We investigate the mathematics of a model of the human mind which has been proposed by the psychologist Jens Mammen. Mathematical realizations of this model consists of what the first author (A.T.) has called Mammen spaces, where a Mammen space is a triple in the Baumgartner–Laver model.Finally, consequences for psychology are discussed.

Yujin Nagasawa presents a new, stronger version of perfect being theism, the conception of God as the greatest possible being. Nagasawa argues that God should be understood, not as omniscient, omnipotent, and omnibenevolent, but rather as a being that has the maximal consistent set of knowledge, power, and benevolence.

Anselmian theists claim (a) that there is a being than which none greater can be conceived; and (b) that it is knowable on purely—solely, entirely—a priori grounds that there is a being than which none greater can be conceived. In this paper, I argue that Anselmian Theism gains traction by conflating different interpretations of the key description ‘being than which no greater can be conceived’. In particular, I insist that it is very important to distinguish between ideal excellence and (...) class='Hi'>maximal possible excellence. At the end of my paper, I illustrate the importance of this distinction by applying my discussion to the recent defence of Anselmian Theism in Nagasawa (Philos Q 58:577–591, 2008). (shrink)

Police officers around the world must often select between equally unappealing, uncertain courses of action in an attempt to achieve the best outcome. Despite the immense importance of such decisions, there remains a lack of understanding in the study of individual differences in police decision-making. Here, using a sample of senior police officers recruited from decision-making training events across the United Kingdom (n = 96), we used the Least-worst Uncertain Choice Inventory For Emergency Responses (LUCIFER) to measure the effect of (...) maximization on both domain-specific (police) and domain-general (military) decisions. In line with a wealth of research on traditional “consumer” decisions, we found that police officers who were “maximizers” found decisions more difficult. Gender and previous military experience also influenced the process of decision-making. Specifically, police officers with military experience took more time to assess the situation but were faster to choose a course of action and commit to it. Female police officers also were slower to assess the situation. As recent events show, the outcomes of police decisions have significant consequences for the public, the officers involved, the police force as a whole, and the wider population, yet psychological research has yet to fully explore the role of individual differences in how such decisions are made. While this study does not seek to identify factors associated with “good” or “better” decision-makers, it provides strong support for the need to factor in perspectives of the individual when creating theory, or applied tools, in support of police decision-making. (shrink)

Fundamental concrete particulars are needed to explain facts about non-fundamental concrete particulars. However, the former can only play this explanatory role if they are properly discernible from the latter. Extant theories of how to discern fundamental concreta primarily concern mereological structure. Those according to which fundamental concreta can bear, but not be, proper parts are motivated by the possibilities that all concreta bear proper parts and that some properties of wholes are not fixed by the properties of their proper parts. (...) In response, theorists who hold that the fundamental concrete particulars can be proper parts may appeal to the possibility that every concrete particular is a proper part—that there is no mereologically maximal whole world, as well as to the intuition that fundamental concreta are qualitatively homogeneous “blocks” from which non-fundamental concreta are built. After motivating the plausibility of gunk and junk, the present essay proposes a constraint on fundamental concrete particulars based on topology instead of mereology: the fundamental concrete particulars must be appropriately connected. This constraint has the unique advantage of consistency with each of gunk, emergence, junk, and the building block intuition. (shrink)

Based on a question posed by global philosophy of religion project regarding the absence of literal attribution of omnibenevolence to God in the Qur’ān, this paper aims to examine how to understand perfect goodness in Islam. I will first discuss the concept of perfect goodness and suggest that perfect goodness is not an independent attribute on its own and it is predicated on other moral attributes of God without which the concept of perfect goodness could hardly (...) be understood. I will examine perfect goodness by a specific emphasis on the attribute of justice as one of the conditions to be satisfied by a perfectly morally good being. In so doing, I will appeal to the distinctions made among great-making properties by Daniel Hill, and Al-Ghazālī’s definition of justice by applying them to God’s moral attributes. I will argue that justice has a crucial role in maximality-optimality balance between great-making properties and it seems quite difficult, if not impossible, to conceive of goodness without justice. Having said that, I will claim that the conceptual frame I suggest sheds light on why omnibenevolence is not literally attributed to God in the Qur’ān. Then, I will briefly show how the divine attributes mentioned in the Qur’ān and the discussions about divine names and attributes in the Islamic tradition supports the understanding of perfect goodness I defend. Consequently, I will try to show that far from indicating that the Islamic concept of God doesn’t involve perfect goodness, the Qur’ān establishes the proper meaning of perfect goodness by focusing on its constitutive attributes, and thus provides us with a sound conception of it. (shrink)

We present three examples of topological semantics for intuitionistic modal logic with one modal operator □. We show that it is possible to treat neighborhood models, introduced earlier, as topological or multi-topological. From the neighborhood point of view, our method is based on differences between properties of minimal and maximal neighborhoods. Also we propose transformation of multitopological spaces into the neighborhood structures.

