Srećko Kovač

We approach the traditional problem of the relationship between opinion and truth, objects and appearances, from the standpoint and tools of logic of belief, combining an informal and technical approach. We describe and comment on some logics of individual concepts and modes of representation, and address the question of how to explain the possibility of a contradictory de re belief integrated with the corresponding non-contradictory de dicto belief. The QB logic is proposed, where the semantic problem is resolved (1) by extending the framework and model domain with the domain of appearances, with a special S-accessibility relation among worlds, and with non-rigid interpretation of appearance names, as well as (2) redefining the concept of satisfaction (and truth), including the satisfaction of identity statements, by relativizing satisfaction to the difference between things and appearances and to the S-accessibility of different epistemic alternatives. In doing so, all worlds are classical within themselves, while inconsistency and incompleteness in beliefs stem from different ways of connecting (merging and separating) worlds in determining the satisfaction of a formula.

The square of opposition (as part of a lattice) is used as a natural way to represent different and opposite ways of who makes decisions, and in what way, in/for a group or a society. Majority logic is characterized by multiple logical squares (one for each possible majority), with the “discursive dilemma” as a consequence. Three-valued logics of majority decisions with discursive dilemma undecided, of veto, consensus, and sequential voting are analyzed from the semantic point of view. For instance, the paraconsistent and paracomplete logics M3, M3veto and C3 are described. The distinction of designated and non-designated values is not used, and instead, the consequence relation is defined as a preservation of the minimum truth-value of the implying set of sentences. The consequence and opposition relations of the logics described are compared, and an ordering of the logics with respect to their opposition relations is established.

After introductory reminder of and comments on Gödel’s ontological proof, we discuss the collapse of modalities, which is provable in Gödel’s ontological system GO. We argue that Gödel’s texts confirm modal collapse as intended consequence of his ontological system. Further, we aim to show that modal collapse properly fits into Gödel’s philosophical views, especially into his ontology of separation and union of force and fact, as well as into his cosmological theory of the nonobjectivity of the lapse of time. As a result, modal collapse should not be conceived in Gödel so much as a deficit, but rather as a kind of the rise of modalities to the “perfect” being. We further show that, in accordance with Gödel’s ontology, the concepts of modality and time should be derived in terms of the “fundamental philosophical concept” of cause. To give an example of how a formalization of such causative Gödelian ontology and ontological proof might look, we propose the transformation of GO into a kind of causally re-interpreted justification logic.

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.

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.

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.

A first-order logic of belief with identity is proposed, primarily to give an account of possible de re contradictory beliefs, which sometimes occur as consequences of de dicto non-contradictory beliefs. A model has two separate, though interconnected domains: the domain of objects and the domain of appearances. The satisfaction of atomic formulas is defined by a particular S-accessibility relation between worlds. Identity is non-classical, and is conceived as an equivalence relation having the classical identity relation as a subset. A tableau system with labels, signs, and suffixes is defined, extending the basic language $\mathscr{L}_{\mathbf{QB}}$ by quasiformulas (to express the denotations of predicates). The proposed logical system is paraconsistent since $\phi \wedge \neg\phi$ does not ``explode'' with arbitrary syntactic consequences.

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.

On the ground of Kant’s reformulation of the principle of con- tradiction, a non-classical logic KC and its extension KC+ are constructed. In KC and KC+, \neg(\phi \wedge \neg\phi), \phi \rightarrow (\neg\phi \rightarrow \phi), and \phi \vee \neg\phi are not valid due to specific changes in the meaning of connectives and quantifiers, although there is the explosion of derivable consequences from {\phi, ¬\phi} (the deduc- tion theorem lacking). KC and KC+ are interpreted as fragments of an S5-based first-order modal logic M. The quantification in M is combined with a “subject abstraction” device, which excepts predicate letters from the scope of modal operators. Derivability is defined by an appropriate labelled tableau system rules. Informally, KC is mainly ontologically motivated (in contrast, for example, to Jaśkowski’s discussive logic), relativizing state of affairs with respect to conditions such as time.

Gödel's philosophical views were to a significant extent influenced by the study not only of Leibniz or Husserl, but also of Kant. Both Gödel and Kant aimed at the secure foundation of philosophy, the certainty of knowledge and the solvability of all meaningful problems in philosophy. In this paper, parallelisms between the foundational crisis of metaphysics in Kant's view and the foundational crisis of mathematics in Gödel's view are elaborated, especially regarding the problem of finding the “secure path of a science” for both mathematics and philosophy. Gödel's temporal subjectivism and metaphysical conceptual objectivism are presented as positively or negatively motivated by Kant's viewpoints. A remark on Gödel's collapse of modalities (in accordance with the collapse of objective time) is added.

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.

We show how some model-theoretical devices (local reasoning, modes of presentation, an additional accessibility relation) can be combined in first-order modal logic to formalize the consequence relation that includes de dicto and de re contradictory beliefs. Instead of special ``sense objects'', appearances of objects in an agent's belief are introduced and presented as ordered pairs consisting of an object and an individual constant. A non-classical identity relation is applied. A relation S on the set of possible worlds is introduced, which models possible distortions in an agent's picture of a (real) world. The application of such models in deontic logic is illustrated by a characteristic example.

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 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.

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.

Quine rejects intensional Platonism and, with it, also rejects attributes (properties) as designations of predicates. He pragmatically accepts extensional Platonism, but conceives of classes as merely auxiliary entities needed to express some laws of set theory. At the elementary logical level, Quine develops an “ontologically innocent” logic of predicates. What in standard quantification theory is the work of variables is in the logic of predicates the work of a few functors that operate on predicates themselves: variables are eliminated. This “predicate functor logic” may be conceived as a peculiar sort of Platonism - ontologically neutral, reduced to schematized linguistic forms.

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).

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.

The reception of Kant began in Croatia at the turn of the 19th century with the writings of J.B. Horváth, whose textbooks were in use at that time in Croatia and Hungary. Unlike Horváth's decidedly negative attitude toward Kant, Šimun Čučić (Simeon Chuchich), in his systematic work Philosophia Critice Elaborata (1815), adopted some aspects of Kantian philosophy. This includes, for example, the formalistic conception of logic, Kantian apriorism and subjectivism, the formalistic approach to the moral law, and the like. However, since these elements were incorporated into a schoolbook type of text, they could not always be consistently implemented. The beginning of the reception of Kant in Croatia coincided with the revival of Croatian national self-awareness. -- Die Kantrezeption setzte in Kroatien an der Wende zum 19. Jahrhundert ein mit den Schriften J. B. Horváths, dessen Lehrbuecher in Kroatien und Ungarn damals in Gebrauch waren. Im Unterschied zu Horváth's ausgeschprochen negativer Einstellung zu Kant hat Šimun Čučić in seinem systematischen Werk Philosophia critice elaborata (1815) einige Aspekte der Kantischen Philosophie uebernommen. Dies betrifft z.B. die formalistische Auffassung der Logik, die Uebernahme des kantianischen Apriorismus und Subjektivismus, die formalistische Auffassungsweise des moralischen Gesetzes und dgl. Da nun diese Momente in ein schulmaeßig konzipiertes Buch aufgenommen wurden, konnten sie nicht immer konsequent durchgefuehrt werden. Der Beginn der Kantrezeption in Kroatien koinzidierte mit der Wiedererweckung des kroatischen nationalen Selbsbewußtseins.

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.