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. (shrink)

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. (shrink)

In this paper, I show that in the Pāli Canon there was a tradition of Buddhist logic, but this tradition was weak, and the proto-logic we can reconstruct on the basis of the early Pāli texts can be evaluated as a predecessor of the Hindu logic. According to the textual analysis of the Pāli texts, we can claim that at the time of the closing of the Pāli Canon there did not exist the Nyāya philosophy known by the Nyāya Sūtra. (...) Meanwhile, we can assume that the Milindapañha, the best logical source of early Pāli literature, was written under influences of the Gandhāran Buddhists and this text preceded the Nyāya philosophy. (shrink)

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. (shrink)

In this paper the non-Archimedean multiple-validity is proposed for basic fuzzy logic BL∀∞ that is built as an ω-order extension of the logic BL∀. Probabilities are defined on the class of fuzzy subsets and, as a result, for the first time the non-Archimedean valued probability logic is constructed on the base of BL∀∞.

In this article, the author attempts to explicate the notion of the best known Talmudic inference rule called qal wa-omer. He claims that this rule assumes a massive-parallel deduction, and for formalizing it, he builds up a case of massive-parallel proof theory, the proof-theoretic cellular automata, where he draws conclusions without using axioms.

In the paper, a new syllogistic system is built up. This system simulates a massive-parallel behavior in the propagation of collectives of parasites. In particular, this system simulates the behavior of collectives of trematode larvae.

In this paper, the four Judaic inference rules: qal wa- ḥ omer, gezerah š awah, heqe š, binyan ’av are considered from the logical point of view and the pragmatic limits of applying these rules are symbolic-logically explicated. According to the Talmudic sages, on the one hand, after applying some inference rules we cannot apply other inference rules. These rules are weak. On the other hand, there are rules after which we can apply any other. These rules are strong. This (...) means that Judaic inference rules have different pragmatic meanings and this fact differs Judaic logic from other ones. The Judaic argumentation theory built up on Judaic logic also contains pragmatic limits for proofs as competitive communication when different Rabbis claim different opinions in respect to the same subject. In order to define these limits we build up a special kind of syllogistics, the so-called Judaic pragmatic-syllogistics, where it is defined whose opinion should be choosen in a dispute. (shrink)

Judaic reasoning is discussed from the standpoint of modern logic. Andrew Schumann defines Judaic logic, traces Aristotelian influence on developing Jewish studies in Judaic reasoning, and shows the non-Aristotelian core of fundamentals of Judaic logic. Further, Schumann proposes some modern approaches to understanding and formalizing Judaic reasoning, including Judaic semantics and (non-Aristotelian) syllogistics.

Knocking on Heaven's Door is the oldest human dream that seems unrealized still. Religious discourse does show the road, but it requires a blind faith in return. In this book logicians try to hear Heaven's Call and to analyze religious discourse. As a result, the notion of religious logic as a part of philosophical logic is introduced. Its tasks are (1) to construct consistent logical systems formalizing religious reasoning that at first sight seems inconsistent (this research is fulfilled within the (...) limits of modal logic, paraconsistent logic and many-valued logic), (2) to carry out an illocutionary analysis of religious discourse (this research is fulfilled in frames of illocutionary logics), and (3) to formalize Ancient and Medieval logical theories used in the theology of an appropriate religion (they could be studied within the limits of unconventional logics, such as non-monotonic logics, non-well-founded logics, etc.). (shrink)

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.

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.). (shrink)

We show that in Kabbalah, the esoteric teaching of Judaism, there were developed ideas of unconventional automata in which operations over characters of the Hebrew alphabet can simulate all real processes producing appropriate strings in accordance with some algorithms. These ideas may be used now in a syllogistic extension of Lindenmayer systems, where we deal also with strings in the Kabbalistic-Leibnizean meaning. This extension is illustrated by the behavior of Physarum polycephalum plasmodia which can implement, first, the Aristotelian syllogistic and, (...) second, a Talmudic syllogistic by qal wa-homer. (shrink)

This paper is devoted to the methodology of history of philosophy. There are considered two approaches: the Hegelian and Schellingian ones. It is shown that the Hegelian approach has many weak points. Both approaches are demonstrated on the material of Indian philosophy. The Schellingian approach was hammered out then by Foucault as archeology of philosophy.

