### Working paper

## Reflection calculus and conservativity spectra

In this early paper C. Wright Mills tries to ground the possibility for the study of thinking (including logical) from the perspective of sociology of knowledge. Following G.H. Mead, he shows that thinking is a social process because every thinker converses with his or her audience using the norms of rationality and logicality common to his or her culture. Language serves as a mediator between thinking and social patterns. Proposing to consider the meaning of language as the common social behavior evoked by it, Mills finds a way to combine three levels of analysis: psychological, social and cultural.

The article is devoted to considering the problem of possible worlds in Leibniz. The author shows that the idea of possible worlds is basic in Leibniz’s theory of «the best of all possible worlds» where it is postulated in the metaphysical justification of the divine creation as a free act and in the solution of the theological problem concerning the existence of evil. Also, Leibniz connects this idea with logic which he interprets as a science about all possible worlds. Leibniz's dichotomy between «truths of reason» and «truths of fact» is investigated in the context of necessity and contingency. Logical and moral reasons for God's choice of the best of possible worlds are examined in detail in both early and mature works by Leibniz.

The article analyses the link between the famous French epistemologists Emile Meyerson and his younger friend and disciple, historian of science Alexandre Koyré. Koyré is known primarily by his research of the history of the scientific thought; his main interest was to grasp the moments of the transformation of rational structures within the history. Nevertheless, in this paper I will argue that Alexandre Koyré in some measure accepts Emile Meyerson’s claim on the immutability of human reason that constitutes the central point of Meyerson’s epistemology. But within the rationality in general he distinguishes the immutable core that is constituted by the logical laws of reasoning and the outer level of “mentality” which is subjected to the historical changes and transformations.

The present manual is written on the basis of the course on inductive logic which is delivered in English to philosophy students of National Research University Higher School of Economics. The manual describes the main approaches to constructing inductive logic; it clarifies its key notions and rules, and it formulates its major problems. This introductory text can be useful for all readers who are interested in contemporary inductive logic.

The present volume is devoted to the 'Open Rusian-Finish Colloquium on Logic' (ORFIC), held at the Saint-Petersburg State University, on June 14-16, 2012. Among the participants there were such prominent Finish logicians as Jaakko Hintikka, Ilkka Niiniluoto ang Gabriel Sandu. The volume covers the most interesting results recently obtained in different areas of research in logic.

This volume is of interest to everyone, concerned in modern logic.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.