Abstract
The aim of this chapter is to apply to some metainferential logics the notions of meta-validity presented in Chapter 2. First, we will explore the global\(_2\) notion (since it is a bit simpler to handle from a technical point of view), and we will show that the pure metainferential logic \(\textbf{ST }\) also recovers classical logic by substructural means, just as the \(\textbf{ST }\)-hierarchy does locally.
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Notes
- 1.
Although in this chapter we will focus on Strong Kleene logics, all the results can be easily adapted to their Weak Kleene counterparts.
- 2.
As we mentioned in Chap. 3, here we are assuming that the constant for the intermediate value is not expressible in the language. Otherwise, independently of the chosen notion for metainferential validity, \(\textbf{ST }\) and \(\textbf{CL }\) will not coincide at the inferential level. We are grateful to an anonymous reviewer for making us reflect on this point.
- 3.
Here, as before, we are assuming that this logic determines the higher levels. So, for instance, the metainferences of level 2 are determined by \(\textbf{TS }\textbf{ST }/\textbf{TS }\textbf{ST }\), those of level 3 by \(\textbf{TS }\textbf{ST }\textbf{TS }\textbf{ST }/\textbf{TS }\textbf{ST }\textbf{TS }\textbf{ST }\), and so on.
- 4.
We became aware of the result for \(\mathbf {TS/ST }\) after an email exchange with Chris Scambler, who first discovered it.
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Pailos, F., Da Ré, B. (2023). Hierarchies of Global and Absolutely Global Metainferential Logics. In: Metainferential Logics. Trends in Logic, vol 61. Springer, Cham. https://doi.org/10.1007/978-3-031-44381-7_7
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