According to the objective Bayesian approach to inductive logic, premisses inductively entail a conclusion just when every probability function with maximal entropy, from all those that satisfy the premisses, satisfies the conclusion. When premisses and conclusion are constraints on probabilities of sentences of a first-order predicate language, however, it is by no means obvious how to determine these maximal entropy functions. This paper makes progress on the problem in the following ways. Firstly, we introduce the concept of (...) a limit in entropy and show that, if the set of probability functions satisfying the premisses contains a limit in entropy, then this limit point is unique and is the maximal entropy probability function. Next, we turn to the special case in which the premisses are categorical sentences of the logical language. We show that if the uniform probability function gives the premisses positive probability, then the maximal entropy function can be found by simply conditionalising this uniform prior on the premisses. We generalise our results to demonstrate agreement between the maximal entropy approach and Jeffrey conditionalisation in the case in which there is a single premiss that specifies the probability of a sentence of the language. We show that, after learning such a premiss, certain inferences are preserved, namely inferences to inductive tautologies. Finally, we consider potential pathologies of the approach: we explore the extent to which the maximal entropy approach is invariant under permutations of the constants of the language, and we discuss some cases in which there is no maximal entropy probability function. (shrink)

This essay intends to define the role of entropy, in particular, the role of the maximum entropy criterion with respect to decision analysis and information economics. By considering the average opportunity loss interpretation, the basic hypothesis for Shannon's derivation can be derived from properties of decision problems. Using the representation Bayes Boundary it is possible to show that selecting a single probability from a set by the Maximum Entropy Criterion corresponds to a minimax criterion for decision-making. Since (...) problems of randomly accessing and storing information as well as communicating information can often be stated in terms of coding problems, this result might be used to develop strategies for minimizing retrieval time or communication costs. (shrink)

Claude Shannon's 1948 entropy formula H = −Σ pᵢ log pᵢ established the mathematical foundation of information theory, quantifying uncertainty in discrete symbol distributions within communication channels. This paper introduces Narrative Entropy (Sn), a formally distinct but epistemologically related metric developed within the Bulut Doctrine framework, which measures the dynamic accumulation of cognitive resistance and structural uncertainty across the temporal dimension of narrative experience. We demonstrate that while Sn draws its foundational intuition from Shannon's H, the two metrics (...) differ across six critical dimensions: domain, temporal structure, measurement object, operational unit, directionality, and engineering function. Sn is not an application of Shannon entropy to literature — it is a structurally independent metric that addresses phenomena Shannon's framework was not designed to capture. We further demonstrate the relationship between Sn, Narrative Gravity (Ng), and biophysical reader output, and provide benchmark calculations for Crime and Punishment, Moby Dick, and Pulp Fiction. (shrink)

The language of entropy is examined for consistency with its mathematics and physics, and for its efficacy as a guide to what entropy means. Do common descriptors such as disorder, missing information, and multiplicity help or hinder understanding? Can the language of entropy be helpful in cases where entropy is not well defined? We argue in favor of the descriptor spreading, which entails space, time, and energy in a fundamental way. This includes spreading of energy spatially (...) during processes and temporal spreading over accessible microstates states in thermodynamic equilibrium. Various examples illustrate the value of the spreading metaphor. To provide further support for this metaphor’s utility, it is shown how a set of reasonable spreading properties can be used to derive the entropy function. A main conclusion is that it is appropriate to view entropy’s symbol S as shorthand for spreading. (shrink)

This book is a critical monograph on the problem of Chinese intellectuals, grounded in the philosophy of science as methodology, thermodynamics and complex systems as analytical tools, and the Lao-Yang Genesis Cosmology (LYGC) as its ontological foundation. The title concept, Sage Entropy (圣人熵), names the book's core original contribution: the ontological error of Confucianism — substituting the subjective will of sages for the laws of the cosmic natural order — accumulates irreversible thermodynamic entropy within civilizational systems, driving the (...) periodic collapse of civilization and the desertification of thought that has recurred across two thousand years of Chinese history. The structural logic of the book follows a seven-layer chain of deduction: Artificial Order Is False (ontological verdict) → Sage Entropy (thermodynamic mechanism) → Knowledge-Power Conspiracy (institutional logic) → Intellectual Desertification (cognitive result) → Civilizational Decline (historical evidence) → Dao Follows Nature (cosmological way out) → Reconstruction of Natural Civilization (practical program) — The Seven-Layer Core Deductive Chain of This Book -/- The book is divided into seven parts (including the Prelude and Part Six, the Constructive section), comprising twenty-five chapters in total, with Appendices A through G and a bibliography. Summary of Each Part -/- [Prelude] Drawing on the Guodian Chu Slips (unearthed 1993), the Mawangdui Silk Manuscripts, and the Taiyi Shengshui as archaeo-philological evidence, this section argues for the Chu civilizational roots of Laozi's thought and establishes a documentary foundation for the LYGC cosmology reaching back two and a half millennia. [Part Zero: Ontological Premises] Using the LYGC cosmic hierarchy (humanity follows Earth → Earth follows Heaven → Heaven follows Dao → Dao follows Nature) as a normative baseline, this part systematically argues that Confucian artificial order (the Five Constants of ren, yi, li, zhi, xin plus the Three Bonds and Five Relationships) constitutes a fourfold subversion of the cosmic natural order: epistemological subversion, hierarchical subversion, energetic subversion, and subversion of knowledge production. [Part One: Diagnosis] A historical-institutional pathology: how Emperor Wu of Han's exclusive promotion of Confucianism established the institutional origin of intellectual serfdom; how the 'system overfitting' of the imperial examination system mass-produced cognitively domesticated intellectuals; why three waves of enlightenment (late Ming/early Qing, May Fourth, the 1980s) each ended with national salvation suppressing awakening; and the contemporary academic degeneration revealed by Li Bozhong's data, alongside Chinese universities' DeepSeek ambitions and DeepFake illusions. [Part Two: Critique of Confucianism] Deconstruction at the level of philosophical ontology: the verdict on the nature of the Five Constants under the proposition 'Artificial Order Is False'; the strict operational definition of Sage Entropy (dS/dt = ρ_r · DAF / ΔCCF) and its historical verification; the structural co-option mechanism behind the myth of 'Confucian-Daoist Complementarity'; and an ontological critique of Tu Weiming's New Confucian thesis of 'Mind as Ontological Foundation.' [Part Three: Roots] The ABCDE five-factor framework for civilizational failure, arguing that A (Confucian artificial order) is the core variable but not the sole one; an extended analysis of the Qian Xuesen case through engineering cybernetics; the Guo Moruo case; and a unified thermodynamic explanation for intellectual degeneration. [Part Four: Comparison] A comparison of the University of California system (self-organizing dissipative structure) versus Confucianized universities (heteronomously organized ritual coercion); an institutional analysis of Chen Yinke as the solitary historical witness to Chinese intellectual independence; and the third path offered by Laozi and Foucault. [Part Five: Transition] The naturalization of scholarship and the university: the convergence of Liang Qichao's 'academic independence' with Dao Follows Nature; a true ontology of the university; and the only possible path for Confucianism to be reborn within the framework of the Natural Way. [Part Six: Construction] Systematic reconstruction of the knowledge system of Natural Civilization: what constitutes genuine knowledge (three conditions: observable, falsifiable, transmissible); the institutional design of the Natural Civilization university; the personality model of the Natural Civilization thinker; and 'Students of the Dao' as the civilizational alternative to two thousand years of 'Disciples of the Sage.' . (shrink)

