Srećko Kovač

Im Folgenden stellen wir drei aktuelle Forschungsbereiche zur formalen Logik bei Kant allgemeinverständlich dar und greifen dabei auf die trans zendentale Logik nur dann zurück, wenn sie ein besseres Verständnis der formalen Logik ermöglicht: Zunächst wird Kants Beitrag zur Rezeption und Weiterentwicklung von Euler-artigen Diagrammen dargestellt. Diese Diagramme wurden in den 1990er Jahren wiederentdeckt, als formales System interpretiert und werden heute insbes. in der Didaktik, in den Kognitionswissenschaften, in der Linguistik, in KI-Bereichen wie der (Logik-basierten) Wissensrepräsentation eingesetzt, und auch in der Philosophie verwendet (Kapitel 2). Mehrere Studien haben in den letzten Jahren gezeigt, dass Kant in weiten Teilen seines Systems Oppositionsverhältnisse verwendet, die der Struktur des logischen Quadrats entsprechen. Das logische Quadrat ist heute Gegenstand intensiver Forschung, da es neben der klassischen, auch die paravollständige und parakonsistente Negation abbildet (Kap. 3). Jüngst wurde zudem die Implementierbarkeit der kantischen Logik in ein formalisiertes deduktives System bewiesen. Die Bedeutung dieses Ergebnisses wird insbesondere im Hinblick auf die formalisierte Umsetzung der Sätze vom Widerspruch, vom zureichenden Grund und der Ausschließung des Mittleren erläutert (Kap. 4). Abschließend möchten wir ein paar Perspektiven für die Forschung zur formalen Logik bei Kant nennen (Kap. 5). -- Note: No financial support acknowledged by the second autor (S. Kovač).

U tekstu se obrađuju Stadlerovo i Bauerovo shvaćanje i zasnivanje formalne logike. Formalna je logika u oba slučaja postavljena u duhu novoskolastičke filozofije druge polovice 19. stoljeća. Subjektivna je strana mišljenja (samosviest, djelatba) spojena s njegovom objektivnom stranom (objektivna vriednost), a logička se načela temelje na ontologijskim. Stadler se i Bauer oddvajaju od onodobnoga formalizma i logicizma, kao i od psihologizma i voluntarizma. Pritom je Bauer izrazitije ušao u razpravljanje s drugačijim shvaćanjima (psihologizam i voluntarizam u Wundta).

U članku se promatraju osnovne karakteristike razvojne dinamike hrvatskoga logičkoga nazivlja od izlazka Pacelove Logike za gimnazije, prve sustavne logike na hrvatskome jeziku, 1868. godine, pa sve do Petrovićeve Logike, također za srednja učilišta, iz 1964., koja je još uviek u uporabi. Nazivlje je u tu svrhu razvrstano u nekoliko tipičih skupina. Općenito, uočava se porast zastupljenosti latinizama (i grecizama) na štetu hrvatskih naziva. U analizi nazivlja autor se ograničuje na knjige namienjene nastavi logike bilo na srednjim učilištima, bilo na sveučilištu, i to na nazivlje tradicionalne logike. Mjestimice su pridodane i napomene o nazivlju moderne (matematičke) logike.

The publication of Gödel’s Max IV contributes to better understanding of the complex development of Gödel’s philosophical thought, and, alongside the other published notebooks, it is a further contribution to modifying a conventional view on the 20th-century philosophy, where Gödel should be recognized as one of the most important and profound philosophers. Moreover, his questions, problem formulations, and ideas, particularly as presented in his philosophical notebooks, transcend the historical distance and can immediately resonate with and inspire the current philosophical research.

The paper aims to show that by abstracting from a specific mythical historical- stylistic context and “ideation” of the notion of the Proto-Slavic deities Perun and Veles, especially in Croatian tradition, symbolic archetypes and abstract notions of form and primordial matter (materia prima) can be extracted from mythical content. We refer to mythical texts and contents according to the reconstructions and materials brought by Radoslav Katičić, and comparative analysis by Mislav Ježić. We distinguish form (1) as that in which identity and possibility of a thing consists, (2) as the reasonableness of judgment and the form of existence, and (3) as the specific structure of a thing, which unites its parts and aspects as a whole and constitutes its internal necessity. These formative notions are relativized and nullified in Veles’ appearances, in which the archetypal notion of primordial matter is encoded. Examples of Platō’s dialogues historically confirm certain aspects of decoding myth as constitutive of philosophy. The above approach also opens a view of the roots of Croatian philosophical thought, which in its genesis can be traced back to Proto-Slavic mythical archetypes. With the implementation of Christianity, the questions of identity, reason and the structure of the whole are raised radically and sharply, as shown by the text of Trpimir’s grant (852).

