Skip to main content
SpringerLink
Log in

Introduction: Non-classical Logics—Between Semantics and Proof Theory (In Relation to Arnon Avron’s Work)

  • Chapter
  • First Online:
Arnon Avron on Semantics and Proof Theory of Non-Classical Logics

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 21))

  • 238 Accesses

Abstract

We recall some of the better known approaches to non-classical logics, with an emphasis on the contributions of Arnon Avron to the subject and in relation to the papers in this volume.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Some other contributions of Avron, which are not related to the theme of this volume, are not covered here. This includes his research on the foundations of mathematics, especially predicative mathematics (e.g., Avron 2008a, 2010; Avron and Cohen 2016), logical frameworks (e.g., Avron et al. 1992; Avron 2008b), as well as purely mathematical results (like Avron 1990a; Avron and Dershowitz 2016).

  2. 2.

    Sadly, Prof. J. Michael Dunn passed away prior to witnessing the publication of this volume. We shall cherish memories of him as a great logician, and as the title of his manuscript suggests, a very kind and humble man.

  3. 3.

    In the case of Grz, he was the first to do so.

References

  • Anderson, A. R., & Belnap, N. D. (1962). The pure calculus of entailment. Journal of Symbolic Logic, 27, 19–52.

    Article  Google Scholar 

  • Anderson, A. R., & Belnap, N. D. (1975). Entailment: The Logic of Relevance and Necessity (Vol. I). Princeton: Princeton University Press.

    Google Scholar 

  • Anderson, A. R., Belnap, N. D., & Dunn, J. M. (1992). Entailment: The Logic of Relevance and Necessity (Vol. II). Princeton: Princeton University Press.

    Google Scholar 

  • Arieli, O., & Avron, A. (1996). Reasoning with logical bilattices. Journal of Logic, Language, and Information, 5(1), 25–63.

    Article  Google Scholar 

  • Arieli, O., & Avron, A. (1998). The value of the four values. Artificial Intelligence, 102(1), 97–141.

    Article  Google Scholar 

  • Arieli, O., & Avron, A. (2000a). Bilattices and paraconsistency. In D. Batens, C. Mortensen, G. Priest, & J. Van Bendegem (Eds.), Frontiers of Paraconsistent Logic (Vol. 8, pp. 11–27). Studies in Logic and Computation. Research Studies Press.

    Google Scholar 

  • Arieli, O., & Avron, A. (2000b). General patterns for nonmonotonic reasoning: from basic entailments to plausible relations. Logic Journal of the IGPL, 8(2), 119–148.

    Google Scholar 

  • Arieli, O., Avron, A., & Zamansky, A. (2011). Ideal paraconsistent logics. Studia Logica, 99(1–3), 31–60.

    Google Scholar 

  • Avron, A. (1984a). On modal systems having arithmetical interpretations. Journal of Symbolic Logic, 49, 935–942.

    Google Scholar 

  • Avron, A. (1984b). Relevant entailment - semantics and formal systems. Journal of Symbolic Logic, 49, 334–342.

    Google Scholar 

  • Avron, A. (1987). A constructive analysis of RM. Journal of Symbolic Logic, 52, 939–951.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Avron, A. (1990a). On strict strong constructibility with a compass alone. Journal of Geometry, 38, 12–15.

    Google Scholar 

  • Avron, A. (1990b). Relevance and paraconsistency - A new approach. Journal of Symbolic Logic, 55, 707–732.

    Google Scholar 

  • Avron, A. (1991). Hypersequents, logical consequence and intermediate logics for concurrency. Annals of Mathematics and Artificial Intelligence, 4, 225–248.

    Article  Google Scholar 

  • Avron, A. (1996a). The method of hypersequents in proof theory of propositional non-classical logics. In Logic: Foundations to Applications (pp. 1–32). Oxford Science Publications.

    Google Scholar 

  • Avron, A. (1996b). The structure of interlaced bilattices. Journal of Mathematical Structures in Computer Science, 6, 287–299.

