Abstract
In a recent paper, Barrio, Tajer and Rosenblatt establish a correspondence between metainferences holding in the strict-tolerant logic of transparent truth ST+ and inferences holding in the logic of paradox LP+. They argue that LP+ is ST+’s external logic and they question whether ST+’s solution to the semantic paradoxes is fundamentally different from LP+’s. Here we establish that by parity of reasoning, ST+ can be related to LP+’s dual logic K3+. We clarify the distinction between internal and external logic and argue that while ST+’s nonclassicality can be granted, its self-dual character does not tie it to LP+ more closely than to K3+.
Highlights
The strict-tolerant logic ST was proposed to deal with paradoxes of vagueness and with the semantic paradoxes [8, 9]
Box 94242, 1090 GE, Amsterdam, The Netherlands namely, it is classical logic for a classical language, but it provides ways of strengthening classical logic to deal with paradoxes in enriched languages
Eduardo Barrio, Lucas Rosenblatt and Diego Tajer show that ST+’s metainferences are closely related to inferences in LP+, the logic LP extended with a transparent truth predicate
Summary
Let L be a propositional language with the usual connectives: ∧, ∨, ⊃, ¬. If being true means taking the value 1 (strict truth), an argument is valid just in case no interpretation gives all premises the value 1 and all conclusions a value less than 1. If being true means taking a value greater than 0 (tolerant truth), an argument is valid just in case no interpretation makes all premises greater than 0 and all conclusions equal to 0. ST+ is a conservative extension of classical logic with a transparent truth-predicate and self-reference. This means that ST+ does not add or subtract valid inferences for the language without T.
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