VERRIUS FLACCUS, HIS ALEXANDRIAN MODEL, OR JUST AN ANONYMOUS GRAMMARIAN? THE MOST ANCIENT DIRECT WITNESS OF A LATIN ARS GRAMMATICA

When dealing with manuscripts transmitting otherwise unknown ancient texts and without a subscriptio, the work of a philologist and literary critic becomes both more difficult and more engrossing. Definitive proof is impossible; at the end there can only be a hypothesis. When dealing with a unique grammatical text, such a hypothesis becomes even more delicate because of the standardization of ancient grammar. But it can happen that, behind crystallized theoretical argumentation and apparently canonical formulas, interstices can be explored that lead to unforeseen possibilities, more exciting—and even more suitable—than those that have already emerged.

Canonical formulas via locally finite reducts and generalized dualities

The method of canonical formulas is a powerful tool for investigating intuitionistic and modal logics. In this talk I will discuss an algebraic approach to this method. I will mostly concentrate on the case of intuitionistic logic. But I will also review the case of modal logic and possible generalizations to substructural logic.

Jankov Formula and Ternary Deductive Term

Grounding on defining relations of a finitely presentable subdirectly irreducible (s.i.) algebra in a variety with a ternary deductive term (TD), we define the characteristic identity of this algebra. For finite s.i. algebras the characteristic identity is equivalent to the identity obtained from Jankov formula. In contrast to Jankov formula, characteristic identity is relative to a variety and even in the varieties of Heyting algebras there are the characteristic identities not related to Jankov formula. Every subvariety of a given locally finite variety with a TD term admits an optimal axiomatization consisting of characteristic identities. There is an algorithm that reduces any finite system of axioms of such a variety to an optimal one. Each variety with a TD term can be axiomatized by characteristic identities of partial algebras, and in certain cases these identities are related to the canonical formulas.

O ESTILO PASTORAL DO VATICANO II E SUA RECEPÇÃO PÓS-CONCILIAR ELABORAÇÃO DE UMA CRITERIOLOGIA E ALGUNS EXEMPLOS SIGNIFICATIVOS

Em diálogo com a leitura de John W. O’Malley sobre o Concílio Vaticano II, o presente artigo analisa seu corpus textual e se pergunta sobre seu “estilo pastoral”. Desde sua preparação, o Concílio se caracterizou por um processo de intercompreensão conciliar e ecumênica. Seu estilo afastou-se das fórmulas canônicas, eliminando os anátemas e aproximando-se do estilo evocativo, bíblico (narrativo, parenético, deliberativo, doxológico). Os preâmbulos das Constituições Dei Verbum, Sacrosanctum Concilium, Lumen Gentium e Gaudium et Spes permitem avaliar esse “estilo pastoral” e as consequências para a recepção visível na exortação Evangelii nuntiandi, na encíclica Ut unum sint e no encontro inter-religioso de Assis em 1986. À guisa de conclusão, recorda que o estilo cria o mundo que o texto habita. Daí o valor heurístico desta abordagem na análise da recepção do Vaticano II e a determinação de uma autêntica fidelidade ao espírito e à letra do Concílio, em contraste com outros textos que revelam o conflito entre esse “estilo pastoral” e a tradição dogmática.ABSTRACT: In dialogue with the reading of John W. O’Malley on the Second Vatican Council this article analyzes its textual corpus and questions its “pastoral style.” Since its preparation, the Council was characterized by a process of conciliar and ecumenical mutual understanding. Its style departed from the canonical formulas, eliminating the anathemas and approaching the evocative style, Biblical (narrative, parenetical, deliberative, doxological). The preambles of the Constitutions Dei Verbum, Sacrosanctum Concilium, Lumen Gentium and Gaudium et Spes assess this “pastoral style” and the consequences for the visible reception in the exhortation Evangelii nuntiandi, in the Encyclical Ut unum sint and in the interreligious meeting in Assisi in 1986. By way of conclusion it points out that the style creates the world in which the text dwells. Hence the heuristic value of this approach in the analysis of the reception of Vatican II and the determination of an authentic fidelity to the spirit and to the literal meaning of the Council, in contrast with other texts that reveal the conflict between this “pastoral style” and the dogmatic tradition.

CANONICAL FORMULAS FOR wK4

We generalize the theory of canonical formulas for K4, the logic of transitive frames, to wK4, the logic of weakly transitive frames. Our main result establishes that each logic over wK4 is axiomatizable by canonical formulas, thus generalizing Zakharyaschev’s theorem for logics over K4. The key new ingredients include the concepts of transitive and strongly cofinal subframes of weakly transitive spaces. This yields, along with the standard notions of subframe and cofinal subframe logics, the new notions of transitive subframe and strongly cofinal subframe logics over wK4. We obtain axiomatizations of all four kinds of subframe logics over wK4. We conclude by giving a number of examples of different kinds of subframe logics over wK4.

Reformulating the Situation Calculus and the Event Calculus in the General Theory of Stable Models and in Answer Set Programming

Circumscription and logic programs under the stable model semantics are two well-known nonmonotonic formalisms. The former has served as a basis of classical logic based action formalisms, such as the situation calculus, the event calculus and temporal action logics; the latter has served as a basis of a family of action languages, such as language A and several of its descendants. Based on the discovery that circumscription and the stable model semantics coincide on a class of canonical formulas, we reformulate the situation calculus and the event calculus in the general theory of stable models. We also present a translation that turns the reformulations further into answer set programs, so that efficient answer set solvers can be applied to compute the situation calculus and the event calculus.

Canonical rules

We develop canonical rules capable of axiomatizing all systems of multiple-conclusion rules over K4 or IPC, by extension of the method of canonical formulas by Zakharyaschev [37]. We use the framework to give an alternative proof of the known analysis of admissible rules in basic transitive logics, which additionally yields the following dichotomy: any canonical rule is either admissible in the logic, or it is equivalent to an assumption-free rule. Other applications of canonical rules include a generalization of the Blok–Esakia theorem and the theory of modal companions to systems of multiple-conclusion rules or (unitary structural global) consequence relations, and a characterization of splittings in the lattices of consequence relations over monomodal or superintuitionistic logics with the finite model property.