Doerig et al. have set several criteria that theories of consciousness need to fulfill. By these criteria, higher-order theories fare better than most existing theories. But they also argue that higher-order theories may not be able to answer both the ‘small network argument’ and the ‘other systems argument’. In response, we focus on the case of the Perceptual Reality Monitoring theory to explain why higher-order theories do just fine.

Instead of the half-century old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The set-theoretic antinomies forced naïve set theory to be reformulated using some iterative notion of a set so that a set would always have higher type or rank than its members. Then the universal u_{F}={x|F(x)} for a property F() could never be self-predicative in the sense of u_{F}∈u_{F}. But the (...) mathematical theory of categories, dating from the mid-twentieth century, includes a theory of always-self-predicative universals--which can be seen as forming the "other bookend" to the never-self-predicative universals of set theory. The self-predicative universals of category theory show that the problem in the antinomies was not self-predication per se, but negated self-predication. They also provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. (shrink)

The original purpose of the present study, 2011, started with a preprint «On the Probable Failure of the Uncountable Power Set Axiom», 1988, is to save from the transfinite deadlock of higher set theory the jewel of mathematical Continuum — this genuine, even if mostly forgotten today raison d’être of all traditional set-theoretical enterprises to Infinity and beyond, from Georg Cantor to David Hilbert to Kurt Gödel to W. Hugh Woodin to Buzz Lightyear.

A logic is called higher order if it allows for quantification over higher order objects, such as functions of individuals, relations between individuals, functions of functions, relations between functions, etc. Higher order logic began with Frege, was formalized in Russell [46] and Whitehead and Russell [52] early in the previous century, and received its canonical formulation in Church [14].1 While classical type theory has since long been overshadowed by set theory as a foundation of mathematics, (...) recent decades have shown remarkable comebacks in the fields of mechanized reasoning (see, e.g., Benzm¨. (shrink)

There is a trade-off between specificity and accuracy in existing models of belief. Descriptions of agents in the tripartite model, which recognizes only three doxastic attitudes—belief, disbelief, and suspension of judgment—are typically accurate, but not sufficiently specific. The orthodox Bayesian model, which requires real-valued credences, is perfectly specific, but often inaccurate: we often lack precise credences. I argue, first, that a popular attempt to fix the Bayesian model by using sets of functions is also inaccurate, since it requires us to (...) have interval-valued credences with perfectly precise endpoints. We can see this problem as analogous to the problem of higher order vagueness. Ultimately, I argue, the only way to avoid these problems is to endorse Insurmountable Unclassifiability. This principle has some surprising and radical consequences. For example, it entails that the trade-off between accuracy and specificity is in-principle unavoidable: sometimes it is simply impossible to characterize an agent’s doxastic state in a way that is both fully accurate and maximally specific. What we can do, however, is improve on both the tripartite and existing Bayesian models. I construct a new model of belief—the minimal model—that allows us to characterize agents with much greater specificity than the tripartite model, and yet which remains, unlike existing Bayesian models, perfectly accurate. (shrink)

