This paper is concerned with a logical system, called Brouwer-Zadeh logic, arising from the BZ poset of all effects of a Hilbert space. In particular, we prove a representation theorem for Brouwer-Zadeh lattices, and we show that Brouwer-Zadeh logic is not characterized by the MacNeille completions of all BZ posets of effects.

In the standard approach to quantum mechanics, closed subspaces of a Hilbert space represent propositions. In the operational approach, closed subspaces are replaced by effects that represent a mathematical counterpart for properties which can be measured in a physical system. Effects are a proper generalization of closed subspaces. Effects determine a Brouwer-Zadeh poset which is not a lattice. However, such a poset can be embedded in a complete Brouwer-Zadeh lattice. From an intuitive point of view, one can say (...) that these structures represent a natural logical abstraction from the structure of propositions of a quantum system. The logic that arises in this way is Brouwer-Zadeh logic. This paper shows that such a logic can be characterized by means of an algebraic and a Kripkean semantics. Finally, a strong completeness theorem for BZL is proved. (shrink)

We investigate certain Brouwer-Zadeh lattices that serve as abstract counterparts of lattices of effects in Hilbert spaces under the spectral ordering. These algebras, called PBZ*-lattices, can also be seen as generalisations of orthomodular lattices and are remarkable for the collapse of three notions of “sharpness” that are distinct in general Brouwer-Zadeh lattices. We investigate the structure theory of PBZ*-lattices and their reducts; in particular, we prove some embedding results for PBZ*-lattices and provide an initial description of the lattice of (...) PBZ*-varieties. (shrink)

We prove that Brouwer-Zadeh logic has the finite model property and therefore is decidable. Moreover, we present a bimodal system (BKB) which turns out to be characterized by the class of all Brouwer-Zadeh frames. Finally, we show that BrouwerZadeh logic can be translated into BKB.

The standard Brouwer–Zadeh poset Σ(H) is the poset of all effect operators on a Hilbert space H, naturally equipped with two types of orthocomplementation. In developing the theory, the question occured if (when) Σ(H) fulfils the de Morgan property with respect to both orthocomplementation operations. In Ref.3 the authors proved that it is the case provided dimH<∞, and they conjectured that if dimH=∞, then the answer is in the negative. In this note, we first give a somewhat simpler proof (...) of the known result for dimH<∞, and then we give a proof to the conjecture: We show that if dimH=∞, then the de Morgan property is not valid. (shrink)

Two different generalizations of Brouwer–Zadeh posets (BZ posets) are introduced. The former (called pre-BZ poset) arises from topological spaces, whose standard power set orthocomplemented complete atomic lattice can be enriched by another complementation associating with any subset the set theoretical complement of its topological closure. This complementation satisfies only some properties of the algebraic version of an intuitionistic negation, and can be considered as, a generalized form of a Brouwer negation. The latter (called degenerate BZ poset) arises from (...) the so-called special effects on a Hilbert space. It is shown that the standard Brouwer negation for effect operators produces a degenerate BZ poset with respect to the order induced from the partial sum operation. (shrink)

On the basis of Mackey's axiomatic approach to quantum physics or, equivalently, of a “state-event-probability” (SEVP) structure, using a quite standard “fuzzification” procedure, a set of unsharp events (or “effects”) is constructed and the corresponding “state-effect-probability” (SEFP) structure is introduced. The introduction of some suitable axioms gives rise to a partially ordered structure of quantum Brouwer-Zadeh (BZ) poset; i.e., a poset endowed with two nonusual orthocomplementation mappings, a fuzzy-like orthocomplementation, and an intuitionistic-like orthocomplementation, whose set of sharp elements is an (...) orthomodular complete lattice. As customary, by these orthocomplementations the two modal-like necessity and possibility operators are introduced, and it is shown that Ludwig's and Jauch-Piron's approaches to quantum physics are “interpreted” in complete SEFP. As a marginal result, a standard procedure to construct a lot of unsharp realizations starting from any sharp realization of a fixed observable is given, and the relationship among sharp and corresponding unsharp realizations is studied. (shrink)

We try to translate the intuitionistic propositional logic INT into Brouwer's modal logic KTB. Our translation is motivated by intuitions behind Brouwer's axiom p →☐◊p The main idea is to interpret intuitionistic implication as modal strict implication, whereas variables and other positive sentences remain as they are. The proposed translation preserves fragments of the Rieger-Nishimura lattice which is the Lindenbaum algebra of monadic formulas in INT. Unfortunately, INT is not embedded by this mapping into KTB.