The aim of this paper is to present a new perspective under which branching-time semantics can be viewed. The set of histories (maximal linearly ordered sets) in a tree structure can be endowed in a natural way with a topological structure. Properties of trees and of bundled trees can be expressed in topological terms. In particular, we can consider the new notion of topological validity for Ockhamist temporal formulae. It will be proved that this notion of validity is equivalent (...) to validity with respect to bundled trees. (shrink)

In a 1987 paper Thomas Morris introduced the phrase ‘perfect being theology’ and argued that in our efforts to construct an adequate theistic conception of God the most fruitful procedure will be for us to engage in reflection and dialogue about what a maximally perfect being would be like . To some of us that approach seems so obvious as to be without a significant alternative, but there are other approaches that have been followed – such as working (...) up a conception of God solely from the statements in a purported repository of divine revelation, such as the Bible . Another approach has been to study a wide range of religious scriptures to see what common conception of the divine can be found in them or most plausibly constructed from them. Those approaches, Morris said, have left us with something unsatisfactorily incomplete, static, and perhaps self-contradictory. Morris argued, a la St. Anselm and René Descartes, that the notion of a maximally perfect being is the north star that has been at the heart of the theistic traditions of the world, and it is that notion that we must explore and elaborate in order that we might construct an adequate conception of God. I agree. (shrink)

Recently the cosmological evolution of the universe has been considered where 3-dimensional spatial topology undergone drastic changes. The process can explain, among others, the observed smallness of the neutrino masses and the speed of inflation. However, the entire evolution is perfectly smooth from 4-dimensional point of view. Thus the raison d’être for such topology changes is the existence of certain non-standard 4-smoothness on R4 already at very early stages of the universe. We show that the existence of such smoothness can (...) be understood as a byproduct of the quantumness of the origins of the universe. Our analysis is based on certain formal aspects of the quantum mechanical lattice of projections of infinite dimensional Hilbert spaces where formalization reaches the level of models of axiomatic set theory. (shrink)

This paper is about David Gauthier’s concept of constrained maximization. Attending to his most detailed and careful account, I try to say how constrained maximization works, and how it might be changed to work better. In section I, that detailed account is quoted along with amplifying passages. Difficulties of interpretation are explained in section II. An articulation, a spelling out, of Gauthier's account is offered in section III to deal with these difficulties. Next, in section IV, constrained maximization thus articulated (...) is tested on several choice problems and shown to be seriously wanting. It appears that there are prisoners’ dilemmas in which constrained maximizers would not cooperate to mutual advantage, but would interact sub-optimally just as straight-maximizers would. ‘Coordination problems’ are described with which constrained maximizers might, especially if transparent to one another, not be able to cope–problems in which they might not be able to make up their minds to do anything at all. And I prove that there are prisoners’ dilemmas that, though possible for real agents and for straight maximizers, are not possible for constrained maximizers, so that agents’ internalising dispositions of constrained maximization could not be of help in connection with such possibly impending dilemmas. Taking constrained maximization as it stands, there are many problems for which it does not afford the ‘moral solutions’ with which Gauthier would have it replace Hobbesian political ones. After displaying these shortcomings of constrained maximization as presently designed, I sketch, in section V, possible revisions that would reduce them, stressing that these revisions would not be cost-free. Whether finishing the job of fixing up and making precise constrained maximization would be worth the considerable trouble it would involve lies beyond the issues taken up in this paper. So, of course, do substantive comparisons of constrained maximization, perfected and made precise, and straight maximization. (shrink)

