Boris Kment takes a new approach to the study of modality that emphasises the origin of modal notions in everyday thought. He argues that the concepts of necessity and possibility originate in counterfactual reasoning, which allows us to investigate explanatory connections. Contrary to accepted views, explanation is more fundamental than modality.

While there are satisfactory answers to the question “How should we teach children mathematics?”, there are no satisfactory answers to the question “What mathematics should we teach children?”. This paper provides an answer to the last question for preschool children (early childhood), although the answer is also applicable to older children. This answer, together with an appropriate methodology on how to teach mathematics, gives a clear conception of the place of mathematics in the children’s world and our role in helping (...) children develop their mathematical abilities. Briefly, children’s mathematics consists of the world of children’s internal activities that they eventually purposefully organize in order to understand and control the outside world and organize their overall activities in it. We need to support a child in mathematical activities that she does spontaneously and in which she shows interest, and we need to teach her mathematics that she is interested in developing through these activities. In doing so, we must be fully aware that the child’s mathematics is part of the child’s world of internal activities and is not outside of it. We help the child develop mathematical abilities by developing them in the context of her world and not outside of it. From the point of view of this conception, the standards established today are limiting and too focused on numbers and geometric figures: these topics are too prominent and elaborated, and other mathematical contents are subordinated to them. Adhering to the standards, we drastically limit the mathematics of the child’s world, hamper the correct mathematical development of a child, and we can turn her away from mathematics. (shrink)

Contemporary semantic description of logic is based on the ontology of all possible interpretations, an insufficiently clear metaphysical concept. In this article, logic is described as the internal organization of language. Logical concepts -- logical constants, logical truths, and logical consequence -- are defined using the internal syntactic and semantic structure of language. For a first-order language, it has been shown that its logical constants are connectives and a certain type of quantifiers for which the universal and existential quantifiers form (...) a functionally complete set of quantifiers. Neither equality nor cardinal quantifiers belong to the logical constants of a first-order language. (shrink)

In this paper I discuss five meta-ethical challenges to Naturalistic Moral Realism, which includes secular moral codes such as Secular Humanism that, in my view, naturalists need to address to keep their commitment to moral realism from looking like special pleading. The five challenges are as follows: 1. The Ontological Problem (OP): How do such moral principles exist? 2. The Epistemic Problem (EP): How does our moral sense/intuition track such principles? 3. The Influence Problem (IP): What authority does the existence (...) of such principles have to supervene on our behavior? 4. The Axiological Problem (AP): What entities have moral value and deserve moral consideration? 5. Methodological Problem (MP): How do we go about arbitrating between moral claims when they conflict? (shrink)

In this article, I will discuss some of the arguments for and against the existence of God, in particular the monotheistic God believed in the Abramahamic religions (Judiasm, Islam, and Christianity) as well as Babism, the Bahai Faith, and Sikhism. Arguments for the existence of God try to argue that either God exists (based on other things people agree with) or that belief in God is reasonable. Arguments against the existence of God try to argue that the existence of God (...) leads to some contradiction or other absurdity. (shrink)

From the comparison of time in inertial frames, possible types of transformations between inertial frames are deduced. This elementary deduction directly relates the properties of time with the type of transformations. When all inertial frames measure the same time (time is absolute), the transformations are Galilean. When each inertial frame has its own time, different from the times of other inertial frames (time is not invariant) the transformations are Lorentz-like with the same positive parameter k. The parameter k is the (...) supremum of possible velocities in an inertial frame, the same for all inertial frames. Einstein's postulate about the invariance of the speed of light says more: there is a uniform motion with the supremum k, which is exactly the motion of light. At the end of the article, attempts to reduce the special theory of relativity to the principle of relativity are criticized. (shrink)

Evidential decision theory (EDT) says that the choiceworthiness of an option depends on its evidential connections to possible outcomes. Causal decision theory (CDT) holds that it depends on your beliefs about its causal connections. While Newcomb cases support CDT, Arif Ahmed has described examples that support EDT. A new account is needed to get all cases right. I argue that an option A's choiceworthiness is determined by the probability that a good outcome ensues at possible A‐worlds that match actuality in (...) the facts causally unaffected by your decision (the “unaffected facts”). Moreover, you should evaluate A on the assumption that A is compossible with the unaffected facts. This view entails that you should use EDT when evaluating A on the assumption that the unaffected facts determine your action, but use CDT when assessing A on the opposite assumption. A's choiceworthiness equals a weighted average of these conditional assessments. The weights are determined by your beliefs about whether the unaffected fact determine your action. This account gets both Newcomb and Ahmed cases right. According to an influential view, whether you take the unaffected facts to determine your action can make a difference to whether you can regard yourself as free and the action as being under your control. While my account is neutral on this issue, it entails that whether you take the unaffected facts to determine your action is important in a different way: it matters to whether you should follow EDT or CDT. (shrink)

