## Notre Dame Journal of Formal Logic

### A General Characterization of the Variable-Sharing Property by Means of Logical Matrices

#### Abstract

As is well known, the variable-sharing property (vsp) is, according to Anderson and Belnap, a necessary property of any relevant logic. In this paper, we shall consider two versions of the vsp, what we label the "weak vsp" (wvsp) and the "strong vsp" (svsp). In addition, the "no loose pieces property," a property related to the wvsp and the svsp, will be defined. Each one of these properties shall generally be characterized by means of a class of logical matrices. In this way, any logic verified by an actual matrix in one of these classes has the property the class generally represents. Particular matrices (and so, logics) in each class are provided.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 53, Number 2 (2012), 223-244.

Dates
First available in Project Euclid: 9 May 2012

https://projecteuclid.org/euclid.ndjfl/1336588252

Digital Object Identifier
doi:10.1215/00294527-1715707

Mathematical Reviews number (MathSciNet)
MR2925279

Zentralblatt MATH identifier
1254.03041

#### Citation

Robles, Gemma; Méndez, José M. A General Characterization of the Variable-Sharing Property by Means of Logical Matrices. Notre Dame J. Formal Logic 53 (2012), no. 2, 223--244. doi:10.1215/00294527-1715707. https://projecteuclid.org/euclid.ndjfl/1336588252

#### References

• Anderson, A. R., and N. D. Belnap, Jr., Entailment. The Logic of Relevance and Necessity. Volume I, Princeton University Press, Princeton, 1975.
• Avron, A., "Relevant entailment–-Semantics and formal systems", The Journal of Symbolic Logic, vol. 49 (1984), pp. 334–42.
• Brady, R. T., editor, Relevant Logics and Their Rivals. Vol. II, Ashgate, Farnham, 2003.
• Brady, R. T., "Completeness proofs for the systems ${\rm RM}3$" and ${\rm BN}4$, Logique et Analyse. Nouvelle Série, vol. 25 (1982), pp. 9–32.
• Brady, R. T., "Depth relevance of some paraconsistent logics", Studia Logica, vol. 43 (1984), pp. 63–73.
• Humberstone, L., and R. K. Meyer, "The relevant equivalence property", Logic Journal of the IGPL, vol. 15 (2007), pp. 165–81.
• Méndez, J. M., "The compatibility of relevance and mingle", Journal of Philosophical Logic, vol. 17 (1988), pp. 279–97.
• Méndez, J. M., "Erratum to: The compatibility of relevance and mingle", Journal of Philosophical Logic, vol. 39 (2010), p. 339.
• Robles, G., and J. M. Méndez, "Deep relevant logics not included in R-mingle". in preparation.
• Routley, R., V. Plumwood, R. K. Meyer, and R. T. Brady, Relevant Logics and Their Rivals. Part I. The Basic Philosophical and Semantical Theory, Ridgeview Publishing Co., Atascadero, 1982.