Language of logic

It is essential to distinguish logic as language from logic as deduction or calculus.
For instance, Frege who began the revival of Leibniz's project, assigns to his Begriffsschrift an aim, that is not merely formalization for checking the correctness of arguments or reasonings, but also to express adequately (not as natural language does) the various thoughts.
He claims that, unlike Boole's, his logic is not not merely a calculus ratiocinator, but a lingua characterica in Leibniz's sense.
When the language of first-order predicate logic and its extensions is treated this way as a foreign language, formalisation, interpretation or representation of English sentences are simply translations in this new language.
This can be done on an intuitive basis, as for any other languages, or in an automatic way, as in Montague or Kamp's discourse representation theory.
Vocabulary
Non-logical vocabulary
* propositional symbols, predicate and function symbols, of given arity.
Logical vocabulary
* variables (free and bound);
*logical constants (including quantifiers, tense, modal, deontic, relevant operators, etc.);
* punctuation marks, like parentheses, brackets, dots...
Syntax
Terms are build up with function symbols and variables;
Atomic formulae are build up with terms and predicate symbols;
Formulae are build up with atomic symbols and logical constants.
Sentences are formulae without free variables.
Examples
Universe of discourse: human beings. <math>L</math> binary predicate symbol meaning ...loves....
<math>\forall x\exists yLxy </math>= Everyone loves someone.
<math>\exists y\forall xLxy </math>= Somebody is loved by everyone.
<math>\forall x\exists yLyx </math>= Everyone is the beloved of someone.
<math>\exists y\forall xLyx </math>= Somebody loves everyone.

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