The powerset operator, [MATHEMATICAL SCRIPT CAPITAL P], is compared with other operators of similar type and logical complexity. Namely we examine positive operators whose defining formula has a canonical form containing at most a string of universal quantifiers. We call them ∀-operators. The question we address in this paper is: How is the class of ∀-operators generated ? It is shown that every positive ∀-operator Γ such that Γ ≠ ∅, is finitely generated from [MATHEMATICAL SCRIPT (...) CAPITAL P], the identity operator Id, constant operators and certain trivial ones by composition, ∪ and ∩. This extends results of [3] concerning bounded positive operators. (shrink)

The target article makes an impressive case that nested set operations (NS) facilitate probability computations by helping make clear the relevant natural frequency partitions; however, NS can also contribute to common errors. That NS constitute a general reasoning process is supported by their role in deductive, modal, causal, and other reasoning. That NS are solely a rule-based process is problematic.

In the present paper we concentrate on fundamental problems concerning ω-operations over partial enumerated sets. The notion of “HOM-lifts” seems to be an adequate tool for this kind of investigations. MSC: 03D45, 18A30.

For regular sets in Euclidean space, previous work has identified twelve ‘basic’ computability notions to which many previous notions considered in literature were shown to be equivalent. With respect to those basic notions we now investigate on the computability of natural operations on regular sets: union, intersection, complement, convex hull, image, and pre-image under suitable classes of functions. It turns out that only few of these notions are suitable in the sense of rendering all those operations uniformly computable.

In this paper, we study when a set-continuous operator has a fixed-point that is the intersection of a directed family. The framework of our study is the Kelley-Morse theory KMC– and the Gödel-Bernays theory GBC–, both theories including an Axiom of Choice and excluding the Axiom of Foundation. On the one hand, we prove a result concerning monotone operators in KMC– that cannot be proved in GBC–. On the other hand, we study conditions on directed superclasses in GBC– in (...) order that their intersection is a fixed-point of a set-continuous operator. Finally, we illustrate our results with a solution to the liar paradox and a construction of maximal bisimulations. (shrink)

A fundamental result from Boolean modal logic states that a first-order definable class of Kripke frames defines a logic that is validated by all of its canonical frames. We generalise this to the level of non-distributive logics that have a relational semantics provided by structures based on polarities. Such structures have associated complete lattices of stable subsets, and these have been used to construct canonical extensions of lattice-based algebras. We study classes of structures that are closed under ultraproducts and whose (...) stable set lattices have additional operators that are first-order definable in the underlying structure. We show that such classes generate varieties of algebras that are closed under canonical extensions. The proof makes use of a relationship between canonical extensions and MacNeille completions. (shrink)

We present an extension of constructive Zermelo{Fraenkel set theory [2]. Constructive sets are endowed with an applicative structure, which allows us to express several set theoretic constructs uniformly and explicitly. From the proof theoretic point of view, the addition is shown to be conservative. In particular, we single out a theory of constructive sets with operations which has the same strength as Peano arithmetic.

Any set x is uniquely specified by the graph of the membership relation on the set obtained by adjoining x to the transitive closure of x. Thus any operation on sets can be looked at as an operation on these graphs. We look at the operations of ordinal arithmetic of sets in this light. This turns out to be simplest for a modified ordinal arithmetic based on the Zermelo ordinals, instead of the usual von Neumann ordinals. In this arithmetic, addition (...) of sets corresponds to concatenating graphs, and multiplication corresponds to replacing each edge of a graph by a copy of another graph. Characterizations for the von Neumann case are also given. (shrink)

We introduce an operational set theory in the style of [5] and [16]. The theory we develop here is a theory of constructive sets and operations. One motivation behind constructive operational set theory is to merge a constructive notion of set ([1], [2]) with some aspects which are typical of explicit mathematics [14]. In particular, one has non-extensional operations (or rules) alongside extensional constructive sets. Operations are in general partial and a limited form of self{application is permitted. The system we (...) introduce here is a fully explicit, nitely axiomatised system of constructive sets and operations, which is shown to be as strong as HA. (shrink)

We show that considerable sets of positive linear operators namely their extensions as closures, adjoints or Friedrichs positive self-adjoint extensions form operator (generalized) effect algebras. Moreover, in these cases the partial effect algebraic operation of two operators coincides with usual sum of operators in complex Hilbert spaces whenever it is defined. These sets include also unbounded operators which play important role of observables (e.g., momentum and position) in the mathematical formulation of quantum mechanics.

