Propositional Functions Research Articles

Utilization of Symmetric Switching Functions in the Symbolic Reliability Analysis of Multi-State k-out-of-n Systems

Symmetric switching functions (SSFs) play a prominent role in the reliability analysis of a binary k-out-of-n: G system, which is a dichotomous system that is successful if and only if at least k out of its n components are successful. The aim of this paper is to extend the utility of SSFs to the reliability analysis of a multi-state k-out-of-n: G system, which is a multi-state system whose multi-valued success is greater than or equal to a certain value j (lying between 1 (the lowest output level) and M (the highest output level)) whenever at least km components are in state m or above for all m such that 1 ≤ m ≤ j. This paper is devoted to the analysis of non-repairable multi-state k-out-of-n: G systems with independent non-identical components. The paper utilizes algebraic techniques of multiple-valued logic (together with known properties of SSFs) to evaluate each of the multiple levels of the system output as an individual binary or propositional function of the system multi-valued inputs. The formula of each of these levels is then written as a probability–ready expression, thereby allowing its immediate conversion, on a one-to-one basis, into a probability or expected value. The symbolic reliability analysis of a commodity-supply system (which serves as a standard gold example of a multi-state k-out-of-n: G system) is completed successfully herein, yielding results that have been checked symbolically, and also were shown to agree numerically with those obtained earlier.

An Optimized Static Propositional Function Model to Detect Software Vulnerability

Due to the lack of appropriate theory to accurately characterize vulnerabilities, the current static detection technologies have two key challenges, i.e., limited applicability, and the problem of state space explosion. In this paper, we put forward a static detection model based on the proposition function. Furthermore, a new program intermediate representation called Vulnerability Executable Path Set (VEPS) is proposed to optimize our model which compresses the program state space distinctly. In addition, in order to confirm the reliability of the static detection model, we conduct three terms of contrast experiments to estimate the results with the vulnerability disclosed by NIST. The results obtained from extensive experiments show that the proposed model effectively detects more Wireshark bugs than NIST, and reveals a higher detection efficiency than FindBugs.

Використання поняття «пропозиціональна функція» в сучасних лінгвістичних дослідженнях

Використання поняття «пропозиціональна функція» в сучасних лінгвістичних дослідженнях

The grammar of syntactic coreference

Since the notions of semantic and syntactic coreference, in the conceptual, terminological and theoretical sense, have long been known to the linguistic public, we consider it appropriate and worthwhile to point out the ways in which this type of connection between the grammatical units in the sentence is materialized. We especially wanted to draw attention to the inherent mechanisms of language which directly signal that the two forms, not necessarily different, are connected with the same meaning, and related to the same semantic role, for which it is directly responsible - the same propositional function they have. Although it may seem that the difference between semantic and syntactic coreference is not so significant, it has been revealed that in the latter case, the syntactic relations are the main language means of expressing propositional functions and corresponding semantic roles. Therefore, the purpose of this study is to highlight the syntagmatic mechanisms of language, with the help of which the structural, informative and informative hierarchy of the sentence members is carried out. The purpose of such hierarchization is not so much to streamline or rethink the structure of the sentence, but rather to ensure the integration of complex content on the one hand, and settle the situational significance of individual participants on the other hand.

Conceptualizing Some Verbs of Propositional Attitudes in English

In this article, we discuss the semantics and syntax of Propositional Attitude Verbs. Our goal is to clarify and illustrate how Propositional Attitudes function and behave and put forth the conceptualization/ formalization of sentences containing this type of Verbs. We will see how intentions and beliefs are useful to study Propositional Attitudes. Also, we will argue that the semantic approach is more adequate than the syntactic one in terms of classifying Propositional Attitude Verbs. This, of course, is carried out in the framework of Conceptual Semantics that was mainly introduced by Jackendoff.

The Singular Relational plus Relativistic Content View

My aim is to defend a peculiar epistemic version of the particularity thesis, which results from a sui generis combination of what I call the ‘singular relational view’ and what I call the ‘relativistic content view.’ Particulars are not represented as part of putative singular content. Instead, we are perceptually acquainted with them in the relevant sense that experience puts us in direct perceptual contact with them. And the content of experience is best modelled as a propositional function, that is, the content of a complex predicate that is true or false only relative to some circumstances of evaluation.