Michel Serres’s philosophy of entropy takes what he famously calls the ‘Northwest Passage’ between the sciences and the humanities. By contextualizing his approach to entropy and affirming the role of a philosophy of difference, this paper explores Serres’s approach by means of ‘entropic differences’. It claims that entropy – or rather, entropies – provide Serres with a paradigmatic case for critical translations between different domains of knowledge. From his early Hermès series, through to The Birth of Physics (...) and later writings on social and ethical themes, he keeps thermodynamical and informational – or ‘hard’ and ‘soft’ – understandings of entropy apart, while simultaneously exploring their relation. By focusing on the systematic significance of Serres’s ‘entropic difference’, this paper shows how it unfolds not necessarily as an ontological difference but as an operative function between the history and philosophy of science, epistemology, and a theory of negentropic (inter)subjectivity. (shrink)

This article reconceptualizes entropy not as a metaphysical substance but as a structural constraint that shapes the formation, energetic cost, and durability of records. It links the coarse-grained—and typically irreversible—flow of time to questions of moral responsibility and divine justice. Drawing on the second law of thermodynamics, information theory, and contemporary cosmology, it advances an analogical and operational framework in which actions are accountable in an analogical sense insofar as they leave energetically costly traces that resist erasure. Within a (...) Qur’ānic metaphysical horizon, concepts such as kitāb (Book), ṣaḥīfa (Record), and tawba (Repentance) function as structural counterparts to informational inscription and revision, without reducing theological meaning to physical process. In contrast to Kantian ethics, which grounds moral law in rational autonomy, the Qurʾān situates responsibility within the irreversible structure of time. Understood in this way, entropy is not a threat to coherence but a condition for accountability. By placing the Qurʾānic vision in dialog with modern science and theology, the article contributes to broader discussions on justice, agency, and the metaphysics of time within the science–religion discourse. (shrink)

Narrative Entropy (Sn) is a proposed quantitative descriptor within the Bulut Doctrine, intended to capture the rate at which a narrative text imposes processing load on a reader. To date the construct has been defined theoretically but not operationalized against real texts. This report documents the first such operationalization (the v2.0 pilot): two narrative scenes — the opening restaurant scene of Quentin Tarantino's Reservoir Dogs screenplay and the opening interior-monologue block of Raymond Carver's short story Cathedral — were coded (...) manually by a single rater and scored with the candidate formula Sn = If × Cb × t. -/- The result was contrary to intuitive expectation. The single-voice monologue block (Sn = 30.0) scored higher than the nine-character rapid-dialogue scene (Sn = 18.8). We treat this divergence not as a result to be explained away but as the central finding of the pilot, and we deliberately refuse post-hoc adjustment of the formula. We present three competing interpretations of the divergence formula incompleteness, genuine high-load prose, and measurement error — and we specify, in advance of data collection, the design that would discriminate among them. This document functions simultaneously as a pilot report (n = 2) and as a pre-registration of the next-stage protocol. It explicitly does not claim that Sn has been validated. -/- Keywords: narrative entropy, operationalization, pilot study, pre-registration, falsifiability, Bulut Doctrine, computational narratology, processing load. (shrink)

Determining a prior probability function via the maximum entropy principle can be a computationally intractable task. However one can easily determine - in advance of entropy maximisation - a list of conditional independencies that the maximum entropy function will satisfy. These independencies can be used to reduce the complexity of the entropy maximisation task. In particular, one can use these independencies to construct a direct acyclic graph in a Bayesian network, and then maximise (...) class='Hi'>entropy with respect to the numerical parameters of this network. This can result in an efficient representation of a prior probability function, and one that may allow efficient updating and marginalisation. The computational complexity of maximising entropy can be further reduced when knowledge of causal relationships is available. Moreover, the proposed simplification of the entropy maximisation task may be exploited to construct a proof theory for probabilistic logic. (shrink)

Both a method of therapy and an exploration of psychic reality, free association is a fundamental element of psychoanalytical practices that refers to the way a patient is asked to describe what comes spontaneously to mind in the therapeutic setting. This paper examines the role of free association from the point of view of psychoanalysis and neuroscience in order to improve our understanding of therapeutic effects induced by psychoanalytic therapies and psychoanalysis. In this regard, we first propose a global overview (...) of the historical origins of the concept of free association in psychoanalysis and examine how Freud established its principles. Then, from Freud’s distinction between primary and secondary processes, we proceed to compare the psychoanalytic model with research originating from cognitive psychology and neuroscience. The notions of entropy and free energy appear particularly relevant at the intersection of these different domains. Finally, we propose the notion of symbolizing transmodality to describe certain specificities of symbolization processes within free association and we summarize the main functions of free association in psychoanalytic practices. (shrink)

A new axiomatic characterization with a minimum of conditions for entropy as a function on the set of states in quantum mechanics is presented. Traditionally unspoken assumptions are unveiled and replaced by proven consequences of the axioms. First the Boltzmann–Planck formula is derived. Building on this formula, using the Law of Large Numbers—a basic theorem of probability theory—the von Neumann formula is deduced. Axioms used in older theories on the foundations are now derived facts.