We build a topological model, based on intuitionistic logic, for multi-agent biological systems (such as _Physarum polycephalum_, bacterial colonies or any other swarm), reacting to external nourishment stimuli. Our construction follows the topological description of brain activity, where particles (neurons) are activated by an external environment, represented by a topological space _X_ with an open cover \(\{U_i:i\in I\}\). The brain builds the model of this external space via the nerve (trace) of a topological space _X_. Here the body of _Physarum (...) polycephalum_ or a swarm made of networks of tubular structures represents a nerve (trace) of _X_ also which means that _Physarum polycephalum_ or a swarm gains orientation in the space of external stimuli even in the absence of any neural system. The logic of living organisms is based on open subsets of _X_ and thus can be represented by Heyting algebra (i.e. intuitionistically). We also consider the generalisation of the nerve construction to a categorical context, where the category is determined by the network structures of multi-agent biological system. This model can be generalised up to simulating the behaviour of any swarm by means of intuitionistic logic. (shrink)

We consider consciousness attributed to systems in space-time which can be purely physical without biological background and focus on the mathematical understanding of the phenomenon. It is shown that the set theory based on sets in the foundations of mathematics, when switched to set theory based on ZFC models, is a very promising mathematical tool in explaining the brain/mind complex and the emergence of consciousness in natural and artificial systems. We formalise consciousness-supporting systems in physical space-time, but this is localised (...) in open domains of spatial regions and the result of this process is a family of different ZFC models. Random forcing, as in set theory, corresponds precisely to the random influence on the system of external stimuli, and the principles of reflection of set theory explain the conscious internal reaction of the system. We also develop the conscious Turing machines which have their external ZFC environment and the dynamics is encoded in the random forcing changing models of ZFC in which Turing machines with oracles are formulated. The construction is applied to cooperating families of conscious agents which, due to the reflection principle, can be reduced to the implementation of certain concurrent games with different levels of self-reflection. (shrink)

In this paper, I show that the idea of logical determinism can be traced back from the Old Babylonian period at least. According to this idea, there are some signs which can explain the appearance of all events. These omens demonstrate the will of gods and their power realized through natural forces. As a result, each event either necessarily appears or necessarily disappears. This idea can be examined as the first version of eternalism – the philosophical belief that each temporal (...) event is actual. In divination lists in Akkadian presented as codes we can reconstruct Boolean matrices showing that the Babylonians used some logical-algebraic structures in their reasoning. The idea of logical contingency was introduced within a new mood of thinking presented by the Greek prose – historical as well as philosophical narrations. In the Jewish genre ’aggādōt, the logical determinism is supposed to be in opposition to the Greek prose. (shrink)

In this paper we consider possibilities for applying p-adic valued logic BL to the task of designing an unconventional computer based on the medium of slime mould, the giant amoebozoa that looks for attractants and reaches them by means of propagating complex networks. If it is assumed that at any time step t of propagation the slime mould can discover and reach not more than attractants, then this behaviour can be coded in terms of p-adic numbers. As a result, this (...) behaviour implements some p-adic valued arithmetic circuits and can verify p-adic valued logical propositions. (shrink)

This book is a collection of rare material regarding logical and analytic-philosophical traditions in Central and Eastern European countries, covering the period from the late nineteenth century to the early twenty-first century. An encyclopedic feature covers the history of logic and analytic philosophy in all European post-Socialist countries.

This book is a collection of rare material regarding logical and analytic-philosophical traditions in Central and Eastern European countries, covering the period from the late nineteenth century to the early twenty-first century. An encyclopedic feature covers the history of logic and analytic philosophy in all European post-Socialist countries.

The Orthodox Christian thought is the most modally rigorous way of inferring. The subject of the book is to investigate possibilities of explicating the Orthodox thought from the viewpoint of analytic philosophy and symbolic logic. The claim that Orthodox thinking is just mystic and illogical is not true. The logical culture of Orthodox Christian thinking is unknown and ununderstandable for the West, although its schemata are very influential in Eastern Europe till now (Marxism-Leninism is just one of their possible instances). (...) This thought can be called totalistic or even totalitarian. For this thought any truth or falsity is necessary. As a result, the whole world is presented as logical and nomothetic and there is no place for contingency. (shrink)