In a preceding paper [1] it was shown that quantum logic, given by the tableaux-calculus Teff, is complete and consistent with respect to the dialogic foundation of logics. Since in formal dialogs the special property of the 'value-definiteness' of propositions is not postulated, the calculus $T_{eff}$ represents a calculus of effective (intuitionistic) quantum logic. Beginning with the tableaux-calculus the equivalence of $T_{eff}$ to calculi which use more familiar figures such as sequents and implications can be investigated. In this paper we (...) present a sequents-calculus of Gentzen-type and a propositional calculus of Brouwer-type which are shown to be equivalent to $T_{eff}$ . The effective propositional calculus provides an interpretation for a lattice structure, called quasi-implicative lattice. If, in addition, the value-definiteness of quantum mechanical propositions is postulated, a propositional calculus is obtained which provides an interpretation for a quasi-modular orthocomplemented lattice which, as is well-known, has as a model the lattice of subspaces of a Hilbert space. (shrink)

The middle of the second until the middle of the first century BCE is one of the most creative periods in the history of human thought, and an important part of this was the interaction between Roman jurists and Hellenistic philosophers. In this highly original book, René Brouwer shows how jurists transformed the study of law into a science with the help of philosophical methods and concepts, such as division, rules and persons, and also how philosophers came to share the (...) jurists' preoccupations with cases and private property. The relevance of this cross-fertilization for present-day law and philosophy cannot be overestimated: in law, its legacy includes the academic study of law and the Western models of dispute resolution, while in philosophy, the method of casuistry and the concept of just property. (shrink)

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Another main result is that connected choice is complete for dimension greater (...) than or equal to two in the sense that it is computably equivalent to Weak Kőnig’s Lemma. While we can present two independent proofs for dimension three and upward that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upward. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater than or equal to one is equivalent to Weak Kőnig’s Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes. (shrink)

We investigate the initial segments of the Medvedev lattice as Brouwer algebras, and study the propositional logics connected to them.

We study the Medvedev degrees of mass problems with distinguished topological properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by the nonzero discrete degrees. We give a complete picture of the relationships of inclusion holding between these sublattices, these filters, and this ideal. We show that the sublattice (...) of the closed Medvedev degrees is not a Brouwer algebra. We investigate the dense degrees of mass problems that are closed under Turing equivalence, and we prove that the dense degrees form an automorphism base for the Medvedev lattice. The results hold for both the Medvedev lattice on the Baire space and the Medvedev lattice on the Cantor space. (shrink)

Practical Mysticism in Islam and Christianity offers a comparative study of the works of the Sufi-poet Jalal al-Din Rumi (1207-1273) and the practical teachings of the German Dominican, Meister Eckhart (c1260-1327/8). Rumi has remained an influential figure in Islamic mystical discourse since the thirteenth century, while also extending his impact to the Western spiritual arena. However, his ideas have frequently been interpreted within the framework of other mystical, philosophical, or religious systems. Through its novel approach, this book aims to reformulate (...) Rumi's practical mysticism by employing four methodological principles: a) mysticism is a coherent structure with mutual interconnection between its parts; b) the imposition of alien structures to interpret any particular mysticism damages its inward coherency; c) practical mysticism consists of two main parts, namely practices and stages; and d) the proper use of comparative methodology enables a deeper understanding of each juxtaposed system. Eckhart's speculative mysticism, which differs from and enjoys similarities with the love-based mysticism of Rumi, provides a "mirror" that highlights the special features of Rumi's practical mysticism. Such comparison also allows a deeper comprehension of Eckhart's practical thought. Offering a critical examination of practical mysticism, this book is a valuable resource for students and scholars of Islamic studies, comparative mysticism, and the intellectual history of Islam. (shrink)

part 1. Fuzzy logic theory 1 -- part 2. Applications and advanced topics of fuzzy logic.