The growth in neoliberal or market-driven higher education has challenged traditional approaches to evaluating faculty members. The involvement of multiple stakeholders (i.e., accreditation bodies, quality assurance officers, administrators, teaching faculty, and students) has led to different and sometimes conflicting needs in faculty evaluation. While extant literature generally suggests that faculty evaluation in contemporary higher education is strongly associated with accountability purposes, little is known about how key agents at the institutional level use evaluation for learning and improvement. Thus, (...) this study attempts to identify the approaches to faculty evaluation that promote learning and improvement in higher education. This study is a qualitative case study of Vietnamese higher education student evaluation of teaching (SET) and voting evaluation practices. It adopts the problem-based methodology (Robinson, 1993; Robinson & Lai, 2006) to investigate the theories of action of some key stakeholders of these faculty evaluation practices in Vietnam. The theories of action comprise the stakeholders’ approaches to faculty evaluation, together with the constraints (that rule in or rule out specific approaches) and the consequences of the approaches. Participants included four quality assurance officers, 18 administrators, and 20 faculty members from seven public higher education institutions in Vietnam. The theories of action about faculty evaluation were constructed based on interviews and key evaluation documents. Overall, the participants took three main approaches to faculty evaluation: (i) complying with the expected evaluation procedures and roles, (ii) taking a harmony-oriented and unilateral approach to problem solving, and (iii) disengaging from evaluation for learning and improvement. The participants generally fulfilled the evaluation policy requirements, but their approaches had a limited impact on learning and improvement at the institutional level. The constraint analysis suggests that the participants’ approaches were not completely passive but were mainly oriented to managerial accountability demands (Røiseland et al., 2015). The participants’ collectivist Confucian harmony-oriented norms hindered their approaches to problem solving and using evaluation for learning and improvement. However, there were a few participants who acted on their own values rather than being confined by political and cultural constraints. In spite of its scarcity, these participants’ self-reliant approach suggests a potential influence of Buddhist principles of self-transformation, contextuality and reflexivity on improving practices (Chu & Vu, 2021; Thich Nhat Hanh, 1987). The study has two key implications. First, it suggests the improvement of faculty evaluation policymaking and implementation by revising several existing constraints and by adding some cultural and individual constraints to the current theories of action. The faculty evaluation problems could be solved by improving the quality of evaluation processes and data, reconceptualising SET and voting evaluation, and treating faculty members’ underperformance as a collective problem. Second, the study suggests a framework to predict the likely success of future intervention on faculty evaluation. For instance, faculty evaluation for learning and improvement will be more feasible if policymakers and implementers prioritise transformational purposes and make joint efforts to foster dialogues and collaboration among individuals and groups. My study also highlights the need to understand faculty evaluation, and possibly other higher education practices in Vietnam, in religious contexts and from individual participants’ spiritual values or philosophical backgrounds. The study implications are applicable to Vietnamese higher education and potentially to other educational settings with similar characteristics. (shrink)

This discussion treats a set of familiar social derelictions as consequences of the perversion of a universalistic moral theory in the service of an ill-considered or insufficiently examined personal agenda.The set includes racism, sexism, anti-Semitism, homophobia, and class elitism, among other similar pathologies, under the general heading of discrimination. The perversion of moral theory from which these derelictions arise, I argue, involves restricting its scope of application to some preferred subgroup of the moral community of human beings. -/- (...) The following analysis of higher-order discrimination suggests that we often select the individuals who constitute such subgroups for reasons that we ourselves would reject on moral grounds were we to examine them carefully, but that we choose instead to put our rational resources in the service of avoiding any such examination at all costs. The implication is that arguments that truncate the scope of moral theory in fact justify bestowing the gift of moral treatment on a select few who deserve it no more than the many from whom we withhold it. Therefore, it would be precipitous to conclude that universalistic moral theory can be legitimately restricted in its practical scope of application in any way at all. (shrink)

An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the `width' of the set theoretic universe, such as Cantor's continuum hypothesis. Within a higher-order framework I show that contingency (...) about the width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the "Leibniz biconditionals" stating that what is possible, in the broadest sense of possible, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions. (shrink)

In this paper, we bring together research on complex problem solving with that on motivational psychology about goal setting. Complex problems require motivational effort because of their inherent difficulties. Goal Setting Theory has shown with simple tasks that high, specific performance goals lead to better performance outcome than do-your-best goals. However, in complex tasks, learning goals have proven more effective than performance goals. Based on the Zurich Resource Model, so-called motto-goals should activate a person’s resources through positive affect. It (...) was found that motto-goals are effective with unpleasant duties. Therefore, we tested the hypothesis that motto-goals outperform learning and performance goals in the case of complex problems. A total of N = 123 subject participated in the experiment. In dependence of their goal condition, subjects developed a personal motto, learning, or performance goal. This goal was adapted for the computer-simulated complex scenario Tailorshop, where subjects worked as managers in a small fictional company. Other than expected, there was no main effect of goal condition for the management performance. An unexpected gender effect revealed better performance for men than women, pointing to a potential stereotype threat. As hypothesized, motto goals led to higher positive and lower negative affect than the other two goal types. Even though positive affect decreased and negative affect increased in all three groups during Tailorshop completion, participants with motto goals reported the lowest rates of negative affect. Exploratory analyses investigated the role of affect in complex problem solving via mediational analyses and the influence of goal type on perceived goal attainment. (shrink)

