# Empty Logics

## Abstract

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.

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## Acknowledgements

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.

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Correspondence to Federico Pailos.

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Pailos, F. Empty Logics. J Philos Logic 51, 1387–1415 (2022). https://doi.org/10.1007/s10992-021-09622-8

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• DOI: https://doi.org/10.1007/s10992-021-09622-8

### Keywords

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