scholarly journals What the Tortoise Said to Achilles: Lewis Carroll’s paradox in terms of Hilbert arithmetic

2021 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Hidden Variables ◽  
Formal Structure ◽  
Lewis Carroll ◽  
First Order ◽  
Logical Paradox ◽  
Completeness Of Quantum Mechanics ◽  
The One ◽  
In Virtue Of ◽  
The Liar ◽  
Completeness Theorems

Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll’s paradox) and that of the arrow (for “Achillesand the Turtle”), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure of both, which can be called “ontological”, on which basis “motion” studied by physics and “conclusion” studied by logic can be unified being able to bridge logic and physics philosophically in a Hegelian manner: even more, the bridge can be continued to mathematics in virtue of (2), which forces the equality (for its property of transitivity) of any two quantities to be postponed analogically ad lib and ad infinitum. The paper shows that Hilbert arithmetic underlies naturally Lewis Carroll’s paradox admitting at least three interpretations linked to each other by it: mathematical, physical and logical. Thus, it can be considered as both generalization and solution of his paradox thereforenaturally unifying the completeness of quantum mechanics (i.e. the absence of hidden variables) and eventual completeness of mathematics as the same and isomorphic to the completeness of propositional logic in relation to set theory as a first-order logic (in the sense of Gödel (1930)’s completeness theorems).

scholarly journals What the Tortoise Said to Achilles: Lewis Carroll’s Paradox in Terms of Hilbert Arithmetic

2021 ◽  
Author(s):  
Vasil Penchev
Keyword(s):  
Hidden Variables ◽  
Formal Structure ◽  
Lewis Carroll ◽  
First Order ◽  
Logical Paradox ◽  
Completeness Of Quantum Mechanics ◽  
The One ◽  
In Virtue Of ◽  
The Liar ◽  
Completeness Theorems

Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll’s paradox) and that of the arrow (for “Achilles and the Turtle”), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure of both, which can be called “ontological”, on which basis “motion” studied by physics and “conclusion” studied by logic can be unified being able to bridge logic and physics philosophically in a Hegelian manner: even more, the bridge can be continued to mathematics in virtue of (2), which forces the equality (for its property of transitivity) of any two quantities to be postponed analogically ad lib and ad infinitum. The paper shows that Hilbert arithmetic underlies naturally Lewis Carroll’s paradox admitting at least three interpretations linked to each other by it: mathematical, physical and logical. Thus, it can be considered as both generalization and solution of his paradox therefore naturally unifying the completeness of quantum mechanics (i.e. the absence of hidden variables) and eventual completeness of mathematics as the same and isomorphic to the completeness of propositional logic in relation to set theory as a first-order logic (in the sense of Gödel (1930)’s completeness theorems).

scholarly journals What the Tortoise Said to Achilles: Lewis Carroll’s paradox in terms of Hilbert arithmetic

2021 ◽  
Author(s):  
Vasil Penchev
Keyword(s):  
The Other ◽  
Formal Structure ◽  
Lewis Carroll ◽  
Other Hand ◽  
Logical Paradox ◽  
The One ◽  
The Liar ◽  
As If

Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll’s paradox) and that of the arrow (for “Achilles and the Turtle”), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure

The Green New Deal

2019 ◽  
Author(s):  
Robert C. Hockett
Keyword(s):  
New Deal ◽  
Large Scale ◽  
Public Investment ◽  
Climate Science ◽  
Time Frame ◽  
White Paper ◽  
Good News ◽  
Carbon Neutrality ◽  
The One ◽  
In Virtue Of

