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

Permanent link to this document

Digital Object Identifier

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.

Export citation


  • [1] Boyer, R., E. Lusk, W. McCune, R. Overbeek, M. Stickel, and L. Wos, ``Set theory in first order logic: clauses for Gödel's axioms,'' Journal of Automated Reasoning, vol. 2 (1986), pp. 287--327.
  • [2] Church, A., Introduction to Mathematical Logic, Volume 1, Princeton University Press, Princeton, 1956.
  • [3] Cohen, P. J., Set Theory and the Continuum Hypothesis, W. A. Benjamin, Reading, 1966.
  • [4] Hamilton, A. G., Logic for Mathematicians, Cambridge University Press, Cambridge, 1988.
  • [5] Hindley, J. R., and D. Meredith, ``Principal type-schemes and condensed detachment,'' The Journal of Symbolic Logic, vol. 55 (1990), pp. 90--105.
  • [6] Kalish, D., and R. Montague, ``On Tarski's formalization of predicate logic with identity,'' Archiv für Mathematische Logik und Grundlagenforschung, vol. 7 (1965), pp. 81--101.
  • [7] Kalman, J. A., ``Condensed detachment as a rule of inference,'' Studia Logica, vol. 42 (1983), pp. 443--451.
  • [8] Kleene, S. C., Introduction to Metamathematics, Van Nostrand, Princeton, 1952.
  • [9] Megill, N. D., and M. W. Bunder, ``Weaker D-complete logics,'' The University of Wollongong Department of Mathematics Preprint Series no. 15/94.
  • [10] Mendelson, E., Introduction to Mathematical Logic, second edition, Van Nostrand, New York, 1979.
  • [11] Meredith, D., ``In memoriam Carew Arthur Meredith (1904--1976),'' Notre Dame Journal of Formal Logic, vol. 18 (1977), pp. 513--516.
  • [12] Mints, G., and T. Tammet, ``Condensed detachment is complete for relevance logic: a computer-aided proof,'' Journal of Automated Reasoning, vol. 7 (1991), pp. 587--596.
  • [13] Monk, J. D., ``Substitutionless predicate logic with identity,'' Archiv für Mathematische Logik und Grundlagenforschung, vol. 7 (1965), pp. 103--121.
  • [14] Nemeti, I., ``Algebraizations of quantifier logics, an overview,'' version 11.4, preprint, Mathematical Institute, Budapest, 1994. A shortened version without proofs appeared in Studia Logica, vol. 50 (1991), pp. 485--569.
  • [15] Peterson, J. G., ``An automatic theorem prover for substitution and detachment systems,'' Notre Dame Journal of Formal Logic, vol. 19 (1978), pp. 119--122.
  • [16] Robinson, J. A., ``A machine-oriented logic based on the resolution principle,'' Journal of the Association for Computing Machinery, vol. 12 (1965), pp. 23--41.
  • [17] Tarski, A., ``A simplified formalization of predicate logic with identity,'' Archiv für Mathematische Logik und Grundlagenforschung, vol. 7 (1965), pp. 61--79.
  • [18] Tarski, A., and S. Givant, A Formalization of Set Theory Without Variables, American Mathematical Society Colloquium Publications, vol. 41, American Mathematical Society, Providence, 1987.
  • [19] Wos, L., Automated Reasoning: 33 Basic Research Problems, Prentice-Hall, Englewood Cliffs, 1987
  • [20] Wos, L., R. Overbeek, E. Lusk and J. Boyle, Automated Reasoning: Introduction and Applications, second edition, McGraw-Hill, New York, 1992.
  • [21] Wos, L. T., and G. A. Robinson, ``Maximal models and refutation completeness: semidecision procedures in automated theorem proving,'' pp. 609--639 in Word Problems: Decision Problems and the Burnside Problem in Group Theory, Studies in Logic and the Foundations of Mathematics, vol. 71, edited by W. W. Boone, F. B. Cannonito, and R. C. Lyndon, North-Holland, Amsterdam, 1973.
  • [22] Zeman, J. J., Modal Logic, Oxford University Press, Oxford, 1973.