Geodesic
The Weak Variable Sharing Property
Bulletin of the Section of Logic, Tome 52 (2023) no. 1, pp. 85-99.

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An algebraic type of structure is shown forth which is such that if it is a characteristic matrix for a logic, then that logic satisfies Meyer's weak variable sharing property. As a corollary, it is shown that RM and all its odd-valued extensions RM2n-1 satisfy the weak variable sharing property. It is also shown that a proof to the effect that the quot;fuzzy quot; version of the relevant logic R satisfies the property is incorrect.
Keywords: characteristic matrix, relevant logics, variable sharing properties 
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Øgaard, Tore Fjetland. The Weak Variable Sharing Property. Bulletin of the Section of Logic, Tome 52 (2023) no. 1, pp. 85-99. http://geodesic.mathdoc.fr/item/BSL_2023_52_1_a4/

[1] A. R. Anderson, N. D. Belnap, Entailment: The Logic of Relevance and Necessity, vol. 1, Princeton University Press, Princeton (1975).

[2] N. D. Belnap, Entailment and Relevance, Journal of Symbolic Logic, vol. 25(2) (1960), pp. 144–146 | DOI

[3] R. T. Brady, Completeness Proofs for the Systems RM3 and BN4, Logique et Analyse, vol. 25(97) (1982), pp. 9–32.

[4] J. Dugundji, Note on a Property of Matrices for Lewis and Langford’s Calculi of Propositions, The Journal of Symbolic Logic, vol. 5(4) (1940), pp. 150–151 | DOI

[5] J. M. Dunn, Algebraic Completeness Results for R-Mingle and its Extensions, Journal of Symbolic Logic, vol. 35(1) (1970), pp. 1–13 | DOI

[6] R. K. Meyer, R-Mingle and Relevant Disjunction, Journal of Symbolic Logic, vol. 36(2) (1971), p. 366 | DOI

[7] T. F. Øgaard, Non-Boolean Classical Relevant Logics II: Classicality Through Truth-Constants, Synthese, vol. 199 (2021), pp. 6169–6201 | DOI

[8] G. Robles, The Quasi-Relevant 3-Valued Logic RM3 and some of its Sublogics Lacking the Variable-Sharing Property, Reports on Mathematical Logic, vol. 51 (2016), pp. 105–131 | DOI

[9] G. Robles, J. M. Méndez, A Companion to Brady’s 4-Valued Relevant Logic BN4: The 4-Valued Logic of Entailment E4, Logic Journal of the IGPL, vol. 24(5) (2016), pp. 838–858 | DOI

[10] E. Schechter, Equivalents of Mingle and Positive Paradox, Studia Logica, vol. 77(1) (2004), pp. 117–128 | DOI

[11] J. Slaney, MaGIC, Matrix Generator for Implication Connectives: Release 2.1 Notes and Guide, Tech. Rep. TR-ARP-11/95, Automated Reasoning Project, Australian National University (1995).

[12] J. K. Slaney, Computers and Relevant Logic: A Project in Computing Matrix Model Structures for Propositional Logics, Ph.D. thesis, Australian National University (1980) | DOI

[13] E. Yang, R and Relevance Principle Revisited, Journal of Philosophical Logic, vol. 42(5) (2013), pp. 767–782 | DOI

[14] E. Yang, Substructural Fuzzy-Relevance Logic, Notre Dame Journal of Formal Logic, vol. 56(3) (2015), pp. 471–491 | DOI