Consider a variant of the usual story about the iterative conception of sets. As usual, at every stage, you find all the (bland) sets of objects which you found earlier. But you also find the result of tapping any earlier-found object with any magic wand (from a given stock of magic wands). -/- By varying the number and behaviour of the wands, we can flesh out this idea in many different ways. This paper's main Theorem is that any loosely constructive (...) way of fleshing out this idea is synonymous with a ZF-like theory. -/- This Theorem has rich applications; it realizes John Conway's (1976) Mathematicians' Liberation Movement; and it connects with a lovely idea due to Alonzo Church (1974). (shrink)

In this contribution, I explore the possibility of characterizing the emergence of time in causal set theory (CST) in terms of the growing block universe (GBU) metaphysics. I show that although GBU seems to be the most intuitive time metaphysics for CST, it leaves us with a number of interpretation problems, independently of which dynamics we choose to favor for the theory —here I shall consider the Classical Sequential Growth and the Covariant model. Discrete general covariance of the (...) CSG dynamics does not allow us to individuate a single history of the universe (defined by a causal history of different causal sets), thereby making the claim that ‘the past exists’ at best problematic. In addition, because the evolution of the universe in CSG dynamics leads to an outward branching causal tree, it becomes impossible to determine a proper ‘line of becoming’, thereby blurring the presentists’ claim that only the present exists. Similarly, the covariant approach runs into the same, if not even more severe problems, since each configuration of the universe would amount to a set of possible causal sets, thereby making the individuation of a single configuration of the universe —and thus the physical interpretation of the theory— implausible. (shrink)

Ambitious higher-order theories of consciousness aim to account for conscious states when these are understood in terms of what-it-is-like-ness. This paper considers two arguments concerning this aim, and concludes that ambitious theories fail. The misrepresentation argument against HO theories aims to show that the possibility of radical misrepresentation—there being a HO state about a state the subject is not in—leads to a contradiction. In contrast, the awareness argument aims to bolster HO theories by showing that subjects are aware of (...) all their conscious states. Both arguments hinge on how we understand two related notions which are ubiquitous in discussions of consciousness: those of what-it-is-like-ness and there being something it is like for a subject to be in a mental state. This paper examines how HO theorists must understand the two crucial notions if they are to reject the misrepresentation argument but assert the awareness argument. It shows that HO theorists can and do adopt an understanding—the HO reading—which seems to give them what they want. But adopting the HO reading changes the two arguments. On this reading, the awareness argument tells us nothing about those states there is something it is like to be in, and so offers no support to ambitious HO theories. And to respond to the misrepresentation understood according to the HO reading is to simply ignore the argument presented, and so to give no response at all. As things stand, we should deny that HO theories can account for what-it-is-like-ness. (shrink)

What is so special and mysterious about the Continuum, this ancient, always topical, and alongside the concept of integers, most intuitively transparent and omnipresent conceptual and formal medium for mathematical constructions and the battle field of mathematical inquiries ? And why it resists the century long siege by best mathematical minds of all times committed to penetrate once and for all its set-theoretical enigma ? -/- The double-edged purpose of the present study is to save from the transfinite deadlock of (...) higher set theory the jewel of mathematical Continuum -- this genuine, even if mostly forgotten today raison d'etre of all set-theoretical enterprises to Infinity and beyond, from Georg Cantor to W. Hugh Woodin to Buzz Lightyear, by simultaneously exhibiting the limits and pitfalls of all old and new reductionist foundational approaches to mathematical truth: be it Cantor's or post-Cantorian Idealism, Brouwer's or post-Brouwerian Constructivism, Hilbert's or post-Hilbertian Formalism, Goedel's or post-Goedelian Platonism. -/- In the spirit of Zeno's paradoxes, but with the enormous historical advantage of hindsight, we claim that Cantor's set-theoretical methodology, powerful and reach in proof-theoretic and similar applications as it might be, is inherently limited by its epistemological framework of transfinite local causality, and neither can be held accountable for the properties of the Continuum already acquired through geometrical, analytical, and arithmetical studies, nor can it be used for an adequate, conceptually sensible, operationally workable, and axiomatically sustainable re-creation of the Continuum. -/- From a strictly mathematical point of view, this intrinsic limitation of the constative and explicative power of higher set theory finds its explanation in the identified in this study ultimate phenomenological obstacle to Cantor's transfinite construction, similar to topological obstacles in homotopy theory and theoretical physics: the entanglement capacity of the mathematical Continuum. (shrink)

