We propose a set theory, called NRFST, in which the Cantorian axiom of infinity is negated, and a new notion of infinity is introduced via non standard methods, i. e. via adequate notions of standard and internal, two unary predicates added to the language of ZF. After some initial results on NRFST, we investigate its relative consistency with respect to ZF and Kawai's WNST.

In this paper we extend to non-classical set theories the standard strategy of proving independence using Boolean-valued models. This extension is provided by means of a new technique that, combining algebras (by taking their product), is able to provide product-algebra-valued models of set theories. In this paper we also provide applications of this new technique by showing that: (1) we can import the classical independence results to non-classical set theory (as an example we prove the independence of $\mathsf (...) {CH}$ ); and (2) we can provide new independence results. We end by discussing the role of non-classical algebra-valued models for the debate between universists and multiversists and by arguing that non-classical models should be included as legitimate members of the multiverse. (shrink)

In ${\mathbf{H}}$ , a set theory with the comprehension principle within Łukasiewicz infinite-valued predicate logic, we prove that a statement which can be interpreted as “there is an infinite descending sequence of initial segments of ω” is truth value 1 in any model of ${\mathbf{H}}$ , and we prove an analogy of Hájek’s theorem with a very simple procedure.

A theoretical development is carried to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then combined into a “geometric technique” used to prove several results about countable non-standard models of set theory. In particular, back-and-forth constructions are carried out to establish various generalizations and refinements of Friedman’s theorem on the existence of rank-initial embeddings between countable non- (...) class='Hi'>standard models of the fragment \ + \-Separation of \; and Gaifman’s technique of iterated ultrapowers is employed to show that any countable model of \ can be elementarily rank-end-extended to models with well-behaved automorphisms whose sets of fixed points equal the original model. These theoretical developments are then utilized to prove various results relating self-embeddings, automorphisms, their sets of fixed points, strong rank-cuts, and set theories of different strengths. Two examples: The notion of “strong rank-cut” is characterized in terms of the theory \, and in terms of fixed-point sets of self-embeddings. (shrink)

Non-well-founded set theories allow set-theoretic exotica that standard ZFC will not allow, such as a set that has itself as its sole member. We can distinguish plenitudinous non-well-founded set theories, such as Boffa set theory, that allow infinitely many such sets, from restrictive theories, such as Finsler-Aczel or AFA, that allow exactly one. Plenitudinous non-well-founded set theories face a puzzle: nothing seems to explain the identity or distinctness of various of the sets they countenance. In this paper I (...) aim to sharpen this puzzle, make clear who it does and does not apply to and, ultimately, to argue in favor of a plenitudinous theory like Boffa. (shrink)

We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be in manifest (...) conflict with each other. However, our conclusion is that it is very difficult to see any real difference between the two. We analyze a phenomenon we call internal categoricity which extends the familiar categoricity results of second order logic to Henkin models and show that set theory enjoys the same kind of internal categoricity. Thus the existence of non-standard models, which is usually taken as a property of first order set theory, and categoricity, which is usually taken as a property of second order axiomatizations, can coherently coexist when put into their proper context. We also take a fresh look at complete second order axiomatizations and give a hierarchy result for second order characterizable structures. Finally we consider the problem of existence in mathematics from both points of view and find that second order logic depends on what we call large domain assumptions, which come quite close to the meaning of the axioms of set theory. (shrink)

Paraconsistent logics are logics that can be used to base inconsistent but non-trivial systems. In paraconsistent set theories, we can quan- tify over sets that in standard set theories, if consistent, would lead to contradictions, such as the Russell set, R = fx : x =2 xg. Quasi-set theories are mathematical systems built for dealing with collections of indiscernible elements. The basic motivation for the development of quasi-set theories came from quantum physics, where indiscernible entities need to be considered. (...) Usually, the way of dealing with indiscernible objects within clas- sical logic and mathematics is by restricting them to certain structures, in a way so that the relations and functions of the structure are not sufficient to individuate the objects; in other words, such structures are not rigid. In quantum physics, this idea appears when symmetry conditions are introduced, say by choosing symmetric and anti-symmetric functions in the relevant Hilbert spaces. But in standard mathematics, such as that built in Zermelo-Fraenkel set theory, any structure can be extended to a rigid structure. That means that, although we can deal with certain objects as they were indiscernible, we realize that from out- side of these structures these objects are no more indiscernible, for they can be individualized in the extended rigid structures: ZF is a theory of individuals, distinguishable objects. In quasi-set theory, it seems that there are structures that cannot be extended to rigid ones, so it seems that they provide a natural mathematical framework for expressing quantum facts without certain symmetry suppositions. There may be situations, however, in which we may need to deal with inconsistent bits of infor- mation in a quantum context, even if these informations are concerned with ways of speech. Furthermore, some authors think that superposi- tions may be understood in terms of paraconsistent logics, and even the notion of complementarity was already treated by such a means. This is, apparently, a nice motivation to try to merge these two frameworks. In this work, we develop the technical details, by basing our quasi-set theory in the paraconsistent system C1. We also elaborate a new hierarchy of paraconsistent calculi, the paraconsistent calculi with indiscernibility. For the finalities of this work, some philosophical questions are outlined, but this topic is left to a future work. (shrink)