We study the extension of Feferman’s operational set theory provided by adding operational versions of unbounded existential quantification and power set and determine its proof-theoretic strength in terms of a suitable theory of sets and classes.

We study the behavior of the game operator $$ on Wadge classes of Borel sets. In particular we prove that the classical Moschovakis results still hold in this setting. We also characterize Wadge classes ${\bf{\Gamma }}$ for which the class has the substitution property. An effective variation of these results shows that for all $1 \le \eta < \omega _1^{{\rm{CK}}}$ and $2 \le \xi < \omega _1^{{\rm{CK}}}$, is a Spector class while is not.

It, is shown that Birkhoff –von Neumann quantum logic (i.e., an orthomodular lattice or poset) possessing an ordering set of probability measures S can be isomorphically represented as a family of fuzzy subsets of S or, equivalently, as a family of propositional functions with arguments ranging over S and belonging to the domain of infinite-valued Łukasiewicz logic. This representation endows BvN quantum logic with a new pair of partially defined binary operations, different from the order-theoretic ones: Łukasiewicz intersection and union (...) of fuzzy sets in the first case and Łukasiewicz conjunction and disjunction in the second. Relations between old and new operations are studied and it is shown that although they coincide whenever new operations are defined, they are not identical in general. The hypothesis that quantum-logical conjunction and disjunction should be represented by Łukasiewicz operations, not by order-theoretic join and meet is formulated and some of its possible consequences are considered. (shrink)

I formulate the Zermelo-Russell paradox for naive set theory. A sketch is given of Zermelo’s solution to the paradox: the cumulative type structure. A careful analysis of the set formation process shows a missing component in this solution: the necessity of an assumed imaginary jump out of an infinite universe. Thus a set is formed by a suitable combination of concrete and imaginary operations all of which can be made or assumed by a Turing machine. Some consequences are drawn from (...) this improved analysis of the concept of set, for the theory of sets and for the philosophy and foundations of mathematics. (shrink)

New conjunctionlike and disjunctionlike operations on orthomodular lattices are defined with the aid of formal Mackey decompositions of not necessarily compatible elements. Various properties of these operations are studied. It is shown that the new operations coincide with the lattice operations of join and meet on compatible elements of a lattice but they necessarily differ from the latter on all elements that are not compatible. Nevertheless, they define on an underlying set the partial order relation that coincides with the original (...) one. The new operations are in general nonassociative: if they are associative, a lattice is necessarily Boolean. However, they satisfy the Foulis-Holland-type theorem concerning associativity instead of distributivity. (shrink)

We introduce a way of relativizing operational set theory that also takes care of application. After presenting the basic approach and proving some essential properties of this new form of relativization we turn to the notion of relativized regularity and to the system OST that extends OST by a limit axiom claiming that any set is element of a relativized regular set. Finally we show that OST is proof-theoretically equivalent to the well-known theory KPi for a recursively inaccessible universe.

In Aczel [1], the existence of largest fixed points of set continuous operators is proved assuming the schema version of dependent choices in Zermelo-Fraenkel set theory without the axiom of Foundation. In the present paper, we study whether the existence of largest fixed points of set continuous operators is provable without the schema version of dependent choices, using Boffa's weak antifoundation axioms.

“Small” large cardinal notions in the language of ZFC are those large cardinal notions that are consistent with V = L. Besides their original formulation in classical set theory, we have a variety of analogue notions in systems of admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics and recursive ordinal notations (as used in proof theory). On the face of it, it is surprising that such distinctively set-theoretical notions have analogues in such disaparate and (...) relatively constructive contexts. There must be an underlying reason why that is possible (and, incidentally, why “large” large cardinal notions have not led to comparable analogues). My long term aim is to develop a common language in which such notions can be expressed and can be interpreted both in their original classical form and in their analogue form in each of these special constructive and semi-constructive cases. This is a program in progress. What is done here, to begin with, is to show how that can be done to a considerable extent in the settings of classical and admissible set theory (and thence, admissible recursion theory). The approach taken here is to expand the language of set theory to allow us to talk about (possibly partial) operations applicable both to sets and to operations and to formulate the large cardinal notions in question in terms of operational closure conditions; at the same time only minimal existence axioms are posited for sets. The resulting system, called Operational Set Theory, is a partial adaptation to the set-theoretical framework of the explicit mathematics framework Feferman (1975). The.. (shrink)

