3.2 Free and bound variables

Free and bound variables are crucial concepts in first-order logic. They determine how formulas are interpreted and evaluated. Free variables act as placeholders, while bound variables are tied to quantifiers, affecting the formula's meaning and truth value.

Understanding these distinctions is key to grasping formula semantics. It impacts how we build complex formulas, define functions, and determine satisfiability. This knowledge forms the foundation for working with more advanced logical concepts and proofs.

Free vs Bound Variables

Defining Free and Bound Variables

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• Free variables remain unbound by quantifiers in first-order logic formulas
• Bound variables fall within the scope of quantifiers
• Variables can appear both free and bound in different parts of a formula depending on their position relative to quantifiers
• Free variables act as placeholders assignable with different values, affecting the formula's truth value
• Bound variables function as "dummy" variables, renamable without altering the formula's meaning if done consistently within the quantifier's scope
• Distinguishing between free and bound variables proves crucial for understanding first-order logic formulas' semantics and interpretation

Types of Formulas

• Sentences or closed formulas contain only bound variables
• Open formulas include at least one free variable
• Open formulas represent predicates or relations rather than propositions
• Open formulas' truth values depend on the interpretation of their free variables, making them context-dependent
• Free variables in formulas express general patterns or templates instantiable with specific values
• Instantiation substitutes terms for free variables as a fundamental operation in first-order logic

Significance of Free Variables

• Free variables allow formulas to serve as building blocks for more complex formulas (quantification, function definition)
• Free variables play a crucial role in defining the concept of satisfiability for first-order logic formulas
• Evaluating formulas with free variables requires assigning specific values from the domain of discourse to determine truth values
• The interplay between free and bound variables in a formula determines its logical strength and satisfiability conditions

Scope of Quantifiers

Understanding Quantifier Scope

• Quantifier scope extends from the quantifier to the formula's end or a closing parenthesis
• Nested quantifiers create nested scopes, with inner quantifiers taking precedence over outer ones for variable binding
• Variables can be bound multiple times in a formula, with each binding instance applying only within its respective scope
• Quantifier order affects variable binding, determining which quantifier binds which variable occurrences
• Scope ambiguities arise in natural language, but formal logic uses parentheses or notational conventions for explicit scopes
• Understanding quantifier scope proves essential for correctly interpreting and manipulating first-order logic formulas, especially with multiple quantifiers

Types of Quantifiers and Their Effects

• Universal quantification ($\forall$) binds variables requiring the subformula to be true for all possible values of the bound variable
• Existential quantification ($\exists$) binds variables requiring the subformula to be true for at least one value of the bound variable
• Evaluating formulas with multiple quantifiers requires careful consideration of evaluation order and variable dependencies
• The order of quantifiers matters when determining variable binding, affecting which quantifier binds which occurrences of variables

Formulas with Free Variables

Characteristics of Formulas with Free Variables

• Open formulas with free variables represent predicates or relations rather than propositions
• Truth values of open formulas depend on the interpretation of their free variables, making them context-dependent
• Free variables in formulas express general patterns or templates instantiable with specific values
• Instantiation substitutes terms for free variables as a fundamental operation in first-order logic
• Free variables allow formulas to serve as building blocks for more complex formulas (quantification, function definition)

Working with Free Variables

• Evaluating formulas with free variables requires assigning specific values from the domain of discourse to determine truth values
• Free variables play a crucial role in defining the concept of satisfiability for first-order logic formulas
• The presence of free variables in a formula indicates its potential use as a building block for more complex formulas
• The interplay between free and bound variables in a formula determines its logical strength and satisfiability conditions

Truth Values of Formulas

Evaluating Formulas with Free and Bound Variables

• Evaluating formulas with free variables requires assigning specific values from the domain of discourse to determine truth values
• Bound variables, being local to their quantifiers, do not affect the overall truth value when their names change consistently within their scope
• Universal quantification ($\forall$) binds variables requiring the subformula to be true for all possible values of the bound variable
• Existential quantification ($\exists$) binds variables requiring the subformula to be true for at least one value of the bound variable
• The interplay between free and bound variables in a formula determines its logical strength and satisfiability conditions

Applying Concepts to Formula Evaluation

• Evaluating formulas with multiple quantifiers requires careful consideration of evaluation order and variable dependencies
• Understanding the distinction between free and bound variables proves essential for correctly applying inference rules and performing logical deductions in first-order logic
• The process of instantiation, substituting terms for free variables, plays a crucial role in evaluating and manipulating formulas
• Analyzing the scope of quantifiers helps in determining the binding of variables and their effect on the formula's truth value