Edward Titchener, one of the great champions of introspectionist psychology, declared that 'the term Introspection, as we find it used today, is highly equivocal, and that the procedure which it connotes may be scientifically illegitimate, or even wholly imaginary' . He made the point because he wanted to insulate his preferred method of doing psychological research from criticisms that were directed against forms of introspection that he conceded to be unreliable. The point, however, is not just that we can introspect (...) more or less carefully. It is that there are 'gross differences in the meaning of the word' . 'Introspection' is Janus-faced. It splinters into several different species, involving different underlying mechanisms. (shrink)

Although many researchers theorize that primitive numerosity processing abilities may lay the foundation for whole number concepts, other classes of numbers, like fractions, are sometimes assumed to be inaccessible to primitive architectures. This research presents evidence that the automatic processing of nonsymbolic magnitudes affects processing of symbolic fractions. Participants completed modified Stroop tasks in which they selected the larger of two symbolic fractions while the ratios of the fonts in which the fractions were printed and the (...) overall sizes of the compared fractions were manipulated as irrelevant dimensions. Participants were slower and less accurate when nonsymbolic dimensions of printed fractions were incongruent with the symbolic comparison decision. Results indicated a robust basic sensitivity to nonsymbolic ratios that exceeds prior conceptions about the accessibility of fraction values. Results also indicated a congruity effect for overall fraction size, contrary to findings of prior research. These findings have implications for extending theory about the nature of human number sense and mathematical cognition more generally. (shrink)

This paper is dedicated to extending and adapting to modal logic the approach of fractional semantics to classical logic. This is a multi-valued semantics governed by pure proof-theoretic considerations, whose truth-values are the rational numbers in the closed interval $[0,1]$. Focusing on the modal logic K, the proposed methodology relies on three key components: bilateral sequent calculus, invertibility of the logical rules, and stability (proof-invariance). We show that our semantic analysis of K affords an informational refinement with respect to the (...) standard Kripkean semantics (a new proof of Dugundji’s theorem is a case in point) and it raises the prospect of a proof-theoretic semantics for modal logic. (shrink)

In the paper we construct a new kind of fractional dynamical model, i.e. the fractional relativistic Yamaleev oscillator model, and explore its dynamical behaviors. We will find that the fractional relativistic Yamaleev oscillator model possesses Lie algebraic structure and satisfies generalized Poisson conservation law. We will also give the Poisson conserved quantities of the model. Further, the relation between conserved quantities and integral invariants of the model is studied and it is proved that, by using the Poisson conserved quantities, we (...) can construct integral invariants of the model. Finally, the stability of the manifold of equilibrium states of the fractional relativistic Yamaleev oscillator model is studied. The paper provides a general method, i.e. fractional generalized Hamiltonian method, for constructing a family of fractional dynamical models of an actual dynamical system. (shrink)

Consider syllogisms in which fraction (percentage) quantifiers are permitted in addition to universal and particular quantificrs, and then include further quantifiers which are modifications of such fractions (such as "almost ½ the S are P" and "Much more than ½ the S are P"). Could a syllogistic system containing such additional categorical forms be coherent? Thompson's attempt (1986) to give rules for determining validity of such syllogisms has failed; cf. Carnes & Peterson (forthcoming) for proofs of the unsoundness and (...) incompleteness of Thompson's rules. Building on Peterson (1985), the coherence of such a syllogistic can, however, be demonstrated with an algebra which provides its semantics; e.g., "almost ¾ the S arc P'" is represented as " -(3(SP) » SP)". (shrink)

This paper presents a novel fractional-order PID controller tuning strategy based on Bode’s optimal loop shaping which is commonly used for LTI feedback systems. Firstly, the controller parameters are achieved based on flat phase property and Bode’s optimal reference model, so that the controlled system is robust to gain variations and can achieve desirable transient performance according to various control requirements. Then, robustness analysis of the controlled system is carried out to support the results. Furthermore, the parameter setting is analyzed (...) to demonstrate the superiority of the proposed controller. At last, some simulation examples are shown to verify the accuracy and usefulness of the proposed control strategy. The proposed fractional-order PID controller does not have any restriction on the controlled plant, so it can be widely applied on both integer-order and fractional-order systems. (shrink)

