Free variable

Definition

A free variable is a variable that is not bound by a quantifier in a logical expression, meaning its value can vary independently. In predicates, free variables allow for the expression of statements without specifying particular individuals, enabling the formulation of general assertions. This contrasts with bound variables, which are limited to specific values within the context of quantifiers.

5 Must Know Facts For Your Next Test

1. In logical expressions, free variables allow for the representation of multiple possible instances, as their values are not fixed by quantification.
2. When a free variable is introduced in a predicate, it enables the creation of general statements about potential objects without committing to specific ones.
3. The distinction between free and bound variables is crucial for understanding the scope and interpretation of logical statements.
4. Free variables can be thought of as placeholders that require further specification to evaluate their truth value in a logical context.
5. In the context of formal proofs, free variables play an important role in expressing hypotheses and deriving conclusions.

Review Questions

• How do free variables differ from bound variables in logical expressions?
  • Free variables differ from bound variables in that free variables are not restricted by any quantifiers and can take on multiple values independently. In contrast, bound variables are confined within the scope of quantifiers, meaning their values are determined by those quantifiers. This distinction is essential for understanding how logical expressions operate, as it affects how we interpret statements involving different types of variables.
• Explain how free variables contribute to the formulation of general statements in predicates.
  • Free variables contribute to the formulation of general statements in predicates by allowing assertions to be made without specifying exact values. By utilizing free variables, predicates can express relationships and properties applicable to any number of potential objects or individuals. This flexibility enables mathematicians and logicians to articulate broad truths while leaving certain aspects open-ended for interpretation.
• Evaluate the role of free variables in formal reasoning and how they impact logical proofs.
  • Free variables play a crucial role in formal reasoning by serving as flexible components that can represent various entities in logical proofs. Their use allows for hypotheses to be expressed without committing to specific instances, facilitating the derivation of conclusions applicable across multiple cases. However, this also requires careful management during proofs, as the introduction and handling of free variables must align with rules governing logical expressions to ensure valid interpretations and results.

Related terms

bound variable: A bound variable is a variable that is restricted in value by a quantifier in a logical expression, indicating that its interpretation depends on the scope of that quantifier.

predicate: A predicate is a statement or function that expresses a property or relation among one or more variables, which can be true or false depending on the values assigned to those variables.

quantifier: A quantifier is a symbol used in logic to indicate the quantity of instances of a variable that satisfies a given condition, such as 'for all' (universal quantifier) or 'there exists' (existential quantifier).