reduction class

logic

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July 13, 2026

Quick Summary

In traditional logic, reduction is a method of rearranging the premises of a syllogism to fit a preferred pattern, typically the first figure where the middle term appears as the subject of the major premise and the predicate of the minor premise. This was historically done because only syllogisms of the first figure were considered self-evident.1

In metalogic, a reduction class refers to a set of logical formulas for which the decision problem is unsolvable. This means there is no general algorithm that can determine, for any formula in the class, whether it is a theorem or valid. The concept is linked to Alan Turing's work on undecidability and the halting problem, demonstrating that even for simplified forms of logical sentences, the decision problem remains unsolvable.2

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metalogic
The undecidability theorem and reduction classes
Given the completeness theorem, it follows that the task of deciding whether any sentence is a theorem of the predicate calculus is equivalent to that of deciding whether any sentence is valid or whether its negation is satisfiable.
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Given the completeness theorem, it follows that the task of deciding whether any sentence is a theorem of the predicate calculus is equivalent to that of deciding whether any sentence is valid or whether its negation is satisfiable.
Read More