Do people represent space, time, number, and other conceptual domains using a generalized magnitude system (GMS)? To answer this question, numerous studies have used the spatial‐numerical association of response codes (SNARC) task and its variants. Yet, for a combination of reasons, SNARC‐like effects cannot provide evidence for a GMS, even in principle. Rather, these effects support a broader theory of how people use space metaphorically to scaffold their understanding of myriad non‐spatial domains, whether or not these domains (...) exhibit variation in magnitude. (shrink)

People implicitly associate different emotions with different locations in left-right space. Which aspects of emotion do they spatialize, and why? Across many studies people spatialize emotional valence, mapping positive emotions onto their dominant side of space and negative emotions onto their non-dominant side, consistent with theories of metaphorical mental representation. Yet other results suggest a conflicting mapping of emotional intensity (a.k.a., emotional magnitude), according to which people associate more intense emotions with the right and less intense emotions with the (...) left — regardless of their valence; this pattern has been interpreted as support for a domain-general system for representing magnitudes. To resolve the apparent contradiction between these mappings, we first tested whether people implicitly map either valence or intensity onto left-right space, depending on which dimension of emotion they attend to (Experiments 1a, b). When asked to judge emotional valence, participants showed the predicted valence mapping. However, when asked to judge emotional intensity, participants showed no systematic intensity mapping. We then tested an alternative explanation of findings previously interpreted as evidence for an intensity mapping (Experiments 2a, b). These results suggest that previous findings may reflect a left-right mapping of spatial magnitude (i.e., the size of a salient feature of the stimuli) rather than emotion. People implicitly spatialize emotional valence, but, at present, there is no clear evidence for an implicit lateral mapping of emotional intensity. These findings support metaphor theory and challenge the proposal that mental magnitudes are represented by a domain-general metric that extends to the domain of emotion. (shrink)

According to proponents of the generalized magnitude system proposal (GMS), SNARC-like effects index spatial mappings of magnitude and provide crucial evidence for the existence of a GMS. Casasanto and Pitt (2019) have argued that these effects, instead, reflect mappings of ordinality, which people compute on the basis of differences among stimuli that vary either qualitatively (e.g., musical pitches) or quantitatively (e.g., dots of different sizes). In response to our paper, Prpic et al. (2021) argued that both (...) magnitude and ordinality play a role in SNARC-like effects. Here, we address each of their arguments and conclude that magnitude is relevant to these effects only insofar as it serves as a basis for ordinality. For this reason and others, SNARC or SNARC-like effects cannot provide evidence for the putative generalized magnitude system. (shrink)

Cognitive Science, Volume 46, Issue 2, February 2022.

Accumulating evidence suggests that different magnitudes (e.g., number, size, and duration) are spatialized in the mind according to a common left–right metric, consistent with a generalized system for representing magnitude. A previous study conducted by two of us (Holmes & Lourenco, ) provided evidence that this metric extends to the processing of emotional magnitude, or the intensity of emotion expressed in faces. Recently, however, Pitt and Casasanto () showed that the earlier effects may have been driven (...) by a left–right mapping of mouth size rather than emotional magnitude, and they found no evidence for an emotional magnitude mapping when using words as stimuli. Here, we report two new experiments that further examine these conclusions. In Experiment 1, using face stimuli with mouths occluded, we replicate the original finding: Less emotional faces were associated with the left and more emotional faces with the right. However, we also find that people can reliably infer the sizes of the occluded mouths, and that these inferred mouth sizes can explain the observed left–right mapping. In Experiment 2, we show that comparative judgments of emotional words yield a left–right mapping of emotional magnitude not attributable to stimulus confounds. Based on these findings, we concur with Pitt and Casasanto that faces pose challenges for isolating the forces driving spatialization, but we suggest that emotional magnitude, when assessed using unconfounded stimuli in a sufficiently sensitive task, may indeed be spatialized as originally proposed. Suggestions for further research on the spatialization of emotional magnitude are discussed. (shrink)