A close interrelationship between logic and religious faith is confirmed in many places of the Bible. In the paper, special attention is paid to the dialogue of Jesus and a Samaritan woman (John 4). In a proposed outline of a logical formalization, religious faith is described as a pragmatic function through which the linguistic and logical content is contextually realized. In the continuation of the paper, Gödel's ontological proof is commented, which in a logically rigorous way describes the ontology where we can detect the systematic place of religious faith as an ontological force.

The structure "home - world - ideals" is presented as the structure of "philosophical striving" (F. Marković). It could be formally described as a model consisting of a domain, relations and a valuation. On that basis, the identity, openness, and the significance of Croatian philosophy is investigated. The programme of the renewal of Croatian philosophy (as proposed 1882 by Franjo Marković) is re-examined, and some unsolved historical-cultural discontinuities within the programme are described. The written beginnings of Croatian philosophical thought are detected in the Charter of Duke Trpimir (852).

The first beginnings of modern logic in Croatia are recognizable as early as in the middle of the 19th century in Vatroslav Bertić. At the turn of the 20th century, Albin Nagy, who was teaching in Italy, made contributions to algebraic logic and to the philosophy of logic. At that time, a distinctive author Mate Meršić stood out, also working on algebraic logic. In the Croatian academic philosophy, until the publication of Gajo Petrović's textbook (1964) and the contributions by Heda Festini, a critical or indifferent attitude toward modern logic prevailed. The first comprehensive modern logic textbook in Croatian was written by mathematician Vladimir Devidé (1964), known, for example, for his axiomatizations of natural numbers and classical propositional logic. In the Croatian philosophy of the 1980s and 1990s, influential logicians are Goran Švob and mathematician Zvonimir Šikić. At the turn of the 21st century, logical investigations are being intensified and extended to various branches of the contemporary logic.

Different forms and transformations of philosophy in the modern world are analyzed according to the view of the Croatian philosopher Vanja Sutlić (1925-1989). A key aspect is the "realization of philosophy" in "work science", which encompasses the entire "world of work" established in modern times. This universal and real science is also an important precondition for "historical thinking". The uniformity of this science, problematized by Sutlić, can be found today in refined logical and semantic structures, which allow for a renewed and richer articulation of philosophical concepts than was traditionally possible.

The Zagreb Royal Academy, the successor of the former Jesuit Neoacademy, was founded in 1776 as the central institution of higher education in Croatia as part of the educational reform in the Austro-Hungarian Empire. After presenting the basic characteristics of the reform concept, the paper deals with the teaching of theoretical philosophy at the Zagreb Academy. Philosophy was taught by E. Raffay, A. Minković, G. Valičić, S. Čučić, S. Pogledić, S. Moyses, and S. Muzler until the abolition of the Academy in 1850. In the transitional period of 1773-1776, theoretical philosophy was taught by V. Kalafatić (ethics by F. Sebastijanović). Philosophy, along with other subjects (primarily mathematics, physics, and history), was studied for two years, including logic, metaphysics, moral philosophy, and empirical psychology. Textual documentation of theoretical (as well as practical) philosophy is represented by printed examination theses that were publicly defended at the Academy. The most significant systematic work of theoretical philosophy, however, is Čučić's "Philosophia critice elaborata". A new, updated bibliography of these theses is also presented in the supplement. [Zagrebačka kraljevska akademija, nasljednica prijašnje isusovačke Neoacademije, osnovana je, u sklopu školske reforme u Austrijskoj Monarhiji, 1776. godine kao središnja visokoškolska ustanova u Hrvatskoj. Nakon izlaganja osnovnih karakteristika reformske koncepcije, rad se bavi nastavom teorijske filozofije na zagrebačkoj akademiji. Filozofiju su do ukinuća Akademije 1850. godine predavali E. Raffay, A. Minković, G. Valičić, S. Čučić, S. Pogledić, S. Moyses i S. Muzler. U prijelaznom razdoblju 1773-76. god. teorijsku je filozofiju predavao V. Kalafatić (etiku F. Sebastijanović). Filozofija se, uz ostale predmete (prvenstveno matematiku, fiziku i povijest), slušala dvije godine i to logika, metafizika i moralna filozofija i empirijska psihologija. Tekstovnu dokumentaciju teorijske (kao i praktičke) filozofije predstavljaju tiskane ispitne teze koje su se na Akademiji javno branile. U dodatku ,se donosi i nova, dopunjena bibliografija tih tezarija. Teorijski je međutim najznačajnije Čučićevo sustavno djelo "Philosophia critice elaborata«.]