To generalize as in [7] the constructions of [2] and [5], let L be a nondegenerate join-semilattice with unit. With A · B = {x ∨ y ∈ L : x ∈ A, y ∈ B} and A⊥ = {y ∈ L : x ∨ y = 1 for all x ∈ A}, the structure , ·,∪, ⊥ , L, ∅) is the algebra of subsets for L. Let R be the maximal ideals of L. Interpreting L as the (...) totality of elementary situations, and 1 ∈ L as the impossible one , the members of R will be called realizations, and R itself – the logical space associated with L. (shrink)

IMTL logic was introduced in [12] as a generalization of the infinitely-valued logic of Lukasiewicz, and in [11] it was proved to be the logic of left-continuous t-norms with an involutive negation and their residua. The structure of such t-norms is still not known. Nevertheless, Jenei introduced in [20] a new way to obtain rotation-invariant semigroups and, in particular, IMTL-algebras and left-continuous t-norm with an involutive negation, by means of the disconnected rotation method. In order to give an algebraic interpretation (...) to this construction, we generalize the concepts of perfect, bipartite and local algebra used in the classification of MV-algebras to the wider variety of IMTL-algebras and we prove that perfect algebras are exactly those algebras obtained from a prelinear semihoop by Jenei's disconnected rotation. We also prove that the variety generated by all perfect IMTL-algebras is the variety of the IMTL-algebras that are bipartite by every maximal filter and we give equational axiomatizations for it. (shrink)

The present paper gives a topological solution to representability problems related to multi-utility, in the field of Decision Theory. Necessary and sufficient topologies for the existence of a semicontinuous and finite Richter–Peleg multi-utility for a preorder are studied. It is well known that, given a preorder on a topological space, if there is a lower semicontinuous Richter–Peleg multi-utility, then the topology of the space must be finer than the Upper topology. However, this condition fails to be sufficient. Instead of (...) search for properties that must be satisfied by the preorder, we study finer topologies which are necessary or/and sufficient for the existence of semicontinuous representations. We prove that Scott topology must be contained in the topology of the space in case there exists a finite lower semicontinuous Richter–Peleg multi-utility. However, the existence of this representation cannot be guaranteed. A sufficient condition is given by means of Alexandroff’s topology, for that, we prove that more order implies less Alexandroff’s topology, as well as the converse. Finally, the paper is implemented with a topological study of the maximal elements. (shrink)

The class of points in a set-presented formal topology is a set, if all points are maximal. To prove this constructively a strengthening of the dependent choice principle to infinite well-founded trees is used.

AssumeGis a group definable in a modelMof a stable theoryT. We prove that the semigroupSG(M) of completeG-types overMis an inverse limit of some semigroups type-definable inMeq. We prove that the maximal subgroups ofSG(M) are inverse limits of some definable quotients of subgroups ofG. We consider the powers of types in the semigroupSG(M) and prove that in a way every type inSG(M) is profinitely many steps away from a type in a subgroup ofSG(M).

Abstract. The aim of this paper is to present a topological method for constructing discretizations (tessellations) of conceptual spaces. The method works for a class of topological spaces that the Russian mathematician Pavel Alexandroff defined more than 80 years ago. Alexandroff spaces, as they are called today, have many interesting properties that distinguish them from other topological spaces. In particular, they exhibit a 1-1 correspondence between their specialization orders and their topological structures. Recently, a special type of Alexandroff spaces was (...) used by Ian Rumfitt to elucidate the logic of vague concepts in a new way. According to his approach, conceptual spaces such as the color spectrum give rise to classical systems of concepts that have the structure of atomic Boolean algebras. More precisely, concepts are represented as regular open regions of an underlying conceptual space endowed with a topological structure. Something is subsumed under a concept iff it is represented by an element of the conceptual space that is maximally close to the prototypical element p that defines that concept. This topological representation of concepts comes along with a representation of the familiar logical connectives of Aristotelian syllogistics in terms of natural settheoretical operations that characterize regular open interpretations of classical Boolean propositional logic. In the last 20 years, conceptual spaces have become a popular tool of dealing with a variety of problems in the fields of cognitive psychology, artificial intelligence, linguistics and philosophy, mainly due to the work of Peter Gärdenfors and his collaborators. By using prototypes and metrics of similarity spaces, one obtains geometrical discretizations of conceptual spaces by so-called Voronoi tessellations. These tessellations are extensionally equivalent to topological tessellations that can be constructed for Alexandroff spaces. Thereby, Rumfitt’s and Gärdenfors’s constructions turn out to be special cases of an approach that works for a more general class of spaces, namely, for weakly scattered Alexandroff spaces. This class of spaces provides a convenient framework for conceptual spaces as used in epistemology and related disciplines in general. Alexandroff spaces are useful for elucidating problems related to the logic of vague concepts, in particular they offer a solution of the Sorites paradox (Rumfitt). Further, they provide a semantics for the logic of clearness (Bobzien) that overcomes certain problems of the concept of higher2 order vagueness. Moreover, these spaces help find a natural place for classical syllogistics in the framework of conceptual spaces. The crucial role of order theory for Alexandroff spaces can be used to refine the all-or-nothing distinction between prototypical and nonprototypical stimuli in favor of a more fine-grained gradual distinction between more-orless prototypical elements of conceptual spaces. The greater conceptual flexibility of the topological approach helps avoid some inherent inadequacies of the geometrical approach, for instance, the so-called “thickness problem” (Douven et al.) and problems of selecting a unique metric for similarity spaces. Finally, it is shown that only the Alexandroff account can deal with an issue that is gaining more and more importance for the theory of conceptual spaces, namely, the role that digital conceptual spaces play in the area of artificial intelligence, computer science and related disciplines. Keywords: Conceptual Spaces, Polar Spaces, Alexandroff Spaces, Prototypes, Topological Tessellations, Voronoi Tessellations, Digital Topology. (shrink)

