Skip to main content
Springer Nature Link
Log in

Fragments of R-Mingle

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

The logic RM and its basic fragments (always with implication) are considered here as entire consequence relations, rather than as sets of theorems. A new observation made here is that the disjunction of RM is definable in terms of its other positive propositional connectives, unlike that of R. The basic fragments of RM therefore fall naturally into two classes, according to whether disjunction is or is not definable. In the equivalent quasivariety semantics of these fragments, which consist of subreducts of Sugihara algebras, this corresponds to a distinction between strong and weak congruence properties. The distinction is explored here. A result of Avron is used to provide a local deduction-detachment theorem for the fragments without disjunction. Together with results of Sobociński, Parks and Meyer (which concern theorems only), this leads to axiomatizations of these entire fragments — not merely their theorems. These axiomatizations then form the basis of a proof that all of the basic fragments of RM with implication are finitely axiomatized consequence relations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  1. Anderson, A. R., and N. D. Belnap, Jnr., ‘Enthymemes’, The Journal of Philosophy 58 (1961), 713–723.

    Google Scholar 

  2. Anderson, A. R., and N. D. Belnap, Jnr., Entailment: The Logic of Relevance and Necessity, Volume 1, Princeton University Press, 1975.

  3. Anderson, A. R., N. D. Belnap, Jnr., and J. M. Dunn, Entailment: The Logic of Relevance and Necessity, Volume 2, Princeton University Press, 1992.

  4. Avron, A., ‘Relevant entailment—semantics and the formal systems’, The Journal of Symbolic Logic 49 (1984), 334–432.

    Google Scholar 

  5. Avron, A., ‘On an implication connective of RM’, Notre Dame Journal of Formal Logic 27 (1986), 201–209.

    Google Scholar 

  6. Avron, A., ‘A constructive analysis of RM’, The Journal of Symbolic Logic 52 (1987), 939–951.

    Google Scholar 

  7. Avron, A., ‘The semantics and proof theory of linear logic’, Theoretical Computer Science 57 (1988), 161–184.

    Google Scholar 

  8. Avron, A., ‘Relevance and paraconsistency—a new approach’, The Journal of Symbolic Logic 55 (1990), 707–732.

    Google Scholar 

  9. Avron, A., ‘Relevance and paraconsistency—a new approach. Part II: The formal systems’, Notre Dame Journal of Formal Logic 31 (1990), 169–202.

    Google Scholar 

  10. Avron, A., ‘Axiomatic systems, deduction and implication’, Journal of Logic and Computation 2 (1992), 51–98.

    Google Scholar 

  11. Avron, A., ‘Whither relevance logic?’, Journal of Philosophical Logic 21 (1992), 243–281.

    Google Scholar 

  12. Avron, A., ‘Multiplicative conjunction as an extensional conjunction’, Logic Journal of the IGPL 5 (1997), 181–208.

    Google Scholar 

  13. Avron, A., ‘Implicational F-structures and implicational relevance logics’, The Journal of Symbolic Logic 65 (2000), 788–802.

    Google Scholar 

  14. Berman, J., and W. J. Blok, ‘The Fraser-Horn and apple properties’, Transactions of the American Mathematical Society 302 (1987), 427–465.

    Google Scholar 

  15. Blok, W. J., and W. Dziobiak, ‘On the lattice of quasivarieties of Sugihara algebras’, Studia Logica 45 (1986), 275–280.

    Google Scholar 

  16. Blok, W. J., P. Köhler, and D. Pigozzi, ‘The algebraization of logic’, unpublished manuscript, 1983.

  17. Blok, W. J., and D. Pigozzi, ‘On the structure of varieties with equationally definable principal congruences I’, Algebra Universalis 15 (1982), 195–227.

    Google Scholar 

  18. Blok, W. J., and D. Pigozzi, ‘A finite basis theorem for quasivarieties’, Algebra Universalis 22 (1986), 1–13.

    Google Scholar 

  19. Blok, W. J., and D. Pigozzi, ‘Protoalgebraic logics’, Studia Logica 45 (1986), 337–369.

    MathSciNet  MATH  Google Scholar 

  20. Blok, W. J., and D. Pigozzi, ‘Local deduction theorems in algebraic logic’, in H. Andreka, J. D. Monk and I. Nemeti, (eds.), Algebraic Logic, Colloquia Mathematica Societatis János Bolyai 54 Algebraic Logic, Budapest (Hungary), 1988, pp. 75–109.

    Google Scholar 

  21. Blok, W. J., and D. Pigozzi, ‘Algebraizable Logics’, Memoirs of the American Mathematical Society, Number 396, Amer. Math. Soc., Providence, 1989.

