Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-29T08:16:42.703Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantifying over propositions in relevance logic: nonaxiomatisability of primary interpretations of ∀p and ∃p*

Published online by Cambridge University Press:  12 March 2014

Philip Kremer
Philip Kremer*
Affiliation:
Department of Philosophy, University of Pittsburgh, Pittsburg, Pennsylvania 15260, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

A typical approach to semantics for relevance (and other) logics: specify a class of algebraic structures and take a model to be one of these structures, α, together with some function or relation which associates with every formula A a subset of α. (This is the approach of, among others, Urquhart, Routley and Meyer and Fine.) In some cases there are restrictions on the class of subsets of α with which a formula can be associated: for example, in the semantics of Routley and Meyer [1973], a formula can only be associated with subsets which are closed upwards. It is natural to take a proposition of α to be such a subset of α, and, further, to take the propositional quantifiers to range over these propositions. (Routley and Meyer [1973] explicitly consider this interpretation.) Given such an algebraic semantics, we call (following Routley and Meyer [1973], who follow Henkin [1950]) the above-described interpretation of the quantifiers the primary interpretation associated with the semantics.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 1 , March 1993 , pp. 334 - 349
Copyright
Copyright © Association for Symbolic Logic 1993

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.)

Footnotes

*

I am much indebted to Nuel Belnap for his constant help and encouragement, and, not least of all, for rather closely inspecting the proofs. I thank Aldo Antonelli for asking me whether the systems studied here fail to be arithmetical (in the recursion theoretic sense of Hinman [1978], Odifreddi [1989] and others) as well as recursively enumerable. Not only are the systems here nonarithmetical, they are recursively isomorphic to full second-order logic. Finally I thank a referee and Richard Shore for pointing in the right direction toward proving this stronger result.

References

REFERENCES

Anderson, Alan Ross and Belnap, Nuel D. Jr. [1975], Entailment, the logic of relevance and necessity, vol. 1, Princeton University Press, Princeton, NJ.Google Scholar
Anderson, Alan Ross, Belnap, Nuel D. Jr., and Dunn, J. Michael [1992], Entailment, the logic of relevance and necessity, vol. 2, Princeton University Press, Princeton, NJ.Google Scholar
Church, Alonzo [1956], Introduction to mathematical logic, Princeton University Press, Princeton, NJ.Google Scholar
Fine, Kit [1970], Propositional quantifiers in modal logic, Theoria, vol. 36, pp. 331–346.CrossRefGoogle Scholar
Fine, Kit, [1974], Models for entailment, this Journal, vol. 3, pp. 347–372.Google Scholar
Henkin, Leon [1950], Completeness in the theory of types, this Journal, vol. 15, pp. 81–91.Google Scholar
Hinman, Peter G. [1978], Recursion-Theoretic hierarchies, Springer-Verlag, New York.CrossRefGoogle Scholar
Hughes, G. E. and Cresswell, M. J. [1968], An introduction to modal logic, Methuel, London.Google Scholar
Kaplan, David [1970], S5 with quantifiable propositional variables, this Journal, vol. 35, p. 355.Google Scholar
Kripke, Saul [1959], A completeness theorem in modal logic, this Journal, vol. 24, pp. 1–14.Google Scholar
Kripke, Saul, [1963], Semantical analysis of intuitionistic logic I, Formal systems and recursive functions. Proceedings of the eighth logic colloquium (Crossley, J. N. and Dummett, M. A. E., editors), North-Holland, Amsterdam, pp. 92–130.Google Scholar
Myhill, Robert [1955], Creative sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 1, pp. 97–108.CrossRefGoogle Scholar
Nerode, Anil and Shore, Richard A. [1980], Second order logic and first order theories of reducibility orderings, The Kleene symposium (Barwise, Jon, Kiesler, H. Jerome, and Kunen, Kenneth, editors), North-Holland, Amsterdam, pp. 181–200.CrossRefGoogle Scholar
Odifreddi, Piergiorgio [1989], Classical recursion theory: The theory of functions and sets of natural numbers, North-Holland, Amsterdam.Google Scholar
Post, Emil L. [1944], Recursively enumerable sets of positive integers and their decision problems, Bulletin of the American Mathematical Society, vol. 50, pp. 284–316.CrossRefGoogle Scholar
Routley, R. and Meyer, R. K. [1972a], The semantics of entailment II, Journal of Philosophical Logic, vol. 1, pp. 53–73.CrossRefGoogle Scholar
Routley, R. and Meyer, R. K., [1972b], The semantics of entailment III, Journal of Philosophical Logic, vol. 1, pp. 192–208.CrossRefGoogle Scholar
Routley, R. and Meyer, R. K., [1973], The semantics of Entailment I, Truth, syntax and modality. Proceedings of the Temple University conference on alternative semantics (Leblanc, H., editor), North-Holland, Amsterdam, pp. 199–243.CrossRefGoogle Scholar
Routley, R. with Myer, R. K., Plumwood, V., and Brady, R. [1982], Relevant logics and their rivals 1. The basic philosophical and semantical theory, Ridgeview, Atascadero, CA.Google Scholar
Urquhart, Alasdair [1972], Semantics for relevant logics, this Journal, vol. 37, pp. 159–169.Google Scholar
Urquhart, Alasdair, [1973], The semantics of entailment, Doctoral dissertation, University of Pittsburg, University Microfilms, Ann Arbor, MI.Google Scholar
Urquhart, Alasdair, [1992], Semilattice semantics for relevance logics, Entailment, the logic of relevance and necessity, vol. 2 (Anderson, Alan Ross, Belnap, Nuel D. Jr., and Dunn, J. Michael, editors), Princeton University Press, Princeton, NJ, 1992.Google Scholar