According to Kant’s Universal Law Formula, maxims that cannot be conceived as universal laws denote duties of perfect obligation. In the recent literature, two versions of the Contradiction in Conception test have received the most attention. When acting on a maxim would violate a perfect duty, according to the Logical Contradiction Interpretation (LCI), universalizing the maxim would make it literally impossible to perform the action as described in the original maxim. According to the Practical Contradiction Interpretation (PCI), the (...) locus of the contradiction is as follows: the agent acts on the maxim in order to achieve some purpose, but were the agent’s maxim universalized, the purpose would be unattainable. Having examined the most widely accepted versions of both interpretations, I argue that i) PCI cannot generate contradictions for the maxims of any actions that violate perfect duties beyond those generated by LCI, ii) despite claims of its proponents, PCI cannot solve the vexing problem of relevant descriptions that Kant’s account faces, and iii) other arguments in favor of PCI are at best inconclusive. I therefore conclude that since LCI better coheres with the text, sympathetic interpreters should focus their efforts on LCI, not PCI. (shrink)

The topology optimization of computer communication network is studied based on improved genetic algorithm, a network optimization design model based on the establishment of network reliability maximization under given cost constraints, and the corresponding improved GA is proposed. In this method, the corresponding computer communication network cost model and computer communication network reliability model are established through a specific project, and the genetic intelligence algorithm is used to solve the cost model and computer communication network reliability model, respectively. It has (...) been proved that GA can solve the complex problems of computer working environment better, which is 80% higher than the general algorithm, and can select the optimal scheme pertinently. (shrink)

Maximizers in isolated Prisoner's Dilemmas are doomed to frustration. But in Braybrooke's view maximizers might do better in a series, securing Pareto-optimal arrangements if not from the very beginning, at least eventually. Given certain favourable special conditions, it can be shown according to Braybrooke and shown even without question-begging motivational or value assumptions, that in a series of Dilemmas maximizers could manage to communicate a readiness to reciprocate, generate thereby expectations of reciprocation, and so give rise to optimizing reciprocations which, (...) in due course, would reinforce expectations, the net result of all this being an increasingly stable practice to mutual benefit. In this way neighbours could learn to be good neighbours: they could learn to respect each other's property, to engage in reciprocal assistance, and perhaps even to make and keep promises covering a range of activities. So maximizers are, Braybrooke holds, capable of a society of sorts. Out of the ashes they might build something. But even under favourable conditions they could not build much and most conditions would defeat them almost entirely, for many-person Dilemmas whether isolated or in series would be quite beyond them, question-begging motivational assumptions aside. In settings at all like those in which we live, utility maximization assuming only ‘traditional’ motivation is self-defeating without remedy, and most certainly not an adequate basis for social life.' The probable inference from as a conception of individual rationality: for surely truly rational agents would be if not perfectly well designed at least better designed for communal living than utility maximizers would be assuming only ‘traditional’, non-question-begging values or motivation. Or, if this inference is denied and maximization is not challenged as a conception of rationality, we can conclude instead, Braybrooke has more recently suggested, that the advice, ‘Be rational,’ is incomplete, and when addressed to groups should be supplemented by, ‘And be somewhat ‘untraditional’ in your values; respect at least some rules that would make coordination and cooperation possible even in many-person interaction problems.’. (shrink)