We study constructive set theories, which deal with operations applying both to sets and operations themselves. Our starting point is a fully explicit, finitely axiomatized system ESTE of constructive sets and operations, which was shown in 10 to be as strong as PA. In this paper we consider extensions with operations, which internally represent description operators, unbounded set quantifiers and local fixed point operators. We investigate the proof theoretic strength of the resulting systems, which turn out to be (...) impredicative . © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

Stem cell tourism—the flow of patients from home countries to destination countries to obtain stem cell treatment—is a growing business in China. Many concerns have been raised regarding fraudsters that operate unsafe stem cell therapies and an officialdom that turns a blind eye to the questionable technology. The Chinese regulatory approach to stem cell research is based on Guidelines and Administrative Measures, rather than legislation, and may have no binding force on certain institutions, such as military hospitals. There is no (...) liability and traceability system and no visible set of penalties for non-compliance in the stem cell legal framework. In addition to the lack of safety and efficacy systems in the regulations, no specific expert authority has been established to monitor stem cell therapy to date. Recognizing the global nature of stem cell tourism, this article argues that resolving stem cell tourism issues may require not only the Chinese government but also an international mechanism for transparency and ethical oversight. A stringent set of international regulations that govern stem cell therapies can encourage China to improve stem cell regulation and enforcement to fulfill its obligations. Through an international consensus, a minimum standard for clinical stem cell research and a central enforcement system will be provided. As a result, rogue clinics that conduct unauthorized stem cell therapies can be penalized, and countries that are reluctant to implement the reconciled regulations should be sanctioned. (shrink)

This article is a preliminary draft for initiating and commencing a new pioneer dimension of expression. To deal with higher-dimensional data or information flowing in this modern era of information technology and artificial intelligence, some innovative super algebraic structures are essential to be formulated. In this paper, we have introduced such matrices that have multiple layers and clusters of layers to portray multi-dimensional data or massively dispersed information of the plithogenic universe made up of numerous subjects their attributes, and sub-attributes. (...) For grasping that field of parallel information, events, and realities flowing from the micro to the macro level of universes, we have constructed hypersoft and hyper-super-soft matrices in a Plithogenic Fuzzy environment. These Matrices classify the non-physical attributes by accumulating the physical subjects and further sort the physical subjects by accumulating their non-physical attributes. We presented themasPlithogenicAttributiveSubjectivelyWholeHyper-Super-Soft-Matrix(PASWHSS-Matrix)andPlithogenic Subjective Attributively Whole-Hyper-Super-Soft-Matrix (PSAWHSS-Matrix). Several types of views and level-layers of these matrices are described. In addition, some local aggregation operators for Plithogenic Fuzzy Hypersoft Set (PPFHS-Set) are developed. Finally, few applications of these matrices and operators are used as numerical examples of COVID-19 data structures. (shrink)

Review article referring to my prior work in many contexts with the upshot that: Subject to an /extremely/ limited set of exceptions, /all/ sting operations are /per se/ gravely and deeply immoral for the simplest and plainest of reasons: They are calculated and deliberate attempts to bring out the worst in a fellow human being, to play to their weaknesses, and to pander to their blind spots. Whether performed by the government, the media, or other private organizations (for-profit or not-for-profit), (...) makes no ethical difference whatsoever, except one: When the government does it, everyone begins to think that such egregious behavior is just fine, although /it is anything but/. (shrink)

In [2], Jockusch and Shore have introduced a new hierarchy of sets and operators called the REA hierarchy. In this note we prove analogues of the Friedberg Jump Theorem and the Sacks Jump Theorem for many REA operators. MSC: 03D25, 03D55.

In many applications we have a set of objects together with their properties. Since the available information is usually incomplete and/or imprecise, the true knowledge about subsets of objects can be determined approximately only. In this paper, we discuss a fuzzy generalisation of two pairs of relation-based operators suitable for fuzzy set approximations, which have been recently investigated by Düntsch and Gediga. Double residuated lattices, introduced by Orlowska and Radzikowska, are taken as basic algebraic structures. Main properties of these (...) operators are presented. (shrink)

We discuss a generalization of the standard notion of probability space and show that the emerging framework, to be called operational probability theory, can be considered as underlying quantal theories. The proposed framework makes special reference to the convex structure of states and to a family of observables which is wider than the familiar set of random variables: it appears as an alternative to the known algebraic approach to quantum probability.