This white paper lays out the guiding vision behind the Green New Deal Resolution proposed to the U.S. Congress by Representative Alexandria Ocasio-Cortez and Senator Bill Markey in February of 2019. It explains the senses in which the Green New Deal is 'green' on the one hand, and a new 'New Deal' on the other hand. It also 'makes the case' for a shamelessly ambitious, not a low-ball or slow-walked, Green New Deal agenda. At the core of the paper's argument lies the observation that only a true national mobilization on the scale of those associated with the original New Deal and the Second World War will be up to the task of comprehensively revitalizing the nation's economy, justly growing our middle class, and expeditiously achieving carbon-neutrality within the twelve-year time-frame that climate science tells us we have before reaching an environmental 'tipping point.' But this is actually good news, the paper argues. For, paradoxically, an ambitious Green New Deal also will be the most 'affordable' Green New Deal, in virtue of the enormous productivity, widespread prosperity, and attendant public revenue benefits that large-scale public investment will bring. In effect, the Green New Deal will amount to that very transformative stimulus which the nation has awaited since the crash of 2008 and its debt-deflationary sequel.

La otredad: ¿real o inventada? El caso de las brujas

1999 ◽  
pp. 21-33
Author(s):  
Elia Nathan Bravo
Keyword(s):  
Early Modern ◽  
Social Inequalities ◽  
Strict Sense ◽  
Empirical Content ◽  
Extreme Situation ◽  
Modern Times ◽  
The Witch ◽  
Early Modern Times ◽  
The One ◽  
In Virtue Of

The purpose of this paper is two-fold. On the one hand, it offers a general analysis of stigmas (a person has one when, in virtue of its belonging to a certain group, such as that of women, homosexuals, etc., he or she is subjugated or persecuted). On the other hand, I argue that stigmas are “invented”. More precisely, I claim that they are not descriptive of real inequalities. Rather, they are socially created, or invented in a lax sense, in so far as the real differences to which they refer are socially valued or construed as negative, and used to justify social inequalities (that is, the placing of a person in the lower positions within an economic, cultural, etc., hierarchy), or persecutions. Finally, I argue that in some cases, such as that of the witch persecution of the early modern times, we find the extreme situation in which a stigma was invented in the strict sense of the word, that is, it does not have any empirical content.

Pluralism and the Liar

2017 ◽  
Author(s):  
Cory Wright ◽  
Bradley Armour-Garb
Keyword(s):  
Alternative Treatment ◽  
Liar Paradox ◽  
Recent Attempt ◽  
In Virtue Of ◽  
The Liar ◽  
The Liar Paradox

Pluralists maintain that there is more than one truth property in virtue of which bearers are true. Unfortunately, it is not yet clear how they diagnose the liar paradox or what resources they have available to treat it. This chapter considers one recent attempt by Cotnoir (2013b) to treat the Liar. It argues that pluralists should reject the version of pluralism that Cotnoir assumes, discourse pluralism, in favor of a more naturalized approach to truth predication in real languages, which should be a desideratum on any successful pluralist conception. Appealing to determination pluralism instead, which focuses on truth properties, it then proposes an alternative treatment to the Liar that shows liar sentences to be undecidable.

Gaslighting, First- and Second-Order

Hypatia ◽  
2020 ◽  
pp. 1-21
Author(s):  
Paul-Mikhail Catapang Podosky
Keyword(s):  
Second Order ◽  
First Order ◽  
The World ◽  
In Virtue Of

Abstract In what sense do people doubt their understanding of reality when subject to gaslighting? I suggest that an answer to this question depends on the linguistic order at which a gaslighting exchange takes place. This marks a distinction between first-order and second-order gaslighting. The former occurs when there is disagreement over whether a shared concept applies to some aspect of the world, and where the use of words by a speaker is apt to cause hearers to doubt their interpretive abilities without doubting the accuracy of their concepts. The latter occurs when there is disagreement over which concept should be used in a context, and where the use of words by a speaker is apt to cause hearers to doubt their interpretive abilities in virtue of doubting the accuracy of their concepts. Many cases of second-order gaslighting are unintentional: its occurrence often depends on contingent environmental facts. I end the article by focusing on the distinctive epistemic injustices of second-order gaslighting: (1) metalinguistic deprivation, (2) conceptual obscuration, and (3) perspectival subversion. I show how each reliably has sequelae in terms of psychological and practical control.

scholarly journals Higher Dimensional Holonomy Map for Rules Submanifolds in Graded Manifolds

2020 ◽  
Vol 8 (1) ◽  
pp. 68-91
Author(s):  
Gianmarco Giovannardi
Keyword(s):  
Characteristic Direction ◽  
Fixed Degree ◽  
One Dimensional ◽  
First Order ◽  
Natural Choice ◽  
Higher Dimensional ◽  
Graded Manifold ◽  
The One ◽  
Graded Manifolds ◽  
Holonomy Map

AbstractThe deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.