When humans are required to perform two or more tasks concurrently, their performance declines as the tasks get closer together in time. Here, we investigated the mechanisms of this cognitive performance decline using a dual-task paradigm in a simulated driving environment, and using drift-diffusion modeling, examined if the two tasks are processed in a serial or a parallel manner. Participants performed a lane change task, along with an image discrimination task. We systematically varied the time difference between the onset of (...) the two tasks and measured its effect on the amount of dual-task interference. Results showed that the reaction times of the two tasks in the dual-task condition were higher than those in the single-task condition. SOA influenced the RTs of both tasks when they were presented second and the RTs of the image discrimination task when it was presented first. Results of drift-diffusion modeling indicated that dual-task performance affects both the rate of evidence accumulation and the delays outside the evidence accumulation period. These results suggest that a hybrid model containing features of both parallel and serial processing best accounts for the results. Next, manipulating the predictability of the order of the two tasks, we showed that in unpredictable conditions, the order of the response to the two tasks changes, causing attenuation in the effect of SOA. Together, our findings suggest higher-level executive functions are involved in managing the resources and controlling the processing of the tasks during dual-task performance in naturalistic settings. (shrink)

The term fuzzy logic is used in this paper to describe an imprecise logical system, FL, in which the truth-values are fuzzy subsets of the unit interval with linguistic labels such as true, false, not true, very true, quite true, not very true and not very false, etc. The truth-value set, , of FL is assumed to be generated by a context-free grammar, with a semantic rule providing a means of computing the meaning of each linguistic truth-value in as a (...) fuzzy subset of [0, 1].Since is not closed under the operations of negation, conjunction, disjunction and implication, the result of an operation on truth-values in requires, in general, a linguistic approximation by a truth-value in . As a consequence, the truth tables and the rules of inference in fuzzy logic are (i) inexact and (ii) dependent on the meaning associated with the primary truth-value true as well as the modifiers very, quite, more or less, etc. (shrink)

This volume, edited by René Brouwer and Emmanuele Vimercati, deals with the debate about fate, providence and free will in the early Imperial age. This debate is rekindled in the 1st century CE during emperor Augustus' rule and ends in the 3rd century CE with Plotinus and Origen, when the different positions in the debate were more or less fully developed. The book aims to show how in this period the notions of fate, providence and freedom were developed and debated, (...) not only within and between the main philosophical schools, that is Stoicism, Aristotelianism, and Platonism, but also in the interaction with other, "religious" movements, here understood in the general sense of groups of people sharing beliefs in and worship of (a) superhuman controlling power(s), such as Gnosticism, Hermetism as well as Judaism and Christianity. (shrink)

It has been argued that metaphor and emotion processing are tightly linked together. Here we explore whether neuroscientific evidence supports this claim.

Medical practice is practiced morality, and clinical research belongs to normative ethics. The present book elucidates and advances this thesis by: 1. analyzing the structure of medical language, knowledge, and theories; 2. inquiring into the foundations of the clinical encounter; 3. introducing the logic and methodology of clinical decision-making, including artificial intelligence in medicine; 4. suggesting comprehensive theories of organism, life, and psyche; of health, illness, and disease; of etiology, diagnosis, prognosis, prevention, and therapy; and 5. investigating the moral and (...) metaphysical issues central to medical practice and research. Many systems of (classical, modal, non-classical, probability, and fuzzy) logic are introduced and applied. Fuzzy medical deontics, fuzzy medical ontology, fuzzy medical concept formation, fuzzy medical decision-making and biomedicine and many other techniques of fuzzification in medicine are introduced for the first time. -/- . (shrink)

This study examined the performances of three groups of participants in interpreting metaphors in three different conditions: congruent gesture-prime conditions, opposite gesture-prime conditions, and no-prime conditions. In congruent gesture-prime conditions, each metaphor was preceded by the congruent gestural representation of metaphor schema. In opposite gesture-prime conditions, each metaphor was preceded by the opposite gestural representation of metaphor schema. The results showed that participants of congruent gesture-prime conditions had the best performance in interpreting metaphors, while participants of opposite gesture-prime conditions had (...) the worst performance. It is suggested that metaphor schema is an important part of metaphorical meaning of a metaphor. Therefore, when this part of meaning is activated by a gestural prime, metaphor comprehension is facilitated. Furthermore, when a schema is used to metaphorically describe a concept or an event, that schema and the neural network that represents it become one part of a larger interconnected neural network that represents the meaning of the metaphor. It could mean that the activation of one part of it could facilitate the activation of the whole of the network. Finally, it is emphasized that a gesture that depicts the schema of a metaphor is in fact the embodied realization of that metaphor. (shrink)