Background and significance of research: Nurses’ awareness of professional values and how those values affect their behaviors is an integral part of nursing care. There is a large body of research on nursing professional values, however, a careful survey of the available literature did not yield any studies investigating the status of professional values in operating rooms. Objective: This study aims to investigate the perception of operating room nurses of university hospitals toward professional values. Research plan: In this cross-sectional study, (...) data were collected using Schank and Weis’s Nurses Professional Values Scale–Revised. The collected data were analyzed in SPSS version 16. Participants and settings: The 513 participants of the study consisted of operating room nurses and nurse anesthetists from six university hospitals. Ethical considerations: This study has been approved by the ethics committee of the university. Findings: The mean total professional values score of the operating room staff was found to be 100.84 ± 15.685, which indicates that the participants had a positive perception toward observance of nursing professional values in practice. The results showed that the participants considered the domains of justice and activism as, respectively, the most and the least important. The operating room staff’s overall professional values scores were not found to correlate significantly with their ages, professional experience, university majors, or attendance at ethics workshops. However, a significant difference was found between the professional values scores of the female and male staff. The professional values scores of the operating room nurses and the nurse anesthetists were not significantly different (p value = 0.494). Conclusion: Operating room staff’s awareness of professional values is essential to providing care to patients based on professional principles. Accordingly, there is need for programs to raise operating room nurses’ awareness of their professional duties and improve their professional performance. (shrink)

I shall be dealing, throughout this book, with a set of related problems: the relationship between morality and reasoning in general, the way in which moral reasoning is properly to be carried on, and why morality is not arbitrary. The solutions to these problems come out of the same train of argument. Morality is not arbitrary, I shall argue, because the acceptance of certain qualities of character as virtues and the rejection of others as vices is forced on us by (...) the co-operative basis of human life. The co-operation in human life is unavoidable; the alternative is a literal Hobbesian state of nature, and that is impossible. It is not that co-operation between people is a good thing or even a very good thing; it is simply unavoidable in human life, and it is impossible unless the qualities of character counted as virtues are encouraged and are at least fairly common. The possibility of human life presupposes a theory of human nature, and working out that theory of human nature is the main job of moral philosophy. These virtues or qualities of character or attitudes lead us towards a theory of reasons. A person with a sense of justice is a person inclined to accept certain sorts of facts as reasons for acting, and if the virtues are presupposed by human life then the acceptance of those sorts of facts as reasons for acting is presupposed by human life. A condition of the life of reasoning beings (a more accurate term here than 'human beings') is that moral reasons are reasons for acting and are at the very basis of reasoning. And from this it follows that properly conducted moral reasoning is ultimately guided by the virtues rather than ultimately guided by a set of rules. (shrink)

Constructive set theory a' la Myhill-Aczel has been extended in (Cantini and Crosilla 2008, Cantini and Crosilla 2010) to incorporate a notion of (partial, non--extensional) operation. Constructive operational set theory is a constructive and predicative analogue of Beeson's Inuitionistic set theory with rules and of Feferman's Operational set theory (Beeson 1988, Feferman 2006, Jaeger 2007, Jaeger 2009, Jaeger 1009b). This paper is concerned with an extension of constructive operational set theory (Cantini and Crosilla 2010) by a uniform operation of Transitive (...) Closure, \tau. Given a set a, \tau produces its transitive closure \tau a. We show that the theory ESTE of (Cantini and Crosilla 2010) augmented by \tau is still conservative over Peano Arithmetic. (shrink)

It is shown, within constructive mathematics, that the unit ball B1 of the set of bounded operators on a Hilbert space H is weak-operator totally bounded. This result is then used to prove that the weak-operator continuity of the mapping T → AT on B1 is equivalent to the existence of the adjoint of A.

The functional capabilities that consciousness seems to provide to biological systems can supply valuable principles in the design of more autonomous and robust technical systems. These functional concepts keep a notable similarity to those underlying the notion of operating system in software engineering, which allows us to specialize the computer metaphor for the mind into that of the operating system metaphor for consciousness. In this article, departing from these ideas and a model-based theoretical framework for cognition, we present an architectural (...) proposal for machine consciousness, called the Operative Mind. According to it, machine consciousness could be implemented as a set of services, in an operative system fashion, based on modeling of the own control architecture, that supervise the adequacy of the system architectural structure to the current objectives, triggering and managing adaptativity mechanisms. (shrink)

We discuss several ontological properties of explicit mathematics and operational set theory: global choice, decidable classes, totality and extensionality of operations, function spaces, class and set formation via formulas that contain the definedness predicate and applications.

