Abstract
In Chapter 2, we examined two different kinds of formalisms whose role is undoubtedly central in the proof theory of substructural logics: sequent calculi, on the one hand, and Hilbert-style systems, on the other. On that occasion, we noticed that there are at least six well-motivated axiomatic calculi — HRW, HR, HRMI, HRM, HLuk, and HLuk 3 — which do not have any sequential counterpart, in that they seem scarcely amenable to a treatment by means of traditional sequents. As we already remarked, Hilbert-style calculi are defmitely not the best one could hope for when it comes to engaging in proof search and theorem proving tasks. As a consequence, it seems desirable to find efficient and manageable formalisms also for the above-mentioned logics.
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Notes
See the next section for a thorough explanation of this terminology.
There is, however, a simpler cut elimination method for hypersequent calculi (see e.g. Ciabattoni and Ferrari 2001); its key feature is that the number of applications of EC is considered as an independent parameter in the induction.
For details, see Avron (1987; 1991b).
Slaney (1990) motivates very well these two different forms of bunching. X; Y is viewed as “the result of taking [the body of information] X as the determinant of available inference and applying it to Y”, whereas X, Y is taken as “the result of pooling information X with information Y”.
More precisely, Avron maintains that the first three properties are indispensable, whereas the last three are simply desirable.
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© 2002 Springer Science+Business Media Dordrecht
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Paoli, F. (2002). Other Formalisms. In: Substructural Logics: A Primer. Trends in Logic, vol 13. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3179-9_4
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DOI: https://doi.org/10.1007/978-94-017-3179-9_4
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