  22. Blok, W. J., and D. Pigozzi, ‘Algebraic semantics for universal Horn logic without equality’, in J.D.H. Smith and A. Romanowska, (eds.), Universal Algebra and Quasigroup Theory, Heldermann Verlag, Berlin, 1992, pp. 1–56.

    Google Scholar 

  23. Blok W. J., and D. Pigozzi, ‘Abstract algebraic logic and the deduction theorem’, The Bulletin of Symbolic Logic, to appear.

  24. Blok, W. J., and J. G. Raftery, ‘Ideals in quasivarieties of algebras’, in X. Caicedo and C. H. Montenegro, (eds.), Models, Algebras and Proofs, Lecture Notes in Pure and Applied Mathematics, Vol. 203, Marcel Dekker, New York, 1999, pp. 167–186.

    Google Scholar 

  25. Blok, W. J., and J. G. Raftery, ‘On assertional logic’, manuscript, 1998.

  26. Brady, R. T., ‘Rules in relevant logic-I: Semantic classification’, Journal of Philosophical Logic 23 (1994), 111–137.

    Google Scholar 

  27. Burris, S., and H. P. Sankappanavar, A Course in Universal Algebra, Graduate Texts in Mathematics, Springer-Verlag, New York, 1981.

    Google Scholar 

  28. Czelakowski, J., ‘Algebraic aspects of deduction theorems’, Studia Logica 44 (1985), 369–387.

    Google Scholar 

  29. Czelakowski, J., ‘Local deduction theorems’, Studia Logica 45 (1986), 377–391.

    Google Scholar 

  30. Czelakowski, J., Protoalgebraic Logics, Kluwer, Dordrecht, 2001.

    Google Scholar 

  31. Czelakowski, J., and W. Dziobiak, ‘Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class’, Algebra Universalis 27 (1990), 128–149.

    Google Scholar 

  32. Czelakowski, J., and W. Dziobiak, ‘The parameterized local deduction theorem for quasivarieties of algebras and its application’, Algebra Universalis 25 (1996), 373–419.

    Google Scholar 

  33. Czelakowski, J., and W. Dziobiak, ‘Deduction theorems within RM and its extensions’, The Journal of Symbolic Logic 64 (1999), 279–290.

    Google Scholar 

  34. Dummett, M., ‘A propositional calculus with a denumerable matrix’, The Journal of Symbolic Logic 24 (1959), 97–106.

    Google Scholar 

  35. Dunn, J.M., ‘Algebraic completeness results for R-mingle and its extensions’, The Journal of Symbolic Logic 35 (1970), 1–13.

    Google Scholar 

  36. Dunn, J.M., ‘Relevance logic and entailment’, in D. Gabbay and F. Guenthner, (eds.), Handbook of Philosophical Logic, Volume III, Reidel, Dordrecht, 1986, pp. 117–226.

    Google Scholar 

  37. Dunn J. M., ‘Partial gaggles applied to logics with restricted structural rules’, in P. Schroeder-Heister and K. Došen, (eds.), Substructural Logics, Clarendon Press, Oxford, 1993, pp. 63–108.

    Google Scholar 

  38. Dunn, J. M., and R. K. Meyer, ‘Algebraic completeness results for Dummett’s LC and its extensions’, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 17 (1971), 225–230.

    Google Scholar 

  39. Dunn, J. M., and G. Restall, ‘Relevance logic and entailment’, in D. Gabbay, (ed.), Handbook of Philosophical Logic, Volume 8, 2nd edn., Kluwer, Dordrecht, 2001, pp. 1–128 (revision of [36]).

    Google Scholar 

  40. Dziobiak, W., ‘Finitely generated congruence distributive quasivarieties of algebras’, Fundamenta Mathematica 133 (1989), 47–57.

    Google Scholar 

  41. Dziobiak, W., ‘Quasivarieties of Sugihara semilattices with involution’, Algebra and Logic 39 (2000), 26–36.

    Google Scholar 

  42. Font, J.M., and G. Rodríguez, ‘Note on algebraic models for relevance logic’, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 36 (1990), 535–540.

    Google Scholar 

  43. Font, J.M., and G. Rodríguez Pérez, ‘A note on Sugihara algebras’, Publicacions Mathemátiques 36 (1992), 591–599.

    Google Scholar 

  44. Freese, R., and E. W. Kiss, ‘Algebra Calculator Program’, available at http://www.math.hawaii.edu/~ralph/software/uaprog or http://www.cs.elte.hu/~ewkiss/software/uaprog.

  45. Freese, R., and J. B. Nation, ‘Congruence lattices of semilattices’, Pacific Journal of Mathematics 49 (1973), 51–58.

