Skip to main content

Empty Logics


TS is a logic that has no valid inferences. But, could there be a logic without valid metainferences? We will introduce TSω, a logic without metainferential validities. Notwithstanding, TSω is not as empty—i.e., uninformative—as it gets, because it has many antivalidities. We will later introduce the two-standard logic [TSω, STω], a logic without validities and antivalidities. Nevertheless, [TSω, STω] is still informative, because it has many contingencies. The three-standard logic [\(\mathbf {TS}_{\omega }, \mathbf {ST}_{\omega }, [{\overline {\emptyset }}{\emptyset }, {\emptyset } {\overline {\emptyset }}]\)] that we will further introduce, has no validities, no antivalidities and also no contingencies whatsoever. We will also present two more validity-empty logics. The first one has no supersatisfiabilities, unsatisfabilities and antivalidities. The second one has no invalidities nor non-valid-nor-invalid (meta)inferences. All these considerations justify thinking of logics as, at least, three-standard entities, corresponding, respectively, to what someone who takes that logic as correct, accepts, rejects and suspends judgement about, just because those things are, respectively, validities, antivalidities and contingencies of that logic. Finally, we will present some consequences of this setting for the monism/pluralism/nihilism debate, and show how nihilism and monism, on one hand, and nihilism and pluralism, on the other hand, may reconcile—at least according to how Gillian Russell understands nihilism, and provide some general reasons for adopting a multi-standard approach to logics.

This is a preview of subscription content, access via your institution.


  1. Barrio, E., Pailos, F., & Szmuc, D. (2018). A recovery operator for nontransitive approaches. Review of Symbolic Logic, pp. 1–25.

  2. Barrio, E., Pailos, F., & Szmuc, D. (2018). Substructural logics, pluralism and collapse. Synthese.

  3. Barrio, E., Pailos, F., & Szmuc, D. (2019). A hierarchy of classical and paraconsistent Logics. Journal of Philosophical Logic, pp. 1–28.

  4. Barrio, E., Pailos, F., & Szmuc, D. (2019). (Meta)inferential levels of entailment beyond the Tarskian paradigm. Synthese.

  5. Barrio, E., Rosenblatt, L., & Tajer, D. (2015). The logics of strict-tolerant logic. Journal of Philosophical Logic, 44(5), 551–571.

    Article  Google Scholar 

  6. Blok, W., & Jónsson, B. (2006). Quivalence of consequence operations. Studia Logica, 83(1), 91–110.

    Article  Google Scholar 

  7. Chemla, E., Egré, P., & Spector, B. (2017). Characterizing logical consequence in many-valued logics. Journal of Logic and Computation, 27(7), 2193–2226.

    Google Scholar 

  8. Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41(2), 347–385.

    Article  Google Scholar 

  9. Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2014). Reaching transparent truth. Mind, 122(488), 841–866.

    Article  Google Scholar 

  10. Cobreros, P., Tranchini, L., & La Rosa, E. (2020). (I can’t get no) antisatisfaction. Synthese pp. 1–15.

  11. Cotnoir, A. (2018). Logical nihilism. In N. Pederson, N. Kellen, & J. Wyatt (Eds.) Pluralisms in truth and logic (pp. 301–329). Basingstoke: Palgrave Macmillan.

  12. Dicher, B. (2020). Requiem for logical nihilism, or: Logical nihilism annihilated. Synthese.

  13. Dicher, B., & Paoli, F. (2019). ST, LP, and tolerant metainferences. In C. Başkent T. Ferguson (Eds.) Graham priest on dialetheism and paraconsistency. Dordrecht: Springer.

  14. Dicher, B. (2021). The original sin of proof-theoretic semantics. Synthese, 198, 615–640.

    Article  Google Scholar 

  15. Estrada-Gonzalez, L. (2011). On the meaning of connectives (apropos of a non-necessitarianist challenge). Logica Universalis, 5, 115–126.

