Concrete Logic Research Articles

On classes of symmetric and asymmetric concrete logics

We refined the axiomatics of asymmetric logics. For logics X(km, k) of family subsets of the km-element set X, which cardinal numbers are multiples of k we completely described the cases in which X(km, k) a) is symmetric or b) is asymmetric. For an infinite set Ω and a natural number n ≥ 2 we constructed the concrete logics EΩn and completely described the cases in which these logics are asymmetric. For asymmetric logics E we determine when both the set A ∈ E and its complement Ac are atoms of the logic E. Let a symmetric logic E of a finite set Ω be not a Boolean algebra, and let A be an algebra of subsets from Ω, and assume that E ⊂ A. Then there exists a measure on E, that does not admit an extension to a measure on A.

The Biopolitics of Parental Access: Cross-Readings of Transnational Adoption and Surrogacy in Denmark and Norway

Abstract This article introduces the notion of the “biopolitics of parental access” as an analytical lens to examine how different forms of reproductive governance support and enable parental access. Through a cross-reading of political and administrative documents relating to the regulation of, respectively, transnational adoption in Denmark and transnational surrogacy in Norway, we examine the logics and techniques that inform the reproductive governance of parental access. Drawing attention to the racialized entanglement of pro- and anti-natalism, the analysis shows how access to parenthood for Danish and Norwegian citizens is continued and secured through the annihilation of the parenthood of surrogate mothers and families losing children to adoption. While the concrete logics and techniques of reproductive governance differ in the two cases, the result—access to parenthood—is similar.

Design of an all-optical multi-logic operation-integrated metamaterial-based terahertz logic gate.

Terahertz logic gates play a vital role in optical signal processing and terahertz digitization. Herein, a strategy to design an all-optical terahertz logic gate device composed of metamaterials with a semiconductor-metal hybrid is proposed; accordingly, a concrete logic gate composed of Ge embedded-in Au stripe supported by a Si board is presented theoretically. Simulation results reveal the dependence of the terahertz transmission spectra on the different illuminations in the device. Based on the illumination-transmission response, the designed device can realize the NOR or OR Boolean operation. The effects of the width of the Ge-Au stripe as well as the Si board on the transmission spectra and logic performance were also investigated.

On the Logic of Belief and Propositional Quantification

We consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as “everything I believe is true” or “there is something that I neither believe nor disbelieve.” Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator (BAOs) validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. Hence, by duality, we also cover the usual method of adding propositional quantifiers to normal modal logics by considering their classes of Kripke frames. In addition, we obtain decidability for all the concrete logics we discuss.

Boolean Properties and Bell-Like Inequalities of Numerical Events

Let S be a set of states of a physical system and p(s) be the probability of the occurrence of an event when the system is in state s. A function p from S to [0,1] is called a numerical event or alternatively an S-probability. If a set P:={p(s)|s in S} is ordered by the order of real functions such that certain plausible requirements are fulfilled, P becomes an orthomodular poset in which properties can be described by the addition and comparison of functions. P is then called an algebra of S-probabilities or algebra of numerical events. We first answer the question under which circumstances it is possible to consider sets of empirically found numerical events as members of an algebra of S-probabilities. Then we discuss the problem to decide whether a given small set Pn of S-probabilities can be embedded into a Boolean subalgebra of an algebra P of S-probabilities, in which case we will call Pn Boolean embeddable. If Pn is not Boolean embeddable, then the physical system at hand will most likely be non-classical. In the case of a concrete logic P, that is a quantum logic which can be represented by sets, we derive criteria for the subset Pn of P to be Boolean embeddable which can be checked by very simple procedures, for arbitrary S-probabilities we provide sets of Bell-like inequalities which characterize the Boolean embeddability of Pn. Finally we will show how these Bell-like inequalities fit into a general framework of Bell inequalities by providing a method for generating Bell inequalities for S-probabilities from elementary Bell valuations.

