Algebraic Analysis of Demodalised Analytic Implication

  • Antonio LeddaEmail author
  • Francesco Paoli
  • Michele Pra Baldi
Open Access


The logic DAI of demodalised analytic implication has been introduced by J.M. Dunn (and independently investigated by R.D. Epstein) as a variation on a time-honoured logical system by C.I. Lewis’ student W.T. Parry. The main tenet underlying this logic is that no implication can be valid unless its consequent is “analytically contained” in its antecedent. DAI has been investigated both proof-theoretically and model-theoretically, but no study so far has focussed on DAI from the viewpoint of abstract algebraic logic. We provide several different algebraic semantics for DAI, showing their equivalence with the known semantics by Dunn and Epstein. We also show that DAI is algebraisable and we identify its equivalent quasivariety semantics. This class turns out to be a linguistic and axiomatic expansion of involutive bisemilattices, a subquasivariety of which forms the algebraic counterpart of Paraconsistent Weak Kleene logic (PWK). This fact sheds further light on the relationship between containment logics and logics of nonsense.


Analytic implication Dependence logic Abstract algebraic logic Plonka sums Regular varieties Relevance logic 



Preliminary versions of this paper were presented at the workshops Nonclassical modalities (Mexico City, September 2018) and Logic in Bochum IV (Bochum, October 2018). Thanks are due to the audiences of these talks for their insightful suggestions. A. Ledda and F. Paoli gratefully acknowledge the support of the Horizon 2020 program of the European Commission: SYSMICS project, number: 689176, MSCA-RISE-2015. All authors express their gratitude for the support of Fondazione di Sardegna within the project Science and its Logics: The Representation’s Dilemma, Cagliari, CUP: F72 F16 003 220 002, and the Regione Autonoma della Sardegna within the projects: Le proprietà d’ordine in matematica e fisica, CUP: F72 F16 002 920 002, Per un’estensione semantica della Logica Computazionale Quantistica - Impatto teorico e ricadute implementative, RAS: SR40341.


  1. 1.
    Bonzio, S., Gil Férez, J., Paoli, F., Peruzzi, L. (2017). On paraconsistent weak Kleene logic: axiomatisation and algebraic analysis. Studia Logica, 105(2), 253–297.CrossRefGoogle Scholar
  2. 2.
    Bonzio, S., Moraschini, T., Pra Baldi, M. Logics of left variable inclusion and Płonka sums of matrices, submitted for publication.Google Scholar
  3. 3.
    Burris S., & Sankappanavar H.P. (2012). A course in universal algebra, the Millennium Edition. Available at
  4. 4.
    Cabrer, L., & Priestley, H. (2015). A general framework for product representations: bilattices and beyond. Logic Journal of IGPL, 23, 816–841.CrossRefGoogle Scholar
  5. 5.
    Carnielli, W. (1987). Methods of proof for relatedness and dependence logic. Reports on Mathematical Logic, 21, 35–46.Google Scholar
  6. 6.
    Ciuni, R., Ferguson, T.M., Szmuc, D. (2018). Relevant logics obeying component homogeneity. Australasian Journal of Logic, 15(2), 301–361.CrossRefGoogle Scholar
  7. 7.
    Ciuni, R., Ferguson, T.M., Szmuc, D. Logics based on linear orders of contaminating values. typescript.Google Scholar
  8. 8.
    Dunn, J.M. (1972). A modification of Parry’s analytic implication. Notre Dame Journal of Formal Logic, 13(2), 195–205.CrossRefGoogle Scholar
  9. 9.
    Epstein, R.D. (1987). The algebra of dependence logic. Reports on Mathematical Logic, 21, 19–34.Google Scholar
  10. 10.
    Epstein R.D. (1995). The semantic foundations of logic: propositional logics, 2nd edn. New York: Oxford University Press.Google Scholar
  11. 11.
    Fariñas del Cerro, L., & Lugardon, V. (1991). Sequents for dependence logic. Logique et Analyse, 34(133-134), 57–71.Google Scholar
  12. 12.
    Ferguson, T.M. (2014). A computational interpretation of conceptivism. Journal of Applied Non-Classical Logics, 24(4), 333–367.CrossRefGoogle Scholar
  13. 13.
    Ferguson, T.M. (2017). Meaning and proscription in formal logic. Berlin: Springer.CrossRefGoogle Scholar
  14. 14.
    Fine, K. (1986). Analytic implication. Notre Dame Journal of Formal Logic, 27(2), 169–179.CrossRefGoogle Scholar
  15. 15.
    Font, J.M. (2016). Abstract algebraic logic: an introductory textbook. London: College Publications.Google Scholar
  16. 16.
    Font, J.M., Jansana, R., Pigozzi, D. (2003). A survey of abstract algebraic logic. Studia Logica, 74, 13–97.CrossRefGoogle Scholar
  17. 17.
    Kielkopf, C.F. (1977). Formal sentential entailment. Washington: University Press of America.Google Scholar
  18. 18.
    Meyer, R.K. (1974). New axiomatics for relevant logics I. Journal of Philosophical Logic, 3, 53–86.CrossRefGoogle Scholar
  19. 19.
    Parry, W.T. (1932). Implication. PhD Thesis, Harvard University.Google Scholar
  20. 20.
    Parry, W.T. (1968). The logic of C. I. Lewis. In Schilpp, P.A. (Ed.) , The Philosophy of C. I. Lewis (pp. 115–154). London: Cambridge University Press.Google Scholar
  21. 21.
    Plonka J. (1967). On distributive quasilattices. Fundamenta Mathematicae, 60, 191–200.CrossRefGoogle Scholar
  22. 22.
    Plonka J. (1967). On a method of construction of abstract algebras. Fundamenta Mathematicae, 61(2), 183–189.CrossRefGoogle Scholar
  23. 23.
    Płonka J. (1968). Some remarks on direct systems of algebras. Fundamenta Mathematicae, 62(3), 301–308.CrossRefGoogle Scholar
  24. 24.
    Romanowska, A. (1986). On regular and regularized varieties. Algebra Universalis, 23, 215–241.CrossRefGoogle Scholar
  25. 25.
    Romanowska, A., & Smith, J. (2002). Modes. Singapore: World Scientific.CrossRefGoogle Scholar
  26. 26.
    Sylvan, R. (2000). On reasoning: (Ponible) reason for (and also against) relevance. In Hyde, D., & Priest, G. (Eds.) , Sociative logics and their applications (pp. 175–188). Ashgate: Aldershot.Google Scholar
  27. 27.
    Urquhart, A. (1973). A semantical theory of analytic implication. Journal of Philosophical Logic, 2(2), 212–219.CrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Antonio Ledda
    • 1
    • 2
    Email author
  • Francesco Paoli
    • 1
  • Michele Pra Baldi
    • 1
  1. 1.Dipartimento di Pedagogia, Psicologia, FilosofiaUniversità di CagliariCagliariItaly
  2. 2.Department of MathematicsVanderbilt UniversityNashvilleUSA

Personalised recommendations