    Google Scholar 

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

    Article  Google Scholar 

  • Avron, A. (1999). On the proof theory of natural many-valued logics. Collegium Logicum (Annals of the Kurt Gödel Society), 3, 51–59.

    Google Scholar 

  • Avron, A. (2002). On negation, completeness and consistency. In Handbook of Philosophical Logic (Vol. 9, pp. 287–319). Kluwer.

    Google Scholar 

  • Avron, A. (2003). Transitive closure and the mechanization of mathematics. In Thirty Five Years of Automating Mathematics (pp. 149–171). Springer.

    Google Scholar 

  • Avron, A. (2008a). Constructibility and decidability versus domain independence and absoluteness. Theoretical Computer Science, 394, 144–158.

    Google Scholar 

  • Avron, A. (2008b). A framework for formalizing set theories based on the use of static set terms. In A. Avron, N. Dershowitz, & A. Rabinovich (Eds.), Pillars of Computer Science (Vol. 4800, pp. 87–106). LNCS. Springer.

    Google Scholar 

  • Avron, A. (2010). A new approach to predicative set theory. In R. Schindler (Ed.), Ways of Proof Theory (pp. 31–63). Onto series in mathematical logic. Onto verlag.

    Google Scholar 

  • Avron, A. (2014a). The classical constraint on relevance. Logica Universalis, 8, 1–15.

    Google Scholar 

  • Avron, A. (2014b). What is relevance logic? Annals of Pure and Applied Logic, 165, 26–48.

    Google Scholar 

  • Avron, A. (2015). Semi-implication: A chapter in universal logic. In The Road to Universal Logic (pp. 59–72). Springer.

    Google Scholar 

  • Avron, A. (2016). RM and its nice properties. In K. Bimbó (Ed.), J. Michael Dunn on Information Based Logics (Vol. 8, pp. 15–43). Outstanding Contributions to Logic. Springer.

    Google Scholar 

  • Avron, A., Arieli, O., & Zamansky, A. (2018). Theory of Effective Propositional Paraconsistent Logics (Vol. 75), Studies in Logic. College Publications.

    Google Scholar 

  • Avron, A., & Cohen, L. (2016). Formalizing scientifically applicable mathematics in a definitional framework. Journal of Formalized Reasoning, 9(1), 53–70.

    Google Scholar 

  • Avron, A., & Dershowitz, N. (2016). Cayley’s formula: A page from the book. American Mathematical Monthly, 123(7), 699–700.

    Article  Google Scholar 

  • Avron, A., Honsell, F. A., Mason, I. A., & Pollack, R. (1992). Using typed lambda calculus to implement formal systems on a machine. Journal of Automated Reasoning, 9, 309–354.

    Article  Google Scholar 

  • Avron, A., Konikowska, B., & Zamansky, A. (2013). Cut-free sequent calculi for C-systems with generalized finite-valued semantics. Journal of Logic and Computation, 23(3), 517–540.

    Article  Google Scholar 

  • Avron, A., Konikowska, B., & Zamansky, A. (2015). Efficient reasoning with inconsistent information using C-systems. Information Science, 296, 219–236.

    Article  Google Scholar 

  • Avron, A., & Lahav, O. (2018). A simple cut-free system for a paraconsistent logic equivalent to S5. In Advances in Modal Logic (Vol. 12, pp. 29–42). College Publications.

    Google Scholar 

  • Avron, A., & Lev, I. (2005). Non-deterministic multi-valued structures. Journal of Logic and Computation, 15, 241–261.

    Article  Google Scholar 

  • Avron, A., & Zamansky, A. (2016). A paraconsistent view on B and S5. In Advances in Modal Logic (Vol. 11, pp. 21–37). College Publications.

    Google Scholar 

  • Avron, A., & Zohar, Y. (2019). Rexpansions of nondeterministic matrices and their applications in nonclassical logics. The Review of Symbolic Logic, 12(1), 173–200.