Quantum Logics Defined by Sets of Numerical Events

Various notions of quantum logics commonly known in literature are unified under a single concept called event structure. Event structures are linked to sets of measurements of a physical system obtained through the following process: Let S be a set of states of the physical system and p(s) the probability of an occurrence of an event when the system is in state s ∈ S. The function p from S to [0, 1] is called a numerical event or a multidimensional probability—more precisely, S -probability. Posets of numerical events constitute relational structures which are identified as ranges of event structures and this way justified to be considered as quantum logics. Moreover, these posets are algebraically characterized by means of the states they might have. In particular, so-called generalized fields of events and algebras of S-probabilities are allocated and identified, among them the well-known concrete logics.

From the Logic of the Child to a Natural Logic: Perspectives as Knowledge

The notion of perspective is often used in Human Sciences, and some authors such as Bruner (1996) refers explicitly to a perspectivist epistemology. Yet, the general idea of such epistemology is to consider that several perspectives on the same object or phenomena can produce more relevant knowledge than one only. Yet, such epistemology is rarely theorized, which raises interesting questions: What is a perspective? Can we describe a particular perspective on something? Answering such question requires more than a philosophical standpoint or credo, none less than an empirical epistemology. Piaget and his team approached such questions empirically in Genetic Epistemology, starting from the study of actual thinking. However, formal logic of his time was not fitted for providing description of the construction of meaning. Grize later theorized and extended unconventional uses of logic within a logic of action representing the thinking of actors within irreversible time without abstracting it from particular situations and contents. Grize’s logic is about singularity—a concrete operational logic—and yet about semiotics, about meaning making. In this sense, Natural Logic may provide researchers studying knowledge with an instrument for including perspectives and points of view. As such, it can contribute to a theory of knowledge taking into account the diversity of points of view and the process of meaning making, based on a perspectivist epistemology.

Quantum Measurements Generating Structures of Numerical Events

Let S be a set of states of a physical system and p(s) the probability of an occurrence of an event when the system is in state s∈S. The function p from S to [0,1] is called a numerical event, multidimensional probability or, more precisely, S-probability. If a set of numerical events is ordered by the order of real functions one obtains a partial ordered set P in which the sum and difference of S-probabilities are related to their order within P. According to the structure that arises, this further opens up the opportunity to decide whether one deals with a quantum mechanical situation or a classical one. In this paper we focus on the situation that P is generated by a given set of measurements, i.e. S-probabilities, without assuming that these S-probabilities can be complemented by further measurements or are embeddable into Boolean algebras, assumptions that were made in most of the preceding papers. In particular, we study the generation by S-probabilities that can only assume the values 0 and 1, thus dealing with so called concrete logics. We characterize these logics under several suppositions that might occur with measurements and generalize our findings to arbitrary S-probabilities, this way providing a possibility to distinguish between potential classical and quantum situations and the fact that an obtained structure might not be sufficient for an appropriate decision. Moreover, we provide some explanatory examples from physics.

Commoning the political, politicizing the common: Community and the political in Jean-Luc Nancy, Roberto Esposito and Giorgio Agamben

Setting out from the work of Jean-Luc Nancy, this article engages with post-Heideggerian thought on community, seeking to bring out and to enhance its political thrust for contemporary democracies. It shows how Jean-Luc Nancy, Roberto Esposito and Giorgio Agamben, ‘common the political’, that is, how they reconsider politics in light of a fundamental sense of co-existence which clears the ground for social openness, solidarity, plurality and autonomy. It then responds to a series of pertinent objections by further politicizing the post-Heideggerian vision of the common. I set out to translate the ontologies of the common into more concrete political logics by relating them to actual political practices and by joining them to the political theory of hegemony and antagonism set out by Ernesto Laclau and Chantal Mouffe. Their conjunction of the ‘common’ with the antagonistic politics of hegemony is tension-ridden, as they draw from conflicting understandings of ‘the political’, pitting plurality and horizontal relations against division and uneven power. To mitigate that conflict, these two approaches should be situated at different sites of political action, and the hegemonic framework should be recast so as to bring it more into line with horizontalist ‘common’ politics.