Given a rich environment, how do we decide on what information to use? A view of a single entity (e.g., a group of birds) affords many distinct interpretations, including their number, average size, and spatial extent. An enduring challenge for cognition, therefore, is to focus resources on the most relevant evidence for any particular decision. In the present study, subjects completed three tasks—number discrimination, surface area discrimination, and convex hull discrimination—with the same stimulus set, where these three features were orthogonalized. (...) Therefore, only the relevant feature provided consistent evidence for decisions in each task. This allowed us to determine how well humans discriminate each feature dimension and what evidence they relied on to do so. We introduce a novel computational approach that fits both feature precision and feature use. We found that the most relevant feature for each decision is extracted and relied on, with minor contributions from competing features. These results suggest that multiple feature dimensions are separately represented for each attended ensemble of many items and that cognition is efficient at selecting the appropriate evidence for a decision. (shrink)

A Theory of Magnitude suggests that space, time, and quantities are processed through a generalized magnitude system. ATOM posits that task-irrelevant magnitudes interfere with the processing of task-relevant magnitudes as all the magnitudes are processed by a common system. Many behavioral and neuroimaging studies have found support in favor of a common magnitude processing system. However, it is largely unknown whether such cross-domain monotonic mapping arises from a change in the accuracy of the (...) magnitude judgments or results from changes in precision of the processing of magnitude. Therefore, in the present study, we examined whether large numerical magnitude affects temporal accuracy or temporal precision, or both. In other words, whether numerical magnitudes change our temporal experience or simply bias duration judgments. The temporal discrimination paradigm was used to present numerical magnitudes across varied durations. We estimated temporal accuracy and precision for each numerical magnitude. The results revealed that temporal accuracy for large numerical magnitude was significantly lower than that of small and identical magnitudes. This implies that the temporal duration was overestimated for large numerical magnitude compared to small and identical numerical magnitude, in line with ATOM’s prediction. However, no influence of numerical magnitude was observed on temporal precision. The findings of the present study suggest that task-irrelevant numerical magnitude selectively affects the accuracy of processing of duration but not duration discrimination itself. Further, we argue that numerical magnitude may not directly affect temporal processing but could influence via attentional mechanisms. (shrink)

Le général André Bach dans une réflexion sur l’« horizon de mort dans la violence de guerre » part d’une approche anthropologique du phénomène de violence et de la peur (quasiment biologique) qu’il engendre en soulignant les difficultés des sociétés occidentales à penser la mort. C’est l’État qui donne à la guerre un sens politique et sacré et qui crée les catégories fonctionnelles de la guerre (les concepts de paix et de guerre ne sont pas en eux-mêmes opérationnels). Dans le (...) cadre d’une réflexion clausewitzienne sur la montée aux extrêmes, l’« extrême » pour le soldat est toujours sa propre mort (plus que celle de l’ennemi). Les notions de barbarisation et humanisation ne peuvent en ce sens rendre compte de la violence et de la peur radicales présentes dans toute guerre où le rôle du commandement est toujours de mettre en œuvre un degré de violence supérieur à celui de l’adversaire, sans aucune considération de « proportionnalité » de cette violence. La mise à distance technique ne diminue pas la violence mais l’augmente à proportion de l’éloignement de la peur. Seul le politique peut et doit mettre des limites à cette violence militaire par nature illimitée. La modernisation des armes ne relève jamais d’une humanisation du combat mais d’une recherche d’une violence plus efficace. L’entraînement du soldat en temps de paix n’est jamais entraînement à la violence mais apprentissage de savoir-faire et effort de constitution d’un groupe soudé, doté d’un bon moral. L’ordre et la formation tentent de limiter la sauvagerie mais il existe une constitution endogène de comportements propres au soldat, que la hiérarchie tente de canaliser dans une symbolique guerrière fondée sur le sentiment de l’honneur (destiné à compenser la peur d’être une proie vouée à mourir et à faire préférer la lutte au coude à coude sur l’instinct de conservation). Le couple violence/honneur fait que l’armée dépourvue de contrat social n’est pas soluble dans le politique (aujourd’hui dans les écoles militaires on ne parle plus d’« honneur » mais d’« éthique »…, mais pour dire la même chose). La conscription fut une rupture de la logique endogène de l’armée et introduisit une brèche dans ce système archaïque. Mais religion (au Moyen Âge) ou logique démocratique (de nos jours) s’avèrent surtout des « béquilles idéologiques » qui, en définitive, sont souvent plus efficaces pour accroître la barbarisation que pour l’éviter (exemple de la colonisation algérienne où une armée de paix, fonctionnarisée et rigoureusement canalisée, appartenant à une génération n’ayant jamais connu de violence de guerre, développe des massacres de masse). (shrink)