The basic principles of Cipra's metaphysics (according to his book "Metamorphoses of Metaphysics") are analyzed with respect to Cipra's request for the revision of classical logical principles (of identity, excluded middle and contradiction). In Cipra's metaphysics, the principle of identity holds for being, necessity and past only, the principle of excluded middle does not hold for coming-to-be, possibility and present, and the principle of contradiction does not hold for the actuality, reality (freedom) and future. A propositional and first-order temporal model and semantics are defined for such a concept of metaphysical time. The propositional ockhamistic branching time model is extended, in a first-order model, with a non-empty set of object shapes and an equivalence relation between object shapes in time, so that, for example, different, previously equivalent object shapes can in future contradict one another.

Zlatko Posavac is one of the most thorough and deepest experts on Croatian cultural history, with a focus on aesthetics both as a theory and as realized in artistic practice. On the basis of his improved factography, he examines aesthetic phenomena through the model of meaningful-stylistic structures and epochal change. As one of the fundamental phenomena, he highlights the "prevention of free transmission of tradition" (silencing, falsifying, expulsion of truth). He introduces significant structural shifts in cultural-historical periodization.

The criticism of Hilbert's axiomatic system of geometry by Mate Meršić (Merchich, 1850-1928), presented in his work "Organistik der Geometrie" (1914, also in "Modernes und Modriges", 1914), is analyzed and discussed. According to Meršić, geometry cannot be based on its own axioms, as a logical analysis of spatial intuition, but must be derived as a "spatial concretion" using "higher" axioms of arithmetic, logic, and "rational algorithmics." Geometry can only be one, because space is also only one. It cannot be reduced to manipulation with symbols but has to deal with concepts and provide real knowledge. According to Meršić, aiming at some particular proof of the consistency of the whole axiomatic system is meaningless because the consistency follows from the truth of individual axioms based on spatial intuition. It is impossible, according to Meršić, for the geometric axioms to be mutually independent because they must be interconnected as essential parts of a whole. Although Meršić did not recognize all the advantages of Hilbert's formalization of geometry, he clearly pointed out some crucial limitations of the axiomatic method.

It is shown, first, that Georgius Benignus [Juraj Dragišić, ca 1445-1520] at the end of the 15th century almost fully disposed with the theory of the fourth syllogistic figure. Prantl's view on Benignus' logic is critically examined. Besides, Benignus' doctrines on the supposition of terms and logical consequences are examined with respect to his two versions of logic (1488/1489 and 1519) and to the main influences (e.g., Strodus, Paulus Venetus, Ferebrich, Paulus Pergolensis). A reduction of the rules of consequences of Benignus' logic 1488/1489 to the rules of logic 1519 is reconstructed. Benignus rejects the reasoning ex impossible quodlibet and e quolibet necessarium in what he defines as material consequences.

A reconstruction of a formal logical deduction of Kant's table of judgments and of the other forms of Kant's "general logic" is proposed on the basis of Kant's texts, remarks and lectures. Logical unity of apperception is taken as the starting point and the reconstructed deduction proceeds only by means of analytic judgments. In distinction to Klaus Reich's approach, the forms of judgment are not derived from the definition of a judgment as relation, but "regressively", starting from quantity, in the order displayed in Kant's table of judgments and on the basis of the fourfold unity of apperception (analytic, synthetic, objective, necessary) [this reconstruction is summarized in the author's paper 'Forms of Judgment as a Link between Mind and the Concepts of Substance and Cause (Kant, Gödel)', 2014). The early, negative and positive, reception of Kant's logic in works of Croatian philosophers (especially, I. K. Horváth and Š. Čučić) is presented and discussed.

Mate Meršić (Merchich, 1850-1928) sees the origin of Zeno’s paradox ‘Achilles’ in the ambiguities of the concept of the infinity. According to him (and to the tradition started by Gregory St. Vincent), those ambiguities are resolved by the concept of convergent geometric series. In this connection, Meršić proposes a general ontological theory with the priority of the finite over the infinite, and, proceeding from Newton’s concept of fluxion, he develops a modal interpretation of differential calculus.

[English in PhilArchive, unpublished]. Ruđer Bošković's (Rogerius Joseph Boscovich, 1711-1787) induction is described as a reasoning procedure that combines abductive, generalizing and deductive forms of inference. According to Bošković, the application of inductive reasoning extends beyond natural science. Bošković's critique of the use of the principle of sufficient reason is discussed, and constructive rules of Bošković's inductive logic are proposed from the standpoint of contemporary justification logic. To that end, justification logic could be extended with Bošković's typology of reasons. Hunter's result (1965) is conceived as a confirmation that the inference from the constructive defectus rationis is a necessary component of Bošković's induction. - English version in _PhilArchive_.

