Notre Dame Journal of Formal Logic

A Finitely Axiomatized Formalization of Predicate Calculus with Equality

Norman D. Megill


We present a formalization of first-order predicate calculus with equality which, unlike traditional systems with axiom schemata or substitution rules, is finitely axiomatized in the sense that each step in a formal proof admits only finitely many choices. This formalization is primarily based on the inference rule of condensed detachment of Meredith. The usual primitive notions of free variable and proper substitution are absent, making it easy to verify proofs in a machine-oriented application. Completeness results are presented. The example of Zermelo-Fraenkel set theory is shown to be finitely axiomatized under the formalization. The relationship with resolution-based theorem provers is briefly discussed. A closely related axiomatization of traditional predicate calculus is shown to be complete in a strong metamathematical sense.

Article information

Notre Dame J. Formal Logic, Volume 36, Number 3 (1995), 435-453.

First available in Project Euclid: 17 December 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B10: Classical first-order logic
Secondary: 03B35: Mechanization of proofs and logical operations [See also 68T15]


Megill, Norman D. A Finitely Axiomatized Formalization of Predicate Calculus with Equality. Notre Dame J. Formal Logic 36 (1995), no. 3, 435--453. doi:10.1305/ndjfl/1040149359.

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