Well-formed formulas (WFFs) are the building blocks of propositional logic. They're like the grammar rules for logical statements, telling us how to combine symbols and connectives to create valid expressions.

Understanding WFFs is crucial for constructing and analyzing logical arguments. By mastering the syntax and formation rules, you'll be able to create complex logical statements and evaluate their validity in propositional logic.

Syntax and Formation Rules

Defining Well-Formed Formulas

• A well-formed formula (WFF) refers to a string of symbols that is constructed according to the syntax and formation rules of a formal language
• The syntax of a formal language specifies the set of symbols that can be used in the language and the rules for combining these symbols
• Formation rules are a set of recursive rules that define how the symbols of a formal language can be combined to create WFFs
• A recursive definition is a method of defining a set or a function in terms of itself, where the definition includes a base case and a recursive case

Components of WFFs

• WFFs are composed of atomic propositions, which are the basic building blocks of the language
• Atomic propositions are represented by lowercase letters (p, q, r) and are considered to be either true or false
• Logical connectives are symbols used to combine atomic propositions or other WFFs to form compound WFFs
• The most common logical connectives are negation (¬), conjunction (∧), disjunction (∨), implication (→), and biconditional (↔)
• Parentheses are used to specify the order of operations and to group subformulas within a WFF

Types of WFFs

Atomic and Compound WFFs

• An atomic WFF is a single proposition letter (p, q, r) that cannot be broken down into simpler components
• Atomic WFFs are the simplest type of WFF and serve as the foundation for building more complex formulas
• A compound WFF is a formula that is constructed by combining one or more atomic WFFs using logical connectives
• Examples of compound WFFs include ¬p, p ∧ q, (p ∨ q) → r, and (p ↔ q) ∧ (q ∨ r)

Parsing WFFs

• Parsing is the process of analyzing the structure of a WFF to determine its components and the order in which the logical connectives are applied
• When parsing a WFF, it is essential to follow the order of operations, which specifies the precedence of the logical connectives
• The order of precedence for logical connectives, from highest to lowest, is: negation (¬), conjunction (∧), disjunction (∨), implication (→), and biconditional (↔)
• Parentheses can be used to override the default order of operations and to group subformulas together
• To parse a WFF, start by identifying the main logical connective and then recursively parse the subformulas on either side of the connective