I argue that Kant's claim in the “Transcendental Aesthetic” of the Critique of Pure Reason that space and time are immediately given in intuition as infinite magnitudes is undercut by his general theory of mathematical knowledge. On this general theory, pure intuition does not give objects of any determinate magnitude at all, but only forms of possible objects. Specifically, what pure intuition itself yields is the recognition that any determinate space or time is part of a larger one, but (...) it requires an inference of reason to go from that to the claim that space and time are infinite. I further argue that this result is consistent with Kant's claim in the second-edition “Transcendental Deduction” that the unity of space and time are the products of synthesis, but also means that the unity of space and time as objects cannot be used a premise in the Deduction but can only be regarded as a conclusion of the deduction and the following “System of Principles.”. (shrink)

The ability to represent, discriminate, and perform arithmetic operations on discrete quantities (numerosities) has been documented in a variety of species of different taxonomic groups, both vertebrates and invertebrates. We do not know, however, to what extent similarity in behavioral data corresponds to basic similarity in underlying neural mechanisms. Here, we review evidence for magnitude representation, both discrete (countable) and continuous, following the sensory input path from primary sensory systems to associative pallial territories in the vertebrate brains. We also (...) speculate on possible underlying mechanisms in invertebrate brains and on the role played by modeling with artificial neural networks. This may provide a general overview on the nervous system involvement in approximating quantity in different animal species, and a general theoretical framework to future comparative studies on the neurobiology of number cognition. (shrink)

Based on the notion that time, space, and number are part of a generalized magnitude system, we assume that the dual-systems approach to temporal cognition also applies to numerical cognition. Referring to theoretical models of the development of numerical concepts, we propose that children's early skills in processing numbers can be described analogously to temporal updating and temporal reasoning.

Each year individuals are required to execute millions of authorizations for the release of their health records as a condition of employment, applying for various types of insurance, and submitting claims for benefits. Generally, there are no restrictions on the scope of information released pursuant to these compelled authorizations, and the development of a nationwide system of interoperable electronic health records will increase the amount of health information released. After quantifying the extent of these disclosures, this article discusses why (...) it is important to limit disclosures of health information for nonmedical purposes as well as how it may be possible to do so. (shrink)

Each year individuals are required to execute millions of authorizations for the release of their health records as a condition of employment, applying for various types of insurance, and submitting claims for benefits. Generally, there are no restrictions on the scope of information released pursuant to these compelled authorizations, and the development of a nationwide system of interoperable electronic health records will increase the amount of health information released. After quantifying the extent of these disclosures, this article discusses why (...) it is important to limit disclosures of health information for nonmedical purposes as well as how it may be possible to do so. (shrink)

The ‘bat and ball’ is one of the problems most frequently employed as a testbed for research on the dual-system hypothesis of reasoning. Frederick (J Econ Perspect 19:25–42, 2005) is the first to envisage the possibility that different numerical arrangements of the ‘bat and ball’ problem could lead to different dynamics of activation of the dual-system, and so to different performances of subjects in task accomplishment. This possibility has triggered a strand of research oriented to accomplish ‘sensitivity analyses’ (...) of the ‘bat and ball’ problem. The scope of this paper is to test experimentally the specific hypothesis that numerals are responsible for the selective activation of the two systems of reasoning in this task. In particular, we argue that their role goes beyond and cannot be reduced to that of numbers conceived as magnitudes. To test our hypothesis, we devise an experimental setting in which the role of numbers (as magnitudes) is rendered irrelevant. We find experimental results consistent with our hypothesis. We further provide a link between the literature on mathematical problem-solving and that on mathematical cognition research, in particular that branch labeled embodied mathematical cognition. (shrink)

Boltzmann's gas model representing the second law of thermodynamics is based on the improbability of certain molecular distributions in space. Maxwell argued that a hypothetical ‘being’ with the faculty of seeing individual molecules could bring about such improbable distributions, thus violating the law of entropy. However, it appears that to render the molecules visible for any observer would increase the entropy more than the demon could decrease it, hence ‘Maxwell's Demon cannot operate’ . In the study presented here Maxwell's Demon (...) is interpreted in a general way as a biological observer system within systems which can upset thermodynamic probabilities provided that the relative magnitudes between observer system and observed system are appropriate.Maxwell's Demon within Boltzmann's Gas Model thus appears only as a special case of inappropriate, relative magnitude between the two systems. (shrink)

