Subintuitionistic logics and the implications they prove. (English) Zbl 1437.03044
Summary: This is an investigation of the implications of \(\mathsf{IPC}\) which remain provable when one weakens intuitionistic logic in various ways. The research is concerned with logics with Kripke models as introduced by G. Corsi [Z. Math. Logik Grundlagen Math. 33, 389–406 (1987; Zbl 0645.03004)], and others like G. Restall, Došen, Visser. This leads to conservativity results for \(\mathsf{IPC}\) with regard to classes of implications in some of these logics. Moreover, similar results are reached for some weaker subintuitionistic systems with neighborhood models introduced by the authors [Lect. Notes Comput. Sci. 10148, 333–354 (2017; Zbl 1428.03044)]. In addition, the relationship between two types of neighborhood models introduced in that work is clarified. This clarification leads also to modal companions for weaker logics.
MSC:
| 03B20 | Subsystems of classical logic (including intuitionistic logic) |
| 03B45 | Modal logic (including the logic of norms) |
Keywords:
intuitionistic logic; subintuitionistic logic; conservativity; Kripke models; neighborhood modelsReferences:
| [1] | Bezhanishvili, N.; de Jongh, D., Stable logics in intuitionistic logic, Notre Dame J. Form. Log., 95, (2018) · Zbl 1456.03052 |
| [2] | Celani, S.; Jansana, R., A closer look at some subintuitionistic logics, Notre Dame J. Form. Log., 42, 4, 225-255, (2001) · Zbl 1034.03007 |
| [3] | Chellas, B., Modal logic: an introduction, (1980), Cambridge University Press · Zbl 0431.03009 |
| [4] | Colacito, A.; de Jongh, D.; Vargas, A. L., Subminimal negation, Soft Comput., 21, 1, 165-174, (2017) · Zbl 1396.03009 |
| [5] | Corsi, G., Weak logics with strict implication, Z. Math. Log. Grundl. Math., 33, 389-406, (1987) · Zbl 0645.03004 |
| [6] | de Jongh, D.; Shirmohammadzadeh Maleki, F., Subintuitionistic logics with Kripke semantics, (11th International Tbilisi Symposium on Logic, Language, and Computation, TbiLLC 2015, LNCS, vol. 10148, (2017), Springer), 333-354 · Zbl 1428.03044 |
| [7] | D. de Jongh, F. Shirmohammadzadeh Maleki, Two neighborhood semantics for subintuitionistic logic, in: The Proceedings of the 12th International Tbilisi Symposium on Logic, Language, and Computation, TbiLLC 2017 (in press).; D. de Jongh, F. Shirmohammadzadeh Maleki, Two neighborhood semantics for subintuitionistic logic, in: The Proceedings of the 12th International Tbilisi Symposium on Logic, Language, and Computation, TbiLLC 2017 (in press). |
| [8] | Došen, K., Duality between modal algebras and neighborhood frames, Studia Logica, 48, 219-234, (1989) · Zbl 0685.03013 |
| [9] | Došen, K., Modal translation in K and D, (Diamonds and Defaults, The Series Synthese Library, vol. 229, (1994)), 103-127 |
| [10] | Hansen, H. H., Monotonic modal logics, (2003), University of Amsterdam, (Master thesis) |
| [11] | Kleene, S. C., Disjunction and existence under implication in elementary intuitionistic formalisms, J. Symbolic Logic, 27, 11-18, (1962) · Zbl 0112.24502 |
| [12] | Restall, G., Subintuitionistic logics, Notre Dame J. Form. Log., 35, 1, (1994), Winter · Zbl 0811.03005 |
| [13] | Shirmohammadzadeh Maleki, F.; de Jongh, D., Weak subintuitionistic logics, Logic J. IGPL, 25, 2, 214-231, (2017) · Zbl 1405.03029 |
| [14] | Troelstra, A. S.; Van Dalen, D., Constructivism in mathematics, (2014), Elsevier · Zbl 0653.03040 |
| [15] | Visser, A.; van Benthem, J.; de Jongh, D.; Renardel de Lavalette, G., NNIL, a study in intuitionistic propositional logic, (Ponse, A.; de Rijke, M.; Venema, Y., Modal Logics and Process Algebra, a Bisimulation Perspective, (1995)), 289-326 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