Probability: From classical to fuzzy

The fuzzification of the classical probability has been initiated by L.A. Zadeh: the classical crisp random events are extended to measurable fuzzy random events and the probability measures are extended to probability integrals. The corresponding fuzzy probability (and fuzzy statistics) can be viewed as an integral part of soft computing. We follow a different approach to fuzzification developed primarily by S. Gudder and S. Bugajski: the classical notion of random variable is modified so that new theory models both fuzzy and quantum phenomena. Our goal is to survey the second approach, GB-approach, to the fuzzification of classical probability using elementary category theory. The motivation comes primarily from quantum physics: to some crisp outcome in the sample space there corresponds a whole spectrum (a genuine probability measure) in the state space; the same situation appears in other fields of science (expert systems, fuzzy logic), too. The corresponding generalization of a random variable constitutes a stochastic channel connecting two objects. Due to the duality, the channel can be equivalently described via generalized random variables and observables. In the fuzzified probability theory, an observable can map a crisp random event to a genuine fuzzy random event. First, in a bottom-up style, we recall some constructions and point out some properties of basic notions in GB-approach. Second, in a top-down style, we outline a simple model of fuzzified probability and show that the properties suffice to build the model so that it can be seen as a minimal extension of the classical probability theory. Fuzzy random events become Łukasiewicz logic propositional functions, probability measures and other relevant maps become morphisms, and basic constructions can be described via commutative diagrams.

Uncertain random logic and uncertain random entailment

Probabilistic logic and uncertain logic, as two branches of the multi-valued logic, have been proposed for dealing with knowledge with random factors and with human uncertainty, respectively. As a generalization, this paper proposes an uncertain random logic to deal with complex knowledge containing random factors and human uncertainty simultaneously, and derives a formula to calculate the truth value of an uncertain random proposition. As an inverse problem of the uncertain random logic, this paper also proposes an uncertain random entailment model which calculates the truth value of a function of some uncertain random propositions based on the truth values of some other functions of these uncertain random propositions. As a byproduct, a probabilistic entailment model is also discussed.