This article presents a new semantics for classical propositional logic. We begin by maximally extending the space of sequent proofs so as to admit proofs for any logical formula; then, we extract the new semantics by focusing on the axiomatic structure of proofs. In particular, the interpretation of a formula is given by the ratio between the number of identity axioms out of the total number of axioms occurring in any of its proofs. The outcome is an informational refinement of (...) traditional Boolean semantics, obtained by breaking the symmetry between tautologies and contradictions. (shrink)

This paper introduces a possible alternative model of gravity based on the theory of fractional-dimension spaces and its applications to Newtonian gravity. In particular, Gauss’s law for gravity as well as other fundamental classical laws are extended to a D-dimensional metric space, where D can be a non-integer dimension. We show a possible connection between this Newtonian Fractional-Dimension Gravity (NFDG) and Modified Newtonian Dynamics (MOND), a leading alternative gravity model which accounts for the observed properties of galaxies and other astrophysical (...) structures without requiring the dark matter hypothesis. The MOND acceleration constant $${a_{0}} \simeq 1.2 \times 10^{-10} \text {m}\, \text{s}^{-2}$$ can be related to a natural scale length $$l_{0}$$ in NFDG, i.e., $$a_{0} \approx GM/l_{0}^{2}$$, for astrophysical structures of mass M, and the deep-MOND regime is present in regions of space where the dimension is reduced to $$D \approx 2$$. For several fundamental spherically-symmetric structures, we compare MOND results, such as the empirical Radial Acceleration Relation (RAR), circular speed plots, and logarithmic plots of the observed radial acceleration $$g_{obs}$$ vs. the baryonic radial acceleration $$g_{bar}$$, with NFDG results. We show that our model is capable of reproducing these results using a variable local dimension $$D\left( w\right)$$, where $$w =r/l_{0}$$ is a dimensionless radial coordinate. At the moment, we are unable to derive explicitly this dimension function $$D\left( w\right)$$ from first principles, but it can be obtained empirically in each case from the general RAR. Additional work on the subject, including studies of axially-symmetric structures, detailed galactic rotation curves fitting, and a possible relativistic extension, will be needed to establish NFDG as a viable alternative model of gravity. (shrink)

The theory of nonnegative matrices is an example of a theory motivated in its origins and development by purely mathematical concerns that later proved to have a remarkably broad spectrum of applications to such diverse fields as probability theory, numerical analysis, economics, dynamical programming, and demography. At the heart of the theory is what is usually known as the Perron–Frobenius Theorem. It was inspired by a theorem of Oskar Perron on positive matrices, usually called Perron’s Theorem. This paper is primarily (...) concerned with the origins of Perron’s Theorem in his masterful work on ordinary and generalized continued fractions (1907) and its role in inspiring the remarkable work of Frobenius on nonnegative matrices (1912) that produced, inter alia, the Perron–Frobenius Theorem. The paper is not at all intended exclusively for readers with expertise in the theory of nonnegative matrices. Anyone with a basic grounding in linear algebra should be able to read this article and come away with a good understanding of the Perron–Frobenius Theorem as well as its historical origins. The final section of the paper considers the first major application of the Perron–Frobenius Theorem, namely, to the theory of Markov chains. When he introduced the eponymous chains in 1908, Markov adumbrated several key notions and results of the Perron–Frobenius theory albeit within the much simpler context of stochastic matrices; but it was by means of Frobenius’ 1912 paper that the linear algebraic foundations of Markov’s theory for nonpositive stochastic matrices were first established by R. Von Mises and V.I. Romanovsky. (shrink)

Drinking kills a significant proportion of individuals every year, particularly in low-income communities. An impulsive differential equation system is used to explore the effectiveness of activated charcoal in detoxifying the body after methanol poisoning. Our impression of activated charcoal is shaped by the fractional dynamics of the problem, which leads to speedy and low-cost first aid. The adsorption capacity of activated charcoal is investigated using impulsive differential equations. The ABC fractional operator’s findings paint a more realistic image of first aid (...) in public and primary health centers, which can assist to reduce methanol poisoning deaths. Numerical simulations are provided using generalized Adams–Bashforth–Moulton method. (shrink)