L’objet de la présente recherche est d’interroger les mérites de la théorie utilitariste relative aux faillites personnelles, qui est dominante dans la littérature juridique depuis que le courant Law and Economics a gagné en importance. Nous démontrons que l’argumentation utilitariste, qui est assignée aux doctrines de l’acquittement des dettes personnelles par les spécialistes de Law and Economics travaillant dans le domaine du droit des faillites, soulève des questions normatives de taille et créée des tensions quand il s’agit de décrire le (...) phénomène. Ces deux problèmes invitent les théoriciens à proposer des modèles alternatifs pour décrire les conditions dans lesquelles nos systèmes juridiques soulagent les dettes des individus qui ne sont pas en mesure de les honorer. Dans notre ouvrage à paraître, The Virtue of Bankruptcy, nous embrassons ce projet. Ici, nous proposons une étape préliminaire à ce projet plus ambitieux en démontrant que les théories utilitaristes sur lesquelles s’appuient les contributions les plus importantes de la littérature sur la question à ce jour sont perfectibles lorsqu’il s’agit de comprendre les raisons et les circonstances dans lesquelles il est possible d’offrir un nouveau départ à ceux qui n’ont pas respecté leurs obligations financières. (shrink)

For each integer n ≥ 2, MVn denotes the variety of MV-algebras generated by the MV-chain with n elements. Algebras in MVn are represented as continuous functions from a Boolean space into a n-element chain equipped with the discrete topology. Using these representations, maximal subalgebras of algebras in MVn are characterized, and it is shown that proper subalgebras are intersection of maximal subalgebras. When A ∈ MV3, the mentioned characterization of maximal subalgebras of A can be given (...) in terms of prime filters of the underlying lattice of A, in the form that was conjectured by A. Monteiro. (shrink)

God is typically conceived as perfectly good and necessarily so, in two senses: in terms of always performing the best possible act and in terms of having maximal moral worth. Yet any being that freely performs the best act she can must be accorded greater moral worth for any such action than a being that does so necessarily. I conclude that any being that performs the best possible act of necessity cannot also have maximal moral worth, making the (...) concept of God’s perfect goodness incoherent. (shrink)

This chapter contains section titled: God and the Nature of Morality Perfect Goodness and Consequences Perfect Virtue and Moral Rules Maximal Greatness and Moral Admirability.

This paper explores a variety of perfect being theism that combines Yujin Nagasawa’s maximal God thesis with the view that God is not atemporal. We argue that the original maximal God thesis still implicitly relies on a “static” view of divine perfections. Instead, following the recent re-evaluation of divine immutability by analytic philosophers, we propose that thinking of divine great-making properties as fluctuating but nevertheless remaining maximal either for every time t or across all times strengthens (...) the original maximal God thesis. Furthermore, we show that “temporalising” Nagasawa’s maximal God and adopting what we call the fluctuating maximal God thesis provides more effective ways of responding to objections to perfect being theism, in particular, the argument from evil and some conceptual problems pertaining to the Incarnation. Finally, we demonstrate that our proposal is compatible with Christian Scriptures and coheres with numerous biblical passages better than Nagasawa’s original proposal does. (shrink)

In a series of influential articles published in the 1980s, Thomas Morris argued that the most promising approach to many issues in the philosophy of religion is “perfect being theology.” A philosopher who adopts it begins by construing God as a maximally perfect being and then fills the conception in by using his or her modal intuitions and intuitions concerning what properties are and are not perfections. While I am sympathetic with Morris’s program, two aspects seem problematic. More (...) justification is needed for construing God as a maximally perfect being, and the appeal to intuitions needs more support than Morris provides for it. I will comment on both difficulties. (shrink)

