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.


You can specify the language of a theory completely by saying that it is "the language of predicate calculus over such-and-such signature" -- assuming that your idea of "signature" includes the names of the predicates. On the other hand, the information contained in a "language" must be enough to allow us to reconstruct the signature.

So distinguishing between the language and the signature is just a matter of pedantry. (Of course this doesn't make it a bad thing; judiciously applied pedantry is the building block of mathematics!)

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.

(There's also the technical point that many constructions while working with a formal system are simpler if we can assume that we just know what the possible symbols are instead of needing to figure it out from contextual clues at each step. That could probably be worked around if we really wanted to, though).

Added later: Finally, for ideas such as that of a complete theory to make good sense, one needs to distinguish between symbols that are not part of the language at all and symbols that are in the language but just happen never to be mentioned by the axioms.


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