    Google Scholar 

  46. Gabbay, D. M., A general theory of structured consequence relations, in P. Schroeder-Heister and K. Došen, (eds.), Substructural Logics, Clarendon Press, Oxford, 1993, pp. 109–151.

    Google Scholar 

  47. Gabbay, D. M., (ed.), What is a Logical System? Clarendon Press, Oxford, 1994.

    Google Scholar 

  48. Galatos, N., and J. G. Raftery, ‘Adding Involution to Residuated Structures’, Studia Logica 77 (2004), 181–207.

    Google Scholar 

  49. Gorbunov, V. A., Algebraic Theory of Quasivarieties, Consultants Bureau, New York, 1998.

    Google Scholar 

  50. Hobby, D., and R. McKenzie, The Structure of Finite Algebras, Contemporary Mathematics 76 American Mathematical Society (Providence, RI), 1988.

  51. Hsieh, A., Personal communication, 2002.

  52. JÓnsson B., ‘On finitely based varieties of algebras’, Colloquium Mathematicum 42 (1979), 255–2

    Google Scholar 

  53. Kearnes, K., ‘Type preservation in locally finite varieties with the CEP’, Canadian Journal of Mathematics 43 (1991), 748–769.

    Google Scholar 

  54. Kearnes, K., and R. McKenzie, ‘Commutator theory for relatively modular quasivarieties’, Transactions of the American Mathematical Society 331 (1992), 465–502.

    Google Scholar 

  55. Kearnes, K., and A. Szendrei, ‘The relationship between two commutators’, International Journal of Algebra and Computation 8 (1998), 497–531.

    Google Scholar 

  56. Kearnes, K., and R. Willard, ‘Residually finite, congruence meet-semidistributive varieties of finite type have a finite residual bound’, Proceedings of the American Mathematical Society 127 (1999), 2841–2850.

    Google Scholar 

  57. Köhler, P., and D. Pigozzi, ‘Varieties with equationally definable principal congruences’, Algebra Universalis 11 (1980), 213–219.

    Google Scholar 

  58. Lipparini, P., ‘n-Permutable varieties satisfy nontrivial congruence identities’, Algebra Universalis 33 (1995), 159–168.

    Google Scholar 

  59. Lipparini, P., ‘A characterization of varieties with a difference term, II: Neutral = meet semidistributive’, Canadian Mathematical Bulletin 41 (1998), 1318–327.

    Google Scholar 

  60. Łoś, J., and R. Suszko, ‘Remarks on sentential logics’, Proc. Kon. Nederl. Akad. van Wetenschappen, Series A 61 (1958), 177–183.

    Google Scholar 

  61. Mares, E. D., and R. K. Meyer, ‘Relevant logics’, in L. Goble, (ed.), The Blackwell Guide to Philosophical Logic, Blackwell, Oxford, 2001, pp. 280–308.

    Google Scholar 

  62. Méndez, J. M., ‘The compatibility of relevance and mingle’, Journal of Philosophical Logic 17 (1988), 279–297.

    Google Scholar 

  63. Meyer, R. K., ‘A characteristic matrix for RM’, unpublished manuscript, 1967.

  64. Meyer, R. K., ‘R-mingle and relevant disjunction’, The Journal of Symbolic Logic 36 (1971), 366.

    Google Scholar 

  65. Meyer, R. K., ‘Conservative extension in relevant implication’, Studia Logica 31 (1972), 39–46.

    Google Scholar 

  66. Meyer, R. K., ‘On conserving positive logics’, Notre Dame J. Formal Logic 14 (1973), 224–236.

    Google Scholar 

  67. Meyer, R. K., ‘Intuitionism, entailment, negation’, in H. Leblanc, (ed.), Truth, Syntax and Modality North Holland, Amsterdam, 1973, pp. 168–198.

    Google Scholar 

  68. Meyer, R. K., ‘Improved decision procedures for pure relevant logic’, in C. A. Anderson and M. Zeleny, eds.), Logic, Meaning and Computation, Kluwer, Dordrecht, 2001, pp. 191–217.

    Google Scholar 

  69. Meyer, R. K., and J. M. Dunn, ‘E, R and λ’, The Journal of Symbolic Logic 34 (1969), 460–474.

    Google Scholar 

  70. Meyer, R. K., J. M. Dunn, and H. Leblanc, ‘Completeness of relevant quantification theories’, Notre Dame Journal of Formal Logic 15 (1974), 97–121.

    Google Scholar 

  71. Meyer, R. K., and R. Z. Parks, ‘Independent axioms for the implicational fragment of Sobociński’s three-valued logic’, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 18 (1972), 291–295.