Relational Chase Procedures Interpreted as Resolution with Paramodulation1

1991 ◽  
Vol 15 (2) ◽  
pp. 123-138
Author(s):  
Joachim Biskup ◽  
Bernhard Convent
Keyword(s):  
Alternative Interpretation ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Problem ◽  
First Order ◽  
Inference Problems ◽  
The One ◽  
The Relationship ◽  
Theoretical Foundations ◽  
Insight Into

In this paper the relationship between dependency theory and first-order logic is explored in order to show how relational chase procedures (i.e., algorithms to decide inference problems for dependencies) can be interpreted as clever implementations of well known refutation procedures of first-order logic with resolution and paramodulation. On the one hand this alternative interpretation provides a deeper insight into the theoretical foundations of chase procedures, whereas on the other hand it makes available an already well established theory with a great amount of known results and techniques to be used for further investigations of the inference problem for dependencies. Our presentation is a detailed and careful elaboration of an idea formerly outlined by Grant and Jacobs which up to now seems to be disregarded by the database community although it definitely deserves more attention.

Ring-diagram summations in the finite-temperature effective potential

1993 ◽  
Vol 71 (5-6) ◽  
pp. 227-236 ◽  
Author(s):  
M. E. Carrington
Keyword(s):  
Phase Transition ◽  
Standard Model ◽  
Effective Potential ◽  
Finite Temperature ◽  
Electroweak Phase Transition ◽  
First Order ◽  
Ring Diagram ◽  
First Order Phase Transition ◽  
The Standard Model ◽  
The One

There has been much recent interest in the finite-temperature effective potential of the standard model in the context of the electroweak phase transition. We review the calculation of the effective potential with particular emphasis on the validity of the expansions that are used. The presence of a term that is cubic in the Higgs condensate in the one-loop effective potential appears to indicate a first-order electroweak phase transition. However, in the high-temperature regime, the infrared singularities inherent in massless models produce cubic terms that are of the same order in the coupling. In this paper, we discuss the inclusion of an infinite set of these terms via the ring-diagram summation, and show that the standard model has a first-order phase transition in the weak coupling expansion.

A stochastic process whose successive intervals between events form a first order Markov chain — I

1968 ◽  
Vol 5 (03) ◽  
pp. 648-668
Author(s):  
D. G. Lampard
Keyword(s):  
Markov Chain ◽  
Point Processes ◽  
Electronic Device ◽  
Statistical Properties ◽  
Time Intervals ◽  
First Order ◽  
Stochastic Point Process ◽  
Order Markov Chain ◽  
The One ◽  
Stochastic Point Processes

In this paper we discuss a counter system whose output is a stochastic point process such that the time intervals between pairs of successive events form a first order Markov chain. Such processes may be regarded as next, in order of complexity, in a hierarchy of stochastic point processes, to “renewal” processes, which latter have been studied extensively. The main virtue of the particular system which is studied here is that virtually all its important statistical properties can be obtained in closed form and that it is physically realizable as an electronic device. As such it forms the basis for a laboratory generator whose output may be used for experimental work involving processes of this kind. Such statistical properties as the one and two-dimensional probability densities for the time intervals are considered in both the stationary and nonstationary state and also discussed are corresponding properties of the successive numbers arising in the stores of the counter system. In particular it is shown that the degree of coupling between successive time intervals may be adjusted in practice without altering the one dimensional probability density for the interval lengths. It is pointed out that operation of the counter system may also be regarded as a problem in queueing theory involving one server alternately serving two queues. A generalization of the counter system, whose inputs are normally a pair of statistically independent Poisson processes, to the case where one of the inputs is a renewal process is considered and leads to some interesting functional equations.

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