On a Generalization of all Strong Kleene Generalizations of Classical Logic

Abstract In his 2016 article On All Strong Kleene Generalizations of Classical Logic , Stefan Wintein provides a detailed and comprehensive semantic and tableau-based analysis of the consequence relations that can be defined over the four-valued Belnap–Dunn semantics. These include familiar consequence relations like $$\textsf{FDE}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>FDE</mml:mi> </mml:math> , which takes $$\{t, b\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> as the set of designated values, but also much less familiar relations that don’t follow the designated-value strategy (i.e. defining logical consequence as preservation of a set of values). It turns out that many of the interesting features of these relations are made evident at the level of metainferences and, although Wintein discusses some aspects of metainferences, his work does not provide a systematic way to decide on the validity of metainferences for these logics. In this paper, we extend Wintein’s tableaux from inferences to metainferences. The paper ends with a discussion about different ways in which we can understand the notion of metainference validity.