# Chapter 1. Propositional Logic

Propositional logic studies reasoning with statements that can be either true or false, and it provides a formal language together with precise rules that allow such reasoning to be carried out in a systematic and verifiable way, independent of the specific meaning of the statements involved.

The basic objects of study are propositional variables, which serve as placeholders for statements, and these variables are combined using logical connectives to form more complex expressions called formulas, so that the structure of reasoning can be analyzed through the structure of these formulas.

The first part of the chapter introduces the syntax of the language, where formulas are defined inductively from variables using connectives, and this inductive definition ensures that every formula has a well defined structure that can be analyzed and manipulated in a precise manner.

The second part introduces semantics, where each variable is assigned a truth value and this assignment is extended to all formulas, so that every formula can be evaluated under any given assignment, and this leads to the method of truth tables which systematically describes the behavior of formulas.

Logical equivalence is then introduced to compare formulas based on their truth behavior rather than their syntactic form, and this allows formulas to be transformed and simplified using general laws that preserve truth, which is essential for both theoretical reasoning and practical computation.

The chapter also studies normal forms, which are standard representations of formulas such as conjunctive normal form and disjunctive normal form, and these forms are important because they provide uniform structures that are well suited for algorithmic processing and for connecting logic with computation.

Finally, the chapter introduces proof systems, which provide formal methods for deriving conclusions from assumptions using fixed rules, and a central result is that these syntactic methods of proof correspond exactly to the semantic notion of truth, so that what can be proved is precisely what is true in all interpretations.

These components together form the foundation of propositional logic, and they establish the basic concepts that will be extended in later chapters to more expressive logical systems.