This paper traces the recent debate over the legitimacy of maturity mismatching and fractional reserve banking. It shows that there is common ground between Bagus and Howden :399–406, 2009, 106:295–300, 2012) on the one hand and Evans on the other regarding contractual arrangements that lead to fractional reserve banking, while both agree that fractional reserve banking that arises out of a bailment or storage contract constitutes fraud. Block and Barnett :711–716, 2009, 100:229–238, 2011) stress the illegitimacy of fractional reserve banking (...) for creating more money substitutes than there is actual money. While it is true that fractional reserve banks are capable of creating money, this activity still cannot be regarded as fraudulent using a common law definition of fraud for it can only take place with requisite client collaboration that makes it impossible to identify a victim. (shrink)

In a recent paper, under the auspices of an unorthodox variety of bilateralism, we introduced a new kind of proof-theoretic semantics for the base modal logic \(\mathbf{K}\), whose values lie in the closed interval \([0,1]\) of rational numbers [14]. In this paper, after clarifying our conception of bilateralism – dubbed “soft bilateralism” – we generalize the fractional method to encompass extensions and weakenings of \(\mathbf{K}\). Specifically, we introduce well-behaved hypersequent calculi for the deontic logic \(\mathbf{D}\) and the non-normal modal logics (...) \(\mathbf{E}\) and \(\mathbf{M}\) and thoroughly investigate their structural properties. (shrink)

Interest has been recently devoted to historical and philosophical aspects about fractional calculus and its adoption as additional mathematical tool in different physics areas as well as in other scientific applications. However, potential application of FC towards relativity is still lacking. In relativity theory, Lorentz transformation of time and position coordinates plays a major role while the corresponding outcomes still defy our ‘common sense’. As some problems in physics can be solved by following different mathematical routes leading to the same (...) result, the present paper suggests an alternative method to relativity. Instead of converting time and position data from different inertial reference frames through Lorentz transformation, it is here envisaged to numerically adjust relativistic results with the help of FC. Bearing in mind well established postulates from relativity, we intend to fill up this gap by addressing and discussing the main arguments in favour of the prospective use of FC as emergent and alternative tool applied to relativity. (shrink)

This paper looks at the nature of idealizations and representational structures appealed to in the context of the fractional quantum Hall effect, specifically, with respect to the emergence of anyons and fractional statistics. Drawing on an analogy with the Aharonov–Bohm effect, it is suggested that the standard approach to the effects— the topological approach to fractional statistics—relies essentially on problematic idealizations that need to be revised in order for the theory to be explanatory. An alternative geometric approach is outlined and (...) endorsed. Roles for idealizations in science, as well as consequences for the debate revolving around so-called essential idealizations, are discussed. (shrink)

Hybrid energy storage system has been widely studied as an important technology for electric vehicles. Since the hybrid energy storage system is a nonlinear and complex system, the modeling of the system and the high-precision nonlinear control strategy are technical difficulties for research. The establishment of a high-precision mathematical model of the hybrid energy storage system is the basis for the study of high-quality nonlinear control algorithms. Fortunately, the theory of fractional calculus can help build accurate mathematical models of hybrid (...) energy storage systems. In order to obtain the high-quality nonlinear control strategy of this complex system, this paper, respectively, carried out fractional-order modeling and analysis on the three basic equivalent working states of the hybrid energy storage system of electric vehicles. Among them, the fractional-order average state space model is carried out for the equivalent Buck and Boost mode. Also, the steady-state analysis of the equivalent Dual-Boost mode is carried out by combining the fractional-order calculus theory with the equivalent small parameter variable method. Finally, the effectiveness and precision of the fractional-order model are proved by simulation and experiment. (shrink)

We agree with Caplan & Waters that there are problems with the single-resource theory of sentence comprehension. However, we challenge their dual-resource alternative on theoretical and empirical grounds and point to a more coherent solution that abandons the notion of working memory resources.