    Google Scholar 

  72. Meyer, R. K., and R. Routley, ‘Algebraic analysis of entailment, I’, Logique et Analyse N.S. 15 (1972), 407–428.

    Google Scholar 

  73. Myhill, J., ‘On the interpretation of the sign ‘⊂’’, The Journal of Symbolic Logic 18 (1953), 60–62.

    Google Scholar 

  74. Nurakunov, A. M., ‘Quasivarieties of algebras with definable principal congruences’ (English translation), Algebra and Logic 29 (1990), 26–34.

    Google Scholar 

  75. Ohnishi, M., and K. Matsumoto, ‘A system for strict implication’, Proceedings of the Symposium on the Foundations of Mathematics, Katadu, Japan, 1962, pp. 98–108.

  76. Parks, R. Z., ‘A note on R-mingle and Sobociński’s three-valued logic’, Notre Dame Journal of Formal Logic 13 (1972), 227–228.

    Google Scholar 

  77. Raftery, J. G., and T. Sturm, ‘Tolerance numbers, congruence n-permutability and BCK-algebras’, Czechoslovak Mathematical Journal 42(117) (1992), 727–740.

    Google Scholar 

  78. Rose, A., ‘An alternative formalization of Sobociński’s three-valued implicational propositional calculus’, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 12 (1956), 166–172.

    Google Scholar 

  79. Routley, R., and R. K. Meyer, ‘The semantics of entailment’, in H. Leblanc, (ed.), Truth, Syntax and Modality North Holland, Amsterdam, 1973, pp. 199–243.

    Google Scholar 

  80. Routley, R., R. K. Meyer, V. Plumwood, and R. T. Brady, Relevant Logics and their Rivals 1, Ridgeview Publishing Company, California, 1982.

    Google Scholar 

  81. Scott, D., ‘Rules and derived rules’, in S. Stenlund, (ed.), Logical Theory and Semantical Analysis, Reidel, Dordrecht, 1974, pp. 147–161.

    Google Scholar 

  82. Slaney, J. K., and R. K. Meyer, ‘A structurally complete fragment of relevant logic’, Notre Dame Journal of Formal Logic 33 (1992), 561–566.

    Google Scholar 

  83. Sobociński, B., ‘Axiomatization of a partial system of three-valued calculus of propositions’, The Journal of Computing Systems 1 (1952), 23–55.

    Google Scholar 

  84. Sugihara, T., ‘Strict implication free from implicational paradoxes’, Memoirs of the Faculty of Liberal Arts, Fukui University, Series I 1955, 55–59.

  85. Tamura, S., ‘The implicational fragment of R-mingle’, Proceedings of the Japan Academy 47 (1971), 71–75.

    Google Scholar 

  86. Tokarz, M., ‘Functions definable in Sugihara algebras and their fragments (I)’ and ‘(II)’, Studia Logica 34 (1975), 295–304, and 35 (1976), 279–283.

    Google Scholar 

  87. Tokarz, M., ‘Deduction theorems for RM and its extensions’, Studia Logica 38 (1979), 105–111.

    Google Scholar 

  88. Tokarz, M., Essays in matrix semantics of relevant logics, Polish Academy of Sciences, Institute of Philosophy and Sociology, Warsawa, 1980.

    Google Scholar 

  89. Troelstra, A. S., Lectures on Linear Logic, CSLI Lecture Notes, No 29, 1992.

  90. Urquhart, A., ‘The undecidability of entailment and relevant implication’, The Journal of Symbolic Logic 49 (1984), 1059–1073.

    Google Scholar 

  91. Ursini, A., ‘On subtractive varieties I, Algebra Universalis 31 (1994), 204–222.

    Google Scholar 

  92. Willard, R., ‘A finite basis theorem for residually finite, congruence meet-semidistributive varieties’, The Journal of Symbolic Logic 65 (2000), 187–200.

    Google Scholar 

  93. Wójcicki, R., Theory of Logical Calculi, Kluwer, Dordrecht, 1988.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851S Morgan Street, 60607-7045, Chicago, Illinois, USA

    W. J. Blok

  2. School of Mathematical Sciences, University of KwaZulu-Natal, Howard College Campus, 4041, Durban, South Africa

    J. G. Raftery

Authors
  1. W. J. Blok
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. J. G. Raftery
    View author publications

    You can also search for this author inPubMed Google Scholar

Additional information

Special issue of Studia Logica: “Algebraic Theory of Quasivarieties” Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko

Rights and permissions

Reprints and permissions

About this article

Cite this article

Blok, W.J., Raftery, J.G. Fragments of R-Mingle. Stud Logica 78, 59–106 (2004). https://doi.org/10.1007/s11225-005-0106-8

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-005-0106-8

Keywords