The Anthropocene crisis is frequently described as the rarefaction of resources or resources per capita. However, both energy and minerals correspond to fundamentally conserved quantities from the perspective of physics. A specific concept is required to understand the rarefaction of available resources. This concept, entropy, pertains to energy and matter configurations and not just to their sheer amount. However, the physics concept of entropy is insufficient to understand biological and social organizations. Biological phenomena display both historicity and systemic (...) properties. A biological organization, the ability of a specific living being to last over time, results from history, expresses itself by systemic properties, and may require generating novelties The concept of anti-entropy stems from the combination of these features. We propose that Anthropocene changes disrupt biological organizations by randomizing them, that is, decreasing anti-entropy. Moreover, second-order disruptions correspond to the decline of the ability to produce functional novelties, that is, to produce anti-entropy. (shrink)

This paper seeks to determine a rational agent’s evidential constraints given her beliefs. Rationality is here construed as adherence to a principle of entropy maximisation. I determine the rational agent’s set of probability efunctions compatible with the evidence, E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {E}}$$\end{document}, given the maximum entropy function and given some constraints on the shape of E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {E}}$$\end{document}. I also consider agents employing (...) a centre of mass approach to form their beliefs rather than entropy maximisation. (shrink)

The present paper is a survey of the economics of survival, a branch of ecological economics that stresses the preservation of the opportunities of future generations over an extended time horizon. It outlines the main analytical foundation of the branch – in which the concept of entropy is a major building block –, and its analysis of the interaction between the economic system and the environment. Regarding its outlook of the future, we see that the founders of the branch (...) were mainly concerned with the consequences of a serious depletion of natural resources – particularly the energetic capital of the earth. More recently, however, emphasis is being placed on problems that stem from the fragility of the global ecosystem in face of the disturbances caused by the entropic acceleration imposed by mankind. It is feared that the ongoing expansion of the scale of the economy may bring about irreversible damages to vital environmental functions, such as protection against undesirable consequences of solar radiation, maintenance of temperature within a range that will support life, and preservation of ecosystem resiliency. (shrink)

Neural complexity and brain entropy have gained greater interest in recent years. The dynamics of neural signals and their relations with information processing continue to be investigated through different measures in a variety of noteworthy studies. The BEN of spontaneous neural activity decreases during states of reduced consciousness. This evidence has been showed in primary consciousness states, such as psychedelic states, under the name of “the entropic brain hypothesis.” In this manuscript we propose an extension of this hypothesis to (...) physiological and pathological aging. We review this particular facet of the complexity of the brain, mentioning studies that have investigated BEN in primary consciousness states, and extending this view to the field of neuroaging with a focus on resting-state functional Magnetic Resonance Imaging. We first introduce historic and conceptual ideas about entropy and neural complexity, treating the mindbrain as a complex nonlinear dynamic adaptive system, in light of the free energy principle. Then, we review the studies in this field, analyzing the idea that the aim of the neurocognitive system is to maintain a dynamic state of balance between order and chaos, both in terms of dynamics of neural signals and functional connectivity. In our exploration we will review studies both on acute psychedelic states and more chronic psychotic states and traits, such as those in schizophrenia, in order to show the increase of entropy in those states. Then we extend our exploration to physiological and pathological aging, where BEN is reduced. Finally, we propose an interpretation of these results, defining a general trend of BEN in primary states and cognitive aging. (shrink)

The application of the maximum entropy principle to determine probabilities on finite domains is well-understood. Its application to infinite domains still lacks a well-studied comprehensive approach. There are two different strategies for applying the maximum entropy principle on first-order predicate languages: applying it to finite sublanguages and taking a limit; comparing finite entropies of probability functions defined on the language as a whole. The entropy-limit conjecture roughly says that these two strategies result in the same probabilities. While (...) the conjecture is known to hold for monadic languages as well as for premiss sentences containing only existential or only universal quantifiers, its status for premiss sentences of greater quantifier complexity is, in general, unknown. I here show that the first approach fails to provide a sensible answer for some \-premiss sentences. I discuss implications of this failure for the first strategy and consequences for the entropy-limit conjecture. (shrink)

The paper aspires to provide an (intended new) perspective on the important concept of sustainability rationality and sustainable path/chreode of the societal system. To this end, the concept of social entropy is convoked and examined, so that the social entropy mark can work as an (ordinal) predictor for the sustainability property of the society. The research uses the logical approach, so the discussion is carried on in its most general and abstract dimensions, to provide (possible) solutions for the (...) most different particular historical or geo-political locations. Some new approaches are presented regarding the societal system, social entropy, the very concept of sustainability, and the quantification (including a log-function) of the social entropy mark which, in turn, quantifies the sustainability state. The rapport between the capitalist model of rationality (currently working through the optimality paradigm) and sustainability is examined and concluded, as well as the rapport between sustainability and ethics—the last being, today, an issue intensively debated and un-concluded yet. (shrink)

It has been shown by several authors that in operations involving information a quantity appears which is the negative of the quantity usually defined as entropy in similar situations. This quantity ℜ = − KI has been termed “negentropy” and it has been shown that the negentropy of information and the physical entropy S are mirrorlike representations of the same train of events. In physical terminology the energy is degraded by an increase in entropy due to an (...) increased randomness in the positions or velocities of components, wave functions, complexions in phase space; in informational terminology some information about the same components has been lost or the negentropy has been decreased. In equilibrium the system has for a given energy content maximum randomness. One consequence of this dual aspect was the idea to apply the methods of statistical mechanics to problems of communication and Brillouin showed that Fermi-Dirac statistics or generalized Fermi statistics are applicable for example to a transmission of signals such as telegrams. (shrink)