Concrete Quantum Logics and Δ -Logics, States and Δ -States

By a concrete quantum logic (in short, by a logic) we mean the orthomodular poset that is set-representable. If $L=({\Omega },\mathcal {L})$ is a logic and $\mathcal {L}$ is closed under the formation of symmetric difference, Δ , we call L a Δ -logic. In the first part we situate the known results on logics and states to the context of Δ -logics and Δ -states (the Δ -states are the states that are subadditive with respect to the symmetric difference). Moreover, we observe that the rather prominent logic $\mathcal {E}^{\text {even}}_{\Omega }$ of all even-coeven subsets of the countable set Ω possesses only Δ -states. Then we show when a state on the logics given by the divisibility relation allows for an extension as a state. In the next paragraph we consider the so called density logic and its Δ -closure. We find that the Δ -closure coincides with the power set. Then we investigate other properties of the density logic and its factor.

Topological Representation of Intuitionistic and Distributive Abstract Logics

We continue work of our earlier paper (Lewitzka and Brunner: Minimally generated abstract logics, Logica Universalis 3(2), 2009), where abstract logics and particularly intuitionistic abstract logics are studied. Abstract logics can be topologized in a direct and natural way. This facilitates a topological study of classes of concrete logics whenever they are given in abstract form. Moreover, such a direct topological approach avoids the often complex algebraic and lattice-theoretic machinery usually applied to represent logics. Motivated by that point of view, we define in this paper the category of intuitionistic abstract logics with stable logic maps as morphisms, and the category of implicative spectral spaces with spectral maps as morphisms. We show the equivalence of these categories and conclude that the larger categories of distributive abstract logics and distributive sober spaces are equivalent, too.

Son-Condescension and the Logic of Theology

SummaryIn the following essay, I compare concrete and abstract theological logics, critiquing the latter in favor of a biblical-revelational version of the former. I then move the comparison to the specific issue of biblical interpretation. Abstraction bares its antinomies openly here. The biblical-revelational approach focuses on Son-condescension as the mode of God’s every act with regard to not-God. If Son-condescension is the mode of the Creature/creature relation, then it should also be the logic of understanding it. Therefore, I commend a Son-condescension model for explaining the Creator/creature relation and then more basically as the logic of Christian theism.

Implicit Kripke semantics and ultraproducts in stratified institutions

We propose stratified institutions (a decade old generalised version of the theory of institutions of Goguen and Burstall) as a fully abstract model theoretic approach to modal logic. This allows for a uniform treatment of model theoretic aspects across the great multiplicity of contemporary modal logic systems. Moreover Kripke semantics (in all its manifold variations) is captured in an implicit manner free from the sometimes bulky aspects of explicit Kripke structures, also accommodating other forms of concrete semantics for modal logic systems. The conceptual power of stratified institutions is illustrated with the development of a modal ultraproducts method that is independent of the concrete details of the actual modal logical systems. Consequently, a wide array of compactness results in concrete modal logics may be derived easily.

Compatibility for probabilistic theories

Abstract We define an index of compatibility for a probabilistic theory (PT). Quantum mechanics with index 0 and classical probability theory with index 1 are at the two extremes. In this way, quantum mechanics is at least as incompatible as any PT. We consider a PT called a concrete quantum logic that may have compatibility index strictly between 0 and 1, but we have not been able to show this yet. Finally, we show that observables in a PT can be represented by positive, vector-valued measures.

Quantum cognition based on an ambiguous representation derived from a rough set approximation

Over the last years, in a series papers by Arecchi and others, a model for the cognitive processes involved in decision making has been proposed and investigated. The key element of this model is the expression of apprehension and judgment, basic cognitive process of decision making, as an inverse Bayes inference classifying the information content of neuron spike trains. It has been shown that for successive plural stimuli this inference, equipped with basic non-algorithmic jumps, is affected by quantum-like characteristics. We show here that such a decision making process is related consistently with an ambiguous representation by an observer within a universe of discourse. In our work the ambiguous representation of an object or a stimuli is defined as a pair of maps from objects of a set to their representations, where these two maps are interrelated in a particular structure. The a priori and a posteriori hypotheses in Bayes inference are replaced by the upper and lower approximations, correspondingly, for the initial data sets that are derived with respect to each map. Upper and lower approximations herein are defined in the context of “rough set” analysis. The inverse Bayes inference is implemented by the lower approximations with respect to the one map and for the upper approximation with respect to the other map for a given data set. We show further that, due to the particular structural relation between the two maps, the logical structure of such combined approximations can only be expressed as an orthomodular lattice and therefore can be represented by a quantum rather than a Boolean logic. To our knowledge, this is the first investigation aiming to reveal the concrete logic structure of inverse Bayes inference in cognitive processes.

