(Part I. Formal logic)

Arguments and an informal notion of validity
Consistency, inconsistency and counter-example sets
Sentence-functors and truth-functors
The construction of a formal language suitable for propositional logic
Truth-tables for truth-functors, structures, semantic sequents, inconsistency and tautologies
Basic properties of semantic entailments: truth-functionality, substitution instances; expressive adequacy; disjunctive and conjunctive normal form; interpolation theorem
Testing the correctness of semantic sequents
The construction of formal systems: Axioms, rules of inference, derivations and theorems; soundness and completeness
A formal system for propositional logic - the propositional calculus (at least, one of the following three types of formal systems is required: axiom system, natural deductions, tableaux system)
The construction of a first-order language suitable for predicate logic
A (Frege-Tarskian) semantics appropriate for the established first-order language
Analyses of some ordinary phrases in English: same, at least/most, exactly, more/less, all, some Relations, names, identity, descriptions
A formal system for predicate logic - the predicate calculus (at least, one of the following three types of formal systems is required: axiom system, natural deductions, tableaux system)
Formalization of ordinary statements/arguments in natural language into formulae/sequents of the established propositional/predicate language and check its validity by either constructing a derivation in the established formal system, or providing a counterexample.

(Part II. The philosophy of logic)

Logical forms: Validity vs. logical consequences
Propositions, sentences, statements and beliefs
The meaning of connectives (truth-functors): Model-theoretical account (truth-tables for connectives); proof-theoretical account (rules of inference for connectives, e.g. natural deductions)
Subjects, predicates and quantification
Designators, names, Description, and existence (Frege