Hostname: page-component-cb9f654ff-fg9bn Total loading time: 0 Render date: 2025-09-06T04:40:45.365Z Has data issue: false hasContentIssue false
  • English
  • Français

There exist exactly two maximal strictly relevant extensions of the relevant logic R

Published online by Cambridge University Press:  12 March 2014

Kazimierz Swirydowicz
Kazimierz Swirydowicz*
Affiliation:
Faculty of Mathematics and Computer Science, Of Adam Mickiewicz University, Poznan, Poland E-mail: swirydow@math.amu.edu.pl
Get access Rights & Permissions [Opens in a new window]

Abstract

In [60] N. Belnap presented an 8-element matrix for the relevant logic R with the following property: if in an implication A → B the formulas A and B do not have a common variable then there exists a valuation v such that v (A → B) does not belong to the set of designated elements of this matrix. A 6-element matrix of this kind can be found in: R. Routley, R.K. Meyer, V. Plumwood and R.T. Brady [82], Below we prove that the logics generated by these two matrices are the only maximal extensions of the relevant logic R which have the relevance property: if A → B is provable in such a logic then A and B have a common propositional variable.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 3 , September 1999 , pp. 1125 - 1154
Copyright
Copyright © Association for Symbolic Logic 1999

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

Anderson, A.R. and Belnap, N.D., Entailment, vol 1 , 1975.Google Scholar
Belnap, N.D., Entailment and relevance, this Journal, vol. 25 (1960), pp. 144–146.Google Scholar
Blok, W.J. and Pigozzi, D., Algebraizable logics, Memoirs of the American Mathematical Society, 1989.CrossRefGoogle Scholar
Dziobiak, W., There are logics with the relevance principle between R and RM, Stadia Logica, vol. XLII, 1 (1983), pp. 49–60.CrossRefGoogle Scholar
Font, J.M. and Rodriguez, G., Note on algebraic models for relevance logic, Zeitschrift für Matematische Logik und Grundlagen der Mathematik, vol. 36 (1990), pp. 535–540.CrossRefGoogle Scholar
Maksimowa, L.L., Struktury s implikacjej, Algebra i Logika, vol. 12 (1973), pp. 445–467.Google Scholar
Meyer, R.K., RI-the bounds of finitude, Zeitschrift für Matematische Logik und Grundlagen der Mathematik, vol. 16 (1970), pp. 385–387.CrossRefGoogle Scholar
Meyer, R.K., Intuitionism, entailment, negation, Truth, syntax and modality (Leblanc, H., editor), 1973, pp. 168–198.CrossRefGoogle Scholar
Routley, R., Plumwood, V., Meyer, R.K., and Brady, R.T., Relevant logics and their rivals, Part 1, 1982.Google Scholar
Thistlewaite, P.B., Mcrobbie, M. A., and Meyer, R.K., Automated theorem proving in non-classical logics, London, 1988.Google Scholar
Tokarz, M., Essays in the matrix semantics of relevant logics, Warszawa, 1980.Google Scholar