Logic Supergroup — Online Colloquium, July 3, 2020
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João Marcos — Let gluts and gaps prevail!
One of the all time favorite strategies for defining a non-classical
negation proceeds by considering additional truth-values, besides `the
True' and `the False', with the intent of using the latter to localise
the phenomena of negation-inconsistency and negation-undeterminedness. From a philosophical standpoint, such an approach often translates, with varying degrees of success, into the consideration of `gaps' and `gluts', thought of as truth-values on their own right. From the perspective of the standard Tarskian consequence-theoretic framework, one may claim that the collection of truth-values associated to a given logic constitutes hardly anything beyond a technical expedient used within the so-called `logical matrices' in order to define some convenient notion of entailment. Indeed, at the metalogical level, no more than one or two `logical values' are needed in order to explicate any given consequence relation and the associated one-dimensional `logical theories' that are intended to collect the assertions upon which one happens to be interested. In this talk I will defend the systematic use of a generalized notion of logical consequence that allows for: (i) gappy and glutty reasoning to be naturally captured, at the metalogical level; (ii) a two-dimensional notion of logical theory, containing both assertions and denials, to be explored; (iii) a plurality of inferential mechanisms to cohabit. Furthermore, concerning the choice of logical primitives, I will also argue that truth-values and judgments about logical consequence are advantageously replaced by cognitive attitudes and judgments about logical incompatibility.