Standard approaches to proper names, based on Kripke's views, hold that the semantic values of expressions are (set-theoretic) functions from possible worlds to extensions and that names are rigid designators, i.e.\ that their values are \emph{constant} functions from worlds to entities. The difficulties with these approaches are well-known and in this paper we develop an alternative. Based on earlier work on a higher order logic that is \emph{truly intensional} in the sense that it does not validate the axiom scheme (...) of Extensionality, we develop a simple theory of names in which Kripke's intuitions concerning rigidity are accounted for, but the more unpalatable consequences of standard implementations of his theory are avoided. The logic uses Frege's distinction between sense and reference and while it accepts the rigidity of names it rejects the view that names have direct reference. Names have constant denotations across possible worlds, but the semantic value of a name is not determined by its denotation. (shrink)

The paper is a contribution to formal ontology. It seeks to use topological means in order to derive ontological laws pertaining to the boundaries and interiors of wholes, to relations of contact and connectedness, to the concepts of surface, point, neighbourhood, and so on. The basis of the theory is mereology, the formal theory of part and whole, a theory which is shown to have a number of advantages, for ontological purposes, over standard treatments of topology in (...) set-theoretic terms. One central goal of the paper is to provide a rigorous formulation of Brentano's thesis to the effect that a boundary can exist as a matter of necessity only as part of a whole of higher dimension which it is the boundary of. It concludes with a brief survey of current applications of mereotopology in areas such as natural-language analysis, geographic information systems, machine vision, naive physics, and database and knowledge engineering. (shrink)

This article is to explore whether the achievement of moral character is the ultimate goal of higher education from a cross cultural approach. To discuss this study logically, three major research questions are addressed. First, what are the concepts of moral, ethics, and character? Second, what is the achievement of moral character from the Eastern and the Western perspectives? Third, what is the role of higher education for the achievement of moral character? To defend these research questions, the (...) author uses a descriptive content analysis method, with a cross cultural approach. In order to explore the questions, the researcher in this study sets several limitations. Moral character is generally limited to the ancient Greek philosophy and Judeo-Christianity as well as to the classical Chinese thought and religion. Specifically, the study is mainly focused on not only Plato's "Republic" and Aristotle's "Nicomachean Ethics," but Confucius' "Analects" and Mencius' "Scripture (The Works of Mengzi)." Additionally, this paper also adjusts the lenses on moral theories, especially moral character, cardinal virtues, social harmony, and the common good. Lastly, higher education is focused on the lenses of Canada and South Korea. The significance of this study is to provide basic theories and valuable resources about moral and character education for educational theorists and practitioners, finding the theories of moral and ethics in the Eastern and the Western thoughts and religions. (shrink)

Certain commentators on Russell's “no class” theory, in which apparent reference to classes or sets is eliminated using higher-order quantification, including W. V. Quine and (recently) Scott Soames, have doubted its success, noting the obscurity of Russell’s understanding of so-called “propositional functions”. These critics allege that realist readings of propositional functions fail to avoid commitment to classes or sets (or something equally problematic), and that nominalist readings fail to meet the demands placed on classes by mathematics. I show (...) that Russell did thoroughly explore these issues, and had good reasons for rejecting accounts of propositional functions as extra-linguistic entities. I argue in favor of a reading taking propositional functions to be nothing over and above open formulas which addresses many such worries, and in particular, does not interpret Russell as reducing classes to language. (shrink)

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In (...) defense of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences. (shrink)

Emotional states of consciousness, or what are typically called emotional feelings, are traditionally viewed as being innately programed in subcortical areas of the brain, and are often treated as different from cognitive states of consciousness, such as those related to the perception of external stimuli. We argue that conscious experiences, regardless of their content, arise from one system in the brain. On this view, what differs in emotional and non-emotional states is the kind of inputs that are processed by a (...) general cortical network of cognition, a network essential for conscious experiences. Although subcortical circuits are not directly responsible for conscious feelings, they provide non-conscious inputs that coalesce with other kinds of neural signals in the cognitive assembly of conscious emotional experiences. In building the case for this proposal, we defend a modified version of what is known as the higher-order theory of consciousness. (shrink)

This article presents an overview of the basic philosophical motivations for, and some recent work in, modal set theory.

