Validities, antivalidities and contingencies: A multi-standard approach

Meta-Classical Non-Classical Logics

“…Hence, a meta n inference is antivalid just in case it is never satisfied by a valuation. 12 In [4], some of us suggested that we can understand the antivalidities of a logic as the meta n inferences that the logic rejects:…”

“…There is a sense in which the study of metainferences can be traced back to Gentzen's [39] pioneer works on sequent calculi. 4 However, the more recent interest in metainferences emerged within studies in truth, vagueness, and other paradoxical phenomena. First, they were used as a technical tool to characterize logics ST and TS (see below) as well as the theories based upon them [e.g.…”

“…Ripley's notion does not apply to validity notions but to what the author calls counterexample relations 10. We say 'at least' but not 'at most' because in[4] some of us consider an even more stringent definition, according to which a logic comprises, in addition, notions of antivalidity and contingency for metainferences of each level. We remain neutral with respect to this latter approach.…”

A truth-maker semantics for ST: refusing to climb the strict/tolerant hierarchy

“…And once we have this-at this point admittedly still vagueidea, we can ask: Is there a logic that agrees with classical logic up to level 1, or 2, or 3, ... or perhaps even some arbitrarily high level n? Over the last couple of years, Barrio, Pailos, and Szmuc have shown that the (astonishing) answer is: Yes, there are such logics for any finite n, and we can even push this to the first limit ordinal ω and agree with classical logic at all finite levels, while still having a transparent truth-predicate (Barrio et al, 2019a;Pailos, 2020;Barrio and Pailos, 2022). 4 To see how this hierarchy of logics is constructed, we must define what meta-inferences of arbitrary levels are and how we can define logics governing such objects.…”