The aim of this article is to study the notion of derivability and its semantic counterpart in the context of non-transitive and non-reflexive substructural logics. For this purpose we focus on the study cases of the logics ST and TS. In this respect, we show that this notion doesn’t coincide, in general, with a nowadays broadly used semantic approach towards metainferential validity: the notion of local validity. Following this, and building on some previous work by Humberstone, we prove that in these systems derivability can be characterized in terms of a notion we call absolute global validity. However, arriving at these results doesn’t lead us to disregard local validity. First, because we discuss the conditions under which local, and also global validity, can be expected to coincide with derivability. Secondly, because we show how taking into account certain families of valuations can be useful to describe derivability for different calculi used to present ST and TS.
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We would like to thank two anonymous reviewers for their comments and suggestions which improved the clarity and the content of the article. Also, we would like to thank the audience of the Workshop on Substructural Logics and Metainferences (Buenos Aires, 2020). Our thanks also go to the members of the Buenos Aires Logic Group. This paper could not have been written without the financial aid of the National Scientific and Technical Research Council (CONICET).
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Da Ré, B., Szmuc, D. & Teijeiro, P. Derivability and Metainferential Validity. J Philos Logic 51, 1521–1547 (2022). https://doi.org/10.1007/s10992-021-09619-3
- Non-reflexive logics
- Non-transitive logics
- Local validity
- Absolute global validity