explicitly define your notation before you start using them, even if the only definition you give is "an uninterpreted $n$-ary predicate/function which has no meaning except what the axioms imply". Telling the reader that he's not supposed to use any preexisting knowledge of the symbol is valuable information for him.