We describe an explicit construction of algebraic models for theories with non-standard elements either with classical or constructive logic. The corresponding truthvalue algebra in our construction is a complete algebra of subsets of some concrete decidable set. This way we get a quite finitistic notion of true which reflects a notion of the deducibility of a given theory. It enables us to useconstructive, proof-theoretical methods for theories with non-standard elements. It is especially useful in the case of (...) theories with constructive logic where algorithmic properties are essential. (shrink)

Non-standard models were introduced by Skolem, first for set theory, then for Peano arithmetic. In the former, Skolem found support for an anti-realist view of absolutely uncountable sets. But in the latter he saw evidence for the impossibility of capturing the intended interpretation by purely deductive methods. In the history of mathematics the concept of a nonstandard model is new. An analysis of some major innovations–the discovery of irrationals, the use of negative and complex numbers, the modern concept (...) of function, and non-Euclidean geometry–reveals them as essentially different from the introduction of non-standard models. Yet, non-Euclidean geometry, which is discussed at some length, is relevant to the present concern; for it raises the issue of intended interpretation. The standard model of natural numbers is the best candidate for an intended interpretation that cannot be captured by a deductive system. Next, I suggest, is the concept of a wellordered set, and then, perhaps, the concept of a constructible set. One may have doubts about a realistic conception of the standard natural numbers, but such doubts cannot gain support from non-standard models. Attempts to utilize non-standard models for an anti-realist position in mathematics, which appeal to meaning-as-use, or to arguments of the kind proposed by Putnam, fail through irrelevance, or lead to incoherence. Robinson’s skepticism, on the other hand, is a coherent position, though one that gives up on providing a detailed philosophical account. The last section enumerates various uses of non-standard models. (shrink)

The first edition of this now classical work appeared in 1953, the second heavily revised edition in 1961; this most recent edition is a revision in detail only of the previous one. The book is divided into three parts, the first two dealing with finite and infinite sets, infinite cardinals and their arithmetic, and related remarks on non-standard mathematics and the equivalence of various definitions of finitude. The third part considers ordered sets and isomorphism types, the special case of (...) linearly ordered sets, well-ordered sets in general, the relations of ordinals and cardinals. There is a very large bibliography and also a further list of literature pertaining to the companion volume. Fraenkel intended the book for philosophers as well as for mathematicians, and has kept to studying the most general and interesting problems of set theory; metamathematical results are kept in the background and the author works in an informal system of Zermelo-Fraenkel set theory. There is relatively little symbolism and Fraenkel's arguments are therefore more intuitive than rigorous, but clearly everything can be formalized in a proper language. There are many exercises in the book; these help to elaborate points only lightly touched upon in the main text and provide practice necessary for the growth of one's set-theoretical intuition. There are many alternative proofs of important theorems and alternative definitions for essential concepts; this lack of conceptual rigidity should especially please those who reject the now fairly widely held notion that Z-F set theory is the "true" one, the others proposed being ad hoc. Anyone who works his way through this delightfully written text will come away thoroughly prepared to attack more advanced work in the subject.—P. J. M. (shrink)

A non-monotonic theory of probability is put forward and shown to have applicability in the quantum domain. It is obtained simply by replacing Kolmogorov's positivity axiom, which places the lower bound for probabilities at zero, with an axiom that reduces that lower bound to minus one. Kolmogorov's theory of probability is monotonic, meaning that the probability of A is less then or equal to that of B whenever A entails B. The new theory violates monotonicity, as its (...) name suggests; yet, many standard theorems are also theorems of the new theory since Kolmogorov's other axioms are retained. What is of particular interest is that the new theory can accommodate quantum phenomena (photon polarization experiments) while preserving Boolean operations, unlike Kolmogorov's theory. Although non-standard notions of probability have been discussed extensively in the physics literature, they have received very little attention in the philosophical literature. One likely explanation for that difference is that their applicability is typically demonstrated in esoteric settings that involve technical complications. That barrier is effectively removed for non-monotonic probability theory by providing it with a homely setting in the quantum domain. Although the initial steps taken in this paper are quite substantial, there is much else to be done, such as demonstrating the applicability of non-monotonic probability theory to other quantum systems and elaborating the interpretive framework that is provisionally put forward here. Such matters will be developed in other works. (shrink)

A finitary characterization for non-well-founded sets with finite transitive closure is established in terms of a greatest fixpoint formula of the modal μ-calculus. This generalizes the standard result in the literature where a finitary modal characterization is provided only for wellfounded sets with finite transitive closure. The proof relies on the concept of automaton, leading then to new interlinks between automata theory and non-well-founded sets.