Analysis, Mathematics, and Logic in Russell’s Early Philosophy

russell: the Journal of Bertrand Russell Studies n.s. 36 (winter 2016–17): 163–90 The Bertrand Russell Research Centre, McMaster U. issn 0036–01631; online 1913–8032 c:\users\arlene\documents\rj\type3602\red\rj 3602 134 red.docx 2017-01-09 4:03 PM oeviews ANALYSIS, MATHEMATICS, AND LOGIC IN RUSSELL’S EARLY PHILOSOPHY James Levine Philosophy / Trinity College Dublin 2, Ireland jlevine@tcd.ie Jolen Galaugher. Foreword by Michael Beaney. Russell’s Philosophy of Logical Analysis: 1897–1905. (History of Analytic Philosophy series.) Basingstoke and NewYork: Palgrave Macmillan, 2013. Pp. xii, 218. isbn 13: 978-1-137-302069 . £60; us$110 (hb). n this book Jolen Galaugher discusses a number of key turning-points in Russell’s early philosophy: his break from Idealism in 1898–99; his acceptance of logicism in the period following the International Congress of Philosophy in Paris in August 1900 at which he was impressed by Peano and his students; and his arrival at his theory of descriptions in 1905 in the context of his attempts to grapple with various versions of what Russell called “the Contradiction”. Her overall purpose is to examine the interplay between the view of the analysis of propositions that Russell accepts following his break with Idealism and his developing views within the philosophy of mathematics. The main merits of the book are the way in which it highlights unresolved issues in our understanding of Russell’s early development and brings to bear relevant evidence in attempting to address those issues. However, as Galaugher acknowledges, the book’s “arguments are complicated in places” (p. 2), and it is not an easy read. Like some other books in this series, it appears to be based on a recently completed phd dissertation, and, like some dissertations, it combines a serious engagement with recent scholarship with a manner of presentation that presumes a shared understanding with the reader of a number of central issues and positions, and so does not clearly articulate and defend her understanding of those issues and positions to the uninitiated or sceptical reader. Hence, while the book will be useful to scholars already familiar with the sorts of issues it addresses, it would be hard to recommend it to someone coming to this material for the first time. The book consists of five chapters. In the first, Galaugher outlines some f= 164 Reviews c:\users\arlene\documents\rj\type3602\red\rj 3602 134 red.docx 2017-01-09 4:03 PM central elements of Russell’s position during his Idealist period and some of the steps involved in his dismantling that position. In the second, she argues more specifically, that in rejecting Idealism, Russell was not, as it is sometimes presented, simply following G. E. Moore—that his engagement with Leibniz’s views led Russell to go beyond Moore in certain ways and to adopt his characteristic view that relations are “external”. In the third, she discusses aspects of Russell’s logicism and its development in the period following the Paris Congress. In the fourth, she focuses on Russell’s logicist definition of the cardinal number and the notion of “class” that underpins it, arguing that in the face of the Contradiction, and his changing view of class and propositional function, it becomes questionable whether Russell and Frege should be said to endorse the same definition of cardinal number. In the fifth, she examines aspects of Russell’s coming to reject the theory of denoting concepts in favour of the theory he presents in “On Denoting”, arguing that by doing so Russell reaffirms his commitment to his post-Idealist view of a “decompositional” view of analysis as opposed to Frege’s function/argument style of analysis. Each of the aspects of Russell’s views that Galaugher examines concerns issues on which Russell often changed his mind, sometimes within a quite short period of time. While much of the material relevant to understanding these changes of mind has now been published in the Collected Papers of Bertrand Russell, Russell scholars have not fully assimilated it. Galaugher not only calls attention to a number of features of Russell’s early development that are not yet fully understood...

Lexical flexibility in Oceanic languages

AbstractWhile word classes are language-specific categories, lexical flexibility remains under discussion. This article investigates this phenomenon in a balanced sample of 36 Oceanic languages, a genetic group that has figured prominently in this debate. Based on a systematic survey of the morphosyntactic behavior of a range of semantic word types in three propositional functions, it is shown how lexical flexibility can be measured and compared across languages and constructions. While Oceanic flexibility is pervasive in some respects, lexical categorization in these languages does not qualitatively deviate from relevant typological tendencies.

Logic is a Moral Science

AbstractBecause of the communicative function of propositions, Logic is a moral science about what ought to be said. The association with morality derives from the connection between Logic and Truth, and the social value of speaking the truth. Propositions are used rather than mentioned sentences, and so are not the platonically abstract objects they have often been taken to be. Instead they are social objects formed by a community's employment of language. So it is the public use of sentences that settles what they mean, and thereby their logic. The matter has a particular relevance to Graham Priest's non-classical views.

Can Many-Valued Logic Help to Comprehend Quantum Phenomena?

Following {\L}ukasiewicz, we argue that future non-certain events should be described with the use of many-valued, not 2-valued logic. The Greenberger-Horne-Zeilinger `paradox' is shown to be an artifact caused by unjustified use of 2-valued logic while considering results of future non-certain events. Description of properties of quantum objects before they are measured should be performed with the use of propositional functions that form a particular model of infinitely-valued {\L}ukasiewicz logic. This model is distinguished by specific operations of negation, conjunction, and disjunction that are used in it.

¿Dios es omnivolente?

Como introducción al artículo de Paul Weingarter “Religiöser Fatalismus und das Problem des Übels”, se hacen algunas aclaraciones sobre las funciones proposicionales, especilmente sobre las funciones concernientes a la voluntad. Se ofrecen otras consideraciones culturales sobre estar dispuesto a y permitir.