    Article  Google Scholar 

  • Batens, D., Mortensen, C., Priest, G., & Van Bendegem, J. (Eds.). (2000). In Frontiers of Paraconsistent Logic, Proceedings of the first World Congress on Paraconsistency (Vol. 8). Studies in Logic and Computation. Research Studies Press.

    Google Scholar 

  • Bell, J., DeVidi, D., & Solomon, G. (2001). Logical Options: An Introduction to Classical and Alternative Logics. Broadview Press.

    Google Scholar 

  • Belnap, N. D. (1977a). How a computer should think. In G. Ryle (Ed.), Contemporary Aspects of Philosophy (pp. 30–56). Oriel Press.

    Google Scholar 

  • Belnap, N. D. (1977b). A useful four-valued logic. In J. M. Dunn, & G. Epstein (Eds.), Modern Uses of Multiple-Valued Logics (pp. 7–37). Reidel Publishing Company.

    Google Scholar 

  • Béziau, J. Y., Carnielli, W., & Gabbay, D. (Eds.). (2007). New Directions in Paraconsistent Logic: Proceedings of the 3rd World Congress of Paraconsistent Logic (Vol. 9). Studies in Logic. College Publications.

    Google Scholar 

  • Béziau, J. Y., Chakraborty, M., & Dutta, S. (Eds.). (2015). New Directions in Paraconsistent Logic: Proceedings of the 5th World Congress of Paraconsistent Logic (Vol. 152). Lecture Notes in Mathematics and Statistics. Springer.

    Google Scholar 

  • Bimbó, K. (2006). Relevance logics. In D. Jacquette (Ed.), Philosophy of Logic (pp. 723–789). Elsevier.

    Google Scholar 

  • Bull, R. A., & Segerberg, K. (2001). Basic modal logic. In D. Gabbay, & F. Guenther (Eds.), Handbook of Philosophical Logic (2nd ed., Vol. 3, pp. 1–81). Kluwer.

    Google Scholar 

  • Carnielli, W. A., & Coniglio, M. E. (2016). Paraconsistent Logic: Consistency, Contradiction and Negation. Number 40 in Logic, Epistemology, and the Unity of Science. Springer.

    Google Scholar 

  • Carnielli, W. A., Coniglio, M. E., & D’Ottaviano, I. (Eds.) (2001). Paraconsistency: The Logical Way to the Inconsistent – Proceedings of the second World Congress on Paraconsistency. Number 228 in Lecture Notes in Pure and Applied Mathematics. Marcel Dekker.

    Google Scholar 

  • Carnielli, W. A., Coniglio, M. E., & Marcos, J. (2007). Logics of formal inconsistency. In D. M. Gabbay & F. Guenthner (Eds.), Handbook of Philosophical Logic (2nd ed., Vol. 14, pp. 1–95). Springer.

    Google Scholar 

  • Chellas, B. F. (1980). Modal Logic – An introduction. Cambridge University Press.

    Google Scholar 

  • Cornelis, C., Arieli, O., Deschrijver, G., & Kerre, E. (2007). Uncertainty modeling by bilattice-based squares and triangles. IEEE Transactions on Fuzzy Systems, 15(2), 161–175.

    Article  Google Scholar 

  • da Costa, N. C. A. (1974). On the theory of inconsistent formal systems. Notre Dame Journal of Formal Logic, 15, 497–510.

    Google Scholar 

  • Dunn, J. M. (1966). The Algebra of Intensional Logics. PhD thesis, University of Pittsburgh, Ann Arbor (UMI).

    Google Scholar 

  • Dunn, J. M., & Restall, G. (2002). Relevance logic. In D. Gabbay & F. Guenther (Eds.), Handbook of Philosophical Logic (2nd ed., Vol. 6, pp. 1–136). Kluwer.

