Abstract
We study filters in residuated structures that are associated with congruence relations (which we call \({\mathsf{FL}}\) -filters), and develop a semantical theory for general substructural logics based on the notion of primeness for those filters. We first generalize Stone’s sheaf representation theorem to general substructural logics and then define the primeness of \({\mathsf{FL}}\) -filters as being “points” (or stalkers) of the space, the spectrum, on which the representing sheaf is defined. Prime FL-filters will turn out to coincide with truth sets under various well known semantics for certain substructural logics. We also investigate which structural rules are needed to interpret each connective in terms of prime \({\mathsf{FL}}\) -filters in the same way as in Kripke or Routley-Meyer semantics. We may consider that the set of the structural rules that each connective needs in this sense reflects the difficulty of giving the meaning of the connective. A surprising discovery is that connectives \(\&,\oplus,\otimes\) , ⅋ of linear logic are linearly ordered in terms of the difficulty in this sense.
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References
Abramsky S., Jagadeesan R. (1994) ‘Games and full completeness for multiplicative linear logic’. J. Symbolic Logic 59, 543–754
Abrusci M. (1991) ‘Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic’. J. Symbolic Logic 56, 1403–1451
Allwein G., Dunn M. (1993) ‘Kripke Model for Linear Logic’. J. Symbolic Logic 58, 514–545
Blass A. (1992) ‘A game semantics for linear logic’. Ann. Pure Appl. Logic 56, 183–220
Chang C. (1958) ‘Algebraic analysis of many valued logics’. Trans. Amer. Math. Soc. 88, 467–490
Chang C., (1959) ‘A new proof of the completeness of the Lukasiewicz axioms’. Trans. Amer. Math. Soc. 93, 74–80
Dunn, M., ‘Gaggle theory: an abstract Galois connections and residuation with application to negation and various logical operations’, in Logics in AI, Proc. European Workshop JELIA 1990, Springer-Verlag, Berlin, 1991.
Galatos N., Ono H. (2006) ‘Algebraization, parameterized local deduction theorem and interpolation for substructural logics over FL’. Studia Logica 83, 279–308
Hartonas C. (1997) ‘Duality for lattice-ordered algebras and for normal algebraizable logics’. Studia Logica 58, 403–450
MacLane S., Moerdijk I. (1992) Sheaves in geometry and logic. Springer-Verlag, New York
Ono, H., ‘Substructural logics and residuated lattices – an introduction’, in V. F. Hendricks and J. Malinowski (eds.), 50 Years of Studia Logica, Trends in Logic 21(2003), 193–228.
Restall, G., ‘Relevant and substructural logics’, in D. Gabbay and J. Woods (eds.), Handbook of the History and Philosophy of Logic.
Sato, K., Alternative topos theory and stalkers, preprint.
Sato, K., The 4-valued logic in terms of substructural logic, preprint.
Urquhart A. (1996) ‘Duality for algebras for relevance logic’. Studia Logica 56, 263–276
Yetter D. (1990) ‘Quantales and (noncommutative) linear logic’. J. Symbolic Logic 55, 41–64
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Kentaro, S. Proper Semantics for Substructural Logics, from a Stalker Theoretic Point of View. Stud Logica 88, 295–324 (2008). https://doi.org/10.1007/s11225-008-9106-9
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DOI: https://doi.org/10.1007/s11225-008-9106-9