Cylindric set algebras are algebraizations of certain logical semantics. The topic surveyed here, i.e. probabilities defined on cylindric set algebras, is closely related, on the one hand, to probability logic (to probabilities defined on logical formulas), on the other hand, to measure theory. The set algebras occuring here are associated, in particular, with the semantics of first order logic and with non-standard analysis. The probabilities introduced are partially continous, they are continous with respect to so-called cylindric sums.

Some philosophers have argued that the open-endedness of the set concept has revisionary consequences for the semantics and logic of set theory. I consider (several variants of) an argument for this claim, premissed on the view that quantification in mathematics cannot outrun our conceptual abilities. The argument urges a non-standard semantics for set theory that allegedly sanctions a non-classical logic. I show that the views about quantification the argument relies on turn out to sanction a classical semantics (...) and logic after all. More generally, this article constitutes a case study in whether the need to account for conceptual progress can ever motivate a revision of semantics or logic. I end by expressing skepticism about the prospects of a so-called non-proof-based justification for this kind of revisionism about set theory. (shrink)

In this paper we introduce a theory of finite sets FST with a strong negation of the axiom of infinity asserting that every set is provably bijective with a natural number. We study in detail the role of the axioms of Power Set, Choice, Regularity in FST, pointing out the relative dependences or independences among them. FST is shown to be provably equivalent to a fragment of Alternative Set Theory. Furthermore, the introduction of FST is motivated in view (...) of a non-standard development. MSC: 03E30, 03E35. (shrink)

Internal sets and the Boolean algebras of the collection of the internal sets are of central importance in non-standard analysis. Boolean algebras are the algebraization of propositional logic while the logic applied in non-standard analysis (in non-standard stochastics) is the first order or the higher order logic (type theory). We present here a first order logic algebraization for the collection of internal sets rather than the Boolean one. Further, we define an unusual probability on this algebraization.

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic (...) tends to teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms. (shrink)

We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions.We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau’s solution.We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn–Banach extension principle which does not hold in Colombeau theory. We establish a (...) connection between our theory with non-standard analysis and thus answer, although indirectly, a question raised by Colombeau. This article provides a bridge between Colombeau theory of generalized functions and non-standard analysis. (shrink)

RésuméL'analyse non‐standard fournit une base solide à la théorie des infinitésimaux. L'approche axiomatique qu'en donne Nelson est basée sur un nouveau predicat qui est ajouté au langage de la théorie usuelle des ensembles. Nous interprétons ce prédicat et formulons les axio‐mes de Nelson ?on;une façon qui peut être comparee à la discussion de P. R. Halmos dans son livre Naïve Set Theory .SummaryNon‐standard analysis gives a proper foundation to the theory of infinitesimals. Nelson's axiomatic approach of (...) it uses a new predicate which is added to the classical language of set theory. We interpret this new predicate and formulate Nelson's axioms in a way that can be compared to P. R. Halmos' discussion of set axioms in his book Naïve Set Theory .ZusammenfassungDie nicht‐standard Analyse liefert eine solide Grundlage für die Theorie der infinitesimalen Grössen. Die von Nelson vorgeschlagene Axiomatisierung beruht auf einem neuen Prädikat, das den in der Sprache der Mengentheorie üblichen hinzugefügt wird. Wir interpretieren dieses Prädikat und formulieren Nelsons Axiome auf eine Art, die mit der Diskus‐sion von P.R. Halmos in Naïve Set Theory verglichen werden kann. (shrink)

In this paper I propose a new approach to the foundation of mathematics: non-monotonic set theory. I present two completely different methods to develop set theories based on adaptive logics. For both theories there is a finitistic non-triviality proof and both theories contain (a subtle version of) the comprehension axiom schema. The first theory contains only a maximal selection of instances of the comprehension schema that do not lead to inconsistencies. The second allows for all the instances, also (...) the inconsistent ones, but restricts the conclusions one can draw from them in order to avoid triviality. The theories have enough expressive power to form a justification/explication for most of the established results of classical mathematics. They are therefore not limited by Gödel’s incompleteness theorems. This remarkable result is possible because of the non-recursive character of the final proofs of theorems of non-monotonic theories. I shall argue that, precisely because of the computational complexity of these final proofs, we cannot claim that non-monotonic theories are ideal foundations for mathematics. Nevertheless, thanks to their strength, first order language and the recursive dynamic (defeasible) proofs of theorems of the theory, the non-monotonic theories form (what I call) interesting pragmatic foundations. (shrink)

(a chapter in Laruelle and Non-Philosophy, ed. John Mullarkey and Anthony Paul Smith) Orthodox reverence of transcendental constructs such as 'dialectical materialism' and the inability to reduce them to chôra - mere transcendental material instead of finished conceptual wholes - is what disables the completion of the project of stepping out of philosophy which Marxism initially set for itself (in the Theses on Feuerbach). In order to radicalise its position, argues Laruelle, and place itself outside philosophy, Marxism has to take (...) a step outside itself by virtue of admitting its own transcendental, i.e., philosophical character. It has to adopt the stance of the 'non-' that is situated in the Real that clones itself through concepts. In order to preserve its grain of 'thinking affected by immanence', 18 Marxism ought to become non-Marxism, argues Laruelle. (shrink)