ABSTRACTAmbitious Higher-order theories of consciousness – Higher-order theories that purport to give an account of phenomenal consciousness – face a well-known objection from the possibility of ra...

The merits of set theory as a foundational tool in mathematics stimulate its use in various areas of artificial intelligence, in particular intelligent information systems. In this paper, a study of various nonstandard treatments of set theory from this perspective is offered. Applications of these alternative set theories to information or knowledge management are surveyed.

Biological evolution and technological innovation, while differing in many respects, also share common features. In particular, implementation of a new technology in the market is analogous to the spreading of a new genetic trait in a population. Technological innovation may occur either through the accumulation of quantitative changes, as in the development of the ocean clipper, or it may be initiated by a new combination of features or subsystems, as in the case of steamships. Other examples of the latter type (...) are electric networks that combine the generation, distribution, and use of electricity, and containerized transportation that combines standardized containers, logistics, and ships. Biological evolution proceeds, phenotypically, in many small steps, but at the genetic level novel features may arise not only through the accumulation of many small, common mutational changes, but also when distinct, relatively rare genetic changes are followed by many further mutations. In particular, capabilities of biologically modern man may have been initiated, perhaps some 150 000 years ago, by one or few accidental but distinct combinations of modules and subroutines of gene regulation which are involved in the generation of the neural network in the cerebral cortex. It is even conceivable that it was one primary genetic event that initiated the evolution of biologically modern man, introducing some novel but subtle feature of connectivity into the cerebral cortex which allowed for meta-levels of abstraction and upgraded modes of information processing. This may have set the stage for the evolution of integrated but diverse higher capabilities such as structured language, symbolic thought, strategic thought, and cognition based empathy. (shrink)

Originally introduced by Plato and Aristotle, Moderation Theory in Ethics is the most prevalent theory of ethics among Islamic scholars. Moderation Theory suggests that every virtue or excellence of character lies in the mean between two vices: excess or defect. Every ethical virtue comes from moderation in actions or emotions and every ethical vice comes from excess or defect. This paper suggests that while Islamic scholars have been influenced by this doctrine, they have also developed and re-conceptualized (...) it in innovative ways. Kindī, Miskawayh, Avicenna, Rāghib Isfahānī, Nasīr al-Dīn Ṭusī, and others are among the Islamic contributors to the subject. Some of their innovations in this theory are as follows: bringing together Aristotle's doctrine of the mean with Plato's psychology (by Kindī), dividing virtues into four higher genuses, dividing vices into eight higher genuses, setting various kinds of vices and virtues under these higher genuses (by Miskawayh), adding the vice qualitative criteria to Aristotle's vice quantitative criteria (excess and defect) (by Ṭusī), dividing various conceptualizations of justice (by Avicenna), adding religious and mystical virtues into the existing list of virtues (by Rāghib Isfahānī), and proposing a comprehensive model for curing diseases of the soul. This paper seeks to establish the main contributions of these Muslim scholars to Moderation Theory and elaborate on this theory’s evolution within the Islamic world. (shrink)

A possible world is a junky world if and only if each thing in it is a proper part. The possibility of junky worlds contradicts the principle of general fusion. Bohn (2009) argues for the possibility of junky worlds, Watson (2010) suggests that Bohn‘s arguments are flawed. This paper shows that the arguments of both authors leave much to be desired. First, relying on the classical results of Cantor, Zermelo, Fraenkel, and von Neumann, this paper proves the possibility of junky (...) worlds for certain weak set theories. Second, the paradox of Burali-Forti shows that according to the Zermelo-Fraenkel set theory ZF, junky worlds are possible. Finally, it is shown that set theories are not the only sources for designing plausible models of junky worlds: Topology (and possibly other "algebraic" mathematical theories) may be used to construct models of junky worlds. In sum, junkyness is a relatively widespread feature among possible worlds. (shrink)

In this book set theory INC# based on intuitionistic logic with restricted modus ponens rule is proposed. It proved that intuitionistic logic with restricted modus ponens rule can to safe Cantor naive set theory from a triviality. Similar results for paraconsistent set theories were obtained in author papers [13]-[16].

