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The articles written by Vladimir L. Vasyukov and Anastasia O. Kopylova were published in "Logical Investigations"

The articles written by the researchers of the LLFP were published in "Logical Investigations" ( Vol. 25, No. 1, 2019).

V.L. Vasyukov. Quantum categories for quantum logic

Abstract

The paper is the contribution to quantum toposophy focusing on the abstract orthomodular structures (following Dunn-Moss-Wang terminology). Early quantum topo-sophical approach to "abstract quantum logic" was proposed based on the topos of functors [E, Sets] where E is a so-called orthomodular preorder category — a modification of categor­ically rewritten orthomodular lattice (taking into account that like any lattice it will be a finite co-complete preorder category). In the paper another kind of categorical semantics of quantum logic is discussed which is based on the modification of the topos construction itself — so called quantos — which would be evaluated as a non-classical modification of topos with some extra structure allowing to take into consideration the peculiarity of nega­tion in orthomodular quantum logic. The algebra of subobjects of quantos is not the Heyting algebra but an orthomodular lattice. Quantoses might be apprehended as an abstract re­flection of Landsman's proposal of "Bohrification", i.e., the mathematical interpretation of Bohr's classical concepts by commutative C*-algebras, which in turn are studied in their quantum habitat of noncommutative C*-algebras — more fundamental structures than com­mutative C*-algebras. The Bohrification suggests that topos-theoretic approach also should be modified. Since topos by its nature is an intuitionistic construction then Bohrification in abstract case should be transformed in an application of categorical structure based on an orthomodular lattice which is more general construction than Heyting algebra — orthomod-ular lattices are non-distributive while Heyting algebras are distributive ones. Toposes thus should be studied in their quantum habitat of "orthomodular" categories i.e. of quntoses. Also an interpretation of some well-known systems of orthomodular quantum logic in quan-tos of functors [E, QSets] is constructed where QSets is a quantos (not a topos) of quantum sets. The completeness of those systems in respect to the semantics proposed is proved.

 

DOI: 10.21146/2074-1472­2019-25-1-70-87

 

Anastasia O. Kopylova.  Empty terms in W. Ockham’s logic: what is the reference for chimaeras

Abstract

 The paper is devoted to the problem of supposition of terms in the propositions about imaginary objects and the conditions of their truth values in the doctrine of William of Ockham who was a leading figure of the scholastic nominalism. His rather radical ontological position acknowledges the existence of no more than two types of essences: unitary substances and qualities. Being devoid of the universals, the Ockhamist doctrine implied the transformation of the previously elaborated semantic theories, including the theory of supposition. In the reconstruction of Ockham’s thought that became classical, the supposition closely approached the reference; however, in 2000s C.Dutilh-Novaes proposed the interpretation of supposition as a theory of propositional meanings. This approach brings forth the understanding of supposition as an intensional rather than extensional theory. One of the crucial arguments for this reconstruction is based on the application of supposition in the propositions about imaginary objects. According to our view, this argument is not free from some drawbacks. The term that makes the reference to the imaginary objects can have only simple or material supposition but not a personal one. W. Ockham names imaginary objects impossible objects.Chimaera is an impossible object, because it is considered as something which is combined of parts of different animals.That’s why it should contain several substantial forms, which leads to contradiction with the metaphysical principle of the uniqueness of the substantial form. In Ockham’s doctrine affirmative propositions about imaginary objects are always false since chimeras do not possess real existence. This observation implies that propositions about imaginary objects are more adequately squared with the extensional rather than intensional interpretation of supposition.

DOI: 10.21146/2074-1472-2019-25-1-52-69