We study the class of ω-categorical structures with n-degenerate algebraic closure for some n ε ω, which includes ω-categorical structures with distributive lattice of algebraically closed subsets , and in particular those with degenerate algebraic closure. We focus on the models of ω-categorical universal theories, absolutely ubiquitous structures, and ω-categorical structures generated by an indiscernible set. The assumption of n-degeneracy implies total categoricity for the first class, stability for the second, and ω-stability for the third.

The weak non-finite cover property (wnfcp) was introduced in [1] in connection with "axiomatizability" of lovely pairs of models of a simple theory. We find a combinatorial condition on a simple theory equivalent to the wnfcp, yielding a direct proof that the non-finite cover property implies the wnfcp, and that the wnfcp is preserved under reducts. We also study the question whether the wnfcp is preserved when passing from a simple theory T to the theory TP of lovely pairs of (...) models of T (true in the stable case). While the question remains open, we show, among other things, that if (for a T with the wnfcp) TP is low, then TP has the wnfcp. To study this question, we describe "double lovely pairs", and, along the way, we develop the notion of a "lovely n-tuple" of models of a simple theory, which is an analogue of the notion of a beautiful tuple of models of stable theories [2]. (shrink)

For a supersimple SU-rank 1 theory T we introduce the notion of a generic elementary pair of models of T . We show that the theory T* of all generic T-pairs is complete and supersimple. In the strongly minimal case, T* coincides with the theory of infinite dimensional pairs, which was used in 1184–1194) to study the geometric properties of T. In our SU-rank 1 setting, we use T* for the same purpose. In particular, we obtain a characterization of linearity (...) for SU-rank 1 structures by giving several equivalent conditions on T*, find a “weak” version of local modularity which is equivalent to linearity, show that linearity coincides with 1-basedness, and use the generic pairs to “recover” projective geometries over division rings from non-trivial linear SU-rank 1 structures. (shrink)

We continue the study of the connection between the “geometric” properties of SU -rank 1 structures and the properties of “generic” pairs of such structures, started in [8]. In particular, we show that the SU-rank of the theory of generic pairs of models of an SU -rank 1 theory T can only take values 1 , 2 or ω, generalizing the corresponding results for a strongly minimal T in [3]. We also use pairs to derive the implication from pseudolinearity to (...) linearity for ω -categorical SU -rank 1 structures, established in [7], from the conjecture that an ω -categorical supersimple theory has finite SU -rank, and find a condition on generic pairs, equivalent to pseudolinearity in the general case. (shrink)

We study expansions of an algebraically closed field K or a real closed field R with a linearly independent subgroup G of the multiplicative group of the field or the unit circle group \\), satisfying a density/codensity condition. Since the set G is neither algebraically closed nor algebraically independent, the expansion can be viewed as “intermediate” between the two other types of dense/codense expansions of geometric theories: lovely pairs and H-structures. We show that in both the algebraically closed field and (...) real closed field cases, the resulting theory is near model complete and the expansion preserves many nice model theoretic conditions related to the complexity of definable sets such as stability and NIP. (shrink)

We study definable groups in dense/codense expansions of geometric theories with a new predicate P such as lovely pairs and expansions of fields by groups with the Mann property. We show that in such expansions, large definable subgroups of groups definable in the original language \ are also \-definable, and definably amenable \-definable groups remain amenable in the expansion. We also show that if the underlying geometric theory is NIP, and G is a group definable in a model of T, (...) then the connected component \ of G in the expansion agrees with the connected component \ in the original language. We prove similar preservation results for \^n\), the P-part of G in a lovely pair, and for subgroups of G in pairs where R is a real closed field and G is a subgroup of \ or the unit circle \\) with the Mann property. (shrink)

We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: "weak one-basedness", absence of type definable "almost quasidesigns", and "generic linearity". Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω-categorical geometric theory interprets an infinite vector space over (...) a finite field. (shrink)

We study the theory of lovely pairs of geometric structures, in particular o-minimal structures. We use the pairs to isolate a class of geometric structures called weakly locally modular which generalizes the class of linear structures in the settings of SU-rank one theories and o-minimal theories. For o-minimal theories, we use the Peterzil–Starchenko trichotomy theorem to characterize for a sufficiently general point, the local geometry around it in terms of the thorn U-rank of its type inside a lovely pair.