According to an influential idea in the philosophy of set theory, certain mathematical concepts, such as the notion of a well-order and set, are indefinitely extensible. Following Parsons (1983), this has often been cashed out in modal terms. This paper explores instead an extensional articulation of the idea, formulated in higher-order logic, that flat-footedly formalizes some remarks of Zermelo. The resulting picture is incompatible with the idea that the entire universe can be well-ordered, but entirely consistent with the (...) idea that the sets of any set-theoretic universe can be. (shrink)

In this paper intuitionistic set theory INC# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.

The incompleteness of set theory ZF C leads one to look for natural nonconservative extensions of ZF C in which one can prove statements independent of ZF C which appear to be “true”. One approach has been to add large cardinal axioms.Or, one can investigate second-order expansions like Kelley-Morse class theory, KM or Tarski-Grothendieck set theory T G or It is a nonconservative extension of ZF C and is obtained from other axiomatic set theories by the inclusion (...) of Tarski’s axiom which implies the existence of inaccessible cardinals. See also related set theory with a filter quantifier ZF (aa). In this paper we look at a set theory NC# ∞# , based on bivalent gyper infinitary logic with restricted Modus Ponens Rule In this paper we deal with set theory NC# ∞# based on bivalent gyper infinitary logic with Restricted Modus Ponens Rule. Nonconservative extensions of the canonical internal set theories IST and HST are proposed. (shrink)

Set-theoretic and category-theoretic foundations represent different perspectives on mathematical subject matter. In particular, category-theoretic language focusses on properties that can be determined up to isomorphism within a category, whereas set theory admits of properties determined by the internal structure of the membership relation. Various objections have been raised against this aspect of set theory in the category-theoretic literature. In this article, we advocate a methodological pluralism concerning the two foundational languages, and provide a theory that fruitfully interrelates (...) a `structural' perspective to a set-theoretic one. We present a set-theoretic system that is able to talk about structures more naturally, and argue that it provides an important perspective on plausibly structural properties such as cardinality. We conclude the language of set theory can provide useful information about the notion of mathematical structure. (shrink)

This article addresses the recent reception of Franz Brentano's writings on consciousness. I am particularly interested in the connection established between Brentano's theory of consciousness and higher-order theories of consciousness and, more specifically, the theory proposed by David Rosenthal. My working hypothesis is that despite the many similarities that can be established with Rosenthal's philosophy of mind, Brentano's theory of consciousness differs in many respects from higher-order theories of consciousness and avoids most of the criticisms (...) generally directed to them. This article is divided into eight parts. The first two sections expound the basic outline of Rosenthal's theory, and the third summarizes the principal objections that Rosenthal addresses to Brentano, which I, then, examine in sections 4 and 5. In sections 6 and 7, I discuss Brentano's principle of the unity of consciousness, and in section 8, I consider the scope of the changes that Brentano brings to his theory of consciousness in his later writings, which follow the 1874 publication of Psychology. I then draw the conclusion that Brentano's theory rests on a view of intransitive and intrinsic self-consciousness. (shrink)

The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality between two arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the standard probabilistic interpretation of quantum (...) class='Hi'>theory to define the probability of equality between two arbitrary observables in an arbitrary state. We apply this new interpretation to quantum measurement theory, and establish a logical basis for the difference between simultaneous measurability and simultaneous determinateness. (shrink)

Many approaches to quantum gravity -the theory that should account for quantum and gravitational phenomena under the same theoretical umbrella- seem to point at some form of spacetime emergence, i.e., the fact that spacetime is not a fundamental entity of our physical world. This tenet has sparked many philosophical discussions: from the so-called empirical incoherence problem to different accounts of emergence and mechanisms thereof. In this contribution, I focus on the partial order relation of causal set theory and (...) argue that causation can be characterized as an a-temporal constraint over the kinematic space defined by the theory. The relation constrains the growth of a new element/event with respect to the other elements/events of a given set. Therefrom, the flow of time emerges from the collection of the possible growths of the given set, where each possibility is characterized by a classical probability assigned by the dynamics of the theory. (shrink)

Cognitive Set Theory is a mathematical model of cognition which equates sets with concepts, and uses mereological elements. It has a holistic emphasis, as opposed to a reductionistic emphasis, and it therefore begins with a single universe (as opposed to an infinite collection of infinitesimal points).

