### The Semantics of Entailment Omega

Mariangiola Dezani-Ciancaglini, Robert K. Meyer, and Yoko Motohama
Source: Notre Dame J. Formal Logic Volume 43, Number 3 (2002), 129-145.

#### Abstract

This paper discusses the relation between the minimal positive relevant logic B and intersection and union type theories. There is a marvelous coincidence between these very differently motivated research areas. First, we show a perfect fit between the Intersection Type Discipline ITD and the tweaking BT of B, which saves implication and conjunction but drops disjunction . The filter models of the -calculus (and its intimate partner Combinatory Logic CL) of the first author and her coauthors then become theory models of these calculi. (The logician's "theory" is the algebraist's "filter".) The coincidence extends to a dual interpretation of key particles—the subtype translates to provable , type intersection to conjunction , function space to implication, and whole domain to the (trivially added but trivial) truth T. This satisfying ointment contains a fly. For it is right, proper, and to be expected that type union should correspond to the logical disjunction of B. But the simulation of functional application by a fusion (or modus ponens product) operation on theories leaves the key Bubbling lemma of work on ITD unprovable for the -prime theories now appropriate for the modeling. The focus of the present paper lies in an appeal to Harrop theories which are (a) prime and (b) closed under fusion. A version of the Bubbling lemma is then proved for Harrop theories, which accordingly furnish a model of and CL.

First Page:
Primary Subjects: 03B47, 03B40
Secondary Subjects: 68N18
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1074290712
Digital Object Identifier: doi:10.1305/ndjfl/1074290712
Mathematical Reviews number (MathSciNet): MR2032579
Zentralblatt MATH identifier: 1042.03019