The book is a collection of papers addressing the role of logic in forming and developing philosophy. In particular, on the ground of modern development of logic, it is shown that philosophy can be established (and, in fact, to a large extent is established) as a modern science. The following problems are addressed: general relationship between philosophy and science (especially from a logical viewpoint); the use of logic in ordinary language; names and descriptions; Quine's pragmatic extensional Platonism and predicate-functor logic; philosophical and logical aspects of Gödel's onto-theological system. In the last two chapters, an early critique and emendement of algebraic logic by Croatian philosophers Franjo Marković and Mate Meršić, respectively, is examined.

This book is an introduction to elementary logic (classical propositional and first-order logic), comprising brief summaries of the basics of elementary logic, with the emphasis on typical questions and procedure descriptions and with a large number of corresponding exercises and problems. Solutions are given for each problem and exercise, often with commentaries. The first part, Basics of Logic, deals with (a) formal language, models, Venn diagrams for sentences, and translation from natural into formal language and vice versa, (b) deduction and refutation methods in the format of natural deduction (Fitch and Suppes-Lemmon styles), tableau system and Venn diagrams, and (c) basic logical properties and relations (consistency, validity, consequence, equivalence, various types of opposition, logical completeness and independency, relationship between syntax and semantics). The second part, Logic in Understanding and Analysis, introduces into (a) logical approach to problem solving in concrete situations and events as presented in brief tales and anecdotes, (b) discovering the meaning, and reconstruction of logical interconnections, in ordinary, scholarly and philosophical texts, (c) application of already mastered logical tools in examining and defining further theoretical (logical, scientific) concepts and in building formal languages. The shorter third part, From the Traditional Logic, is dedicated to the traditional logic of concepts, judgements and syllogisms, largely with the use of Venn diagrams.

The Aristotelian origins of formal systems are outlined, together with Aristotle's use of causal terms in describing syllogisms. The precision and exactness of a formalism, based on the projection of logical forms into perceptive signs, is contrasted with foundational, abstract concepts, independent of any formalism, which are presupposed for the understanding of a formal language. The definition of a formal system by means of a Turing machine is put in the context of Wittgenstein's general considerations of a machine understood as a sign. A modification of Łukasiewicz's logic Ł3 with the inclusion of justification terms is proposed in order to formally analyse some features of formalistic reasoning as a mechanical, causal affair within a wider context of "indeterminacy".

There is a long tradition of logic, from Aristotle to Gödel, of understanding a proof from the concepts of necessity and causality. Gödel's attempts to define provability in terms of necessity led him to the distinction of formal and absolute (abstract) provability. Turing's definition of mechanical procedure by means of a Turing machine (TM) and Gödel's definition of a formal system as a mechanical procedure for producing formulas prompt us to understand formal provability as a mechanical causality. We propose a formalism which makes explicit the mechanical causal nature of a TM's work. We claim that Gödel's axiomatised ontotheology and his ontological proof give a clue for the understanding of the concept of absolute provability and the pattern of the corresponding absolute completeness proof, respectively.

Faust Vrančić's (Faustus Verantius, 1551-1617) logic is analyzed in comparison to Renaissance Aristotelianism in logic with regard to the problem of determining logic, the subject of logic, and understanding the method. Vrančić's logic is compared to Markantun de Dominis' understanding of logic and to the understanding of logic in Jacopo Zabarella and in the Jesuit Renaissance tradition (P. Fonseca, F. Toletus, F. Suárez). In addition, the concept of science is discussed. "Censura logicae" published at the end of Vrančić's "Logica nova" (1616) is attributed to de Dominis.

Faust Vrančić's (Faustus Verantius, 1551-1617) logic is analyzed in comparison to Renaissance Aristotelianism in logic with regard to the problem of determining logic, the subject of logic, and understanding the method. Vrančić's logic is compared to Markantun de Dominis' understanding of logic and to the understanding of logic in Jacopo Zabarella and in the Jesuit Renaissance tradition (P. Fonseca, F. Toletus, F. Suárez). In addition, the concept of science is discussed. "Censura logicae" published at the end of Vrančić's "Logica nova" (1616) is attributed to de Dominis. -- German version: 'Faust Vrančić and der Aristotelismus in der Logik', Studia historiae philosophiae Croaticae 2 (1993), 229-252).