The revival of the philosophy of mathematics in the 60s following its post-1931 slump left us with two conflicting positions on arithmetic’s ontological relationship to set theory. W.V. Quine’s view, presented in 'Word and Object' (1960), was that numbers are sets. The opposing view was advanced in another milestone of twentieth-century philosophy of mathematics, Paul Benacerraf’s 'What Numbers Could Not Be' (1965): one of the things numbers could not be, it explained, was sets; the other thing numbers could not (...) be, even more dramatically, was objects. Curiously, although Benacerraf’s article appeared in the heyday of Quine’s influence, it declined to engage the Quinean position squarely, even seemed to think it was not its business to do so. Despite that, in my experience, most philosophers believe that Benacerraf’s article put paid to the reductionist view that numbers are sets (though perhaps not the view that numbers are objects). My chapter will attempt to overturn this orthodoxy. (shrink)

Here, we analyse some recent applications of set theory to topology and argue that set theory is not only the closed domain where mathematics is usually founded, but also a flexible framework where imperfect intuitions can be precisely formalized and technically elaborated before they possibly migrate toward other branches. This apparently new role is mostly reminiscent of the one played by other external fields like theoretical physics, and we think that it could contribute to revitalize the interest in (...) set theory in the future. (shrink)

According to higher-order theories of consciousness, a mental state is conscious only when represented by another mental state. Higher-order theories must predict there to be some brain areas (or networks of areas) such that, because they produce (the right kind of) higher-order states, the disabling of them brings about deficits in consciousness. It is commonly thought that the prefrontal cortex produces these kinds of higher-order states. In this paper, I first argue that this is likely correct, (...) meaning that, if some higher-order theory is true, prefrontal lesions should produce dramatic deficits in visual consciousness. I then survey prefrontal lesion data, looking for evidence of such deficits. I argue that no such deficits are to be found, and that this presents a compelling case against higher-order theories. (shrink)

In the paper we will employ set theory to study the formal aspects of quantum mechanics without explicitly making use of space-time. It is demonstrated that von Neuman and Zermelo numeral sets, previously efectively used in the explanation of Hardy’s paradox, follow a Heisenberg quantum form. Here monadic union plays the role of time derivative. The logical counterpart of monadic union plays the part of the Hamiltonian in the commutator. The use of numerals and monadic union in the classical (...) probability resolution of Hardy’s paradox [1] is supported with the present derivation of a commutator for sets. (shrink)

In this article Russell’s paradox and Cantor’s paradox resolved successfully using intuitionistic logic with restricted modus ponens rule.

In this book set theory INC# based on intuitionistic logic with restricted modus ponens rule is proposed. It proved that intuitionistic logic with restricted modus ponens rule can to safe Cantor naive set theory from a triviality.

In this book the authors introduce and study the following notions: Neutrosophic Crisp Points, Neutrosophic Crisp Relations, Neutrosophic Crisp Sets, Neutrosophic Set Generated by (Characteristic Function), alpha-cut Level for Neutrosophic Sets, Neutrosophic Crisp Continuous Function, Neutrosophic Crisp Compact Spaces, Neutrosophic Crisp Nearly Open Sets, Neutrosophic Crisp Ideals, Neutrosophic Crisp Filter, Neutrosophic Crisp Local Functions, Neutrosophic Crisp Sets via Neutrosophic Crisp Ideals, Neutrosophic Crisp L-Openness and Neutrosophic Crisp L-Continuity, Neutrosophic Topological Region, Neutrosophic Closed Set and Neutrosophic Continuous Function, etc. They compute (...) the distances between neutrosophic sets and extend it to Neutrosophic Hesitancy Degree. The authors also generalize the Crisp Topological Space and Intuitionistic Topological Space to the notion of Neutrosophic Crisp Topological Space. At the end, they present applications to Neutrosophic Database, and show a security scheme based on Public Key Infrastructure (PKI) using Neutrosophic Logic Manipulation. The authors utilize neutrosophic sets in order to analyze social networks data conducted through learning activities, and for the Geographical Information Systems (GIS) they employ fundamental concepts and properties of a Neutrosophic Spatial Region. Keywords: Neutrosophic Crisp Points; Neutrosophic Crisp Relations; Neutrosophic Crisp Sets; Neutrosophic Crisp Continuous Function; Neutrosophic Crisp Compact Spaces; Neutrosophic Crisp Nearly Open Sets; Neutrosophic Crisp Ideals; Neutrosophic Crisp Filter; Neutrosophic Crisp Local Functions; Neutrosophic Crisp Sets via Neutrosophic Crisp Ideals; Neutrosophic Crisp L-Openness and Neutrosophic Crisp L-Continuity; Neutrosophic Topological Region; Neutrosophic Closed Set and Neutrosophic Continuous Function; Neutrosophic Hesitancy Degree; Neutrosophic Crisp Topological Space; Neutrosophic Database; Neutrosophic Logic Manipulation; Neutrosophic Spatial Region. (shrink)

