Quantification Theory Rule: Universal Generalization

The question asks for the specific rule of inference in quantification theory that allows us to conclude a universally quantified statement from a premise that holds true for any arbitrary individual.

Understanding Inference Rules

In quantification theory (also known as predicate logic), inference rules are fundamental for deriving new statements from existing ones. The rule in question deals with generalizing from specific instances to a universal statement.

Analyzing the Options

• Universal Generalization (UG): This rule states that if you can prove a statement 'P(x)' holds for an *arbitrary* individual 'x', then you can infer the universally quantified statement '∀x P(x)'. This matches the question's description precisely.
• Existential Instantiation (EI): Allows inferring a specific instance from an existentially quantified statement (e.g., from '∃x P(x)', infer 'P(c)' for some new individual 'c').
• Universal Instantiation (UI): Allows inferring a specific instance from a universally quantified statement (e.g., from '∀x P(x)', infer 'P(c)' for any specific individual 'c').
• Existential Generalization (EG): Allows inferring an existentially quantified statement from a specific instance (e.g., from 'P(c)', infer '∃x P(x)').

Conclusion

The definition provided in the question directly corresponds to the rule of Universal Generalization. It permits inferring a statement like '∀x (all x are P)' from the fact that the statement holds true for any arbitrarily chosen 'x'.