The paradoxes and Russell’s theory of incomplete symbols

Russell claims in his autobiography and elsewhere that he discovered his 1905 theory of descriptions while attempting to solve the logical and semantic paradoxes plaguing his work on the foundations of mathematics. In this paper, I hope to make the connection between his work on the paradoxes and the theory of descriptions and his theory of incomplete symbols generally clearer. In particular, I argue that the theory of descriptions arose from the realization that not only can a class not be thought of as a single thing, neither can the meaning/intension of any expression capable of singling out one collection (class) of things as opposed to another. If this is right, it shows that Russell’s method of solving the logical paradoxes is wholly incompatible with anything like a Fregean dualism between sense and reference or meaning and denotation. I also discuss how this realization lead to modifications in his understanding of propositions and propositional functions, and suggest that Russell’s confrontation with these issues may be instructive for ongoing research.

Normatividade e Determinação nas Investigações Filosóficas

Boa parte dos esforços empreendidos por Wittgenstein em suas Investigações Filosóficas se orientam no sentido contrário à ideia de que a normatividade da lógica atua sobre os falantes de uma língua como uma força compulsora. A lógica e suas regras seriam, segundo essa concepção, a maquinaria subjacente à nossa linguagem: sua condição de possibilidade e, ao mesmo tempo, nosso instrumento de controle sobre os usos linguísticos. Retomamos o método wittgensteiniano para avaliar a função normativa de proposições gramaticais em casos particulares, visando mostrar os excessos mais comuns da filosofia da lógica. Daí resulta a visão minimalista que Wittgenstein tem da lógica, da gramática e do conceito de “normatividade”: a lógica exibe perspicuamente a linguagem ordinária justamente porque pretende apresentar-se como o parâmetro de correções para os próximos usos.

All About the Meaning of E, a Symbol of Being an Element of a Set, as Being Part of that Which Has No Part

In the study of Set Theory, few have ever brought forth the question on the meaning of ϵ, the symbol denoting the most central relation between two objects in the domain. This is similar to the study of Geometry in which few have ever questioned the meaning of the most central notion Point. It is also similar to the study of Phenomenology in which few have ever raised the question on the meaning of the most central concept Being until Heidegger in 1927. In this exchange note following “Between ‘A Brief History of Analytic Philosophy: from Russell to Rawls’ and ‘Logico-Philosophicus Serial Booklet’” (abbrev as ‘Between …’ note), we bring forth the question on the meaning of ϵ and make a trying effort to find the answer to it. We first show the difference and the thing in common between Russell's own resolution to Russell’s Paradox and the one proposed by us in the ending part of ‘Between …’ note to directly address Professor Schwartz’s critical comment on the seeming similarity between these two. Then, with the help of Frege’s Concept of Functions, Russell’s Naive Set and Zermelo’s Definition of ‘Definite’, we demonstrate in order that (1) the two resolutions to Russell’s Paradox are both equal and necessary to each other; (2) the Russell’s Paradox, the Naive Set in particular, essentially reveals the lack of our understanding on ϵ being argument of a propositional function; (3) ϵ, the mathematical symbol denoting being an element of [a set into a whole] according to the words of Russia-born Georg Cantor, is meant for being part of [that which has no part] in the language of Euclid of Alexandria. In the end, we extend the discussion to show that the final outcome of our analysis in this article are also tightly related with (1) the Dasein concept investigated by Heidegger (2) the Point & Line notion explored by Euclid (3) Continuum Hypothesis initiated by Georg Cantor and (4) ‘Logico-Philosophicus Serial Booklet’, an article series authored by us that revisit the central issues in Philosophy of Language and Logic (Meaning of a Name), Philosophy of Phenomenology (Being of a Thing) and Classic Axioms in Science (Exact Meaning and Logic of Definitions and Postulates) respectively, with the hope of shedding light on a few critical concepts in the study of Science and Philosophy.

What Is Existence?