Accretive and monotone operator theory are central branches of nonlinear functional analysis and constitute the abstract study of certain set-valued mappings between function spaces. This paper deals with the computational properties of these accretive and (generalized) monotone set-valued operators. In particular, we develop (and extend) for this field the theoretical framework of proof mining, a program in mathematical logic that seeks to extract computational information from prima facie “non-computational” proofs from the mainstream literature. To this end, we establish logical (...) metatheorems that guarantee and quantify the computational content of theorems pertaining to accretive and (generalized) monotone set-valued operators. On the one hand, our results unify a number of recent case studies, while they also provide characterizations of central analytical notions in terms of proof theoretic ones on the other, which provides a crucial perspective on needed quantitative assumptions in future applications of proof mining to these branches. (shrink)

With the current pandemic, many scholars have contended that clinical criteria offer the best way to implement triage. Further, they dismiss the criteria of social value as a good one for triage. In this paper, I respond to refute this perspective. In particular, I present two sets of arguments. Firstly, I argue that the objections to the social value criteria they present apply to the clinical criteria they favor. Secondly, they exaggerate the negative aspects of the social value criteria, while (...) I suggest it is reasonable to use this. I end the article by recommending how operative public values can be a good way to make triaging decisions. (shrink)

Let K = (p, q...; &, ∨, ~) be a zeroth-order formal language with sentence variables p, q..., two place connectives & (and), ∨ (or) and negation sign ~, and let F be the formula algebra (set of well-formed formulas in K defined in the standard way by induction from the sentence variables). If v is an assignment of truth values 1(true), 0(f alse) to the sentence variables p, q..., then classical propositional logic is characterized by extending v by induction (...) from p, q... to the formula algebra F in such a manner that the extension (called interpretation and also denoted by v) be a “homomorphism” from F into the two-element Boolean algebra B 2 ≡ (1, 0, ⋂, ⋃, ⊥), where “homomorphism” means that the following hold υ(q&p)=υ(q)∩υ(p)υ(q∨p)=υ(q)∪υ(p)υ(∼q)=υ(q)⊥. (shrink)

In the October 1937 number of Philosophy of Science Lindsay has made certain criticisms of the adequacy of the “operational method” of analyzing and giving meaning to the concepts of physics, documenting his criticisms chiefly from my own writings. In these criticisms he has made statements as to the method which I would by no means accept. This is not characteristic of his paper only, for I have seldom indeed seen a printed discussion of the method which I would accept (...) as being an adequate or sometimes even fair representation of what I understand by it. It will perhaps pay therefore if I attempt to state what I conceive it to be all about, particularly since I have never attempted such a comprehensive statement and since my own ideas on the subject have been developing since I first wrote in the Logic of Modern Physics. For one reason I have hesitated to do this, for fear of seeming to subscribe to the not uncommon idea that we are dealing with some elaborate and profound new theory of the nature of knowledge or of meaning. I believe that I myself have never talked of “operationalism” or “operationism”, but I have a distaste for these grandiloquent words which imply something more philosophic and esoteric than the simple thing that I see. What we are here concerned with is an observation and description of methods which at least some physicists had already, perhaps unconsciously, adopted and found successful—the practise of the methods already existed. What I have attempted is to analyze these successful methods, not to set up a philosophical system and a theory of the properties that any method must have if it hopes to be successful. Since I was concerned with a technique already extant, my principal method of getting others to see what the technique involved has been to exhibit examples of the technique in action, rather than to attempt any exhaustive characterization of the technique itself. (shrink)

A simple model accounting for semantic properties of propositional attitude operators in negative contexts with no reference to possible worlds is proposed. Verbs occurring in such operators denote relations between individuals and specific sets of sentences (of a given natural language) and their negation is defined as the complement within a specific set of cognitively determined sentences. This approach avoids in particular the problem of intensionality of propositional attitude operators and allows to use many tools from the (...) generalised quantifier theory. In that way the negation giving rise to factive presuppositions and to the neg-raising is defined in a natural way. (shrink)