Empty Logics

Abstract

TS is a logic that has no valid inferences. But, could there be a logic without valid metainferences? We will introduce TS_ω, a logic without metainferential validities. Notwithstanding, TS_ω is not as empty—i.e., uninformative—as it gets, because it has many antivalidities. We will later introduce the two-standard logic [TS_ω, ST_ω], a logic without validities and antivalidities. Nevertheless, [TS_ω, ST_ω] 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.