The paper aims to discuss the results of testing a neural network which classifies the vowels of the vocalic system with the [ATR] (Advanced Tongue Root) contrast based on the data of Akebu (Kwa family). The acoustic nature of the [ATR] feature is yet understudied. The only reliable acoustic correlate of [ATR] is the magnitude of the first formant (F1) which can be also modulated by tongue height, resulting in significant overlap between high [-ATR] vowels and mid [+ATR] (...) vowels. Other acoustic metrics which had been associated with the [ATR], such as F1 bandwidth (B1), relative intensity of F1 to F2 (A1-A2), etc., are typically inconsistent across vowel types and speakers. The values of four metrics – F1, F2, A1-A2, B1 – were used for training and testing the neural network. We tested four versions of the model differing in the presence of the fifth variable encoding the speaker and the number of hidden layers. The models which included the variable encoding the speaker achieved slightly higher accuracy, whereas the precision and recall metrics of the three-layer model were generally higher than those with two hidden layers. (shrink)

The paper aims to discuss the results of testing a neural network which classifies the vowels of the vocalic system with the [ATR] (Advanced Tongue Root) contrast based on the data of Akebu (Kwa family). The acoustic nature of the [ATR] feature is yet understudied. The only reliable acoustic correlate of [ATR] is the magnitude of the first formant (F1) which can be also modulated by tongue height, resulting in significant overlap between high [-ATR] vowels and mid [+ATR] (...) vowels. Other acoustic metrics which had been associated with the [ATR], such as F1 bandwidth (B1), relative intensity of F1 to F2 (A1-A2), etc., are typically inconsistent across vowel types and speakers. The values of four metrics – F1, F2, A1-A2, B1 – were used for training and testing the neural network. We tested four versions of the model differing in the presence of the fifth variable encoding the speaker and the number of hidden layers. The models which included the variable encoding the speaker achieved slightly higher accuracy, whereas the precision and recall metrics of the three-layer model were generally higher than those with two hidden layers. (shrink)

. In this paper, the significance of using general logic-systems and finite consequence operators defined on non-organized languages is discussed. Results are established that show how properties of finite consequence operators are independent from language organization and that, in some cases, they depend only upon one simple language characteristic. For example, it is shown that there are infinitely many finite consequence operators defined on any non-organized infinite language L that cannot be generated from any finite logic-system. On the other (...) hand, it is shown that for any nonempty language L, a set map is a finite consequence operator if and only if it is defined by a general logic-system. Simple logic-system examples that determine specific consequence operator properties are given. (shrink)

We present a general method for inserting proofs in Frege systems for classical logic that produces systems that can internalize their own proofs.

In this paper we define and study a generalized notion of a logical system that covers on an equal formal basis sentential, equational and sequential systems. We develop a general theory of equivalence between generalized logics that provides, first, a conception of algebraizable logic , second, a formal concept of equivalence between sequential systems and, third, a notion of equivalence between sentential and sequential systems. We also use our theory of equivalence for developing a general algebraic approach (...) to conjunctive non-pseudo-axiomatic self-extensional sentential logics. Finally, we consider within the framework of the mentioned approach various sequential formulations for some well-known sentential logics. (shrink)

The aim of this article is to describe the basic parameters of a value-oriented approach to assessing the education results as a possible basis for the methodology for assessment of the educational work in the general system of education. The key methods we used were content analysis of text sources, cross-reference analysis, comparative analysis, and humanitarian examination of juristic documents. The interpretation of education as a unity of teaching and upbringing for the state as a key subject of education, (...) which forms the requirements for the results and organization of the educational process, sets the task of assessing personal, subject and metasubject educational results. The philosophy of education faces a challenge regarding the determination of the expediency of assessing educational results, the conceptual basis of assessment, fixing the orientation of assessment on the conditions of educational organzations' activities, and on the results achieved by students. A practical managerial task is to develop an attitude towards using formalized procedures and methods for assessing educational results. The article proposes a value-oriented approach to assessing the educational work in the general education system of the Russian Federation based on an analysis of key regulatory documents of the Russian education system, the interpretation of upbringing as a process of forming value-semantic attitudes. The approach is based on identifying three main groups of value orientations to build a possible system of indicators that fix educational results: "Value orientations related to life, health and safety"; "Value orientations of social interaction"; "Value orientations of personal development". Values are an essential element in regulating human behavior, which guides the process of defining goals and choosing the means to achieve them. Values acquire a functional character in value orientations and can serve as indicators of education results at the level of students as a collective subject. It seems impossible to single out the contribution of individual actors to the output of education and to avoid contradictions between the value-semantic attitudes that exist in the space of modern culture. For the education system, the subject of assessing the quality of upbringing as a purposeful process can be both the quality of the organization of upbringing work and the outcomes of upbringing efforts. The results of education as a manifestation of value orientations in the activities of students can be considered based on quantitative and qualitative indicators, which is significant for making managerial decisions at different levels. (shrink)