This paper contributes to a theory of rational choice for decision-makers with incomplete preferences due to partial ignorance, whose beliefs are representable as sets of acceptable priors. We focus on the limiting case of `Complete Ignorance' which can be viewed as reduced form of the general case of partial ignorance. Rationality is conceptualized in terms of a `Principle of Preference-Basedness', according to which rational choice should be isomorphic to asserted preference. The main result characterizes axiomatically a new choice-rule called (...) `Simultaneous Expected Utility Maximization'. It can be interpreted as agreement in a bargaining game (Kalai-Smorodinsky solution) whose players correspond to the (extremal) `acceptable priors' among which the decision maker has suspended judgment. An essential but non-standard feature of Simultaneous Expected Utility choices is their dependence on the entire choice set. This is justified by the conception of optimality as compromise rather than as superiority in pairwise comparisons. (shrink)

*These notes were folded into the published paper "Probability and nonclassical logic*. Revising semantics and logic has consequences for the theory of mind. Standard formal treatments of rational belief and desire make classical assumptions. If we are to challenge the presuppositions, we indicate what is kind of theory is going to take their place. Consider probability theory interpreted as an account of ideal partial belief. But if some propositions are neither true nor false, or are half (...) true, or whatever—then it’s far from clear that our degrees of belief in it and its negation should sum to 1, as classical probability theory requires (?, cf.). There are extant proposals in the literature for generalizing (categorical) probability theory to a non-classical setting, and we will use these below. But subjective probabilities themselves stand in functional relations to other mental states, and we need to trace the knock-on consequences of revisionism for this interrelationship (arguably, degrees of belief only count as kinds of belief in virtue of standing in these functional relationships). (shrink)

I provide an examination and comparison of modal theories for underwriting different non-modal theories of sets. I argue that there is a respect in which the `standard' modal theory for set construction---on which sets are formed via the successive individuation of powersets---raises a significant challenge for some recently proposed `countabilist' modal theories (i.e. ones that imply that every set is countable). I examine how the countabilist can respond to this issue via the use of regularity axioms and raise (...) some questions about this approach. I argue that by comparing them with the `standard' uncountabilist theory, a new approach that brings in arbitrariness rather than the strict controls of forcing is desirable. (shrink)

Appeal to standard set theory in minimalist syntax is shown to be in conflict with the goal of analyzing dependency formation, a.k.a. movement, as involving genuine constituent copies. The underlying tension is due to extensionality, which—other things being equal—favors a perspective on dependencies in terms of multidominance. The above argument is developed against the backdrop of a recent exposition of minimalist syntax :229–261, 2019), which can be seen as exemplary. The resulting critical assessment should be taken as removing (...) obstacles on the way toward proper minimalist foundations for syntactic theory. (shrink)

Through a number of illuminating cases studies which draw in large part on the largely under-utilized data of Romance dialectal varieties, the present article sets out to highlight the importance that Romance data, especially those of non-standard varietites, can play in testing and enriching currents theories of syntax. In particular, we shall show that dialectal varieties, although frequently overlooked in the past, offer an immensely fertile and still relatively unexplored experimental territory in which to profitably investigate new ideas about (...) language structure, change and variation. At the same time, we champion the mutual benefits for theoretical linguists and Romanists of adopting a more integrated and reciprocally-informed approach in their respective treatments of linguistic evidence, highlighting how a familiarity with the key tools and ideas established in both fields has the potential to enrich linguistic and empirical analyses considerably. (shrink)

This paper articulates the structure of a two species of weakly aggregative necessity in a common idiom, neighbourhood semantics, using the notion of a k-filter of propositions. A k-filter on a non-empty set I is a collection of subsets of I which contains I, is closed under supersets on I, and contains ∪{Xi ≤ Xj : 0 ≤ i < j ≤ k} whenever it contains the subsets X0,…, Xk. The mathematical content of the proof that weakly aggregative modal logic (...) is complete relative to k-ary frame theory, the standard semantic idiom for weakly aggregative modal logic is presented in language-independent terms as a representation theorem for k-filters: every non-trivial k-filter is included in the union of ≤ k non-trivial filters. The elementary theory of k-filters is developed and then applied in the form of an ultrafilter extension result for k-ary frame theory. (shrink)

We present a case study for the debate between the American and the Australian plans, analyzing a crucial aspect of negation: expressivity within a theory. We discuss the case of non-classical set theories, presenting three different negations and testing their expressivity within algebra-valued structures for ZF-like set theories. We end by proposing a minimal definitional account of negation, inspired by the algebraic framework discussed.