This paper has a negative and a positive claim. The negative claim is that the Frege-Russell account of existence as a higher-order predicate is mistaken and should be abandoned, even with respect to general statements of existence such as “Flying mammals exist” (where statements of this sort are supposed to be best accommodated by the account). The Frege-Russell view seems to be supported by two ideas. First, the idea that existence is entirely expressed by the existential quantifier of standard predicate logic. Second, the idea that the existential quantifier is a higher-order predicate, a predicate of predicates, not of individuals. I think that both ideas are wrong but will focus on the latter. By construing prima facie first-order statements such as “Flying Mammals exist” as higher-order predications such as “The Fregean Concept Flying Mammal maps at least one individual onto the True”, the Frege-Russell view commits one merely on the basis of the meaning it assigns to the existence predicate – to abstract objects such as concepts (Gottlob Frege), or propositional functions (Bertrand Russell), or classes (Rudolf Carnap), or properties, kinds, and so on. This cannot be right, I think. The positive claim of the present paper is that, at least in the context of first-order discourse, the existence predicate is just what it seems to be: a bona fide first-order predicate (pace Kant, Hume, Frege, Russell and others). Three important ideas about existence are shared with the Frege-Russell conception of existence, though.

A Combinatorial Classifier for Error-Data in Joining Processes with Diverse-Granular Computing

In this paper, we present a novel hybrid classification model with fuzzy clustering and design a newly combinatorial classifier for error-data in joining processes with diverse-granular computing, which is an ensemble of a naïve Bayes classifier with fuzzy c-means clustering. And we apply it to improve classification performance of traditional hard classifiers in more complex real-world situations. The fuzzy c-means clustering is applied to a fuzzy partition based on a given propositional function to augment the combinatorial classifier. This strategy would work better than a conventional hard classifier without fuzzy clustering. Proper scale granularity of objects contributes to higher classification performance of the combinatorial classifier. Our experimental results show the newly combinatorial classifier has improved the accuracy and stability of classification.

The Versatility of Universality in Principia Mathematica

In this article, I examine the ramified-type theory set out in the first edition of Russell and Whitehead's Principia Mathematica. My starting point is the ‘no loss of generality’ problem: Russell, in the Introduction (Russell, B. and Whitehead, A. N. 1910. Principia Mathematica, Volume I, 1st ed., Cambridge: Cambridge University Press, pp. 53–54), says that one can account for all propositional functions using predicative variables only, that is, dismissing non-predicative variables. That claim is not self-evident at all, hence a problem. The purpose of this article is to clarify Russell's claim and to solve the ‘no loss of generality’ problem. I first remark that the hierarchy of propositional functions calls for a fine-grained conception of ramified types as propositional forms (‘ramif-types’). Then, comparing different important interpretations of Principia’s theory of types, I consider the question as to whether Principia allows for non-predicative propositional functions and variables thereof. I explain how the distinction between the formal system of the theory, on the one hand, and its realizations in different epistemic universes, on the other hand, makes it possible to give us a more satisfactory answer to that question than those given by previous commentators, and, as a consequence, to solve the ‘no loss of generality’ problem. The solution consists in a substitutional semantics for non-predicative variables and non-predicative complex terms, based on an epistemic understanding of the order component of ramified types. The rest of the article then develops that epistemic understanding, adding an original epistemic model theory to the formal system of types. This shows that the universality sought by Russell for logic does not preclude semantical considerations, contrary to what van Heijenoort and Hintikka have claimed.

Propositional Functions in Extension

In his “The Foundations of Mathematics”, Ramsey attempted to marry the Tractarian idea that all logical truths are tautologies and vice versa, and the logicism of the Principia. In order to complete his project, Ramsey was forced to introduce propositional functions in extension (PFEs): given Ramsey's definitions of 1 and 2, without PFEs even the quantifier-free arithmetical truth that 1 ≠ 2 is not a tautology. However, a number of commentators have argued that the notion of PFEs is incoherent. This response was first given by Wittgenstein but has been best developed by Sullivan. While I agree with Wittgenstein and Sullivan's common conclusion, I believe that even the most compelling of Sullivan's arguments is importantly mistaken and that Wittgenstein's remarks are too opaque to be left as the end of the matter. In this article I uncover the fault in Sullivan's argument and present an alternative criticism of PFEs which is Wittgensteinian in spirit without being too mystifying.