Andrew Schumann

In this paper an algebraic system of the new type is proposed (namely, a vectorial lattice). This algebraic system is a lattice for the language of Aristotle’s syllogistic and as well as a lattice for the language of Vasiľév’s syllogistic. A lattice for the language of Aristotle’s syllogistic is called a vectorial lattice on cap-semilattice and a lattice for the language of Vasiľév’s syllogistic is called a vectorial lattice on closure cap-semilattice. These constructions are introduced for the first time

In the paper we build up the ontology of Leśniewski’s type for formalizing synthetic propositions. We claim that for these propositions an unconventional square of opposition holds, where a, i are contrary, a, o (resp. e, i) are contradictory, e, o are subcontrary, a, e (resp. i, o) are said to stand in the subalternation. Further, we construct a non-Archimedean extension of Boolean algebra and show that in this algebra just two squares of opposition are formalized: conventional and the square that we invented. As a result, we can claim that there are only two basic squares of opposition. All basic constructions of the paper (the new square of opposition, the formalization of synthetic propositions within ontology of Leśniewski’s type, the non-Archimedean explanation of square of opposition) are introduced for the first time

Fichte’s Foundations of the Entire Wissenschaftslehre 1794 is one of the most fundamental books in classical German philosophy. The use of laws of thought to establish foundational principles of transcendental philosophy was groundbreaking in the late eighteenth and early nineteenth century and is still crucial for many areas of theoretical philosophy and logic in general today. Nevertheless, contemporaries have already noted that Fichte’s derivation of foundational principles from the law of identity is problematic, since Fichte lacked the tools to correctly present the formal parts of Foundations. In this paper, however, we argue that Fichte’s approach intuitively offers an important contribution to transcendental philosophy, and especially to philosophy of logic. We first point out the difficulties of Fichte’s logic in the Foundations and improve it in a second part on the basis of a formal system in which both propositional logic and syllogistic are combined.

The Yogācāra school of logic developed a theory of sign-inferences that has many features of the Stoic and Epicurean logical teachings with small inclusions of Aristotelian ideas. In the Nyāyabindu of Dharmakīrti, we can find the following schemes of formal reasoning: modus Barbara (Figure I) and modus Camenes (Figure IV) of the Aristotelian syllogistic, and all the inference rules of the Stoic logic: modus ponens, modus tollens, modus ponendo tollens, modus tollendo ponens I, modus tollendo ponens II. The three premises of Yogācāra inference (if p, then q; there is an example that p and q occur together; p; then q) corresponds to the Epicurean doctrine of three premises with a necessary exemplification. Thus, the paper is a comparative study of Greek and Buddhist theory of sign-inferences, showing a deep similarity between them.

The Yogācāra school of logic developed a theory of sign-inferences that has many features of the Stoic and Epicurean logical teachings with small inclusions of Aristotelian ideas. In the Nyāyabindu of Dharmakīrti, we can find the following schemes of formal reasoning: modus Barbara (Figure I) and modus Camenes (Figure IV) of the Aristotelian syllogistic, and all the inference rules of the Stoic logic: modus ponens, modus tollens, modus ponendo tollens, modus tollendo ponens I, modus tollendo ponens II. The three premises of Yogācāra inference (if p, then q; there is an example that p and q occur together; p; then q) corresponds to the Epicurean doctrine of three premises with a necessary exemplification. Thus, the paper is a comparative study of Greek and Buddhist theory of sign-inferences, showing a deep similarity between them.