Given a lovely pair P ≺ M of models of a simple theory T, we study the structure whose universe is P and whose relations are the traces on P of definable (in ℒ with parameters from M) sets in M. We give a necessary and sufficient condition on T (which we call weak lowness) for this structure to have quantifier-elimination. We give an example of a non-weakly-low simple theory.

We study the theory of the structure induced by parameter free formulas on a “dense” algebraically independent subset of a model of a geometric theory T. We show that while being a trivial geometric theory, inherits most of the model theoretic complexity of T related to stability, simplicity, rosiness, the NIP and the NTP2. In particular, we show that T is strongly minimal, supersimple of SU‐rank 1, has the NIP or the NTP2 exactly when has these properties. We show that (...) if T is superrosy of thorn rank 1, then so is, and that the converse holds if T satisfies acl = dcl. (shrink)

We introduce the notion of a lovely pair of models of a simple theory T, generalizing Poizat's “belles paires” of models of a stable theory and the third author's “generic pairs” of models of an SU-rank 1 theory. We characterize when a saturated model of the theory TP of lovely pairs is a lovely pair , finding an analog of the nonfinite cover property for simple theories. We show that, under these hypotheses, TP is also simple, and we study forking (...) and canonical bases in TP. We also prove that assuming only that T is low, the existentially universal models of the universal part of a natural expansion TP+ of TP, are lovely pairs, and “simple Robinson universal domains”. (shrink)

Based on the work done in [][] in the o‐minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of forking‐independent elements that is dense inside a partial type, which we call H‐structures. We show that any two such expansions have the same theory and that under some technical conditions, the saturated models of this common theory are again H‐structures. We prove that under these assumptions the expansion is supersimple and (...) characterize forking and canonical bases of types in the expansion. We also analyze the effect these expansions have on one‐basedness and CM‐triviality. In the one‐based case, when T has SU‐rank and the SU‐rank is continuous, we take to be the type of elements of SU‐rank and we describe a natural “geometry of generics modulo H” associated with such expansions and show it is modular. (shrink)

We introduce proof nets and sequent calculus for the multiplicative fragment of non-commutative logic, which is an extension of both linear logic and cyclic linear logic. The two main technical novelties are a third switching position for the non-commutative disjunction, and the structure of order variety.

The emergence of Hasidism as a mystical form of the Jewish tradition during the eighteenth century caused organizational divisions in Judaism. This split, which swept half of the Jewish world of Central and Eastern Europe, was largely caused by the extraordinary success of the new religious movement among the common people.Many researchers point to the attractiveness of Hasidism, which was, first and foremost, the over-accessibility of truths proclaimed by the followers of Boal-Shem- This is the founder of the movement. At (...) least the use of moral and instructive sentiments, aphorisms, and even "legendary anecdotes" to express the ideas of Hasidic teachings and their outspoken propaganda among the commoners have made this new movement of the Jewish tradition popularized. Figuratively speaking, the Hasidim sought to fill the inner world of man with properly crafted mystical ideas. (shrink)

The twentieth century brought enormous change to subjects such as language, metaphysics, ethics and epistemology. This volume covers the major developments in these areas and more.

In a previous article, the writer explored the geometric foundation of the generally covariant spinor calculus. This geometric reasoning can be extended quite naturally to include the Lie covariant differentiation of spinors. The formulas for the Lie covariant derivatives of spinors, adjoint spinors, and operators in spin space are deduced, and it is observed that the Lie covariant derivative of an operator in spin space must vanish when taken with respect to a Killing vector. The commutator of two Lie covariant (...) derivatives is calculated; it is noted that the result is consistent with the geometric interpretation of the Jacobi identity for vectors. Lie current conservation is seen to spring from the result that the operator of spinor affine covariant differentiation commutes with the operator of spinor Lie covariant differentiation with respect to a Killing vector. It is shown that differentiations of the spinor field defined geometrically are Lorentz-covariant. (shrink)

The first part of the paper discusses the origins and meaning of democracy relative to the development of Christian political thought through the modern period; it is important here that democracy means something different in the ancient world than it does in the modern. The second part discusses the view of democracy proposed in the formative period of modern Catholic social doctrine in especially from the pontificate of Leo XIII to the Second Vatican Council. The third part analyzes the political (...) thought of St. John Paul II which seems to be the apogee of Catholic thinking about democracy. The fourth part discusses some remaining tensions and problems related to democracy that are articulated partly also in John Paul II’s thought, but in a sharper way in the thought of Pope Benedict XVI and one quite prominent challenge to the Catholic view of democracy in the phenomenon of pluralism. One can see in this history that the Church has gradually come to appreciate democracy not simply as an acceptable form of government, one that is not intrinsically at odds with Christianity, but in a positive sense, as an opportunity for human beings to achieve a level of moral development not available in other regimes. But there remain challenges associated with democracy to government and social life consistent with the natural moral law and to Christian faith. (shrink)