I reply to comments by David Miguel Gray and Grant Gillett concerning my paper, ‘The Generality Constraint and the Structure of Thought’.

Carey has argued that there is a system of core numerical cognition – the analog magnitude system – in which cardinal numbers are explicitly represented in iconic format. While the existence of this system is beyond doubt, this paper aims to show that its representations cannot have the combination of features attributed to them by Carey. According to the argument from abstractness, the representation of the cardinal number of a collection of individuals as such requires the (...) representation of individuals as such, and this in turn requires non-iconic format, from which it is concluded that the explicit representation of the cardinal number of some individuals requires non-iconic representational format. In support of the first premise, an account is given of what approximate cardinal numbers might be, and in support of the second, a direct argument is articulated and defended. Finally, in response to an objection, a second argument for the central thesis is provided. While the discussion is couched in the terms of Carey’s work, the considerations it adduces are perfectly general, and the conclusion should therefore be taken into consideration by all those aiming to characterize the AM system. (shrink)

Following the idea of L -fuzzy generalized neighborhood systems as introduced by Zhao et al., we will give the join-complete lattice structures of lower and upper approximation operators based on L -fuzzy generalized neighborhood systems. In particular, as special approximation operators based on L -fuzzy generalized neighborhood systems, we will give the complete lattice structures of lower and upper approximation operators based on L -fuzzy relations. Furthermore, if L satisfies the double negative law, then there exists an (...) order isomorphic mapping between upper and lower approximation operators based on L -fuzzy generalized neighborhood systems; when L -fuzzy generalized neighborhood system is serial, reflexive, and transitive, there still exists an order isomorphic mapping between upper and lower approximation operators, respectively, and both lower and upper approximation operators based on L -fuzzy relations are complete lattice isomorphism. (shrink)

The author is interested in discussing various aspects of the propositional calculus; in particular, the relationships among the various propositional connectives in various systems of logic such as Intuitionistic and modal are scrutinized. The first three chapters survey the notation to be used and describe the general notion of logistic system; the author then describes the concept of a deductive system in exceptional generality, then treats the connexions of equivalence and independence among such deductive systems in what are (...) essentially algebraic terms. Building on the latter work, he then treats the idea of one system's being an extension of the other, going into the details concerning the fine structure of exactly what sort of extension it is—conservative, consistent, and thus arrives at Lindenbaum's lemma. The equivalence relations of congruence and equipollence among propositional systems form the central core around which Porte's discussion of axiomatizability of systems of connectives constituting various propositional calculi is based. Porte is one of the best of the several young logicians now working in France and this book is useful both as a technical treatise containing a sophisticated treatment of sentential logic, and as a guide to at least one direction and trend of contemporary French logical thought.—P. J. M. (shrink)

In this paper we argue that the formalisms for decoherence originally devised to deal just with closed or open systems can be subsumed under a general conceptual framework, in such a way that they cooperate in the understanding of the same physical phenomenon. This new perspective dissolves certain conceptual difficulties of the einselection program but, at the same time, shows that the openness of the quantum system is not the essential ingredient for decoherence. †To contact the authors, please write (...) to: Mario Castagnino, CONICET-IAFE, Universidad Nacional de Buenos Aires, Casilla de Correos 67, Sucursal 28, 1428 Buenos Aires, Argentina; Roberto Laura, IFIR-Universidad Nacional de Rosario, Av. Pellegrini 250, 2000 Rosario, Argentina; Olimpia Lombardi, CONICET-Universidad Nacional de Buenos Aires, C. Larralde 3440, 6°D, 1430, Buenos Aires; e-mail: [email protected]. (shrink)