Among the main concerns of 20th century philosophy was that of the foundations of mathematics. But usually not recognized is the relevance of the choice of a foundational approach to the other main problems of 20th century philosophy, i.e., the logical structure of language, the nature of scientific theories, and the architecture of the mind. The tools used to deal with the difficulties inherent in such problems have largely relied on set theory and its “received view”. There are specific (...) issues, in philosophy of language, epistemology and philosophy of mind, where this dependence turns out to be misleading. The same issues suggest the gain in understanding coming from category theory, which is, therefore, more than just the source of a “non-standard” approach to the foundations of mathematics. But, even so conceived, it is the very notion of what a foundation has to be that is called into question. The philosophical meaning of mathematics is no longer confined to which first principles are assumed and which “ontological” interpretation is given to them in terms of some possibly updated version of logicism, formalism or intuitionism. What is central to any foundational project proper is the role of universal constructions that serve to unify the different branches of mathematics, as already made clear in 1969 by Lawvere. Such universal constructions are best expressed by means of adjoint functors and representability up to isomorphism. In this lies the relevance of a category-theoretic perspective, which leads to wide-ranging consequences. One such is the presence of functorial constraints on the syntax–semantics relationships; another is an intrinsic view of (constructive) logic, as arises in topoi and, subsequently, in more general fibrations. But as soon as theories and their models are described accordingly, a new look at the main problems of 20th century’s philosophy becomes possible. The lack of any satisfactory solution to these problems in a purely logical and set-theoretic setting is the result of too circumscribed an approach, such as a static and punctiform view of objects and their elements, and a misconception of geometry and its historical changes before, during, and after the foundational “crisis”, as if algebraic geometry and synthetic differential geometry – not to mention algebraic topology – were secondary sources for what concerns foundational issues. The objectivity of basic geometrical intuitions also acts against the recent version of structuralism proposed as ‘the’ philosophy of category theory. On the other hand, the need for a consistent and adequate conceptual framework in facing the difficulties met by pre-categorical theories of language and scientific knowledge not only provides the basic concepts of category theory with specific applications but also suggests further directions for their development (e.g., in approaching the foundations of physics or the mathematical models in the cognitive sciences). This ‘virtuous’ circle is by now largely admitted in theoretical computer science; the time is ripe to realise that the same holds for classical topics of philosophy. (shrink)

The study was conducted to unveil social connectedness of the older adults in non-urban societies in Ghana, and the ecological and social inclusion theories underpinned the study. The descriptive cross-sectional survey was undertaken based on the positivist school of thought. Older adults who were 60 years or more, participated in the study. Older Adults’ Social Connectedness Questionnaire was designed to gather data to answer the research questions and test the hypothesis. Frequency counts and percentages, mean and standard deviation and (...) independent samples t-test were the analytical tools employed for the data analysis. The study unveiled, inter alia, that the older adults tend to enjoy moderate level of social connectedness and the major groups of people they frequently socially interact with are relatives, religious group members, neighbours and friends. The study also unveiled that religious activities, phone calls, cultural and social activities are the principal conduits for the older adults’ social interaction with significant others. Finally, the study establ ished a difference in the level of social connectedness of the male and female older adults. It is reckoned that the older adults in the rural settings need more and intensive social interactions to serve as impetus for longevity. (shrink)

This paper examines from a special perspective the problem of methodological limita­tions of the computational approach in the philosophy of mind and empirical sciences. The main goal is to consistently substantiate the dependence of philosophical “metaphori­cal dictionaries” on advances in the field of computer science, historical contexts of epis­temology, formal and methodological limitations of algorithmic and computational proce­dures. The key idea is that despite the success of computational models in empirical re­search, their conceptual level does not allow us to correctly (...) formulate the question of the ontology of consciousness. Computationalism in philosophical theories of con­sciousness is presented as a practice of word usage, which posits systems of consistent descriptions of the information properties of consciousness and cognitive processes within the framework of certain methodological rules. The limitations of the computa­tional approach are associated with the lack of a scientific theory of subjectivity, the fun­damental irreducibility of external properties of consciousness to internal states, and fun­damental restrictions on the completeness and consistency of computer mathematics. A classification of several anti-computational programs has been carried out. The article discusses various limitations of computational approaches, such as: the complexity of for­malizing nonlinear and hidden processes, the dependence of emotions and intuition on the situational context and individual differences, the difficulty of predicting the emerg­ing properties of autonomous and autopoietic systems, and the limitations associated with non-standard computational processes of quantum and dynamic phenomena. A special role is given to the problem of set-theoretic reductive realism within the framework of mathe­matical structuralism and some aspects of p-adic number systems as a possible alternative to a “universal language” for the computational philosophy of consciousness. (shrink)

This paper presents an axiomatization of the principle of maximizing expected utility that does not rely on the independence axiom or sure-thing principle. Perhaps more importantly the new axiomatization is based on an ex ante approach, instead of the standard ex post approach. An ex post approach utilizes the decision maker's preferences among risky acts for generating a utility and a probability function, whereas in the ex ante approach a set of preferences among potential outcomes are on the input (...) side of the theory and the decision maker's preferences among risky acts on the output side. (shrink)