With reference to two specific modalities of sensation, the taste of saltiness of chloride salts, and the loudness of steady tones, it is shown that the laws of sensation (logarithmic and power laws) are expressions of the entropy per mole of the stimulus. That is, the laws of sensation are linear functions of molar entropy. In partial verification of this hypothesis, we are able to derive an approximate value for the gas constant, a fundamental physical constant, directly from (...) psychophysical measurements. The significance of our observation lies in the linking of the phenomenon of “sensation” directly to a physical measure. It suggests that if the laws of physics are universal, the laws of sensation and perception are similarly universal. It also connects the sensation of a simple, steady physical signal with the molecular structure of the signal: the greater the number of microstates or complexions of the stimulus signal, the greater the magnitude of the sensation (saltiness or loudness). The hypothesis is currently tested on two sensory modalities. (shrink)

For understanding and controlling spreading in complex networks, identifying the most influential nodes, which can be applied to disease control, viral marketing, air traffic control, and many other fields, is of great importance. By taking the effect of the spreading rate on information entropy into account, we proposed an improved information entropy method. Compared to the benchmark methods in the six different empirical networks, the IIE method has been found with a better performance on Kendall’s Tau and imprecision (...) function under the Susceptible Infected Recovered model. Especially in the Facebook network, Kendall’s Tau can grow by 120% as compared with the original IE method. And, there is also an equally good performance in the comparative analysis of imprecise functions. The imprecise functions’ value of the IIE method is smaller than the benchmark methods in six networks. (shrink)

The objective of the consistent-amplitude approach to quantum theory has been to justify the mathematical formalism on the basis of three main assumptions: the first defines the subject matter, the second introduces amplitudes as the tools for quantitative reasoning, and the third is an interpretative rule that provides the link to the prediction of experimental outcomes. In this work we introduce a natural and compelling fourth assumption: if there is no reason to prefer one region of the configuration space over (...) another, then they should be “weighted” equally. This is the last ingredient necessary to introduce a unique inner product in the linear space of wave functions. Thus, a form of the principle of insufficient reason is implicit in the Hilbert inner product. Armed with the inner product we obtain two results. First, we elaborate on an earlier proof of the Born probability rule. The implicit appeal to insufficient reason shows that quantum probabilities are not more objective than classical probabilities. Previously we had argued that the consistent manipulation of amplitudes leads to a linear time evolution; our second result is that time evolution must also be unitary. The argument is straightforward and hinges on the conservation of entropy. The only subtlety consists of defining the correct entropy; it is the array entropy, not von Neumann's. After unitary evolution has been established we proceed to introduce the useful notion of observables and we explore how von Neumann's entropy can be linked to Shannon's information theory. Finally, we discuss how various connections among the postulates of quantum theory are made explicit within this approach. (shrink)

The maximum entropy principle is widely used to determine non-committal probabilities on a finite domain, subject to a set of constraints, but its application to continuous domains is notoriously problematic. This paper concerns an intermediate case, where the domain is a first-order predicate language. Two strategies have been put forward for applying the maximum entropy principle on such a domain: applying it to finite sublanguages and taking the pointwise limit of the resulting probabilities as the size n of (...) the sublanguage increases; selecting a probability function on the language as a whole whose entropy on finite sublanguages of size n is not dominated by that of any other probability function for sufficiently large n. The entropy-limit conjecture says that, where these two approaches yield determinate probabilities, the two methods yield the same probabilities. If this conjecture is found to be true, it would provide a boost to the project of seeking a single canonical inductive logic—a project which faltered when Carnap's attempts in this direction succeeded only in determining a continuum of inductive methods. The truth of the conjecture would also boost the project of providing a canonical characterisation of normal or default models of first-order theories. Hitherto, the entropy-limit conjecture has been verified for languages which contain only unary predicate symbols and also for the case in which the constraints can be captured by a categorical statement of quantifier complexity. This paper shows that the entropy-limit conjecture also holds for categorical statements of complexity, for various non-categorical constraints, and in certain other general situations. (shrink)

The current study examined the effects of cardiorespiratory fitness, identified with a continuous graded cycle ergometry, and aerobic exercise on cognitive functioning and entropy of the electroencephalogram (EEG) in 30 adolescents between the ages of 13 and 14 years. Higher and lower fit participants performed an executive function task after a bout of acute exercise and after rest while watching a film. EEG entropy, using the sample entropy measure, was repeatedly measured during the 1500ms post-stimulus interval (...) to evaluate changes in entropy over time. Analysis of the behavioural data for lower and higher fit groups revealed an interaction between fitness levels and acute physical exercise. Notably, lower fit, but not higher fit, participants had higher error rates for NoGo relative to Go trials in the rest condition, whereas in the acute exercise condition there were no differences in error rates between groups; higher fit participants also had significantly faster reaction times in the exercise condition in comparison with the rest condition. Analysis of EEG data revealed that higher fit participants demonstrated lower entropy post-stimulus than lower fit participants in the left frontal hemisphere, possibly indicating increased efficiency of early stage stimulus processing and more efficient allocation of cognitive resources to the task demands. The results suggest that EEG entropy is sensitive to stimulus processing demands and varies as a function of physical fitness levels, but not acute exercise. Physical fitness, in turn, may enhance cognition in adolescence by facilitating higher functionality of the attentional system in the context of lower levels of frontal EEG entropy. (shrink)

We investigate quantum entanglement of gravitational configurations in 3D AdS gravity using the AdS/cft correspondence. We derive explicit formulas for the holographic entanglement entropy (EE) of the BTZ black hole, conical singularities and regularized AdS3. The leading term in the large temperature expansion of the holographic EE of the BTZ black hole reproduces exactly its Bekenstein-Hawking entropy SBH, whereas the subleading term behaves as ln SBH. We also show that the leading term of the holographic EE for the (...) BTZ black hole can be obtained from the large temperature expansion of the partition function of a broad class of 2D CFTs on the torus. This result indicates that black hole EE is not a fundamental feature of the underlying theory of quantum gravity but emerges when the semiclassical notion of spacetime geometry is used to describe the black hole. (shrink)

