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ed text is available (free access) at https://brill.com/view/journals/scri/14/1/article-p63_6.xml

sis International. Philosophy, n. 24) [Sesto San Giovanni (Milano)]: Mimesis International, 2018, 283–299: https://www.amazon.co.uk/Relations-Ontology-Philosophy-Daniele-Bertini/dp/8869771261 The most formal Byzantine explanation of paraconsistent logic applied to the Trinity. NB! I am unable to add here the picture in a good resolution, alas.

nature of numbering in the Trinity.

hole journal issue is available at http://www.collegepublications.co.uk/downloads/ifcolog00035.pdf Abstract The intuition of number implied in the Byzantine notion of Holy Trinity is inconsistent and, more specifically, paraconsistent. The corresponding paraconsistent numbers can be called ‘quasi-ordinals’ taking in mind their ‘duals’, the numbers introduced for paracomplete quantum logics by Newton da Costa et al., which are called ‘quasi-cardianls’.

гических интерпретаций христианского представления о троичности Бога. Все они могут быть разбиты на четыре типа: (1) 3 = 3 («тривиализирующая» интерпретация, в которой единичность Бога мыслится не онтологически, а собирательно: «тритеизм Иоанна Филопона и едва ли не все предложенные на сегодняшний день триадологии аналитической теологии); (2) «3» = 1 (реальны не ипостаси как сами по себе сущие, а только черты их различия: учение Дамиана Александрийского и «мейнстрима» латинской схоластики, в частности, томизма); (3) card(3) = 1 (ипостаси реальны сами по себе, но бесконечны, а единственность Бога — своеобразный аналог общей кардинальности этих трех разных, но упорядоченных бесконечностей: традиция Дунса Скота в латинской схоластике, но отчасти и более ранний рационализм — например, у Бабая Великого); (4) 3 =1 (настоящий парадокс, параконсистентная логика: основная позиция византийских богословов). В триадологии четвертого типа вызывает вопрос само понятие числа, которое уже не является ни натуральным (как в типах (1) и (2)), ни хотя бы просто консистентным (как в типе (3)). Очевидно, что понятие числа в типе (4) не будет противоречить интуитивистской трактовке числа у Пуанкаре, но несовместимо с теоретико-множественными представлениями о числе, которые все используют понятие пары. В византийской триадологии никакие пары невозможны: это такой счет до трех, при котором пропускается два. В качестве аналога пары выступает неразложимая на элементы триада, в логической структуре которой операцией логического сложения служит не итерация обычной бинарной эксклюзивной дизъюнкции, а другой коннектив — эксклюзивное тернарное «или», впервые описанный Эмилем Постом в 1941 году и до сих пор еще малоизвестный логикам. Число три, подразумеваемое в Троице — минимальное из соответствующего ряда параконсистентных чисел. Описываются также некоторые свойства этих чисел, в частности, неэкстенсиональность. Summary: A classification of the presently available logical interpretations of the Christian understanding of the triunity of God is proposed. There are four main types: (1) 3 = 3 (“trivializing” interpretation, where the unity is understood as one or other kind of collectivity, that is, not on the same reality level as each of the three: “Tritheism” of John Philoponus and all the modern “analytical” Triadologies put forward so far); (2) “3” = 1 (not the hypostases as beings existing per se but only their individualising features / relationes are real: Damian of Alexandria and the mainstream of the Latin scholasticism, including Thomism); (3) card(3) = 1 (the hypostases are real as three beings existing per se but infinite, whereas the unity of God is a kind of the common cardinality of these three different but ordered infinities: Duns Scotus’s line in the Latin scholasticism but also some earlier rationalising theologies, e.g., Babai the Great); 3 = 1 (true paradox and paraconsistent logic: the mainstream of the Byzantine theology since the 4th to the 15th cent.). The type (4) poses a problem of the very notion of number. It is obvious that this number is no longer natural (as in types (1) and (2)) nor consistent at all (as in type (3)). It is obvious that the notion of number implied in type (4) would be perfectly fitting with Poincaré’s intuitivist understanding of number but incompatible with the set-theoretical interpretations/definitions of number, because they imply the notion of pair. In the Byzantine Triadology, no pair is permissible: it implies such counting to three where two is skipped. Instead of the pair, this Triadology implies a triad, which is indissoluble to the elements, and where the logical addition operation, instead of the iteration of the ordinary (binary) exclusive disjunction, is a different connective called ternary exclusive OR, first described by Emil Post in 1941 and even now little known to the logicians. The number three implied in the Trinity is the minimal one in the series of such paraconsistent numbers. Some properties of these numbers are described, especially non-extensionality.

ical Analysis of a Byzantine Approach to the Filioque Nicephorus Blemmydes’ symmetric formula “through each other” proposed instead of the Filioque in the 1250s belongs to the Cappadocian tradition of Triadology. This conclusion is not demonstrable from a habitual (in patrology) philological analysis but can be reached through a logical analysis. The implied theory of numbers is sketched.

. They were driven to a standstill with no logically consistent way out. Thus, Patriarch Nicephorus and Theodore the Studite adopted two different strategies. Nicephorus’ Christology became illogical in its core, whereas Theodore dared to explain openly the basic paraconsistence of the patristic Byzantine usage of the Aristotelean categories. Namely, he constructed his Christology as a transposition of Maxim the Confessor’s teaching on the deification of the man.

Faculty of Philosophy in Kosovska Mitrovica 2014.

g the axioms K + D + CD. The three-stage hierarchies together with the negation of the being (through the evil) as the fourth accessible state form the full logical square (being = possibility, well-being = perfection = necessity, illumination = internal negation of being, non-being = impossibility = external negation of being). The evil has no existence at all because it is a logical connective (external negation of being). This logic has much in common but also much at odds with Leibniz’s and his Spanish Jesuit predecessors’ *deontic* logic of the “moral necessity” imposed on the divine Providence. Both have, however, the unique accessibility relation (in Kripke frames). In Dionysius, no deontic logic could be imposed on God, but the deontic logic for humans who try to be saved (= deified), that is the logic of ascetics (different from the deontic logic of Church law) has shift reflexive accessibility relation. The main differences between Leibniz and Spanish Jesuits, on the one side, and Dionysius and the Eastern patristics, on the other side, result from their different theological attitutdes. The West did not know the Eastern doctrine of deification, and, moreover, these Western thinkers were Augustinians who deliberately tried to avoid what they called “Semi-Pelagianism” but what, in fact, was the normative teaching of the Eastern Fathers on the free will.”