Inferences and Metainferences in ST
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Inferences and Metainferences in ST Pablo Cobreros1
· Paul Egre´ 2 · David Ripley3 · Robert van Rooij4
Received: 6 February 2019 / Accepted: 17 January 2020 / © The Author(s) 2020
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+ . Keywords Strict-tolerant logic · Metainferences · Proof theory · Internal vs external logic · Paradoxes The strict-tolerant logic ST was proposed to deal with paradoxes of vagueness and with the semantic paradoxes [8, 9]. There is something very distinctive about ST: Pablo Cobreros
[email protected] Paul Egr´e
[email protected] David Ripley
[email protected] Robert van Rooij
[email protected] 1
Department of Philosophy, University of Navarra, 31009 Pamplona, Spain
2
D´epartement de Philosophie & D´epartement d’Etudes Cognitives de l’ENS, Institut Jean-Nicod (CNRS-EHESS-ENS), PSL University, 29, rue d’Ulm, 75005, Paris, France
3
Philosophy Department SoPHIS, Building 11, Monash University, VIC 3800, Australia
4
Institute for Logic, Language and Computation, Universiteit van Amsterdam, P.O. Box 94242, 1090 GE, Amsterdam, The Netherlands
P. Cobreros et al.
namely, it is classical logic for a classical language, but it provides ways of strengthening classical logic to deal with paradoxes in enriched languages. For example, the logic ST+ (ST for a language with a transparent truth predicate T and self-referential sentences) is an inference-preserving extension of classical logic. That is, ST+ is not only non-trivial, but it extends all the valid inferences of classical logic to cover the full (T -involving) language [10, 23]. How is this possible? Well, because ST+ preserves all classically valid inferences but not some classical metainferences. The question then arises of exactly which are the metainferences of ST+ . In a recent paper, 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. In this paper we review their result and put the connection in a broader context. In particular, we show that in much the same way in which ST+ is related to the paraconsistent logic LP+ , it can be related to the paracomplete logic K3+ when we look at the logic’s meta-anti-inferences. The results in this paper are either results already proved in Barrio et al. [4] or corollaries based on duality considerations. On the technical side, the contributi
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