The application of the maximum entropy principle to determine probabilities on finite domains is well-understood. Its application to infinite domains still lacks a well-studied comprehensive approach. There are two different strategies for applying the maximum entropy principle on first-order predicate languages: (i) applying it to finite sublanguages and taking a limit; (ii) comparing finite entropies of probability functions defined on the language as a whole. The entropy-limit conjecture roughly says that these two strategies result in the same (...) probabilities. While the conjecture is known to hold for monadic languages as well as for premiss sentences containing only existential or only universal quantifiers, its status for premiss sentences of greater quantifier complexity is, in general, unknown. I here show that the first approach fails to provide a sensible answer for some Σ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _2$$\end{document}-premiss sentences. I discuss implications of this failure for the first strategy and consequences for the entropy-limit conjecture. (shrink)

A graph’s entropy is a functional one, based on both the graph itself and the distribution of probability on its vertex set. In the theory of information, graph entropy has its origins. Hex-derived networks have a variety of important applications in medication store, hardware, and system administration. In this article, we discuss hex-derived network of type 1 and 2, written as HDN 1 n and HDN 2 n, respectively of order n. We also compute some degree-based entropies such (...) as Randić, ABC, and G A entropy of HDN 1 n and HDN 2 n. (shrink)

A remarkable thesis prevails in the physics of information, saying that the logical properties of operations that are carried out by computers determine their physical properties. More specifically, it says that logically irreversible operations are dissipative by klog2 per bit of lost information. (A function is logically irreversible if its input cannot be recovered from its output. An operation is dissipative if it turns useful forms of energy into useless ones, such as heat energy.) This is Landauer's dissipation thesis, (...) hereafter LDT. LDT underlies and motivates numerous researches in physics and computer science. Nevertheless, this paper shows that is it plainly wrong. This conclusion is based on a detailed study of LDT in terms of the various notions of entropy used in main stream statistical mechanics. It is supported by a counter example for LDT. Further support is found in an analysis of the phase space representation on which LDT relies. This analysis emphasises the constraints placed on the choice of probability distribution by the fact that it has to be the basis for calculating phase averages corresponding to thermodynamic properties of individual systems. An alternative representation is offered, in which logical irreversibility has nothing to do with dissipation. The strong connection between logic and physics, that LDT implies, is thereby broken off. (shrink)

The maximum entropy principle is widely used to determine non-committal probabilities on a finite domain, subject to a set of constraints, but its application to continuous domains is notoriously problematic. This paper concerns an intermediate case, where the domain is a first-order predicate language. Two strategies have been put forward for applying the maximum entropy principle on such a domain: (i) applying it to finite sublanguages and taking the pointwise limit of the resulting probabilities as the size n (...) of the sublanguage increases; (ii) selecting a probability function on the language as a whole whose entropy on finite sublanguages of size n is not dominated by that of any other probability function for sufficiently large n. The entropy-limit conjecture says that, where these two approaches yield determinate probabilities, the two methods yield the same probabilities. If this conjecture is found to be true, it would provide a boost to the project of seeking a single canonical inductive logic—a project which faltered when Carnap's attempts in this direction succeeded only in determining a continuum of inductive methods. The truth of the conjecture would also boost the project of providing a canonical characterisation of normal or default models of first-order theories. Hitherto, the entropy-limit conjecture has been verified for languages which contain only unary predicate symbols and also for the case in which the constraints can be captured by a categorical statement of quantifier complexity. This paper shows that the entropy-limit conjecture also holds for categorical statements of complexity, for various non-categorical constraints, and in certain other general situations. (shrink)

This paper offers a critical analysis of contemporary global governance from the perspective of scientific philosophy and cosmology, with particular attention to elite dialogue mechanisms represented by the World Economic Forum in Davos. It argues that the root cause of today’s global crises is not institutional or technological failure, but rather the loss of awareness of the cosmic generative order in governance practice. The Davos model functions as a high-entropy governance system: its dialogues and institutions remain confined to short-term (...) interests, localized negotiation, and technical optimization, while lacking cognition of long-term, low-entropy civilizational pathways. -/- The paper proposes a civilizational transition “from Davos to Dao,” structured in three stages: first, the reconstruction of cosmic order cognition; second, the design of institutions aligned with generative laws; and finally, political action oriented toward low-entropy civilization. By introducing the Lao–Yang Genesis Cosmology, the study articulates the concept of Zhi-Zhi (governance through generative knowledge), which repositions civilization within cosmic evolutionary logic and enables sustainable, low-entropy global governance. The conclusion demonstrates that only by conforming to the cosmic generative order can governance achieve genuine self-correction, sustain long-term civilizational stability, and provide fundamental solutions to global crises. (shrink)

I begin with the definition of power, and find that it is finalistic inasmuch as work directs energy dissipation in the interests of some system. The maximum power principle of Lotka and Odum implies an optimal energy efficiency for any work; optima are also finalities. I advance a statement of the maximum entropy production principle, suggesting that most work of dissipative structures is carried out at rates entailing energy flows faster than those that would associate with maximum power. This (...) is finalistic in the sense that the out-of-equilibrium universe, taken as an isolated system, entrains work in the interest of global thermodynamic equilibration. I posit an evolutionary scenario, with a development on Earth from abiotic times, when promoting convective energy flows could be viewed as the important function of dissipative structures, to biotic times when the preservation of living dissipative structures was added to the teleology. Dissipative structures are required by the equilibrating universe to enhance local energy gradient dissipation. (shrink)

This paper proposes a scientific-philosophical model of “natural-civilization prime ministers” based on historical cases, systems theory, and entropy analysis. By examining the human nature and governance strategies of Sun Shu’ao and Yan Ying during the Spring and Autumn period, the study identifies their low-disturbance, decentralized, and non-performance-driven traits, which maintained civilizational stability. Sun Shu’ao and Yan Ying are not moral exemplars but “low-entropy governance personalities,” functioning as steady-state regulators within political systems. A comparative analysis with the Warring States (...) period demonstrates the structural disappearance of such personalities due to accelerated civilization, militarization, and bureaucratic quantification. The paper further unifies this model with the Lao–Yang Genesis Cosmology: prime ministers as “Earth,” rulers as “Heaven,” and civilization as a natural system, providing a scientifically grounded framework for individual governance, institutional design, and social self-organization. Finally, the study discusses implications for modern governance, emphasizing the scarcity and systemic importance of low-entropy personalities in highly engineered societies. This research offers a cross-temporal perspective connecting historical politics, civilizational evolution, and contemporary governance. (shrink)