    Google Scholar 

  • Fitting, M. (1991). Bilattices and the semantics of logic programming. Journal of Logic Programming, 11(2), 91–116.

    Article  Google Scholar 

  • Fitting, M. (1993). The family of stable models. Journal of Logic Programming, 17, 197–225.

    Article  Google Scholar 

  • Fitting, M. (2002). Fixpoint semantics for logic programming a survey. Theoretical Computer Science, 278(1–2), 25–51.

    Article  Google Scholar 

  • Fitting, M. (2006). Bilattices are nice things. In T. Bolander, V. Hendricks & S. A. Pedersen (Eds.), Self Reference (Vol. 178, pp. 53–77). CSLI Lecture Notes. CLSI Publications.

    Google Scholar 

  • Gabbay, D. (1986). What is negation in a system? In F. R. Drake & J. K. Truss (Eds.), Logic Colloquium ’86 (pp. 95–112). North Holland: Elsevier Science Publishers.

    Google Scholar 

  • Gabbay, D., & Wansing, H. (Eds.). (1999). What is Negation? Applied Logic Series (Vol. 13). Springer.

    Google Scholar 

  • Ginsberg, M. (1988). Multi-valued logics: A uniform approach to reasoning in AI. Computer Intelligence, 4, 256–316.

    Google Scholar 

  • Jaśkowski, S. (1999). A propositional calculus dor inconsistent deductive systems. Logic, Language and Philosophy, 7, 35–56. Translation of the original paper in Studia Societatis Scienctiarun Torunesis, Sectio A, I(5), 57–77 (1948).

    Google Scholar 

  • Lukasiewicz, J. & Wedin, V. (1971). On the principle of contradiction in aristotle. The Review of Metaphysics, pp. 485–509.

    Google Scholar 

  • Mares, E. D. (2004). Relevant Logic. Cambridge University Press.

    Google Scholar 

  • Metcalfe, G., Olivetti, N., & Gabbay, D. (2009). Proof Theory for Fuzzy Logics. Springer.

    Google Scholar 

  • Nelson, D. (1959). Negation and separation of concepts in constructive systems. In A. Heyting (Ed.), Constructivity in Mathematics (pp. 208–225). North Holland.

    Google Scholar 

  • Priest, G. (2012). An Introduction to Non-Classical Logic. Cambridge University Press.

    Google Scholar 

  • Read, S. (1988). Relevant Logic. Basil Blackwell.

    Google Scholar 

  • Schechter, E. (2005). Classical and Nonclassical Logics: An Introduction to the Mathematics of Propositions. Princeton University Press.

    Google Scholar 

  • van Benthem, J., Heinzmann, M., Rebuschi, G., & Visser, H. (2009). The Age of Alternative Logics (2nd ed.). Springer.

    Google Scholar 

  • Vasilev, N. A. (1993). Imaginary (non-aristotelian) logic. Axiomathes, 4(3), 353–355.

    Google Scholar 

  • Wansing, H. (2014). Connexive logic. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/logic-connexive/.

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, Tel-Aviv Academic College, Tel-Aviv, Israel

    Ofer Arieli

  2. Department of Information Systems, University of Haifa, Haifa, Israel

    Anna Zamansky

Authors
  1. Ofer Arieli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anna Zamansky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ofer Arieli .

Editor information

Editors and Affiliations

  1. School of Computer Science, Academic College of Tel Aviv-Yafo, Tel Aviv, Israel

    Ofer Arieli

  2. Information Systems Department, University of Haifa, Haifa, Israel

    Anna Zamansky

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Arieli, O., Zamansky, A. (2021). Introduction: Non-classical Logics—Between Semantics and Proof Theory (In Relation to Arnon Avron’s Work). In: Arieli, O., Zamansky, A. (eds) Arnon Avron on Semantics and Proof Theory of Non-Classical Logics. Outstanding Contributions to Logic, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-030-71258-7_1

Download citation

Publish with us

Policies and ethics

Search

Navigation