Intermediate prepositional logics we consider here describe the setI() of regular informational types introduced by Yu. T. Medvedev [7]. He showed thatI() is a Heyting algebra. This algebra gives rise to the logic of infinite problems from [13] denoted here asLM 1. Some other definitions of negation inI() lead to logicsLM n (n ). We study inclusions between these and other systems, proveLM n to be non-finitely axiomatizable (n ) and recursively axiomatizable (n ). We also show that formulas (...) in one variable do not separateLM from Heyting's logicH, andLM n (n ) from Scott's logic (H+S). (shrink)

There is a Turing computable embedding $\Phi $ of directed graphs $\mathcal {A}$ in undirected graphs. Moreover, there is a fixed tuple of formulas that give a uniform effective interpretation; i.e., for all directed graphs $\mathcal {A}$, these formulas interpret $\mathcal {A}$ in $\Phi $. It follows that $\mathcal {A}$ is Medvedev reducible to $\Phi $ uniformly; i.e., $\mathcal {A}\leq _s\Phi $ with a fixed Turing operator that serves for all $\mathcal {A}$. We observe that there is a graph (...) G that is not Medvedev reducible to any linear ordering. Hence, G is not effectively interpreted in any linear ordering. Similarly, there is a graph that is not interpreted in any linear ordering using computable $\Sigma _2$ formulas. Any graph can be interpreted in a linear ordering using computable $\Sigma _3$ formulas. Friedman and Stanley [4] gave a Turing computable embedding L of directed graphs in linear orderings. We show that there is no fixed tuple of $L_{\omega _1\omega }$ -formulas that, for all G, interpret the input graph G in the output linear ordering $L$. Harrison-Trainor and Montalbán [7] have also shown this, by a quite different proof. (shrink)

Let ${\mathcal{P}_s}$ be the lattice of degrees of non-empty ${\Pi_1^0}$ subsets of 2 ω under Medvedev reducibility. Binns and Simpson proved that FD(ω), the free distributive lattice on countably many generators, is lattice-embeddable below any non-zero element in ${\mathcal{P}_s}$ . Cenzer and Hinman proved that ${\mathcal{P}_s}$ is dense, by adapting the Sacks Preservation and Sacks Coding Strategies used in the proof of the density of the c.e. Turing degrees. With a construction that is a modification of the one by (...) Cenzer and Hinman, we improve on the result of Binns and Simpson by showing that for any ${\mathcal{U} < _s \mathcal{V}}$ , we can lattice embed FD(ω) into ${\mathcal{P}_s}$ strictly between ${deg_s(\mathcal{U})}$ and ${deg_s({\mathcal V})}$ . We also note that, in contrast to the infinite injury in the proof of the Sacks Density Theorem, in our proof all injury is finite, and that this is also true for the proof of Cenzer and Hinman, if a straightforward simplification is made. (shrink)

Монография предназначена для специалистов в области социальной философии, социологии и истории, а также для широкого круга читателей.

1. There can be no doubt as to the fact that the natural numbers of the ancients were numbers having neither a positive nor a negative sign. We find it necessary to insist upon the recognition of this truth, because articles and books on the history of elementary mathematics yield much information on how the notion of negative numbers gradually arose, but it is impossible to find any indication as to how the notion of positive numbers came about, and also (...) as to the role that was played by numbers without signs in the history of mathematics. (shrink)

Despite tendencies to compete for a prime place in moral theory, neither virtue ethics nor the four principles approach should claim to be superior to, or logically prior to, the other. Together they provide a more adequate account of the moral life than either can offer on its own. The virtues of principlism are clarity, simplicity and (to some extent) universality. These are well illustrated by Ranaan Gillon’s masterly analysis of the cases he has provided. But the vices of this (...) approach are the converse of its virtues: neglect of emotional and personal factors, oversimplification of the issues, and excessive claims to universality. Virtue ethics offers a complementary approach, providing insights into moral character, offering a blend of reason and emotion, and paying attention to the context of decisions. The cases provided can be more adequately understood if we combine the approaches. Both should foster the virtues of humility and magnanimity. (shrink)

t. 1. Vlastʹ kli︠u︡cheĭ -- t. 2. Na vesakh Iova.

Издание адресовано научным работникам, преподавателям, студентам и аспирантам, а также практикам, интересующимся вопросами истории философии, эстетики, религиоведения и смежными проблемами.

Книга представляет собой исследование, где предпринимается попытка на основе детальной проработки первоисточников и воспроизведения основных ходов мысли и интуиций Гегеля и позднего Шеллинга провести сравнительный анализ их учений об абсолюте.