The relation between entropy and time is central to debates on thermodynamic irreversibility and the arrow of time. This paper clarifies that relation by distinguishing several roles often associated with entropy in such debates: temporal ordering, temporal orientation, temporal flow and measurement, and thermodynamic asymmetry. The paper does not deny that entropy increase, together with a low-entropy past and suitable coarse-graining, may explain the thermodynamic arrow or help orient an already ordered sequence of states. It also (...) does not deny that thermodynamic or statistical structure may contribute to the selection or measurement of physically meaningful temporal flow in special frameworks. It addresses a narrower question: whether standard entropy notions can themselves supply temporal ordering or serve as general temporal parameters. Using thermodynamic, Boltzmann, Gibbs, and coarse-grained entropy within a minimal dynamical-systems framework, we show that they do not satisfy this role in general. Entropy functionals may be non-injective along trajectories; fine-grained Gibbs entropy is invariant under Hamiltonian flow; coarse-grained entropy depends on descriptive partitions; and entropy monotonicity depends on boundary conditions rather than an intrinsic temporal orientation. An open-system example is included only to illustrate that subsystem entropy may decrease while the dynamical time parameter continues to order the evolution. The novelty is therefore not in the bare claim that entropy and time are non-identical, nor in the attribution of a crude entropy-equals-time thesis to the literature, but in the explicit role-separation argument showing why entropy can characterize asymmetry, help orient an already ordered history, or contribute to temporal-flow selection only after suitable dynamical, statistical, or ordering structure is already given. Entropy remains central to statistical-mechanical accounts of irreversibility, but under standard definitions, it cannot itself supply temporal ordering. (shrink)

We investigate to what extent theorems about quantum mechanical or classical entropy can be generalized to functionals of the type ρ→Tr f(ρ), or ψ→∫f(ψ)dμ, respectively, wheref is an arbitrary concave function.

Entropy-based algorithms have been suggested as robust estimators of electroencephalography (EEG) predictability or regularity. This study aimed to examine possible disturbances in EEG complexity as a means to elucidate the pathophysiological mechanisms in chronic stroke, before and after a brain computer interface (BCI)-motor observation intervention. Eleven chronic stroke subjects and nine unimpaired subjects were recruited to examine the differences in their EEG complexity. The BCI-motor observation intervention was designed to promote functional recovery of the hand in stroke subjects. Fuzzy (...) approximate entropy (fApEn), a novel entropy-based algorithm designed to evaluate complexity in physiological systems, was applied to assess the EEG signals acquired from unimpaired subjects and stroke subjects, both before and after training. The results showed that stroke subjects had significantly lower EEG fApEn than unimpaired subjects (p < 0.05) in the motor cortex area of the brain (C3, C4, FC3, FC4, CP3, CP4) in both hemispheres before training. After training, motor function of the paretic upper limb, assessed by the Fugl-Meyer Assessment-Upper Limb (FMA-UL), Action Research Arm Test (ARAT), and Wolf Motor Function Test (WMFT) improved significantly (p < 0.05). Furthermore, the EEG fApEn in stroke subjects increased considerably in the central area of the contralesional hemisphere after training (p < 0.05). A significant correlation was noted between clinical scales (FMA-UL, ARAT, and WMFT) and EEG fApEn in C3/C4 in the contralesional hemisphere (p < 0.05). This finding suggests that the increase in EEG fApEn could be an estimator of the variance in upper limb motor function improvement. In summary, fApEn can be used to identify abnormal EEG complexity in chronic stroke, when used with BCI-motor observation training. Moreover, these findings based on the fApEn of EEG signals also expand the existing interpretation of training-induced functional improvement in stroke subjects. Entropy-based analysis might serve as a novel approach to understanding the abnormal cortical dynamics of stroke and the neurological changes induced by rehabilitation training. (shrink)

After the birth of thermodynamics’ second principle—outlined in Carnot's Réflexions sur la puissance motrice du feu —several studies provided new arguments in the field. Mainly, they concerned the thermodynamics’ first principle—including energy conceptualisation—, the analytical aspects of the heat propagation, the statistical aspects of the mechanical theory of heat. In other words, the second half of nineteenth century was marked by an intense interdisciplinary research activity between physics and chemistry: new disciplines applied to the heat developed in the form of (...) analytical, mechanical and statistical theories. Inside all these theories, entropy—the brand-new function that Clausius coined in his Mechanical theory of heat—started to play a central epistemic role. In the present paper, we analyse some steps of the historical process of conceptualisation of such function from 1850 to 1902. Particularly, we retrace the historical–foundational path that—starting from Clausius’ Second Law—lead Boltzmann and Gibbs to their distinguished formulations of statistical entropy. As usual, our research has been unrolled through the analyses of primary sources and by leaning on critical readings of the secondary literature. As for the methodological approach, text analysis of historical documents constituted our privileged modus operandi. This paper is the expression of a collaborative historical research program focused on the thermodynamic foundations of physics–chemistry relationship; early results have already been published by the same authors upon the concepts of reversibility––and––thermal equilibrium. (shrink)

Edwin Jaynes’ principle of maximum entropy holds that one should use the probability distribution with maximum entropy, from all those that fit the evidence, to draw inferences, because that is the distribution that is maximally non-committal with respect to propositions that are underdetermined by the evidence. The principle was widely applied in the years following its introduction in 1957, and in 1978 Jaynes took stock, writing the paper ‘Where do we stand on maximum entropy?’ to present his (...) view of the state of the art. Jaynes’ principle needs to be generalised to a principle of maximal entropy if it is to be applied to first-order inductive logic, where there may be no unique maximum entropy function. The development of this objective Bayesian inductive logic has also been very fertile and it is the task of this chapter to take stock. The chapter provides an introduction to the logic and its motivation, explaining how it overcomes some problems with Carnap’s approach to inductive logic and with the subjective Bayesian approach. It also describes a range of recent results that shed light on features of the logic, its robustness and its decidability, as well as methods for performing inference in the logic. (shrink)