### References

[1] Anderson, A. R., and N. D. Belnap, Jr., Entailment. Vol. I. The Logic of Relevance and Necessity., Princeton University Press, Princeton, 1975.
Mathematical Reviews (MathSciNet): MR53:10542
Zentralblatt MATH: 0323.02030
[2] Anderson, A. R., N. D. Belnap, Jr., and J. M. Dunn, Entailment. The Logic of Relevance and Necessity. Vol. II, Princeton University Press, Princeton, 1992.
Mathematical Reviews (MathSciNet): MR94b:03042
Zentralblatt MATH: 0921.03025
[3] Barbanera, F., M. Dezani-Ciancaglini, and U. de'Liguoro, "Intersection and union types: Syntax and semantics", Information and Computation, vol. 119 (1995), pp. 202--30.
Mathematical Reviews (MathSciNet): MR96c:68111
Zentralblatt MATH: 0832.68065
Digital Object Identifier: doi:10.1006/inco.1995.1086
[4] Barendregt, H. P., The Lambda Calculus. Its Syntax and Semantics, revised edition, vol. 103 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1984.
Mathematical Reviews (MathSciNet): MR86a:03012
Zentralblatt MATH: 0551.03007
[5] Barendregt, H., M. Coppo, and M. Dezani-Ciancaglini, "A filter lambda model and the completeness of type assignment", The Journal of Symbolic Logic, vol. 48 (1983), pp. 931--40.
Mathematical Reviews (MathSciNet): MR85j:03014
Zentralblatt MATH: 0545.03004
[6] Curry, H. B., Foundations of Mathematical Logic, McGraw-Hill Book Co., New York, 1963.
Mathematical Reviews (MathSciNet): MR26:6036
Zentralblatt MATH: 163.24209
[7] Curry, H. B., and R. Feys, Combinatory Logic. Vol. I, Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1958.
Mathematical Reviews (MathSciNet): MR20:817
Zentralblatt MATH: 0081.24104
[8] Dezani-Ciancaglini, M., A. Frisch, E. Giovannetti, and Y. Motohama, "The relevance of semantic subtyping", in ITRS 2002, vol. 70 (1) of Electric Notes in Theoretical Computer Science, edited by S. van Bakel, 2002. \hrefhttp://www.elsevier.nl/locate/entcs/volume70.htmlhttp://www.elsevier.nl/locate/entcs/volume70.html
[9] Dezani-Ciancaglini, M., and J. R. Hindley, "Intersection types for combinatory logic", Theoretical Computer Science, vol. 100 (1992), pp. 303--24.
Mathematical Reviews (MathSciNet): MR93i:03018
Zentralblatt MATH: 0771.03004
Digital Object Identifier: doi:10.1016/0304-3975(92)90306-Z
[10] Dunn, J. M., The Algebra of Intensional Logics, Ph.D. thesis, University of Pittsburgh, 1966.
[11] Frisch, A., G. Castagna, and V. Benzaken, "Semantic subtyping", pp. 137--46 in Seventeenth IEEE Symposium on Logic in Computer Science, edited by G. Plotkin, IEEE Computer Society Press, 2002.
Zentralblatt MATH: 1082.68581
[12] Harrop, R., "Concerning formulas of the types $A\rightarrow B\bigvee C,\,A\rightarrow (Ex)B(x)$" in intuitionistic formal systems, The Journal of Symbolic Logic, vol. 25 (1960), pp. 27--32.
Mathematical Reviews (MathSciNet): MR24:A686
Zentralblatt MATH: 0098.24201
Digital Object Identifier: doi:10.2307/2964334
[13] Hindley, J. R., and G. Longo, "Lambda-calculus models and extensionality", Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 26 (1980), pp. 289--310.
Mathematical Reviews (MathSciNet): MR81m:03019
Zentralblatt MATH: 0453.03015
[14] Hindley, J. R., and J. P. Seldin, Introduction to Combinators and $\lambda$-Calculus, vol. 1 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1986.
Mathematical Reviews (MathSciNet): MR88j:03009
Zentralblatt MATH: 0614.03014
[15] Kripke, S. A., "Semantical analysis of modal logic. I. Normal modal propositional calculi", Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 9 (1963), pp. 67--96.
Mathematical Reviews (MathSciNet): MR26:3579
Zentralblatt MATH: 0118.01305
Digital Object Identifier: doi:10.1002/malq.19630090502
[16] Leblanc, H., "On dispensing with things and worlds", pp. 103--19 in Existence, Truth, and Provability, edited by H. Leblanc, State University of New York Press, Albany, 1982. Originally published in Logic and Ontology, pp. 241--59, edited by M. K. Munitz, New York University Press, New York, 1973.
Mathematical Reviews (MathSciNet): MR82k:01040
Zentralblatt MATH: 0502.03003
[17] Meyer, R. K., and R. Routley, "Algebraic analysis of entailment. I", Logique et Analyse, Nouvelle Série, vol. 15 (1972), pp. 407--28.
Mathematical Reviews (MathSciNet): MR48:5857
Zentralblatt MATH: 0336.02020
[18] Routley, R., and R. K. Meyer, "The semantics of entailment. III", Journal of Philosophical Logic, vol. 1 (1972), pp. 192--208.
Mathematical Reviews (MathSciNet): MR53:12878
Zentralblatt MATH: 0317.02019
Digital Object Identifier: doi:10.1007/BF00650498
[19] Routley, R., V. Plumwood, R. K. Meyer, and R. T. Brady, Relevant Logics and Their Rivals. Part I, Ridgeview Publishing Co., Atascadero, 1982.
Mathematical Reviews (MathSciNet): MR85k:03013
Zentralblatt MATH: 0579.03011
[20] van Bakel, S., M. Dezani-Ciancaglini, U. de'Liguoro, and Y. Motohama, "The minimal relevant logic and the call-by-value lambda calculus", Technical Report TR-ARP-05-2000, Australian National University, 2000.
[21] Venneri, B., "Intersection types as logical formulae", Journal of Logic and Computation, vol. 4 (1994), pp. 109--24.
Mathematical Reviews (MathSciNet): MR95b:68068
Zentralblatt MATH: 798.03013