Logical pluralism and non-classical category theory


V.L. Vasyukov


Logical pluralism prove to be much more intruiging phenomenon if we envisage its impact on elementary logical theories. Breaking the tenet of the unique (namely, classical) logical basis for those we find ourself in the realm of non-classical elementary logical theories based on the various non-classical logics. It is expecially important if we take into account that such theories underlie non-classical mathematics according to the apt slogan “there are as many mathematics as logics” — suffish it to recall relevant arithmetic, quantum set theory, fuzzy set theory, paraconsistent mathematics etc. In the paper non-classical axiomatic category theories are approached which are based on some non-classical categorical constructions.






Васюков В.Л. Интерпретация релевантной логики в топосах // Логика и В.Е.К. М., 2004. С. 112-121.
Васюков В.Л. Квантовая логика. М.: Per Se, 2005.
Васюков В.Л. Паранепротиворечивые категории для паранепротиворечивой логики // Логические исследования, вып. 17, М.-СПб.: Центр гуманитарных инициатив, 2011. С. 69-83.
Голдблатт Р. Топосы. Категорный анализ логики. М., 1983.
Blanc G., Donnadieu M. R. Axiomatisation de la categorie des categories // Cah. Topol. Geom. Different. XVII, 2, 1976. P. 1-38.
Beall J.C. and Restall G. Logical Pluralism // Australasian Journal of Philosophy, 78, 2000. P. 475-493.
N.C.A. da Costa. On Paraconsistent Set Theory // Logique et Analyse 115, 1986. P. 361-371.
N.C.A. da Costa. Paraconsistent Mathematics // Frontiers of Paraconsistent Logic/D.Batens, C.Mortensen, G.Priest and J.-P. van Bendegem (eds.), Research Studies Press Ltd., Baldock, Hartfordshire, England, 2000. P. 166-179.
N.C.A. da Costa, Bueno O., Volkov A. Outline of a Paraconsistent Category Theory // Alternative Logics. Do Science Need Them?/Weingartner P. (Ed.), Springer, Berlin, Heidelberg, New York, 2004. P. 95-114.
Meyer R.K. Relevant Arithmetic // Bulletin of the Section of Logic of the Polish Academy of Sciences, 5, 1976. P. 133-137.
Mortensen K. Inconsistent Mathematics. Dordrecht: Kluwer, 1995.
Takeuti G. Quantum Set Theory // Current Issues on quantum logic / Beltrametti S., Fraassen B. Van (eds.). New York; London: Plenum, 1981. P. 303-322.
Takeuti G. and Titani S.Fuzzy Logic and Fuzzy Set Theory // Arch. Math. Log., 1992. P. 17-18.
Vasyukov V.L. Paraconsistency in Categories // Frontiers of Paraconsistent Logic / D. Batens, C. Mortensen, G. Priest and J.-P. van Bendegem (eds.), Research Studies Press Ltd., Baldock, Hartfordshire, England, 2000. P. 263-278.
Vasyukov V.L. Quantum Categories for Quantum Logic (в печати).