1.2 Basic concepts and terminology

Algebraic logic combines math and logic, using symbols to represent ideas and relationships. It's a powerful tool for analyzing complex arguments and solving problems in various fields, from computer science to philosophy.

At its core, algebraic logic uses variables, constants, and operators to create expressions. These building blocks allow us to represent and manipulate logical statements, helping us understand the structure and validity of arguments.

Fundamental Concepts and Logical Connectives

Fundamentals of algebraic logic

• Variables represent unknown or changeable values typically denoted by lowercase letters (p, q, r)
• Constants are fixed values in logical expressions including truth values true (1) and false (0)
• Operators symbolize logical operations encompassing basic (AND, OR, NOT) and advanced (implication, equivalence) operations
• Expressions combine variables, constants, and operators forming meaningful logical statements that can be evaluated to determine truth values

Types of logical connectives

• Conjunction (AND) symbolized by $\land$ performs multiplication algebraically and is true only when both operands are true
• Disjunction (OR) symbolized by $\lor$ performs addition algebraically and is true when at least one operand is true
• Negation (NOT) symbolized by $\neg$ performs complementation algebraically and reverses the truth value of its operand
• Implication symbolized by $\rightarrow$ is algebraically represented as $p \rightarrow q \equiv \neg p \lor q$ and is false only when antecedent is true and consequent is false
• Equivalence symbolized by $\leftrightarrow$ is algebraically represented as $(p \rightarrow q) \land (q \rightarrow p)$ and is true when both operands have the same truth value

Syntax, Semantics, and Algebraic Manipulation

Syntax and semantics of formulas

• Syntax rules govern atomic formulas (variables and constants), compound formulas (combinations of atomic formulas and connectives), and parentheses for grouping and precedence
• Semantics involve truth tables for evaluating complex expressions, interpretation of variables and constants, and meaning of logical connectives in different contexts
• Well-formed formulas (WFFs) follow proper syntax rules and can be evaluated to determine truth values ($p \land q$, $\neg(p \lor q) \rightarrow r$)
• Construction techniques involve building complex formulas from simpler ones, using parentheses to clarify operator precedence, and translating natural language statements into logical expressions

Principles of algebraic manipulation

• Commutativity applies to conjunction and disjunction ($p \land q \equiv q \land p$, $p \lor q \equiv q \lor p$)
• Associativity allows grouping of multiple operations of the same type ($(p \land q) \land r \equiv p \land (q \land r)$, $(p \lor q) \lor r \equiv p \lor (q \lor r)$)
• Distributivity involves distribution of conjunction over disjunction ($p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$) and distribution of disjunction over conjunction ($p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)$)
• De Morgan's Laws govern negation of conjunction and disjunction ($\neg(p \land q) \equiv \neg p \lor \neg q$, $\neg(p \lor q) \equiv \neg p \land \neg q$)
• Absorption Laws simplify redundant terms ($p \land (p \lor q) \equiv p$, $p \lor (p \land q) \equiv p$)
• Double Negation allows cancellation of double negation ($\neg \neg p \equiv p$)