Representational transformation is a technique for deepening our understanding of mathematical concepts and facilitating the process of solving mathematical problems. This article suggests that representational transformation can contribute to thought processes in four ways. Firstly, representational transformation helps people employ a wider range of cognitive resources. When one representation of a concept or a problem is transformed into another representation, the sensory-motor resources that are involved in the processing of the base representation are employed to process the target representation. Secondly, (...) representational transformation helps in the process of organizing scattered information into suitable units. This gives structure to various parts of information and makes them much easier to handle. Thirdly, representational transformation helps in the process of suppressing irrelevant information and unnecessary elements of problems. This is a key step in organizing and structuring information. Fourthly, representational transformation helps in the process of discovering general rules and turning them into explicit knowledge. (shrink)

This study measured hand and head action strengths of eight typical emotional states using an authentic but implicit emotion elicitation task. Participants listened to and then retold five stories in which eight typical emotional states were experienced by the narrators. The number of hand and head gestures that occur naturally while experiencing an emotional state was used as an index to determine the hand and head action strength of that emotional state. Results showed a larger number of head gestures than (...) hand gestures, suggesting that head action strengths of the eight emotional states are stronger than their hand action strengths. These findings are consistent with the data extracted from Lancaster Sensorimotor Norms (LSN), although the two sets of data were gathered in two completely different experimental conditions and in two different languages. Furthermore, our data showed a prototypical directionality effect for the typical emotional states, specifically, happiness, anger and pride were primarily accompanied by upward gestures but downward gestures for sadness and shame; surprise was primarily accompanied by forward gestures but backward gestures for fear and disgust. (shrink)

Luitzen Egburtus Jan Brouwer founded a school of thought whose aim was to include mathematics within the framework of intuitionistic philosophy; mathematics was to be regarded as an essentially free development of the human mind. What emerged diverged considerably at some points from tradition, but intuitionism has survived well the struggle between contending schools in the foundations of mathematics and exact philosophy. Originally published in 1981, this monograph contains a series of lectures dealing with most of the fundamental topics such (...) as choice sequences, the continuum, the fan theorem, order and well-order. Brouwer's own powerful style is evident throughout the work. (shrink)

Social representations research is often undertaken by scholars who seek to ‘give voice’ to knowledge that are held by socially disenfranchised individuals and groups. However, this endeavour poses a number of problems in practice, not least because it assumes that the ‘voices’ voiced by individuals and/or groups in social research will be unambiguous and uniform, and unchanged by the research encounter. Despite the growth of attention to the critical potential of social representations theory, there remains a lack of scholarship on (...) the relationship between the theory's emphasis on the relational, or dialogical, nature of social life and the implications of this for critical research. In this article, I argue that the dialogical epistemology from which social representations research must depart incites reflection upon the design of empirical studies, which must equally attend to both ‘Ego’ and ‘Alter’, and the researcher-researched relationship, which may itself be best viewed as an Ego-Alter interaction. I make the case for adopting dialogical epistemology in critical social psychology, and argue that this is essential to undertaking ethical social research. I conclude by suggesting that claims to ‘giving voice’ have little place in critical social psychology in general, and social representations scholarship in particular. (shrink)

The rate of human health care-associated infections caused by Acinetobacter baumannii has increased significantly in recent years for its remarkable resistance to desiccation and most antibiotics. Phospholipases, capable of destroying a phospholipid substrate, are heterologous group of enzymes which are believed to be the bacterial virulence determinants. There is a need for in silico studies to identify potential vaccine candidates. A. baumannii phospholipase D role has been proved in increasing organism’s resistance to human serum, destruction of host epithelial cell and (...) pathogenesis in murine model. In this in silico study high potentials of A. baumannii PLD in elicitation of humoral and cellular immunities were elucidated. Thermal stability, long half-life, non-similarity to human and gut flora proteome and non-allergenicity were in a list of A. baumannii PLD positive properties. Potential epitopic sequences were also identified that could be used as peptide vaccines against A. baumannii and various other human bacterial pathogens. (shrink)

