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The completeness of a predicate-functor logic

Published online by Cambridge University Press:  12 March 2014

John Bacon
John Bacon*
Affiliation:
The University of Sydney, Sydney, New South Wales, Australia 2006
*
120 East 90th Street, Apt. 7A, New York, New York 10128
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Extract

Predicate-functor logic, as founded by W. V. Quine ([1960], [1971], [1976], [1981]), is first-order predicate logic without individual variables. Instead, adverbs or predicate functors make explicit the permutations and replications of argument-places familiarly indicated by shifting variables about. For the history of this approach, see Quine [1971, 309ff.]. With the evaporation of variables, individual constants naturally assimilate to singleton predicates or adverbs, leaving no logical subjects whatever of type 0. The orphaned “predicates” may then be taken simply as terms in the sense of traditional logic: class and relational terms on model-theoretic semantics, schematic terms on Quine's denotational or truth-of semantics. Predicate-functor logic thus stands forth as the pre-eminent first-order term logic, as distinct from propositional-quantificational logic. By the same token, it might with some justification qualify as “first-order combinatory logic”, with allowance for some categorization of the sort eschewed in general combinatory logic, the ultimate term logic.

Over the years, Quine has put forward various choices of primitive predicate functors for first-order logic with or without the full theory of identity. Moreover, he has provided translations of quantificational into predicate-functor notation and vice versa ([1971, 312f.], [1981, 651]). Such a translation does not of itself establish semantic completeness, however, in the absence of a proof that it preserves deducibility.

An axiomatization of predicate-functor logic was first published by Kuhn [1980], using primitives rather like Quine's. As Kuhn noted, “The axioms and rules have been chosen to facilitate the completeness proof” [1980, 153]. While this expedient simplifies the proof, however, it limits the depth of analysis afforded by the axioms and rules. Mindful of this problem, Kuhn ([1981] and [1983]) boils his axiom system down considerably, correcting certain minor slips in the original paper.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 50 , Issue 4 , December 1985 , pp. 903 - 926
Copyright
Copyright © Association for Symbolic Logic 1985

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References

REFERENCES

Anderson, Alan Ross [1960], Completeness theorems for the systems E of entailment and EQ of entailment with quantification, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 6, pp. 201–216.CrossRefGoogle Scholar
Artin, Emil [1926], Theorie der Zöpfe, Collected papers (Lang, S. and Tate, J., editors), Addison-Wesley, Reading, Mass., 1965, pp. 416–441.CrossRefGoogle Scholar
Bacon, John [1982], Logic based on syllogism, Proceedings of the Russellian Society, University of Sydney, vol. 7, pp. 49–57.Google Scholar
Bacon, John [198?], Sommers and modern logic, The new syllogistic: critical essays on the logic of Fred Sommers (Englebretsen, G., editor), North-Holland, Amsterdam (to appear).Google Scholar
Bernays, Paul [1959], Über eine natürliche Erweiterung des Relationenkalkuls, Constructivity in mathematics (Heyting, A., editor), North-Holland, Amsterdam, pp. 1–14.Google Scholar
Burnside, W. [1897], Note on the symmetric group, Proceedings of the London Mathematical Society, ser. 1, vol. 28, pp. 119–129.Google Scholar
Coxeter, H. S. M. and Moser, W. O. J. [1957], Generators and relations for discrete groups, 3rd ed., Springer-Verlag, Berlin, 1972.CrossRefGoogle Scholar
Egli, Urs [1979], Anhang to Knöpfler [1979].Google Scholar
Fitch, Frederic Brenton [1952], Symbolic logic: an introduction, Ronald, New York.Google Scholar
Grünberg, Teo [1983], A tableau system of proof for predicate-functor logic with identity, this Journal, vol. 48, pp. 1140–1144.Google Scholar
Grünberg, Teo [198?], Predicate-functor logic with operation symbols (to appear).Google Scholar
Knöpfler, Siegfried [1979], Linguistische und formallogische Untersuchung zur Prädikat-Funktor-Logik (PFL), xeroxed, Sonderforschungsbereich 99 “Linguistik”, University of Konstanz.Google Scholar
Kuhn, Steven T. [1980], Quantifiers as modal operators, Studio Logica, vol. 39, pp. 145–158.CrossRefGoogle Scholar
Kuhn, Steven T. [1981], Notes on a problem of W. V. Quine, forthcoming.Google Scholar
Kuhn, Steven T. [1983], An axiomatization of predicate functor logic, Notre Dame Journal of Formal Logic, vol. 24, pp. 233–241.CrossRefGoogle Scholar
Lockwood, Michael [1982], The logical syntax of TFL, Appendix G of Sommers [1982], pp. 426–456.Google Scholar
McIntosh, Clifton [1982], Traditional formal logic, Appendix F of Sommers [1982], pp. 387–425.Google Scholar
Moore, Eliakim Hastings [1897], Concerning the abstract group of order κ! and ½κ! holohedrically isomorphic with the symmetric and the alternating substitution-groups on κ letters, Proceedings of the London Mathematical Society, ser. 1, vol. 28, pp. 357–366.Google Scholar
Noah, Aris [1980], Predicate-functors and the limits of decidability in logic, Notre Dame Journal of Formal Logic, vol. 21, pp. 701–707.CrossRefGoogle Scholar
Quine, W. V. [1960], Variables explained away, Selected logic papers, Random House, New York, 1966, pp. 227–235.Google Scholar
Quine, W. V. [1971], Predicate-functor logic, Proceedings of the Second Scandinavian Logic Symposium (Fenstad, J., editor), North-Holland, Amsterdam, pp. 309–315.CrossRefGoogle Scholar
Quine, W. V. [1976], Algebraic logic and predicate functors, The ways of paradox and other essays, 2nd ed., Harvard University Press, Cambridge, Massachusetts, pp. 283–307.Google Scholar
Quine, W. V. [1981], Predicate functors revisited, this Journal, vol. 46, pp. 649–652.Google Scholar
Schönfinkel, Moses [1924], Über die Bausteine der mathematischen Logik, Mathematische Annalen, vol. 92, pp. 305–316.CrossRefGoogle Scholar
Shiman, Paul L. [1983], Toward a cononical term functor logic, Paper presented at the annual meeting of the Association for Symbolic Logic, Boston, 12 29; abstract published in this Journal, vol. 49 (1984), p. 1447.Google Scholar
Sommers, Fred [1970], The calculus of terms, Mind, vol. 79, pp. 1–39.CrossRefGoogle Scholar
Sommers, Fred [1982], The logic of natural language, Clarendon Press, Oxford.Google Scholar
Suppes, Patrick [1976], Elimination of quantifiers in the semantics of natural language by use of extended relation algebras, Revue Internationale de Philosophie, vol. 117-118, pp. 243–259.Google Scholar
Suppes, Patrick [1979], Logical inference in English: a preliminary analysis, Studia Logica, vol. 38, pp. 375–391.CrossRefGoogle Scholar
Zimmermann, Thomas [1979], Anhang to Knöpfler [1979].Google Scholar