Due to the variation, contingency and complexity of living systems, biology is often taken to be a science without fundamental theories, laws or general principles. I revisit this question in light of the quest for design principles in systems biology and show that different views can be reconciled if we distinguish between different types of generality. The philosophical literature has primarily focused on generality of specific models or explanations, or on the heuristic role of abstraction. This paper takes a different (...) approach in emphasizing a theory-constituting role of general principles. Design principles signify general dependency-relations between structures and functions, given a set of formally defined constraints. I contend that design principles increase our understanding of living systems by relating specific models to general types. The categorization of types is based on a delineation of the scope of biological possibilities, which serves to identify and define the generic features of classes of systems. To characterize the basis for general principles through generic abstraction and reasoning about possibility spaces, I coin the term constraint-based generality. I show that constraint-based generality is distinct from other types of generality in biology, and argue that general principles play a unifying role that does not entail theory reduction. (shrink)

Understanding biological systems in terms of scientific materialism has arguably reached a frontier, leaving fundamental questions about their complexity unanswered. In 1998, Friedrich Cramer proposed a general resonance theory as a way forward. His theory builds on the extension of the quantum physical duality of matter and wave to the macroscopic world. According to Cramer’ theory, agents constituting biological systems oscillate, akin to musical soundwaves, at specific eigenfrequencies. Biological system dynamics can be described as “Symphonies of Life” emerging from (...) the resonance (and dissonance) of eigenfrequencies within the interacting collective. His theory has potential for studying biological problems of increasing complexity in a fast‐changing Anthropocene from a new and transdisciplinary angle. Despite data becoming increasingly available for analyses, Cramer's theory remains ignored and therefore untested a quarter century after its publication. This paper discusses how the theory can move to quantitative assessments and application. Cramer's general resonance theory deserves revival. (shrink)

The purpose of this paper is to present in a uniform way the commutator theory for k-deductive system of arbitrary positive dimension k. We are interested in the logical perspective of the research — an emphasis is put on an analysis of the interconnections holding between the commutator and logic. This research thus qualifies as belonging to abstract algebraic logic, an area of universal algebra that explores to a large extent the methods provided by the general theory of deductive (...) systems. In the paper the new term ‘commutator formula’ is introduced. The paper is concerned with the meanings of the above term in the models provided by the commutator theory and clarifies contexts in which these meanings occur. The work is presented in an abstracted form: main ideas are outlined but proofs are deferred to the second part of the paper. (shrink)

The article analyzes the possibilities and limitations of artificial intelligence. The article considers the subjectivity of artificial intelligence, determines its necessity for solving intellectual problems depending on the possibility of representing the real world as a deterministic system. Methodological limitations of artificial intelligence, which is based on the use of big data technologies, are stated. These limitations cause the impossibility of forming a holistic representation of the objects of cognition and the world as a whole. As a tool for (...) deterministic description of the universe it is proposed to use empirical-metaphysical general theory of systems, which is an extension of existing general theories of systems due to ontological justification of the phenomenon of isomorphism and definition of a limited set of laws, rules, patterns and primitives of forms and relations of objects in the universe. The distinction of natural (human) and artificial intelligence is considered, including the realization of multisystem integration of intelligence in physical, biological, social and spiritual systems. A philosophically grounded approach to ensuring the evolutionary properties of artificial intelligence is formulated, based on the inclusion of non-equilibrium mechanisms through which stability is realized. (shrink)

Numbers are symbols manipulated in accord with the axioms of arithmetic. They sometimes represent discrete and continuous quantities, but they are often simply names. Brains, including insect brains, represent the rational numbers with a fixed-point data type, consisting of a significand and an exponent, thereby conveying both magnitude and precision.

Dynamical systems are mathematical structures whose aim is to describe the evolution of an arbitrary deterministic system through time, which is typically modeled as (a subset of) the integers or the real numbers. We show that it is possible to generalize the standard notion of a dynamical system, so that its time dimension is only required to possess the algebraic structure of a monoid: first, we endow any dynamical system with an associated graph and, second, we prove (...) that such a graph is a category if and only if the time model of the dynamical system is a monoid. In addition, we show that the general notion of a dynamical system allows us not only to define a family of meaningful dynamical concepts, but also to distinguish among a cluster of otherwise tangled notions of reversibility, whose logical relationships are finally analyzed. (shrink)