Topics:

Propositional Logic - Syntax

• Well-formed formulas
• Unique readability

Propositional Logic - Semantics

• Interpretations and satisfaction
• Validity, satisfiability, and unsatisfiability
• Logical implication and material implication
• Completeness of connectives

Propositional Logic - Decision Procedures

• Truth tables
• Complexity (NP-completeness)
• Compactness

First-Order Logic - Syntax and Semantics

• Terms and formulas
• Structures, assignments, and semantics
• Satisfiability, validity, and logical implication

First-Order Logic - Definability

• Preservation under isomorphism
• Elementary equivalence vs. isomorphism
• Definability: classes of structures, relations
• Compactness and applications to definability

First-Order Logic - A Formal System

• Deductions
• Propositional Reasoning
• The Deduction Theorem
• Soundness

First-Order Logic - Completeness and Consequences

• Completeness
• Lowenheim-Skolem Theorm
• Compactness Theorem
• Models and theories

First-Order Logic - Undecidability and Incompleteness

• Recursive enumerability of validity
• Non-recursive enumerability of satisfiability
• Godel's Incompleteness Theorem

Second-Order Logic

• Second-order languages
• Skolemization

References:

• H.B. Enderton: A Mathematical Introduction to Logic, Academic Press, 1972, Chapters 1,2,4.1-4.2