Since logic, like other branches of mathematics, has trouble dealing with ambiguous statements, and since English is a notoriously ambiguous language, logicians (and other mathematicians) usually work in a language of their own, in which every term and symbol has a clear, well-defined meaning -- the "language of first-order logic", or FOL as the textbook calls it.
We'll start by examining relations, properties that can be either true or false of specific objects. Relations can be combined using Boolean connectives such as "not", "and", "or", "implies", and "iff", and we'll discuss how to use, prove, and disprove statements involving those connectives. Then we'll add quantifiers to the language, so we can talk about "all" objects or "at least one" object, and discuss how to use, prove, and disprove statements involving quantifiers. Finally, we'll visit some topics of particular importance in computer science: mathematical induction and computer theorem-proving (which underlies the Prolog programming language).