Andrew Schumann

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.

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.

At the end of the nineteenth century, Leo Tolstoy and Vladimir Solovyov presented two diametrically opposed modes of thinking and reasoning. The question of which of these two figures emerged as the greatest thinker of the time remains an existential one, with significant ontological, epistemological, and metaphysical implications. This opposition is evident in their differing attitudes towards key topics such as political reality, violence, and morality. While Tolstoy’s philosophical contributions are often underestimated and Solovyov is generally regarded as the “winner” of this intellectual rivalry, this paper seeks to reevaluate Tolstoy’s legacy and his role as a philosopher in the latter half of the nineteenth century.

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.

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 (e.g. on a database containing a contradiction) 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 (TIPS) 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.

The paper considers main features of two groups of logics for biological devices, called Physarum Chips, based on the plasmodium. Let us recall that the plasmodium is a single cell with many diploid nuclei. It propagates networks by growing pseudopodia to connect scattered nutrients. As a result, we deal with a kind of computing. The first group of logics for Physarum Chips formalizes the plasmodium behaviour under conditions of nutrient-poor substrate. This group can be defined as standard storage modification machines. The second group of logics for Physarum Chips covers the plasmodium computing under conditions of nutrient-rich substrate. In this case the plasmodium behaves in a massively parallel manner and propagates in all possible directions. The logics of the second group are unconventional and deal with non-well-founded data such as infinite streams.

The logical reasoning first appeared within the Babylonian legal tradition established by the Sumerians in the law codes which were first over the world: Ur-Nammu ; Lipit-Ishtar, and later by their successors, the Akkadians: Hammurabi. In these codes the casuistic law formulation began first to be used: “If/when this or that occurs, this or that must be done” allowed the Akkadians to build up a theory of logical connectives: “... or…”, “… and…”, “if…, then…”, “not…” that must have been applied in their jurisprudence. So, a trial decision looked like an inference by modus pones and modus tollens or by other logical rules from some facts and an appropriate article in the law code represented by an ever true implication. The law code was announced by erecting a stele with the code or by engraving the code on a stone wall. It was considered a set of axioms announced for all. Then the trial decisions are regarded as claims logically inferred from the law code on the stones. The only law code of the Greeks that was excavated is the Code of Gortyn. It is so similar to the Babylonian codes by its law formulations; therefore, we can suppose that the Greeks developed their codes under a direct influence of the Semitic legal tradition: the code was represented as the words of the stele and the court was a logic application from these words. In this way the Greek logic was established within a Babylonian legal tradition, as well. Hence, we can conclude that, first, logic appeared in Babylonia and, second, it appeared within a unique legal tradition where all trial decisions must have been transparent, obvious, and provable. The symbolic logic appeared first not in Greece, but in Mesopotamia and this tradition was grounded in the Sumerian/Akkadian jurisprudence.

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 (symbolic meanings or symbolic values). 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.

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

The Stoics are traditionally regarded as the founders of propositional logic. However, this is not entirely correct. They developed a theory of inference from signs (omens). And their theory became a continuation of the logical technique of Babylonian divination (in particular, of Babylonian medical forecasting). The Stoic theory was not so much propositional logic as it was a technique of propositional logic for databases consisting of IF-THEN expert rules. In the Babylonian divination, each event has a positive or negative value and all events are connected to each other. The Stoics also developed this idea and proposed a special modal logic in which logical determinism is considered an axiom. The paper reconstructs the sign-inference of the Stoics, as well as their modal logic. In particular, two Stoic squares of oppositions are proposed (for signs and for modal operators), which differ markedly from Aristotle’s square.

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.

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.”