Hostname: page-component-6b989bf9dc-wj8jn Total loading time: 0 Render date: 2024-04-14T19:47:09.530Z Has data issue: false hasContentIssue false
  • English
  • Français

Principal type-schemes and condensed detachment

Published online by Cambridge University Press:  12 March 2014

J. Roger Hindley  and
David Meredith
J. Roger Hindley
Affiliation:
Mathematics Division, University College, Swansea SA2 8PP, Wales
David Meredith
Affiliation:
Old Blood Road, Merrimack, New Hampshire 03054
Get access Rights & Permissions [Opens in a new window]

Extract

The condensed detachment rule, or ruleD, was first proposed by Carew Meredith in the 1950's for propositional logic based on implication. It is a combination of modus ponens with a “minimal” amount of substitution. We shall give a precise detailed statement of rule D. (Some attempts in the published literature to do this have been inaccurate.)

The D-completeness question for a given set of logical axioms is whether every formula deducible from the axioms by modus ponens and substitution can be deduced instead by rule D alone. Under the well-known formulae-as-types correspondence between propositional logic and combinator-based type-theory, rule D turns out to correspond exactly to an algorithm for computing principal type-schemes in combinatory logic. Using this fact, we shall show that D is complete for intuitionistic and classical implicational logic. We shall also show that D is incomplete for two weaker systems, BCK- and BCI-logic.

In describing the formulae-as-types correspondence it is common to say that combinators correspond to proofs in implicational logic. But if “proofs” means “proofs by the usual rules of modus ponens and substitution”, then this is not true. It only becomes true when we say “proofs by rule D”; we shall describe the precise correspondence in Corollary 6.7.1 below.

This paper is written for readers in propositional logic and in combinatory logic. Since workers in one field may not feel totally happy in the other, we include short introductions to both fields.

We wish to record thanks to Martin Bunder, Adrian Rezus and the referee for helpful comments and advice.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 1 , March 1990 , pp. 90 - 105
Copyright
Copyright © Association for Symbolic Logic 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Anderson, A. R. and Belnap, N. D. [1975], Entailment, Vol. 1, Princeton University Press, Princeton, New Jersey.Google Scholar
Bunder, M. W. [1986], Author's review [and correction] of Bunder, M. W. and Meyer, R. K., A result for combinators, BCK logics and BCK algebras (Logique et Analyse, vol. 28 (1985), pp. 33–40), Zentralblatt für Mathematik und ihre Grenzgebiete, vol. 574, pp. 10–11 (review 03004).Google Scholar
Church, A. [1951], The weak theory of implication, Kontrolliertes Denken. Untersuchungen zum Logikkalkül und zur Logik der Einzelwissen-schaften (Festgabe zum 60. Geburtstag von Prof. W. Britzelmayr) (Menne, A.et al., editors), Komissionsverlag Karl Alber, München (Sonderdruck), pp. 22–37.Google Scholar
Curry, H. B. [1942], The combinatory foundations of mathematical logic, this Journal, vol. 7, pp. 49–64.Google Scholar
Curry, H. B. [1952], On the definition of substitution, replacement and allied notions in an abstract formal system, Revue Philosophique de Louvain, vol. 50, pp. 50–251.CrossRefGoogle Scholar
Curry, H. B. [1969], Modified base functionality in combinatory logic, Dialectica, vol. 23, pp. 23–83.CrossRefGoogle Scholar
Curry, H. B. and Feys, R. [1958], Combinatory logic. Vol. 1, North-Holland, Amsterdam.Google Scholar
Curry, H. B., Hindley, J. R., and Seldin, J. P. [1972], Combinatory logic. Vol. II, North-Holland, Amsterdam.Google Scholar
Herbrand, J. [1930], Recherches sur la théorie de la démonstration, thesis, University of Paris, Paris; English translation in his Logical writings (W. Goldfarb, editor), Harvard University Press, Cambridge, Massachusetts, 1971, pp. 44–202.Google Scholar
Hindley, J. R. [1969], The principal type-scheme of an object in combinatory logic, Transactions of the American Mathematical Society, vol. 146, pp. 29–60.Google Scholar
Hindley, J. R. [1989], BCK-combinators and linear λ-terms have types, Theoretical Computer Science, vol. 64, pp. 64–97.CrossRefGoogle Scholar
Hindley, J. R. and Seldin, J. P. [1986], Introduction to combinators and λ-calculus, Cambridge University Press, Cambridge.Google Scholar
Jaśkowski, S. [1963], Über Tautologien, in welchen keine Variable mehr als zweimal vorkommt, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 9, pp. 9–219.CrossRefGoogle Scholar
Kalman, J. A. [1983], Condensed detachment as a rule of inference, Stadia Logica, vol. 42, pp. 42–443.Google Scholar
Lemmon, E. J., Meredith, C. A., Meredith, D., Prior, A. N., and Thomas, I. [1957], Calculi of pure strict implication, mimeographed; published in Philosophical logic (Davis, J. W.et al., editors), Reidel, Dordrecht, 1969, pp. 215–250.Google Scholar
Meredith, C. A. and Prior, A. N. [1963], Notes on the axiomatics of the propositional calculus, Notre Dame Journal of Formal Logic, vol. 4, pp. 4–171.CrossRefGoogle Scholar
Meredith, C. A. and Prior, A. N. [1964], Investigations into implicational S5, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 10, pp. 10–203.CrossRefGoogle Scholar
Meredith, D. [1977], In memoriam Carew Arthur Meredith, Notre Dame Journal of Formal Logic, vol. 18, pp. 18–513.CrossRefGoogle Scholar
Meredith, D. [1978], Positive logic and λ-constants, Studio Logica, vol. 37, pp. 37–269.CrossRefGoogle Scholar
Meyer, R. K. and Bunder, M. W. [1990], Condensed detachment and combinators, Journal of Automated Reasoning (to appear).Google Scholar
Mints, G. and Tammet, T. [1990], Condensed detachment is complete for relevant logic: proof using computer, Journal of Automated Reasoning (to appear).Google Scholar
Morris, J. H. [1968], Lambda-calculus models of programming languages, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts.Google Scholar
Prior, A. N. [1955], Formal logic, Clarendon Press, Oxford.Google Scholar
Rezus, A. [1982], On a theorem of Tarski, preprint no. 227, Mathematics Department, University of Utrecht, Utrecht; published in Libertas Mathematica, vol. 2, pp. 2–63.Google Scholar
Robinson, J. A. [1965], A machine-oriented logic based on the resolution principle, Journal of the Association for Computing Machinery, vol. 12, pp. 12–23.CrossRefGoogle Scholar
Schönfinkel, M. [1924], Über die Bausteine der mathematischen Logik, Mathematische Annalen, vol. 92, pp. 305–316; English translation, From Frege to Gödel: a source-book in mathematical logic (J. van Heijenoort, editor), Harvard University Press, Cambridge, Massachusetts, 1967, pp. 355–366.CrossRefGoogle Scholar