On Formalizing Logical Modalities

This paper is in the scope of the philosophy of modal logic; more precisely, it concerns the semantics of modal logic, when the modal elements are interpreted as logical modalities. Most authors have thought that the logic for logical modality—that is, the one to be used to formalize the notion of logical truth (and other related notions)—is to be found among logical systems in which modalities are allowed to be iterated. This has raised the problem of the adequacy, to that formalization purpose, of some modal schemes, such as S4 and S5 . It has been argued that the acceptance of S5 leads to non-normal modal systems, in which the uniform substitution rule fails. The thesis supported in this paper is that such a failure is rather to be attributed to what will be called “Condition of internalization.” If this is correct, there seems to be no normal modal logic system capable of formalizing logical modality, even when S5 is rejected in favor of a weaker system such as S4, as recently proposed by McKeon.

Interpolation rational integral fraction of nth order on a continuum set of nodes

The paper constructs and investigates an integral rational interpolant of the nth order on a continuum set of nodes, which is the ratio of a functional polynomial of the first degree to a functional polynomial of the (n-1)th degree. Subintegral kernels are determined from the corresponding continuum conditions. Additionally, we obtain an integral equation to determine the kernel of the numerator integral. This integral equation, using elementary transformations, is reduced to the standard form of the integral Volterra equation of the second kind. Substituting the obtained solution into expressions for the rest of the kernels, we obtain expressions for all kernels included in the integral rational interpolant. Then, in order for a rational functional of the nth order to be interpolation on continuous nodes, it is sufficient for this functional to satisfy the substitution rule. Note that the resulting interpolant preserves any rational functional of the obtained form.

On Proof Complexities Relations in Some Systems of Propositional Calculus

The number of linear proofs steps for some sets of formulas is compared in the folowing systems of propositional calculus: PK – seguent system with cut rule, PK— - the same system without cut rule, SPK – the same system with substitution rule, QPK – the same system with quantifier rules. The number of steps of tree-like proofs in the same systems for some considered set of formulas is compared from Alessandra Carbone in [1] and some distinctive property of the system QPK is revealed: QPK has an exponential speed-up over the systems SPK and PK, which, in their turn, have an exponential speed-up over the system PK—. This result drew the heavy interest for the study of the system QPK. In this work for linear proofs steps in the same systems the other relations are received: it is showed that the system QPK has no preference over the system SPK, it is showed also that for the considered formula sets the system PK has no preference over the system PK—, which, in its turn, has no preference over the monotone system PMon. It is proved also, that the same results are reliable for some other sets of formulas and for other systems as well.

Change in Soccer Substitutions Rule due to COVID-19: Why only Five Substitutions?

Each sport has its specific rules, which determine what is allowed (or not) impacting directly on the sport demands. Studies involving physiological and time-motion measurements have shown that soccer is a highly demanding sport. The new coronavirus disease 2019 (COVID-19) has been a world health crisis. Soccer seasons were interrupted worldwide to avoid spreading the virus. Leagues resumed the season (no fans at the arenas) after several weeks of interruption, causing overlay of schedule. This overlay (e.g., games every Sunday and Wednesday) will cause accumulated fatigue on players, raising the risk of injuries. Considering this condensed calendar, the Fédération Internationale de Football Association (FIFA) has changed (temporarily) up to five substitutions during elite games (instead of three as the regular rule allows). Considering the already published scientific evidence, clearly, the change in the soccer substitution rule due to COVID-19 is insufficient. Implementing unlimited substitutions may benefit soccer players' health, coaches’ jobs, more entertainment for fans and sponsors (e.g., keeping intensity during all game, including on the second half) and eventually prolonging the useful life of the players. A real game-changer!

SPACE-FILLING CURVES OF SELF-SIMILAR SETS (III): SKELETONS

Skeleton is a new notion designed for constructing space-filling curves of self-similar sets. In a previous paper by Dai and the authors [Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule, Nonlinearity 32(5) (2019) 1772–1809] it was shown that for all the connected self-similar sets with a skeleton satisfying the open set condition, space-filling curves can be constructed. In this paper, we give a criterion of existence of skeletons by using the so-called neighbor graph of a self-similar set. In particular, we show that a connected self-similar set satisfying the finite-type condition always possesses skeletons: an algorithm is obtained here.

New Polyalphabetic Substitution Scheme for Secure Communication

The internet is a very powerful and useful tool for communication, information and connectivity. So it is very important to keep yourself safe and secure online. The best way of secure information is encryption; there are many cryptographic algorithms available for encryption. These cryptographic algorithms are classified according to their encrypting process; as substitution cipher or transposition cipher. In Polyalphabetic ciphers, the substitution rule changes continuously from character to character according to the keyword and plaintext. Vigenere cipher is considered to be the most efficient Polyalphabetic substitution cipher. But it is vulnerable to attacks, due to its repeating nature of the keyword. To overcome this vulnerability, here we are presenting a new Polyalphabetic substitution scheme which uses infinite number of 26 x 26 random tables for encryption. During encryption, whenever the keyword repeats, this proposed Polyalphabetic substitution cipher generates a 26 x 26 alphabetical random table. Instead of using the same Vigenere Table here we are using an infinite number of alphabetical tables depending on the length of the plaintext and keyword. Each random table will be completely independent from the previous table. This will reduces the repeating sequences in the ciphertext. The repeating nature of the keyword does not help the crackers to break this code. So this proposed Polyalphabetic substitution cipher is considered as an unbreakable cryptosystem. The Proposed Polyalphabetic cipher can provide security for many applications such as web transactions, web transactions, personal emails, secret information transmitted between public or private organization, military application etc.

MOVEMENT TRANSFORMATION IN GAYO SYNTAX

The purpose of this study is to present syntactic descriptions of how movement transformational rules apply in Gayo syntax and to examine the status of movement transformational rules in Gayo language (henceforth GL) in the theoretical framework of Transformational Linguistics (TL) proposed by Chomsky (1965, 1981) and Suhadi (2018). In this theory there are three kinds of syntactic rules: Movement Rule, Deletion Rule and Substitution Rule. The discussion focuses on Movement Transformational Rules in GL. Transformation is the inter-related process between the deep structure and the surface structure of a sentence by the application of one or more transformational rules. The method used in this study was descriptive qualitative approach as noted by Martin (2004). Descriptive research is to portray accurately the characteristics of a particular situation or group or individual with or without special initial hypotheses about the nature of these characteristics. Thus, descriptive research design was applied to give a detail description of a certain case accurately. The data were analyzed from two angles: the application and the status of movement rules in GL, which can be compulsory, optional, and restricted. The data of this research derived from some sentences in the folklore story written in GL and the invention of the writer herself as the native speaker of the language. The finding shows that all the twelve kinds of movement transformational rules proposed by Suhadi (2018) are relevant to apply in GL. After the application of movement rules, the main finding is on the status of movement transformational rules in GL in which it is found that four movement rules are compulsory, eight are optional and there is no restricted rule in the language.