Мета. У статті передбачено: 1) визначити місце і роль апріорних топосів знання в загальному процесі еволюції людської свідомості, з’ясувати співвідношення між вродженими і набутими здібностями людини; 2) на основі еволюційного підходу визначити творчий потенціал апріорних і набутих здібностей людини; 3) з’ясувати базові інтенціональності людини, які дозволили їй вийти на новий рівень адаптаційної поведінки. Теоретичний базис. Досліджено генезу і значення ідеї апріорі в традиційній теорії пізнання та її сучасні імплікації в еволюційній теорії свідомості. Еволюційний підхід дозволяє переосмислити досить широку сферу того, (...) що потрапляє під означення апріорності: від аналітики, нормативності, різноманітних установок свідомості до поведінкових реакцій людини. У процесі аналізу апріоризму Канта, який застосував його для обґрунтування можливості існування теоретичних наук, здійснено спробу визначити природу апріорності в контексті еволюційної теорії свідомості. Хоча ідею апріорності критикували як опоненти Канта (спираючись на історичний досвід), так і його послідовники (які виступали проти формалізму апріорного знання), особливу роль у цій критиці і подальшій розробці цієї теми відіграла еволюційна теорія свідомості, яка сприяла більш детальному розгляду питання генезису так званих апріорних форм. Зрештою, завдяки поєднанню різних форм навчання і сукупності соціокультурних форм освоєння світу людина зуміла отримати доступ до "потенційного інтелекту" і новий "простір для творення". Наукова новизна. Обґрунтовано тезу, що еволюційна теорія свідомості дозволяє пояснити як генезу апріорної сфери, так і визначити механізм дії штучних утворень, або напрацювань людської цивілізації. Оскільки люди отримали здатність генерувати й тестувати гіпотези щодо реальності замість того, щоб безпосередньо мати справу з конкретними загрозами, виникла й постійно зростає ймовірність хибних рішень і помилкових уподобань. Тож на загал маємо нагальну потребу в подвійній рефлексії – як щодо меж застосування апріорних форм пізнання, так і щодо інтенційних установок людського габітусу. Спираючись на здобутки еволюційної теорії свідомості, автор вказує на можливість значно ширшого підходу до проблеми апріорного, у якому виділяє не тільки різні форми знання, а й установки, упередженості, інтенціональності. Тобто епістемологічне апріорі є лише підрозділом загальної сфери вроджених і набутих здібностей людини. Відтак можна стверджувати, що базові інтенціональності – це теж своєрідні апріорі, але які розповсюджені як у сфері етології, так і в поведінці людей. Висновки. Спираючись на ідеї природного добору Дарвіна, дослідження в галузі етології, антропології, дані нейронаук, сучасні уявлення про свідомість тощо, представники еволюційної теорії свідомості запропонували більш детальний підхід до вивчення сфери апріорного. З одного боку, була підважена віра в аподиктичну достовірність апріорних форм, а з іншого – з’ясована складна природа тих форм пізнання, які Кант вважав апріорними. Зрештою, ідея апріорності виводить нас на питання щодо здатності людей до навчання, формулювання завдань, гіпотез, теорій тощо. Зокрема, еволюційна теорія свідомості дозволяє поглянути на пізнавальні спроможності людини під кутом зору адаптації до викликів навколишнього середовища. (shrink)

The introduction of ethics courses in the middle classes of the school in the present circumstances seems to be useful and important. Since the ideological neutral ethics course, which would not boil down to purely formal, spiritual, meaningless norms, is difficult to imagine now, the idea of ​​a system of electives that, in their totality, would realize the task of ethical education on the basis of world-view pluralism, naturally arises. The general principle behind this approach can be formulated as an (...) affirmation of the ethical education obligation in general through the optionality of the specific courses offered. At the same time, the choice for each family must be real. A student who chooses such an elective will not be required to take any of the other similarly offered ethics courses. (shrink)