Entropy is one of the physical bases for the fractal dimension definition, and the generalized fractal dimension was defined by Renyi entropy. Using the fractal dimension, we can describe urban growth and form and characterize spatial complexity. A number of fractal models and measurements have been proposed for urban studies. However, the precondition for fractal dimension application is to find scaling relations in cities. In the absence of the scaling property, we can make use of the entropy (...) function and measurements. This paper is devoted to researching how to describe urban growth by using spatial entropy. By analogy with fractal dimension growth models of cities, a pair of entropy increase models can be derived, and a set of entropy-based measurements can be constructed to describe urban growing process and patterns. First, logistic function and Boltzmann equation are utilized to model the entropy increase curves of urban growth. Second, a series of indexes based on spatial entropy are used to characterize urban form. Furthermore, multifractal dimension spectra are generalized to spatial entropy spectra. Conclusions are drawn as follows. Entropy and fractal dimension have both intersection and different spheres of application to urban research. Thus, for a given spatial measurement scale, fractal dimension can often be replaced by spatial entropy for simplicity. The models and measurements presented in this work are significant for integrating entropy and fractal dimension into the same framework of urban spatial analysis and understanding spatial complexity of cities. (shrink)

The extension of quantum mechanics to a general functional space (“rigged Hilbert space”), which incorporates time-symmetry breaking, is applied to construct extract dynamical models of entropy production and entropy flow. They are illustrated by using a simple conservative Hamiltonian system for multilevel atoms coupled to a time-dependent external force. The external force destroys the monotonicity of the ℋ-function evolution. This leads to a model of the entropy flow that allows a steady nonequilibrium structure of the emitted (...) field around the unstable particles. (shrink)

In this study, we propose a new approach based on fuzzy TODIM (Portuguese acronym for interactive and multicriteria decision-making) for decision-making problems in uncertain environments. Our method incorporates group utility and individual regret, which are often ignored in traditional multicriteria decision-making (MCDM) methods. To enhance the analysis and application of fuzzy sets in decision-making processes, we introduce novel entropy and distance measures for q-rung picture fuzzy sets. These measures include an entropy measure based on the sine function (...) and a distance measure derived from the Jensen-Shannon divergence. In our methodology, incorporating the sine function into the entropy measure stands out as a distinctive decision, grounded in a profound understanding of the inherent characteristics of fuzzy sets. Utilizing the sine function proves especially advantageous when handling fuzzy sets that exhibit cyclical variations or fluctuations in their membership degrees. We effectively weight the criteria for an improved evaluation by using this new entropy measure. The introduced distance measure finds application in the TODIM approach, allowing the execution of TODIM method steps within a fuzzy environment until the determination of one alternative’s dominance over another—an advancement beyond traditional approaches. We apply our enhanced fuzzy TODIM method to a real-life construction project management problem from the literature and compare the results with those in the literature and obtained from other MCDM methods. Our proposed measures are robust, as demonstrated by the sensitivity analysis that varied the weights of group utility and individual regret, with the results visualized in a 3D sensitivity plot. The findings demonstrate the superiority of our method in providing a more comprehensive evaluation of alternatives, making it a useful tool for decision-makers facing complex and uncertain decision-making problems. (shrink)

The principle of maximum entropy is a method for assigning values to probability distributions on the basis of partial information. In usual formulations of this and related methods of inference one assumes that this partial information takes the form of a constraint on allowed probability distributions. In practical applications, however, the information consists of empirical data. A constraint rule is then employed to construct constraints on probability distributions out of these data. Usually one adopts the rule that equates the (...) expectation values of certain functions with their empirical averages. There are, however, various other ways in which one can construct constraints from empirical data, which makes the maximum entropy principle lead to very different probability assignments. This paper shows that an argument by Jaynes to justify the usual constraint rule is unsatisfactory and investigates several alternative choices. The choice of a constraint rule is also shown to be of crucial importance to the debate on the question whether there is a conflict between the methods of inference based on maximum entropy and Bayesian conditionalization. (shrink)

Objective Bayesian epistemology invokes three norms: the strengths of our beliefs should be probabilities, they should be calibrated to our evidence of physical probabilities, and they should otherwise equivocate sufficiently between the basic propositions that we can express. The three norms are sometimes explicated by appealing to the maximum entropy principle, which says that a belief function should be a probability function, from all those that are calibrated to evidence, that has maximum entropy. However, the three (...) norms of objective Bayesianism are usually justified in different ways. In this paper we show that the three norms can all be subsumed under a single justification in terms of minimising worst-case expected loss. This, in turn, is equivalent to maximising a generalised notion of entropy. We suggest that requiring language invariance, in addition to minimising worst-case expected loss, motivates maximisation of standard entropy as opposed to maximisation of other instances of generalised entropy. Our argument also provides a qualified justification for updating degrees of belief by Bayesian conditionalisation. However, conditional probabilities play a less central part in the objective Bayesian account than they do under the subjective view of Bayesianism, leading to a reduced role for Bayes’ Theorem. (shrink)

We have refined entropy theory to explore the meaning of the increasing sequence data on nucleic acids and proteins more conveniently. The concept of selection constraint was not introduced, only the analyzed sequences themselves were considered. The refined theory serves as a basis for deriving a method to analyze non-coding regions (NCRs) as well as coding regions. Positions with maximal entropy might play the most important role in genome functions as opposed to positions with minimal entropy. This (...) method was tested in the well-characterized coding regions of 12 strains of Classical Swine Fever Virus (CSFV) and non-coding regions of 20 strains of CSFV. It is suitable to analyze nucleic acid sequences of a complete genome and to detect sensitive positions for mutagenesis. As such, the method serves to formulate the basis for elucidating the functional mechanism. (shrink)