This paper restores the historical context of Milindapaha. The text is unique, because it is one of the very few documents of Ancient India, in which one of the authors is considered a Greek as a participant in the dialog. To reconstruct the context of the book, the basic archeological data about the Indo-Greek Kingdom, including epigraphics, are summed up, as well as there are analyzed some references to the kingdom given in the Mahāvaṃsa, the earliest chronicle of Sri Lanka. These mentions in the Mahāvaṃsa are matched with the numismatics of Ceylon. From this analysis it is concluded that the document of Milindapaha was most likely created in Gāndhārī in the interval from the beginning of the 1st century B.C. to the end of the 1st century A.D., i.e. during the period of the domination of the syncretic culture of the North of India, combining Buddhism with certain elements of Hellenism. The treatise of Milindapaha was then preserved in Sri Lanka's tradition by continuing good political contacts with the Roman Empire after 400 A.D., that is, after the collapse of the Kushan and the Western Kshatrapas, the last dynasties that had previously preserved elements of Hellenism in the Indian subcontinent. The philosophical meaning of the treatise is then considered and it is concluded that in the text we can find direct references to the proto-Nyāya with the requirement to verify premises by examples. But the logical teaching of Milindapaha is far more archaic than the teaching of Nyāyasūtra, because only two sources of true knowledge are implicitly used: paccakkha and anumāna, and instead of the two verification methods called udāhārana and upamā, only one verification method called opamma is offered.

Spatial and diagrammatic reasoning is a significant part not only of logical abilities, but also of logical studies. The authors of this paper consider some novel trends in studying this type of reasoning. They show that there are the following two main trends in spatial logic: (i) logical studies of the distribution of various objects in space (logic of geometry, logic of colors, etc.); (ii) logical studies of the space algorithms applied by nature itself (logic of swarms, logic of fungi colonies, etc.).

The paper substantiates the prospects of normativism in law, namely the possibility of using purely logical means to make a judgment as a logical conclusion. The main criticism of normativism is based on the possibility of conflicts of judicial decisions and especially on conflicts of the norms themselves, when a court under similar circumstances can make opposing decisions which are formally valid. Critics of normativism argue that logic is helpless in resolving the conflicts and that higher justice must be guided by the discretion of judges who share common values or a common ideology, especially in the case of totalitarian societies. However, we show that there are two types of logical argumentation in the elimination of these conflicts: (a) Aristotelian, when a more general basis is sought, from which only one of the contradictions follows; (b) non-Aristotelian, when a compromise is reached between two contrary statements. In the second case, this technique is used mainly for civil cases. Moreover, this technique is quite formal and does not require an appeal to paraconsistent or other non-classical logics for eliminating the conflict. Using two recent court cases, we demonstrate how the proposed two types of logical argumentation work.

There are two different modal logics: the logic T assuming contingency and the logic K = assuming logical determinism. In the paper, I show that the Aristotelian treatise On Interpretation has introduced some modal-logical relationships which correspond to T. In this logic, it is supposed that there are contingent events. The Nāgārjunian treatise Īśvara-kartṛtva-nirākṛtiḥ-viṣṇoḥ-ekakartṛtva-nirākaraṇa has introduced some modal-logical relationships which correspond to K =. In this logic, it is supposed that there is a logical determinism: each event happens necessarily or it does not happen necessarily. The Nāgārjunian approach was inherited by the Yogācārins who developed, first, the doctrine of causality of all real entities and, second, the doctrine of momentariness of all real entities. Both doctrines were a philosophical ground of the Yogācārins for the logical determinism. Hence, Aristotle implicitly used the logic T in his modal reasoning. The Madhyamaka and Yogācāra schools implicitly used the logic K = in their modal reasoning.

It is a Preface to Volume 9:3/4 that has brought a renewed focus to the role of truth conceptions in frameworks of semantics and logic. Jan Woleński is known due to his works on epistemological aspects of logic and his systematization of semantic truth theory. He became the successor and the worthy continuer of prominent Polish logicians: Alfred Tarski and Kazimierz Ajdukiewicz. This volume is collected on the 80th anniversary of Woleński’s birth and draws together new research papers devoted to judgments and truth. These papers take measure of the scope and impact of Woleński’s views on truth conceptions, and present new contributions to the field of philosophy and logic.

In this paper reflexive games are defined as a way to act beyond equilibria to control our opponents by our hiding motives. The task of a reflexive game is to have the opponent’s actions become transparent for us, while our actions remain obscure for the competitor. In case a reflexive game is carried out between agents belonging to the same organisation (corporation, company, institute), success in a reflexive game can be reached by a purposeful modification of some components of a controlled system. Such a modification for the guaranteed victory in a reflexive game is called reflexive management. This kind of management uses reflexive games to control a knowledge structure of agents in a way their actions unconsciously satisfy the centre’s goals.