A basic way of evaluating metaphysical theories is to ask whether they give satisfying answers to the questions they set out to resolve. I propose an account of “third-order” virtue that tells us what it takes for certain kinds of metaphysical theories to do so. We should think of these theories as recipes. I identify three good-making features of recipes and show that they translate to third-order theoretical virtues. I apply the view to two theories—mereological universalism and plenitudinous platonism—and draw (...) out their third-order virtues and vices. One lesson is that there is an important difference between essentially and non-essentially third-order vicious theories. I also argue that if a theory is essentially third-order vicious, it cannot be assessed for more standard “second-order” theoretical virtues and vices, like parsimony. This motivates the idea that third-order virtues are distinct from second-order ones. Finally, I suggest that the relationship between truth, progress, and third-order virtue is more complex than it seems. (shrink)

In this paper I propose a fundamental modification of standard type theory, produce a new kind of type theoretic language, and couch in this language a comprehensive theory of abstract individuals and abstract properties and relations of every type. I then suggest how to employ the theory to solve the four following philosophical problems: the identification and ontological status of Frege's Senses; the deviant behavior of terms in propositional attitude contexts; the non-identity of necessarily equivalent propositions, (...) and the paradox of analysis. We can roughly describe these solutions as follows: the senses of English names and descriptions which denote individuals will be modelled as abstract individuals; the senses of English relation denoting expressions of a given type will be modelled as abstract relations of that type. Inside de dicto attitude contexts, these English expressions denote their senses. Relations and propositions will not be identified with their extensions, nor with functions or sets of any kind. They will be taken as primitive, and precise being and identity conditions will be proposed consistent with the view that necessarily equivalent relations and propositions may be distinct. With the modelling described in, the expressions being a brother and being a male sibling may both denote the same property, though their senses may differ. Just as Frege predicts, being a brother just is being a male sibling is an informative identity statement because the terms flanking the identity sign have the same denotation, though distinct senses. (shrink)

Sorkin’s recent proposal for a realist interpretation of quantum theory, the anhomomorphic logic or coevent approach, is based on the idea of a “quantum measure” on the space of histories. This is a generalisation of the classical measure to one which admits pair-wise interference and satisfies a modified version of the Kolmogorov probability sum rule. In standard measure theory the measure on the base set Ω is normalised to one, which encodes the statement that “Ω happens”. Moreover, (...) the Kolmogorov sum rule implies that the measure of any subset A is strictly positive if and only if A cannot be covered by a countable collection of subsets of zero measure. In quantum measure theory on the other hand, simple examples suffice to demonstrate that this is no longer true. We propose an appropriate generalisation, the quantum cover, which in addition to being a cover of A, satisfies the property that if the quantum measure of A is non-zero then this is also the case for at least one of the elements in the cover. Our work implies a non-triviality result for the coevent interpretation for Ω of finite cardinality, and allows us to cast the Peres-Kochen-Specker theorem in terms of quantum covers. (shrink)

Registered nurses in psychiatric-mental health nursing continuously balance the ethical principles of duty to do good (beneficence) and no harm (non-maleficence) with the duty to respect patient choices (autonomy). However, the problem of nurses’ level of control versus patients’ choices remains a challenge. The aim of this article is to discuss how nurses accomplish their simultaneous responsibility for balancing patient safety (beneficence and non-maleficence) with patient choices (autonomy) through the theory of protective empowering. This is done by reflecting on (...) interview excerpts about caring from 17 registered nurses taking part in a grounded theory study conducted in three acute urban psychiatric hospital settings in Canada. The interplay between the protective and empowering dimensions of the theory of protective empowering was found to correspond with international, national, and local nursing codes of ethics and standards. The overall core process of protective empowering, and its associated reflective questions, is offered as a new lens for balancing patient safety with choices. (shrink)

Alexius Meinong's Gegenstandstheorie is subject to a formal semantic paradox. The theory of defective objects originally developed by Meinong in response to Ernst Mally's paradox about self-referential thought is rejected as a general solution to paradox in the object theory. The intentionality thesis is also refuted by the counter-example of the unapprehended mountain. It is argued that despite these difficulties, an object theory is required in order to make intuitively correct sense of ontological commitment. ;A version of (...) Meinong's theory is formalized. The logic has non-standard truth value semantics and an objectual interpretation of quantifiers for a domain of existent and non-existent objects. Theories of intentional, referential, and extensional identity are offered, together with two theories of definite description, lambda abstraction, eight systems of alethic modal logic, and formal metatheory. The logic avoids the incompleteness and undecidability limitations of Godel and Church, and all logical, semantic, and set theoretical paradoxes, without type theory or Tarskian hierarchy of metalanguages. It is naively complete and mechanically decidable. ;Wittgenstein's private language argument is refuted as an obstacle to defining private mental objects in the logic. The argument is shown to be valid, but unsound. The object theory logic may therefore provide a mixed or impure private language, and its domain may contain private mental objects as well as ordinary public and non-existent objects. This means that it constitutes an object theory logic of intention, and can be used to represent the logical structure of intentional and specifically phenomenological philosophical theories that require the intelligible designation of private mental objects. ;Appendices include Henkin-type consistency, completeness, and compactness proofs ; criticism of Kazimierz Twardowski's arguments for the distinction between content and object in phenomenological psychology; translation of and commentary on Mally's essay, "Uber die Unabhangigkeit der Gegenstande vom Denken"; a discussion of the distinction between nuclear and extra-nuclear properties, and the eliminability of the modal moment; and a philosophical application of the logic to a solution of the paradox of analysis. (shrink)