The word _meṇḍaka_, a derivative of _meṇḍa_ (“ram”), is generally translated as “made of the ram” or “about the ram” or “horned.” However, in the Pāli _Milindapañha_ (_Questions of Milinda_), the word _meṇḍakapañha_, literally, a question about the ram, is also rendered as a logical conclusion that refutes an imaginary dilemma. Hence, in this treatise, the word _meṇḍaka_ is a special logical term which means an imaginary dilemma that can be logically refuted. This raises the question as to why the (...) word _meṇḍaka_ has come to be associated with this logical technique. To answer this question, this paper examines various aspects of the word and its possible connections to a dilemma and its refutation. The discussion ranges from the meaning of this word in a tale in the _Jātaka_ (Birth Stories), within the contextual usage in a _meṇḍaka_ question, to a relatively recent commentarial text (_aṭṭhakathā_) which gives a different perspective on the etymology of the word. The _Milindapañha_ is explicit in defining a _meṇḍaka_ question as knotty, hard to penetrate, and difficult to resolve, some of which an opponent puts forth to undermine certain aspects of the Buddhist system. However, the way certain _meṇḍaka_ questions are framed, though not directly stated, seem to utilize the principles of logic in a dilemmatic form of argument. With that, a _meṇḍaka_ question, at least in the _Milindapañha_, could also mean “a dilemmatic expression put forth by a challenger to undermine an opponent, but which can be logically refuted.”. (shrink)

One of the main assumptions of mathematical tools in science is represented by the idea of measurability and additivity of reality. For discovering the physical universe additive measures such as mass, force, energy, temperature, etc. are used. Economics and conventional business intelligence try to continue this empiricist tradition and in statistical and econometric tools they appeal only to the measurable aspects of reality. However, a lot of important variables of economic systems cannot be observable and additive in principle. These variables (...) can be called symbolic values or symbolic meanings and studied within symbolic interactionism, the theory developed since George Herbert Mead and Herbert Blumer. In statistical and econometric tools of business intelligence we accept only phenomena with causal connections measured by additive measures. In the paper we show that in the social world we deal with symbolic interactions which can be studied by non-additive labels. For accepting the variety of such phenomena we should avoid additivity of basic labels and construct a new probabilistic method in business intelligence based on non-Archimedean probabilities. (shrink)

In this paper, I show that a kind of perfect logical competence is observed in the Babylonian tablets used for forecasting. In these documents, we see an intuition of some algebraic structures that are used for inferring prognoses as logical conclusions. The paper is based mainly on the omen series reconstructed by N. De Zorzi. It is shown that in composing these divination lists there was implicitly used the Boolean algebra.

In this article, the author attempts to explicate the notion of the best known Talmudic inference rule called qal wa- omer. He claims that this rule assumes a massive-parallel deduction, and for formalizing it, he builds up a case of massive-parallel proof theory, the proof-theoretic cellular automata, where he draws conclusions without using axioms.

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. (shrink)

It is a Preface to Volumes 7:3 and 7:4 consisting of articles presented at the International Interdisciplinary Conference Ideas and Society on the 150th anniversary of the birth of Leon Petrażycki, held on November 24, 2017, in Rzeszów, Poland.

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, 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. (shrink)

We have proposed to use some features of swarm behaviours in modelling business processes. Due to these features we deal with a propagation of business processes in all accessible directions. This propagation is involved into our formalization instead of communicating sequential processes. As a result, we have constructed a business process diagram language based on the swarm behavior and an extension of that language in the form of reflexive management language.

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. (shrink)

In decision making quite often we face permanently changeable and potentially infinite databases when we cannot apply conventional algorithms for choosing a solution. A decision process on infinite databases is called troubleshooting. A decision on these databases is called creative reasoning. One of the first heuristic semi-logical means for creative decision making were proposed in the theory of inventive problem solving by Genrich Altshuller. In this paper, I show that his approach corresponds to the so-called content-generic logic established by Soviet (...) philosophers as an alternative to mathematical logic. The main assumption of content-genetic is that we cannot reduce our thinking to a mathematical combination of signs or to a language as such and our thought is ever cyclic and reflexive so that it contains ever a history. (shrink)

In constructing the three-valued logic, Jan Łukasiewicz was highly inspirited by the Aristotelian idea of logical contingency. Nevertheless, we can construct a four-valued logic for explicating the Stoic idea of logical determinacy. In this system, we have the following truth values: 0 (‘possibly false), 1 (‘necessarily false’), 2 (‘possibly true’), 3 (‘necessarily true’), where the designated truth value is represented by the two values: 2 and 3.

It is a Preface to Volume 8:2 (2019) consisting of articles presented at the International Interdisciplinary Conference anniversary of the birth of Jan Łukasiewicz, Rzeszów, Poland.

In this paper, we introduce the subject of the special issue Trends in Argumentation Logic. Here we mainly describe two approaches to argumentation logic with explicating monotonic and non-monotonic, or defeasible, reasoning and explain the role of artificial intelligence in applying argumentation logic. Then we give a short overview of the papers contributed to the special issue.