BackgroundReconciling results obtained using different types of sensory measures is a challenge for autism sensory research. The present study used questionnaire, psychophysical, and neurophysiological measures to characterize autistic sensory processing in different measurement modalities.MethodsParticipants were 46 autistic and 21 typically developing 11- to 14-year-olds. Participants and their caregivers completed questionnaires regarding sensory experiences and behaviors. Auditory and somatosensory event-related potentials were recorded as part of a multisensory ERP task. Auditory detection, tactile static detection, and tactile spatial resolution psychophysical thresholds were (...) measured.ResultsSensory questionnaires strongly differentiated between autistic and typically developing individuals, while little evidence of group differences was observed in psychophysical thresholds. Crucially, the different types of measures appeared to be largely independent of one another. However, we unexpectedly found autistic participants with larger auditory Tb ERP amplitudes had reduced hearing acuity, even though all participants had hearing acuity in the non-clinical range.LimitationsThe autistic and typically developing groups were not matched on cognitive ability, although this limitation does not affect our main analyses regarding convergence of measures within autism.ConclusionOverall, based on these results, measures in different sensory modalities appear to capture distinct aspects of sensory processing in autism, with relatively limited convergence between questionnaires and laboratory-based tasks. Generally, this might reflect the reality that laboratory tasks are often carried out in controlled environments without background stimuli to compete for attention, a context which may not closely resemble the busier and more complex environments in which autistic people’s atypical sensory experiences commonly occur. Sensory questionnaires and more naturalistic laboratory tasks may be better suited to explore autistic people’s real-world sensory challenges. Further research is needed to replicate and investigate the drivers of the unexpected association we observed between auditory Tb ERP amplitudes and hearing acuity, which could represent an important confound for ERP researchers to consider in their studies. (shrink)

For meaning normativism to hold, meaning must have a constitutive part which is obligation-producing. I claim in this paper that linguistic communication is such a constitutive part. I try to show this by means of appeal to Davidson’s triangulation thesis. If I am successful, it may fairly be said that “a Davidsonian can rescue the normativity of meaning”.

Metaphors are cognitive and linguistic tools that allow reasoning. They enable the understanding of abstract domains via elements borrowed from concrete ones. The underlying mechanism in metaphorical mapping is the manipulation of concepts. This article proposes another view on what concepts are and their role in metaphor and reasoning. That is, based on current neuroscientific and behavioural evidence, it is argued that concepts are grounded in perceptual and motor experience with physical and social environments. This definition of concepts is then (...) embedded in the Structure-Mapping Theory, a model for metaphorical processing and reasoning. The blended view of structure-mapping and embodied cognition offers an insight into the processes through which the target domain of a metaphor is embodied or realised in terms of its base domain. The implications of the proposed embodied SMT model are then discussed and future topics of investigation are outlined. (shrink)

A total of 944 expressed sequence tags generated 2212 EST loci mapped to homoeologous group 1 chromosomes in hexaploid wheat. EST deletion maps and the consensus map of group 1 chromosomes were constructed to show EST distribution. EST loci were unevenly distributed among chromosomes 1A, 1B, and ID with 660, 826, and 726, respectively. The number of EST loci was greater on the long arms than on the short arms for all three chromosomes. The distribution of ESTs along chromosome arms (...) was nonrandom with EST clusters occurring in the distal regions of short arms and middle regions of long arms. Duplications of group 1 ESTs in other homoeologous groups occurred at a rate of 35.5%. Seventy-five percent of wheat chromosome 1 ESTs had significant matches with rice sequences, where large regions of conservation occurred between wheat consensus chromosome 1 and rice chromosome 5 and between the proximal portion of the long arm of wheat consensus chromosome 1 and rice chromosome 10. Only 9.5% of group 1 ESTs showed significant matches to Arabidopsis genome sequences. The results presented are useful for gene mapping and evolutionary and comparative genomics of grasses. (shrink)