Abstract
When studying a logical calculus S of any kind, it is extremely important to be in a position to fmd a class of adequate models for it — i.e. a class of algebraic structures which verify exactly the provable formulae of S. Thus, for example, it turns out that the algebraic counterpart of classical propositional logic are Boolean algebras, while intuitionistic propositional logic corresponds to Heyting algebras. As a rule, these correspondences pave the way for a profitable interaction: the investigation of models may yield several fruitful insights on the structure of the given calculus, and, conversely, it may even happen that proof-theoretical techniques be of some avail in proving purely algebraic results (Grishin 1982; Kowalski and Ono 2000).
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Notes
To avoid possible misunderstandings of the previous remark, we point out that we do not intend by any means to underestimate the extremely important contributions of great logicians working in the relevant tradition, like Belnap, Dunn, Meyer and the other people mentioned above, to the algebraic knowledge of substructural logics.
For the sake of precision, the structures just defined are the duals of commutative Girard quantales as usually defined in the literature. Here and in the following, however, we shall feel free to disregard such distinctions.
To be precise once again, these notions coincide with one another up to dualities and up to minor differences in the presentation (w.r.t. e.g. the choice of primitives).
A systematic and thorough treatment of the theory of MV-algebras is contained in the volume by Cignoli et al. (1999); see also Hajek (1998).
Hereafter, when no confusion can arise, we shall fail to mention explicitly the carrier and the operations of the indicated structure. Thus, for example, in this case it is tacitly assumed that A is the carrier of A.
Henceforth, the expression “without loss of generality” will be abbreviated by “w.l.g.”.
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© 2002 Springer Science+Business Media Dordrecht
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Paoli, F. (2002). Algebraic Structures. In: Substructural Logics: A Primer. Trends in Logic, vol 13. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3179-9_5
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DOI: https://doi.org/10.1007/978-94-017-3179-9_5
Publisher Name: Springer, Dordrecht
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