We prove that Standardization fails in every nontrivial universe definable in the nonstandard set theory BST, and that a natural characterization of the standard universe is both consistent with and independent of BST. As a consequence we obtain a formulation of nonstandard class theory in the ∈-language.

This paper introduces a multi-modal polymorphic type theory to model epistemic processes characterized by trust, defined as a second-order relation affecting the communication process between sources and a receiver. In this language, a set of senders is expressed by a modal prioritized context, whereas the receiver is formulated in terms of a contextually derived modal judgement. Introduction and elimination rules for modalities are based on the polymorphism of terms in the language. This leads to a multi-modal non-homogeneous version of (...) a type theory, in which we show the embedding of the modal operators into standard group knowledge operators. (shrink)

A non-expert who struggles to make good decisions and who turns to decision theory for help, might be more than a little surprised by what they find. If they read a standard treatment of the subject, they will find that they are assumed to be logically omniscient: they know all the logical facts about the propositions whose truth they have considered. Their beliefs are also assumed to be logically closed: if they believe each of a set of propositions (...) S, then they believe everything that can be deduced from S. Finally, they are assumed to be maximally opinionated—they have assigned precise probabilities and cardinal utilities to each possible state of the world that can be formulated via S. The normative core of standard decision theory consists of some very weak axioms for these probabilities and car-dinal utilities, plus the advice to maximize their expected utility, which is a function of the probabil-ities and cardinal utilities for each possible state of the world given each possible choice that they can make. The non-expert might understandably react by saying that this theory is too idealized to be useful for human beings. It is this criticism that Richard Bradley addresses with patience, rigour, and ardour in this book. (shrink)

Décio Krause is one of the staunchest defenders of the “Received View” of “identical quantum particles”, i.e. quantum particles of the same kind. According to the Received View identical quantum particles do not possess individuating properties: they are entities without identity. Still, they are “different” from each other in the weak sense that there can be more than one of them. As Décio Krause has pointed out, such identity-less objects must be handled by a non-standard set theory—quasi-set (...) class='Hi'>theory, a subject to which he has made important contributions.In this Chapter we compare and contrast the ideas of the Received View with those of a rival interpretation that has recently started to attract attention. According to this Alternative View quantum particles are emerging entities—at a fundamental level, the world does not consist of particles. But once emerged, quantum particles of the same kind are distinguishable and possess identity, so that they can be dealt with by standard set theory and ordinary mathematics. (shrink)

This dissertation sets out to answer the question ''What fixes the semantic values of context-sensitive referential terms—like names, demonstratives, and pronouns—in context?'' I argue that it is the speaker's intentions that play this role, as constrained by the conventions governing the use of particular sorts of referential terms. These conventions serve to filter the speaker's intentions for just those which meet these constraints on use, leaving only these filtered-for intentions as semantically relevant. By considering a wide range of cases, including (...) many involving confused and deceptive speakers, I argue that this 'constraint theory' provides a better account of linguistic reference than does any extant alternative, whether intentionalist or non-intentionalist. Along the way, I argue that semantics cannot depend on the reactions of idealized listeners, that speaker meaning is far less clear, and helpful, a notion than it is standardly taken to be, and that speakers needn't aim to be cooperative in order to fix the meaning of their terms in context. (shrink)

Integrated information theory is a theory of consciousness that was originally formulated, and is standardly still expressed, in terms of controversial interpretations of its own ontological and epistemological basis. These form the orthodox interpretation of IIT. The orthodox epistemological interpretation is the axiomatic method, whereby IIT is ultimately derived from, justified by, and beholden to a set of phenomenological axioms. The orthodox ontological interpretation is panpsychism, according to which consciousness is fundamental, intrinsic, and pervasive. In this paper it (...) is argued that both components of the orthodox interpretation should be rejected. But IIT should not be rejected since an interpretation-neutral formulation is available. After explaining the neutral formulation, more plausible non-axiomatic epistemologies are defended. The neutral formulation is then shown to be consistent with various contemporary physicalist ontologies of consciousness, including the phenomenal concept strategy, representationalism, and even illusionism. Along the way, instructive connections between interpretations of IIT and interpretations of quantum mechanics are noted. (shrink)