The purpose of this paper is to introduce new types of neutrosophic crisp sets with three types 1, 2, 3. After given the fundamental definitions and operations, we obtain several properties, and discussed the relationship between neutrosophic crisp sets and others. Also, we introduce and study the neutrosophic crisp point and neutrosophic crisp relations. Possible applications to database are touched upon.

Proponents of the higher‐order (HO) theory of consciousness (e.g., Lau and Rosenthal) have recently appealed to brain lesion evidence to support their thesis that mental states are conscious when and only when represented by other mental states. This article argues that this evidence fails to support HO theory, doing this by first determining what kinds of conscious deficit should result when HO state‐producing areas are damaged, then arguing that these kinds of deficit do not occur in the (...) studies to which HO theorists appeal. The article also develops an apparatus that can be used to evaluate whether other lesion evidence confirms or disconfirms HO theory. (shrink)

Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vopěnka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the (...) axioms of ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent model of ZF. The logic (PS3 ,*) coincides (up to language) with da Costa and D'Ottaviano logic J3, a 3-valued paraconsistent logic that have been proposed independently in the literature by several authors and with different motivations such as CluNs, LFI1 and MPT. We propose in this paper a family of algebraic models of ZFC based on LPT0, another linguistic variant of J3 introduced by us in 2016. The semantics of LPT0, as well as of its first-order version QLPT0, is given by twist structures defined over Boolean agebras. From this, it is possible to adapt the standard Boolean-valued models of (classical) ZFC to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. We argue that the implication operator of LPT0 is more suitable for a paraconsistent set theory than the implication of PS3, since it allows for genuinely inconsistent sets w such that [(w = w)] = 1/2 . This implication is not a 'reasonable implication' as defined by Löwe and Tarafder. This suggests that 'reasonable implication algebras' are just one way to define a paraconsistent set theory. Our twist-valued models are adapted to provide a class of twist-valued models for (PS3,*), thus generalizing Löwe and Tarafder result. It is shown that they are in fact models of ZFC (not only of ZF). (shrink)

In this article, we will describe higher order thought theories of consciousness. Then we will describe some examples from synesthesia. Finally, we will explain why the latter may be relevant to the former.

The success of set theory as a foundation for mathematics inspires its use in artificial intelligence, particularly in commonsense reasoning. In this survey, we briefly review classical set theory from an AI perspective, and then consider alternative set theories. Desirable properties of a possible commonsense set theory are investigated, treating different aspects like cumulative hierarchy, self-reference, cardinality, etc. Assorted examples from the ground-breaking research on the subject are also given.

Relevance logic has become ontologically fertile. No longer is the idea of relevance restricted in its application to purely logical relations among propositions, for as Dunn has shown in his (1987), it is possible to extend the idea in such a way that we can distinguish also between relevant and irrelevant predications, as for example between “Reagan is tall” and “Reagan is such that Socrates is wise”. Dunn shows that we can exploit certain special properties of identity within the context (...) of standard relevance logic in a way which allows us to discriminate further between relevant and irrelevant properties, as also between relevant and irrelevant relations. The idea yields a family of ontologically interesting results concerning the different ways in which attributes and objects may hang together. Because of certain notorious peculiarities of relevance logic, however,1 Dunn’s idea breaks down where the attempt is made to have it bear fruit in application to relations among entities which are of homogeneous type. (shrink)