Formal Language vs First Order Language

Question

My book, when speaks about the "First Order Logic" calls it "language". This term can be used also to denote a sub-set of this language or it can only be used to denote things like: -First order language -propositional language -second order language

??

Answer 1

A Formal system is made of:

(i) a language, made of

(i.a) an alphabet: a set (usually finite) of symbols, and

(i.b) a grammar: a set of rules which tell how some expressions are well-formed (i..e. meaningful);

and:

(ii) a proof system (or calculus), made of

(ii.a) a set of "special" formulas: the axioms, and

(ii.b) a set of inference rules.

First-order logic (or predicate calculus) is a proof system based on first-order language, i.e. the language having as logical symbols, in addition to the (propositional) connectives, also the quantifiers and the (individual) variables.

It is called "first-order" because quantification is allowed on individual variables only.

When quantification is allowed also on predicate variables, we call it Higher-order logic.