In this paper, I ask whether there is a defensible philosophical view according to which everybody is beautiful. I review two purely aesthetical versions of this claim. The No Standards View claims that everybody is maximally and equally beautiful. The Multiple Standards View encourages us to widen our standards of beauty. I argue that both approaches are problematic. The former fails to be aspirational and empowering, while the latter fails to be sufficiently inclusive. I conclude by presenting a hybrid ethical–aesthetical (...) view according to which everybody is beautiful in the sense that everybody can be perceived through a loving gaze. I show that this view is inclusive, aspirational and empowering, and authentically aesthetical. (shrink)

Rationalist models of divine agency typically ascribe perfect freedom to God, where this is understood as a freedom from external causal influences and non-rational influences, including desires or preferences not derived from reason alone. Paul Draper has recently developed a rationalist model of God’s agency on which God faces “hard choices” between options differing in moral and non-moral value. He argues that this model is preferable to rival rationalist models because it is compatible with God’s having significant freedom and (...) being maximally worthy of praise and gratitude. I argue that on an alternative model of divine agency, which rejects perfect freedom and holds that God makes hard choices on the basis of brute preferences, God would be more worthy of praise and gratitude. However, a probabilistic problem for theism which Draper identifies for his model also applies to the brute preference model. (shrink)

For each integer n ≥ 2, MVn denotes the variety of MV-algebras generated by the MV-chain with n elements. Algebras in MVn are represented as continuous functions from a Boolean space into a n-element chain equipped with the discrete topology. Using these representations, maximal subalgebras of algebras in MVn are characterized, and it is shown that proper subalgebras are intersection of maximal subalgebras. When A ∈ MV3, the mentioned characterization of maximal subalgebras of A can be given (...) in terms of prime filters of the underlying lattice of A, in the form that was conjectured by A. Monteiro. (shrink)

Maxims play a crucial role in Kant's ethical philosophy, but there is significant disagreement about what maxims are. In this two-part essay, I survey eight different views of Kantian maxims, presenting their strengths, and their weaknesses. Part I: Established Approaches, begins with Rüdiger Bubner's view that Kant took maxims to be what ordinary people of today take them to be, namely pithily expressed precepts of morality or prudence. Next comes the position, most associated with Rüdiger Bittner and Otfried Höffe, that (...) maxims are Lebensregeln, or 'life-rules'– quite general rules for how to conduct oneself based on equally general outlooks on how the world is. These first two interpretations make sense of Kant's claim, made in his anthropological and pedagogical writings, that we have to learn how to act on maxims, but they become less plausible in light of Kant's probable view that people always act on maxims – after all, how can people learn how to act on something they always act on anyway? The next two views, each advanced, at different times, by Onora O'Neill, make better sense of the fact that people always act on maxims, for they hold that maxims are intentions – either specific intentions, such as 'to open the door', or general intentions, such as 'to make guests feel welcome'– and it is perfectly sensible to claim that people always act on intentions. However, they face the same problem as the two previous views, which is that if people always act on maxims, what sense does it make to say they also have to learn how to act on them? Henry Allison, the main representative of the fifth view, claims, on the basis of Kant's doctrine of the 'highest maxim', that maxims are principles organized hierarchically, such that an agent endorses one maxim because she endorses a more general maxim. Unfortunately for Allison, there is little direct textual support for his claim that maxims are organized hierarchically. (shrink)

Gauthier's argument for constrained maximization, presented inMorals by Agreement, is perfected by taking into account the possibility of accidental exploitation and discussing the limitations on the values of the parameters which measure the translucency of the actors. Gauthier's argument is nevertheless shown to be defective concerning the rationality of constrained maximization as a strategic choice. It can be argued that it applies only to a single actor entering a population of individuals who are themselves not rational actors but simple rule-followers. (...) A proper analysis of the strategic choice situation involving two rational actors who confront each other shows that constrained maximization as the choice of both actors can only result under very demanding assumptions. (shrink)

The aim of this article is to determine and analyze the meaning of the transitio and the posset that not only enable the radical modal experience of the Amor Dei intellectualis but which are also central features in the attainment of human perfection and of the highest knowledge. I wish to answer the following questions. What power is attributable to the Amor Dei intellectualis? In other words, what is the power that corresponds to human perfection and to the possession of (...) blessedness? And why does Spinoza explain this relation in terms of love? In this article, I intend to give a possible definition of human nature, a definition whose determination implies a maximal affirmation of modal power. At this point more questions arise: by attributing this maximal power to human nature, are we underlining a specific necessity, i.e., a constitutive relation capable of giving a specific form to this power and to human nature? Does the third kind of knowledge imply modifications in human nature itself, e.g., in the body or soul? Finally, does the correlation between the third kind of knowledge and the Amor Dei intellectualis allow for the determination of a model of human nature that surpasses the one given by Spinoza in the preface to part four of the Ethics? (shrink)

This book examines the divine nature in terms of maximal greatness. It investigates each attribute associated with maximal greatness - omnipotence, omniscience, perfect goodness, eternity, and beauty, arguing that maximal greatness is necessary and sufficient for divinity.

