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A solution to the completeness problem for weakly aggregative modal logic

Published online by Cambridge University Press:  12 March 2014

Peter Apostoli  and
Bryson Brown
Peter Apostoli
Affiliation:
Department of Philosophy, University of Toronto, Toronto, Ontario, Canada, E-mail: apostoli@cs.toronto.edu
Bryson Brown
Affiliation:
Department of Philosophy, University of Lethbridge, Lethbridge, Alberta, Canada, E-mail: brown@hg.uleth.ca
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Extract

We are accustomed to regarding K as the weakest modal logic admitting of a relational semantics in the style made popular by Kripke. However, in a series of papers which demonstrates a startling connection between modal logic and the theory of paraconsistent inference, Ray Jennings and Peter Schotch have developed a generalized relational frame theory which articulates an infinite hierarchy of sublogics of K, each expressing a species of “weakly aggregative necessity”. Recall that K is axiomatized, in the presence of N and RM, by the schema of “binary aggregation”

For each n ≥ 1, the weakly aggregative modal logic Kn is axiomatized by replacing K with the schema of “n-ary aggregation”

which is an n-ary relaxation, or weakening, of K. Note that K1 = K.

In [3], the authors claim without proof that Kn is determined by the class of frames F = (W, R), where W is a nonempty set and R is an (n + 1)-ary relation on W, under the generalization of Kriple's truth condition according to which □α is true at a point w in W if and only if α is true at one of x1,…,xn for all x1,…, xn in W such that Rw, x1,…, xn.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 3 , September 1995 , pp. 832 - 842
Copyright
Copyright © Association for Symbolic Logic 1995

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References

REFERENCES

[1]Chellas, Brian L., Modal logic: an introduction, Cambridge University Press, Cambridge, 1980.CrossRefGoogle Scholar
[2]Goldblatt, R. I., Logics of time and computation, CSLI Lecture Notes, No. 7, Center for the Study of Language and Information, Stanford University, Stanford, California, 1978.Google Scholar
[3]Jennings, R. E. and Schotch, P. K., The preservation of coherence, Studia Logica, vol. 43 (1984), pp. 89–106.CrossRefGoogle Scholar
[4]Jennings, R. E. and Schotch, P. K., Some remakrs on (weakly) weak modal logics, Notre Dame Journal of Formal Logic, vol. 22 (1981), pp. 309–314.CrossRefGoogle Scholar
[5]Jennings, R. E., Schotch, P. K., and Johnston, D. K., The n-adic first order undefinahility of the Geach formula, Notre Dame Journal of Formal Logic, vol. 22 (1981), pp. 375–378.CrossRefGoogle Scholar
[6]Jennings, R. E., Schotch, P. K., and Johnston, D. K., Universal first order definability in modal logic, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 26 (1980), pp. 327–330.CrossRefGoogle Scholar
[7]Johnston, D. K., A generalized relational semantics for modal logic, M. A. Thesis, Department of Philosophy, Simon Fraser University, Barnaby, British Columbia, 1978.Google Scholar
[8]Schotch, P. K. and Jennings, R., Inference and necessity, Journal of Philosophical Logic, vol. 9 (1980), pp. 327–340.CrossRefGoogle Scholar
[9]Schotch, P. K. and Jennings, R., Modal logic and the theory of modal aggregation, Philosophia, vol. 9 (1980), pp. 265–278.CrossRefGoogle Scholar
[10]Schotch, P. K. and Jennings, R., On detonating, Paraconsistent logic (Priest, G. and Routley, R., editors), Philosophia Verlag, Munich, 1989, pp. 306–327.CrossRefGoogle Scholar