    Article  Google Scholar 

  16. French, R. (2016). Structural reflexivity and the paradoxes of self-reference. Ergo, 3(05).

  17. Humberstone, L. (1996). Valuational semantics of rule derivability. Journal of Philosophical Logic, 25(5), 451–461.

    Article  Google Scholar 

  18. Malinowski, G. (1990). Q-consequence operation. Reports on Mathematical Logic, 24(1), 49–59.

    Google Scholar 

  19. Malinowski, G. (2014). Kleene logic and inference. Bulletin of the Section of Logic, 43(1/2), 43–52.

    Google Scholar 

  20. Mortensen, C. (1989). Anything is possible. Erkenntnis, 30(3), 319–337.

    Article  Google Scholar 

  21. Pailos, F. (2019). A family of metainferential logics. The Review of Symbolic Logic, 29(1), 97–120.

    Google Scholar 

  22. Pailos, F. A fully classical truth theory characterized by substructural means. The Review of Symbolic Logic. Forthcoming.

  23. Prawitz, D. (1965). Natural deduction: a proof-theoretical study. Stockholm: Almqvist and Wiksell.

    Google Scholar 

  24. Priest, G. (2006). In Contradiction: A Study of the Transconsistent. Oxford: Oxford University Press.

  25. Restall, G. (2013). Assertion, denial, and non-classical theories. In K. Tanaka, F. Berto, E. Mares, & F. Paoli (Eds.) Paraconsistency: logic and applications. logic, epistemology, and the unity of science., (Vol. 26 pp. 81–100). Dordrecht: Springer.

  26. Ripley, D. (2017). Bilateralism, coherence, warrant. In F. Moltmann M. Textor (Eds.) Act-based conceptions of propositional content (pp. 307–324). Oxford University Press.

  27. Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139–164.

    Article  Google Scholar 

  28. Ripley, D. One step is enough. Manuscript.

  29. Ripley, D. A toolkit for metainferential logics. Manuscript.

  30. Ripley, D. (2012). Conservatively extending classical logic with transparent truth. Review of Symbolic Logic, 5(02), 354–378.

    Article  Google Scholar 

  31. Ripley, D. (2018). On the ‘transitivity’ of consequence relations. Journal of Logic and Computation, 28(2), 433–450.

    Article  Google Scholar 

  32. Rumfitt, I. (2000). “yes” and “no”. Mind, 109(436), 781–823.

    Article  Google Scholar 

  33. Russell, G. (2018). Logical nihilism: Could there be no logic? Philosophical Issues, 28(1), 308–324.

    Article  Google Scholar 

  34. Russell, G. (2018). Varieties of logical consequence by their resistance to logical nihilism. In N. Pederson, N. Kellen, & J. Wyatt (Eds.) Pluralisms in truth and logic. Basingstoke: Palgrave Macmillan.

  35. Scambler, C. (2020). Classical logic and the strict tolerant hierarchy. Journal of Philosophical Logic, 49, 351–370.

    Article  Google Scholar 

  36. Shramko, Y., & Wansing, H. (2010). Truth values. In E. Zalta (Ed.) The stanford encyclopedia of philosophy. Stanford Uniersity Summer 2010 edition.

  37. Smiley, T. (1996). Rejection. Analysis, 56(1), 1–9.

    Article  Google Scholar 

Download references


The ideas included in this article were presented to the audiences of the SeLoI: Seminario de Lógica Iberoamericana (2020), and the Buenos Aires Logic Group WIP Seminar (2020), to which I am also grateful for their feedback. Thanks also go to Dave Ripley, Peter Schroeder-Heister, Luca Tranchini, Luis Estrada-Gonzalez, Elia Zardini, Bodgan Dicher, Ole Hjortland, Joao Marcos, Bruno Da Ré, Damán Szmuc, Paula Teijeiro, Ariel Roffé, Joaquín Toranzo Calderón and the members of the Buenos Aires Logic Group. While writing this paper, I enjoyed a Humboldt Research Fellowship for experienced researchers (March 2020 to July 2021). This research was also supported by the CONICET and the University of Buenos Aires.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Federico Pailos.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pailos, F. Empty Logics. J Philos Logic 51, 1387–1415 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Logic
  • Metainferences
  • Metainferential validity
  • Multi-standard logics
  • Substructural logics
  • Empty logics