By the Galvin–Mycielski–Solovay theorem, a subset X of the line has Borel’s strong measure zero if and only if $M+X\neq \mathbb {R}$ for each meager set M.A set $X\subseteq \mathbb {R}$ is meager-additive if $M+X$ is meager for each meager set M. Recently a theorem on meager-additive sets that perfectly parallels the Galvin–Mycielski–Solovay theorem was proven: A set $X\subseteq \mathbb {R}$ is meager-additive if and only if it has sharp measure zero, a notion akin to strong measure zero.We investigate the (...) validity of this result in Polish groups. We prove, e.g., that a set in a locally compact Polish group admitting an invariant metric is meager-additive if and only if it has sharp measure zero. We derive some consequences and calculate some cardinal invariants. (shrink)

At the turn of the 21st century, topology, the mathematical study of spatial properties that remain the same under the continuous deformation of objects, has come to invest all fields of aesthetics and culture. In particular, the algebraic topology of continuity has added to the digital realm of binary information, the on and off states of 0s and 1s, an invariant property, which now governs the relation between different forms of data. As this invariant function of continual transformation has entered (...) the field of automated computation, the culture of binary digits has shifted towards a new level of calculation derived from the introduction of temporal quantities into finite sets of algorithmic instructions and parameters. This new level of topological computation, it will be argued, defines new operative procedures of control, constantly adding axioms at the limit of calculation through an invariant function that establishes a smooth or uninterrupted connectivity between distinct data. The establishment of a continual function between distinct forms of data is based on homeomorphism or topological isomorphism between data objects, of which parametricism, as the new global style for architecture and design, is a perfect example. (shrink)

In this article, we present a new logical framework to think about surprise. This research does not just aim to better understand, model and predict human behaviour, but also attempts to provide tools for implementing artificial agents. Moreover, these artificial agents should then also be able to reap the same epistemic benefits from surprise as humans do. We start by discussing the dominant literature regarding propositional surprise and explore its shortcomings. These shortcomings are of both an empirical and a conceptual (...) nature. Next, we propose a philosophical solution to the problems that ail these systems, based on the notion of issue of epistemic interest. Finally, we give a formal framework to think about surprise. More specifically, we develop a probabilistic dynamic epistemic logic that succeeds at formalizing the relevant philosophical concepts. This will be done through an issue management system grounded in topology. As an added bonus, the additional expressive power allows us to capture a richer variety of scenarios, and it also enables a more careful analysis of said scenarios. (shrink)

This paper gives a formalization of general topology in second-order arithmetic using countably based MF spaces. This formalization is used to study the reverse mathematics of general topology. For each poset P we let MF denote the set of maximal filters on P endowed with the topology generated by {Np | p ∈ P}, where Np = {F ∈ MF | p ∈ F}. We define a countably based MF space to be a space of the form MF for (...) some countable poset P. The class of countably based MF spaces includes all complete separable metric spaces as well as many nonmetrizable spaces. The following reverse mathematics results are obtained. The proposition that every nonempty Gδ subset of a countably based MF space is homeomorphic to a countably based MF space is equivalent to [Formula: see text] over ACA0. The proposition that every uncountable closed subset of a countably based MF space contains a perfect set is equivalent over [Formula: see text] to the proposition that [Formula: see text] is countable for all A ⊆ ℕ. The proposition that every regular countably based MF space is homeomorphic to a complete separable metric space is equivalent to [Formula: see text] over [Formula: see text]. (shrink)