PurposeTo investigate spontaneous brain activity in patients with dry eye and healthy control using the fractional amplitude of low frequency fluctuation technique with the aim of elucidating the relationship between the clinical symptoms of DE and changes in brain function.Material and MethodsA total of 28 patients with DE and 28 matched healthy volunteers were enrolled. Resting-state functional magnetic resonance imaging scans were performed in both groups. Then all subjects were required to complete a comprehensive Hospital Anxiety and Depression Scale. Receiver (...) operating characteristic curve analysis was used to evaluate the differences in fALFF values between the two groups and their diagnostic value. Linear correlations between HADS and fALFF values in different brain regions of DE patients were analyzed using the Pearson correlation coefficient.ResultsPatients with DE had significantly higher fALFF values in the left calcarine sulcus than the HC group, while fALFF values in the bilateral middle frontal gyrus and right MFG/right inferior frontal gyrus were significantly lower in DE patients than in HC group. fALFF values had a high diagnostic value for differentiating patients with DE from the HC group. Right MFG and right MFG/IFG were significantly correlated with HADS values.ConclusionOur study found that DE mainly involved functional disorders in the brain areas of the left CS, bilateral MFG and right MFG/right IFG, which helped us to find possible clinical features of DE disease and reflected the potential pathological mechanism of DE. (shrink)

The attributable fraction (population) has attracted much attention from a theoretical perspective and has been used extensively to assess the impact of potential health interventions. However, despite its extensive use, there is much confusion about its concept and calculation methods. In this article, we discuss the concepts of and calculation methods for the attributable fraction and related measures in the counterfactual framework, both with and without stratification by covariates. Generally, the attributable fraction is useful when the exposure of interest has (...) a causal effect on the outcome. However, it is important to understand that this statement applies to the exposed group. Although the target population of the attributable fraction (population) is the total population, the causal effect should be present not in the total population but in the exposed group. As related measures, we discuss the preventable fraction and prevented fraction, which are generally useful when the exposure of interest has a preventive effect on the outcome, and we further propose a new measure called the attributed fraction. We also discuss the causal and preventive excess fractions, and provide notes on vaccine efficacy. Finally, we discuss the relations between the aforementioned six measures and six possible patterns using a conceptual schema. (shrink)

To construct fractional rogue waves, this paper first introduces a conformable fractional partial derivative. Based on the conformable fractional partial derivative and its properties, a fractional Schrödinger equation with Lax integrability is then derived and first- and second-order fractional rogue wave solutions of which are finally obtained. The obtained fractional rogue wave solutions possess translational coordination, providing, to some extent, the degree of freedom to adjust the position of the rogue waves on the coordinate plane. It is shown that the (...) obtained first- and second-order fractional rogue wave solutions are steeper than those of the corresponding NLS equation with integer-order derivatives. Besides, the time the second-order fractional rogue wave solution undergoes from the beginning to the end is also short. As for asymmetric fractional rogue waves with different backgrounds and amplitudes, this paper puts forward a way to construct them by modifying the obtained first- and second-order fractional rogue wave solutions. (shrink)

Although the working memory capacity involved in syntactic processing may be separate from the capacity involved in word list recall, other aspects of initial sentence interpretation appear to depend on some of the same capacities tapped by span tasks. Specifically, there appears to a capacity for lexical–semantic retention involved in both sentence comprehension and span measures.

According to what I call the Traditional View, there is a fundamental semantic distinction between counting and measuring, which is reflected in two fundamentally different sorts of scales: discrete cardinality scales and dense measurement scales. Opposed to the Traditional View is a thesis known as the Universal Density of Measurement: there is no fundamental semantic distinction between counting and measuring, and all natural language scales are dense. This paper considers a new argument for the latter, based on a puzzle I (...) call the Fractional Cardinalities Puzzle: if answers to ‘how many’-questions always designate cardinalities, and if cardinalities are necessarily discrete, then how can e.g. ‘2.38’ be a correct answer to the question ‘How many ounces of water are in the beaker?’? If cardinality scales are dense, then the answer is obvious: ‘2.38’ designates a fractional cardinality, contra the Traditional View. However, I provide novel evidence showing that ‘many’ is not uniformly associated with the dimension of cardinality across contexts, and so ‘how many’-questions can ask about other kinds of measures, including e.g. volume. By combining independently motivated analyses of cardinal adjectives, measure phrases, complex fractions, and degrees, I develop a semantics intended to defend the Traditional View against purported counterexamples like this and others which have received a fair amount of recent philosophical attention. (shrink)