The purpose of this contribution is to examine the current state of the de Broglie-Bohm theory (dBB) in light of Bohm’s vision as he explicitly set it out in his book Quantum theory [In Bohm, D., Quantum theory, Courier corporation, (1961b)]. In particular, two programmes that differ in many crucial respects are currently being pursued. On the one hand, the Bohmian mechanics school, founded by Dürr Goldstein and Zanghì, considers the theory to be Galilean invariant, regards (...) particles’ motion as determined by a nomological entity, the universal wave function, upholds the quantum equilibrium hypothesis and explains probabilities in terms of typicality. On the other hand, the Pilot-wave school advocated by Valentini considers the theory to be based on Aristotelian dynamics, regards the wave function as a physical field displaying a contingent nature, and explains quantum equilibrium as the result of a process of relaxation from quantum non-equilibrium. Looking at Bohm’s work [In Bohm, D., Quantum theory, Courier corporation, (1961b)], it is clear that his intention was to construct a theory that was empirically different from standard quantum mechanics, so that it would be testable and falsifiable. Only this way could he defend dBB from the criticism that accused the theory of being 'metaphysical'. These methodological concerns about the falsifiability of the theory constitute, in my opinion, a strong reason for regarding Valentini's programme as methodologically valuable, even if it might turn out to be wrong. Indeed, the programme of the Pilot-wave school aims to be falsifiable with respect to standard quantum mechanics and can in principle defend the empirical status of particle trajectories. (shrink)

It is easy to show that in many natural axiomatic formulations of physical and even mathematical theories, there are many superfluous concepts usually assumed as primitive. This happens mainly when these theories are formulated in the language of standard set theories, such as Zermelo–Fraenkel’s. In 1925, John von Neumann created a set theory where sets are definable by means of functions. We provide a reformulation of von Neumann’s set theory and show that it can be used to (...) formulate physical and mathematical theories with a lower number of primitive concepts very naturally. Our basic proposal is to offer a new kind of set-theoretic language that offers advantages with respect to the standard approaches, since it doesn’t introduce dispensable primitive concepts. We show how the proposal works by considering significant physical theories, such as non-relativistic classical particle mechanics and classical field theories, as well as a well-known mathematical theory, namely, group theory. This is a first step of a research program we intend to pursue. (shrink)

It is well known that the validity of Choice Principles is problematic in non-standard Set Theories which do not abide by the Limitation of Size Principle. In this paper we discuss the consistency of various Choice Principles with respect to the Generalized Positive Comprehension Principle . The Principle GPC allows to take as sets those classes which can be specified by Generalized Positive Formulae, e.g. the universe. In particular we give a complete characterization of which choice principles hold in (...) Hyperuniverses. Hyperuniverses are structures which arose independently in Non-well-founded Set Theory and in Mathematical Semantics of Concurrent Programming Languages and are hitherto the only existing models of GPC. Hyperuniverses are naturally endowed with a κ-compact uniform κ-topology and are uniformly isomorphic to their exponential space, i.e. the space of their closed subsets endowed with the Exponential Uniformity. (shrink)

The way, in which quantum information can unify quantum mechanics (and therefore the standard model) and general relativity, is investigated. Quantum information is defined as the generalization of the concept of information as to the choice among infinite sets of alternatives. Relevantly, the axiom of choice is necessary in general. The unit of quantum information, a qubit is interpreted as a relevant elementary choice among an infinite set of alternatives generalizing that of a bit. The invariance to the axiom (...) of choice shared by quantum mechanics is introduced: It constitutes quantum information as the relation of any state unorderable in principle (e.g. any coherent quantum state before measurement) and the same state already well-ordered (e.g. the well-ordered statistical ensemble of the measurement of the quantum system at issue). This allows of equating the classical and quantum time correspondingly as the well-ordering of any physical quantity or quantities and their coherent superposition. That equating is interpretable as the isomorphism of Minkowski space and Hilbert space. Quantum information is the structure interpretable in both ways and thus underlying their unification. Its deformation is representable correspondingly as gravitation in the deformed pseudo-Riemannian space of general relativity and the entanglement of two or more quantum systems. The standard model studies a single quantum system and thus privileges a single reference frame turning out to be inertial for the generalized symmetry [U(1)]X[SU(2)]X[SU(3)] “gauging” the standard model. As the standard model refers to a single quantum system, it is necessarily linear and thus the corresponding privileged reference frame is necessary inertial. The Higgs mechanism U(1) → [U(1)]X[SU(2)] confirmed enough already experimentally describes exactly the choice of the initial position of a privileged reference frame as the corresponding breaking of the symmetry. The standard model defines ‘mass at rest’ linearly and absolutely, but general relativity non-linearly and relatively. The “Big Bang” hypothesis is additional interpreting that position as that of the “Big Bang”. It serves also in order to reconcile the linear standard model in the singularity of the “Big Bang” with the observed nonlinearity of the further expansion of the universe described very well by general relativity. Quantum information links the standard model and general relativity in another way by mediation of entanglement. The linearity and absoluteness of the former and the nonlinearity and relativeness of the latter can be considered as the relation of a whole and the same whole divided into parts entangled in general. (shrink)