Effective inseparability of pairs of sets is an important notion in logic and computer science. We study the effective inseparability of sets which appear as index sets of subsets of an effectively given topological T0-space and discuss its consequences. It is shown that for two disjoint subsets X and Y of the space one can effectively find a witness that the index set of X cannot be separated from the index set of Y by a recursively enumerable set, if X (...) intersects the topological closure of an effectively enumerable subset of Y. As a consequence of a more general parametric inseparability result a theorem of Rice-Shapiro type is obtained. Moreover, under some additional requirements it follows that nonopen subsets have productive index sets. This implies a generalized Rice theorem: Connected spaces have only trivial completely recursive subsets. As application some decision problems in computable analysis and domain theory are studied. It follows that the complement of the halting problem can be reduced to the problem to decide of a number whether it is a computable irrational. The same is true for the problems to decide whether two numbers are equal, whether one is not greater than the other, and whether a number is equal to a given number. In the case of an effectively given continuous complete partial order the complexity of the last problem depends on whether the given element is the smallest element, in which case the complement of the halting problem is reducible to it, whether it is a base element and maximal, then the decision problem is recursively isomorphic to the halting problem, or whether it is none of these. In this case, both the halting problem and its complement are reducible to the problem. The same is true in nontrivial cases for the problems whether an element belongs to the basis, whether two elements of the partial order are equal, or whether one approximates the other. In general, for any nonempty proper subset of the partial order either the halting problem or its complement can be reduced to the membership problem of the subset. (shrink)

The efficiency argument for profit maximization says that corporations and their managers should maximize profits because this is the course of action that will lead to an ‘economically efficient’ or ‘welfare maximizing’ outcome. In this paper, I argue that the fundamental problem with this argument is not that markets in the real world are less than perfect, but rather that the argument does not properly acknowledge the personal sphere. Morality allows each of us a sphere in which we are (...) free to pursue our personal interests, even if these are not optimal from the social point of view. But the efficiency argument does not come to terms with this feature of social life. (shrink)

We think of the world around us as divided into physical objects like toasters and daisies, rather than solely as a smear of properties like yellow and smooth. How do we single out these objects? One theory of object concepts uses part‐of relations and relations of connectedness. According to this proposal, an object is a connected spatial item of maximal extent: Any other connected item that overlaps (i.e., shares a part with) the object must be a part of that (...) object. This article reports four experiments that test this proposal. Participants see descriptions or diagrams of spatial items that vary across trials in their relative positions. In separate experiments, participants decide whether the items are physical objects, whether they are wholes, or how many objects are present. All experiments find support for connectedness as a contributor to object status, but they find little support for maximality. The results suggest that maximality is not a necessary feature of wholes or of objects. (shrink)

We investigate uncountable maximal antichains of perfect trees and of splitting trees. We show that in the case of perfect trees they must have size of at least the dominating number, whereas for splitting trees they are of size at least \\), i.e. the covering coefficient of the meager ideal. Finally, we show that uncountable maximal antichains of superperfect trees are at least of size the bounding number; moreover we show that this is best possible.

We take the well-known intuitionistic modal logic of Fischer Servi with semantics in bi-relational Kripke frames, and give the natural extension to topological Kripke frames. Fischer Servi’s two interaction conditions relating the intuitionistic pre-order with the modal accessibility relation generalize to the requirement that the relation and its inverse be lower semi-continuous with respect to the topology. We then investigate the notion of topological bisimulation relations between topological Kripke frames, as introduced by Aiello and van Benthem, and show that their (...) topology-preserving conditions are equivalent to the properties that the inverse relation and the relation are lower semi-continuous with respect to the topologies on the two models. The first main result is that this notion of topological bisimulation yields semantic preservation w.r.t. topological Kripke models for both intuitionistic tense logics, and for their classical companion multi-modal logics in the setting of the Gödel translation. After giving canonical topological Kripke models for the Hilbert-style axiomatizations of the Fischer Servi logic and its classical companion logic, we use the canonical model in a second main result to characterize a Hennessy–Milner class of topological models between any pair of which there is a maximal topological bisimulation that preserve the intuitionistic semantics. (shrink)

Patrick Grim and Einar Duenger Bohn have recently argued that there can be no perfectly knowing Being. In particular, they urge that the object of omniscience is logically absurd (Grim) or requires an impossible maximal point of all knowledge (Bohn). I argue that, given a more classical notion of omniscience found in Aquinas and Augustine, we can shift the focus of perfect knowledge from what that being must know to the mode of that being’s understanding. Since Grim and (...) Bohn focus on the object rather than mode of God’s knowledge, this classical approach to omniscience undermines their objections. (shrink)