The present theory leads to a set of subjective weights such that the utility of an uncertain alternative (gamble) is partitioned into three terms involving those weights—a conventional subjectively weighted utility function over pure consequences, a subjectively weighted value function over events, and a subjectively weighted function of the subjective weights. Under several assumptions, this becomes one of several standard utility representations, plus a weighted value function over events, plus an entropy term of the weights. (...) In the finitely additive case, the latter is the Shannon entropy; in all other cases it is entropy of degree not 1. The primary mathematical tool is the theory of inset entropy. (shrink)

This paper develops an expanded structural-perspectival framework that reconciles the time-symmetry of fundamental physical laws with the asymmetric phenomenology of temporal experience and quantum retrocausality. Building on Penrose’s geometric interpretation of low-entropy boundary conditions and Price’s analysis of time-symmetry in entanglement, the argument situates entropy as a global geometric gradient within a block-universe ontology. The apparent “flow” of time is reinterpreted as a cognitive artifact arising from coarse-grained access to globally constrained structure. The paper introduces entity-agnostic realism, a (...) metaphysical stance in which theoretical entities function as modal-structural roles rather than intrinsic objects, thereby avoiding ontological inflation while preserving realism. Retrocausal correlations are explained through orthogonality relations and global consistency constraints in the space of possible configurations, resolving paradoxes such as bilking without invoking backward-in-time causation. The framework integrates epistemic limitations, biological filtering mechanisms, and modal coherence to show how embedded observers generate unidirectional temporal phenomenology from a static spacetime manifold. The result is a unified metaphysics of entropy and time that harmonizes determinism with retrocausality, structural asymmetry with experiential directionality, and global constraint with local uniqueness. (shrink)

In a previous paper, a statistical method of constructing quantum models of classical properties has been described. The present paper concludes the description by turning to classical mechanics. The quantum states that maximize entropy for given averages and variances of coordinates and momenta are called ME packets. They generalize the Gaussian wave packets. A non-trivial extension of the partition-function method of probability calculus to quantum mechanics is given. Non-commutativity of quantum variables limits its usefulness. Still, the general form (...) of the state operators of ME packets is obtained with its help. The diagonal representation of the operators is found. A general way of calculating averages that can replace the partition function method is described. Classical mechanics is reinterpreted as a statistical theory. Classical trajectories are replaced by classical ME packets. Quantum states approximate classical ones if the product of the coordinate and momentum variances is much larger than Planck constant. Thus, ME packets with large variances follow their classical counterparts better than Gaussian wave packets. (shrink)

Objective Bayesian epistemology invokes three norms: the strengths of our beliefs should be probabilities, they should be calibrated to our evidence of physical probabilities, and they should otherwise equivocate sufficiently between the basic propositions that we can express. The three norms are sometimes explicated by appealing to the maximum entropy principle, which says that a belief function should be a probability function, from all those that are calibrated to evidence, that has maximum entropy. However, the three (...) norms of objective Bayesianism are usually justified in different ways. In this paper we show that the three norms can all be subsumed under a single justification in terms of minimising worst-case expected loss. This, in turn, is equivalent to maximising a generalised notion of entropy. We suggest that requiring language invariance, in addition to minimising worst-case expected loss, motivates maximisation of standard entropy as opposed to maximisation of other instances of generalised entropy. Our argument also provides a qualified justification for updating degrees of belief by Bayesian conditionalisation. However, conditional probabilities play a less central part in the objective Bayesian account than they do under the subjective view of Bayesianism, leading to a reduced role for Bayes’ Theorem. (shrink)

Boltzmann’s H-theoremH-theorem(1872) was explicitlyLille UniversityconceivedIEMN in order to proofProof Clausius’ Second LawLaw (1850s) and for cyclicCyclicprocessesProcess only (1854). Secondary literature used to join the function EE function, in Boltzmann’s 1872–paper, to the development of EntropyEntropy. However, Clausius’ Second Law for cyclical processes did not deal with Entropy; moreover, in a cyclic processProcess, entropy variation, being a state function, is necessarily equal to zero. Later, Clausius introduced the Entropy quantity in order to settle the framework (...) of non-cyclic processes mathematically (1865; 1872). In this paper we will shed light on this issue through the textual analysisAnalysis of Clausius and Boltzmann’s seminal works; a critical discussion of the main historiographyHistoriography of the H-theoremH-theorem and its interplay with EntropyEntropy will be presented as well. (shrink)

This work defines various stakeholder roles to overcome the barriers to implementing Education 4.0, which were recently identified in the domain literature. The stakeholder roles are evaluated against these barriers, and such evaluation is structured as a multicriteria sorting problem. To this end, an integrated entropy-based CRITIC-CODAS-SORT under a Fermatean fuzzy environment addresses epistemic uncertainties inherent in decision-making. The FF CRITIC assigns the priority weights of the barriers, while the FF CODAS-SORT determines the high-priority stakeholder roles. A case of (...) an HEI evaluating 57 possible roles of 5 stakeholders is demonstrated here. Findings suggest the lack of collaboration, apprehensive stakeholders, cybersecurity threats, health issues, and cost as crucial barriers to the HEI. The sorting process yields 13 high-priority roles, encompassing those within the influence of the government, university management, human resource function, and educators. Various managerial insights are offered as inputs to the design of initiatives in EDUC4 implementation. (shrink)

A continuous family of static spherically symmetric solutions of Einstein’s vacuum field equations with a spatial singularity at the origin \ is found. These solutions are parametrized by a real valued parameter \ and such that the radial horizon’s location is displaced continuously towards the singularity ) as \ increases. In the extreme limit \, the location of the singularity and horizon merges leading to a null singularity. In this extreme case, any infalling observer hits the null singularity at the (...) very moment he/she crosses the horizon. This fact may have important consequences for the resolution of the fire wall problem and the complementarity controversy in black holes. An heuristic argument is provided how one might avoid the Hawking particle emission process in this extreme case when the singularity and horizon merges. The field equations due to a delta-function point-mass source at \ are solved and the Euclidean gravitational action corresponding to those solutions is evaluated explicitly. It is found that the Euclidean action is precisely equal to the black hole entropy. This result holds in any dimensions \. (shrink)