Absurdity, Contradictions, and Logical Formalisms

In this paper, we examine logical formalisms with respect to contradictions and related notions. We deal with virtually any kind of logical formalisms, including those that fail to have any connective. We investigate various properties, especially paraconsistency. Among the topics discussed are ways of specifying contradictions and, more generally, inconsistency, as well as a few criteria for paraconsistency. We further explore logical formalisms that overlap with fragments of classical logic, and provide a couple of formal results for them. A number of options about inconsistency are also discussed throughout the text, insisting on how contradictions can be specified either directly, or by way of reference to a fragment of classical logic. Importantly, the concept of undesirable conclusions is given a formal account, which is applied to most of the issues involved in this paper. A noticeable point is that we take into account not only features of the language, depending on the formalism, but also some aspects of inference, most notably the so-called Identity.