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🤹🏼formal logic ii review

Closed Formula

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

A closed formula is a type of logical expression in first-order logic (FOL) that does not contain any free variables. It is complete and can be evaluated for truth without needing any additional information or assignment of values. This makes closed formulas particularly significant when discussing the semantics of logical statements and their interpretations, as they represent definitive propositions that can be true or false in a given model.

5 Must Know Facts For Your Next Test

  1. Closed formulas are also known as sentence forms since they express complete thoughts that do not depend on variable assignments.
  2. In contrast to open formulas, closed formulas can be evaluated solely based on their internal structure and the meanings assigned during interpretation.
  3. All closed formulas are either true or false in any given model, making them essential for establishing logical proofs and reasoning.
  4. The scope of quantifiers is particularly important in determining whether a formula is closed; if every variable in the formula is bound, it results in a closed formula.
  5. The validity of a closed formula can be tested using truth assignments across different interpretations to confirm its consistency across models.

Review Questions

  • How does the distinction between free and bound variables impact the classification of a formula as closed?
    • The distinction between free and bound variables is crucial because a closed formula contains no free variables; all variables must be bound by quantifiers. If even one variable remains free, the formula cannot be considered closed, which means its truth value could change depending on how that free variable is interpreted. Thus, understanding which variables are bound helps in identifying whether a formula qualifies as closed or open.
  • Discuss how closed formulas relate to interpretations in first-order logic.
    • Closed formulas are inherently tied to interpretations in first-order logic because they provide definitive propositions that can be evaluated for truth within any given model. Each closed formula can be assigned a truth value based solely on the structure of the formula itself and the meanings assigned to its symbols. As such, their analysis is central to understanding how logical expressions behave under different interpretations and how they contribute to logical reasoning.
  • Evaluate the significance of closed formulas in proving logical statements within formal systems.
    • Closed formulas are significant in proving logical statements because they represent complete assertions that do not rely on external context for their evaluation. Their binary nature—being either true or false—allows them to serve as foundational components in formal proofs. By demonstrating the truth of various closed formulas through established logical rules and interpretations, we can build coherent arguments and derive further conclusions within formal systems.