States on systems of sets that are closed under symmetric difference

We consider extensions of certain states. The states are defined on the systems of sets that are closed under the formation of the symmetric difference (concrete quantum logics). These systems can be viewed as certain set-representable quantum logics enriched with the symmetric difference. We first show how the compactness argument allows us to extend states on Boolean algebras over such systems of sets. We then observe that the extensions are sometimes possible even for non-Boolean situations. On the other hand, a difference-closed system can be constructed such that even two-valued states do not allow for extensions. Finally, we consider these questions in a σ-complete setup and find a large class of such systems with rather interesting state properties.

Representation of Concrete Logics and Concrete Generalized Orthomodular Posets

In the present paper, we deal with the question when an effect algebra, resp. a generalized effect algebra, can be represented in the projection lattice of a Hilbert space. We show that such representability is closely related to the existence of a rich set of two-valued and Jauch–Piron states, resp. generalized two-valued and Jauch–Piron states. AMS Classification : Primary 81Q10, Secondary 03G12.

체현 인지에 기반한 개념 메타포에 관한 연구

The existing interface study has been focused on rhetorical metaphor based on a cognitive viewpoint in traditional objectivism. But the existing study viewpoint about the interface shows logical limitation by the changes in media environment. This study suggests conceptual metaphor based on embodied cognition for an alternative to solve the limitation, and research logical suitability of the existing interface environment. The result of this study is as follow; The first, the embodied cognitive viewpoint provides the interface as a space for holistic recognition communication, and the conceptual metaphor based on embodied cognition provides a well-structured metaphor logic and is a suitable metaphor to explain complicated metaphor structure. The second, the container image schema provides concrete and detailed metaphor structure than previous metaphor when it comes to explaining desktop metaphor by offering subsidiary image schema from `IN-OUT-BOUNDARY` image schema. The third, path image schema is possible to make a concrete logic and works off the laxity and confusion of logical structure in existing metaphor by offering subsidiary image schema from `SOURCE-PATH-GOAL` image schema. The conceptual metaphor based on embodied cognition is expected to be a basic theory to explain interfaces for various media environment by thinking of human physical experience as container and path image schema.

The Theory of Social Systems as a Framework for Understanding Family Businesses

There is a long tradition of systemic approaches in family business research, but recent developments have been widely ignored. This article gives a brief overview of these approaches and models and then introduces the modern theory of social systems. This theory no longer considers individuals as parts of the system but assumes that the basic elements of a social system are acts of communication. In each system (family, business, ownership), a specific communication pattern evolves and each system operates within a specific functional logic. Families process relationship communication, businesses process decision communication, and ownership operates on the basis of legally secured communication. Each of these three structurally coupled systems provides a specific context. The functional logics are context markers and assign meaning to the communicative acts. To understand how a family business functions, it is important to understand the concrete functional logics and the structural coupling of the three systems.

A dissimilarity-based framework for generating inconsistency-tolerant logics

Many commonly used logics, including classical logic and intuitionistic logic, are trivialized in the presence of inconsistency, in the sense that inconsistent premises cause the derivation of any formula. It is thus often useful to define inconsistency-tolerant variants of such logics, which are faithful to the original logic with respect to consistent theories but also allow for nontrivial inconsistent theories. A common way of doing so is by incorporating distance-based considerations for concrete logics. So far this has been done mostly in the context of two-valued semantics. Our purpose in this paper is to show that inconsistency-tolerance can be achieved for any logic that is based on a denotational semantics. For this, we need to trade distances for the more general notion of dissimilarities. We then examine the basic properties of the entailment relations that are obtained and exemplify dissimilarity-based reasoning in various forms of denotational semantics, including multi-valued semantics, non-deterministic semantics, and possible-worlds (Kripke-style) semantics. Moreover, we show that our approach can be viewed as an extension of several well-studied forms of reasoning in the context of belief revision, database integration, consistent query answering, and inconsistency maintenance in knowledge-based systems.