:First of all, please see a short disclaimer on my . 
 :# What is formal logic in the first sentence? Is this a way to leave aside the common-sense logic, or does it mean here non-mathematical interpretation of mathematical logic? Is the term "formal logic" an original invention? Anyway, i cannot propose anything better. 
 :# Formal language is a very general concept, see the article where it is linking. For example, the set of all words using only the letter "a" ("a", "aa", "aaa", "aaaa",...) is a formal language. It seems that here a formal first-order language is meant. However, sentential letters, i think, are not used in first-order languages. 
 :# Be aware that sometimes (in mathematical logic at least) the empty domain is allowed. For example, the sentence  is satisfied even if the domain is empty, and the sentence  is satisfied only if the domain is empty. 
 :# It is not good to say that "The term interpretation is synonymous with the term structure". I would suggest a milder formulation: "In mathematical context, the term interpretation may be thought of as a synonym of the term structure". Or better, to remove this altogether. First of all, structure is linking to a mathematical notion, while the context of this article is supposed to be more general than mathematical logic. Second, i do not believe that interpretations of sentential letters are ever used in structures. Third, in any case, interpretation is no more synonymous to structure, than relation is synonymous to subset of Cartesian power, which some people prefer to treat differently. 
 :# The term interpretation is definitely not a synonym of signature, i am removing this. 
 :# I do not understand the sentence about model for languages being synonymous to interpretation for formal languages. What are these non-formal languages anyway? 
 :# I do not understand the sentence about predicate/property and extensional/intensional. I am simply not familiar with the terms. 
 :All in all, it seems that the introduction deals now with propositional and first-order logics in mathematical sense (and with their non-mathematical interpretations 0 ), while mixing them together. 
 :Hope my comments are helpful.