Approximate coherentism suggests that imperfectly rational agents should hold approximately coherent credences. This norm is intended as a generalization of ordinary coherence. I argue that it may be unable to play this role by considering its application under learning experiences. While it is unclear how imperfect agents should revise their beliefs, I suggest a plausible route is through Bayesian updating. However, Bayesian updating can take an incoherent agent from relatively more coherent credences to relatively less coherent credences, depending on the (...) data observed. Thus, comparative rationality judgments among incoherent agents are unduly sensitive to luck. (shrink)

The problem of evil is one of the most significant challenges to theism and Christianity in particular, asking why there seems to be so much evil if an omnibenevolent (all good), loving God exists. The problem of evil, as posed by many atheists and agnostics today, (following Epicurus) often asserts that the following premises cannot all be true: 1. God exists, and is omnipotent (all-powerful) 2. God exists, and is omnibenevolent (all-good) 3. God exists, and is omniscient (all-knowing) 4. Evil (...) exists Christian thinkers over the past two thousand years have given various responses to the problem of evil in the form of various theodicies, or attempts to explain why evil exists (or appears to exist) given the existence of an Omniscient, Omnipotent, Omnibenevolent God. This document serves as a brief overview of different theodicies offered by Christian thinkers throughout history. (shrink)

Anti-intellectualism involves general mistrust of scholars, academics, and ex- perts, often as pretentious or power-motivated. While scholars have described currents of anti-intellectualism in American public life, evangelical Christianity, in responses to COVID, and rural identity, to my knowledge none have looked at how anti-intellectualism specifically manifests in the New Atheism movement. In this work, we explore the way anti-intellectualism is commonly found and expressed in New Atheism and the modern Skeptical Movement, including scientific skepticism more generally.

On the received view, counterfactuals are analysed using the concept of closeness between possible worlds: the counterfactual 'If it had been the case that p, then it would have been the case that q' is true at a world w just in case q is true at all the possible p-worlds closest to w. The degree of closeness between two worlds is usually thought to be determined by weighting different respects of similarity between them. The question I consider in the (...) paper is which weights attach to different respects of similarity. I start by considering Lewis's answer to the question and argue against it by presenting several counterexamples. I use the same examples to motivate a general principle about closeness: if a fact obtains in both of two worlds, then this similarity is relevant to the closeness between them if and only if the fact has the same explanation in the two worlds. I use this principle and some ideas of Lewis's to formulate a general account of counterfactuals, and I argue that this account can explain the asymmetry of counterfactual dependence. The paper concludes with a discussion of some examples that cannot be accommodated by the present version of the account and therefore necessitate further work on the details. (shrink)

In this handbook, I put into practice my philosophical views on children's mathematics. The handbook contains brief instructions and examples of mathematical activities. In the INSTRUCTIONS section, instructions are given on how, and in part why that way, to help preschool children in their mathematical development. In the ACTIVITIES section, there are examples of activities through which the child develops her mathematical abilities.

During the last quarter of a century, a number of philosophers have become attracted to the idea that necessity can be analyzed in terms of a hyperintensional notion of essence. One challenge for proponents of this view is to give a plausible explanation of our modal knowledge. The goal of this paper is to develop a strategy for meeting this challenge. My approach rests on an account of modality that I developed in previous work, and which analyzes modal properties in (...) terms of the notion of a metaphysical law. I discuss what information about the metaphysical laws is required for modal knowledge. Moreover, I describe two ways in which we might be able to acquire this information. The first way employs inference to the best explanation. The metaphysical laws, including the essential truths, play a crucial role in causal and grounding explanations and we can gain knowledge of these laws by abductive inferences from facts of which we have perceptual or a priori knowledge. The second way of gaining information about the metaphysical laws rests on knowledge that is partly constitutive of competence with the concepts that are needed to express the relevant information. Finally, I consider how knowledge of the metaphysical laws can be used to establish modal claims, paying special attention to the much-discussed connection between conceiving and possibility. (shrink)

Many philosophers have argued that statistical evidence regarding group characteristics (particularly stereotypical ones) can create normative conflicts between the requirements of epistemic rationality and our moral obligations to each other. In a recent article, Johnson-King and Babic argue that such conflicts can usually be avoided: what ordinary morality requires, they argue, epistemic rationality permits. In this article, we show that as data get large, Johnson-King and Babic’s approach becomes less plausible. More constructively, we build on their project and develop a (...) generalized model of reasoning about stereotypes under which one can indeed avoid normative conflicts, even in a big data world, when data contain some noise. In doing so, we also articulate a general approach to rational belief updating for noisy data. (shrink)

This paper analyzes the historical context and systematic importance of Kant's hypothetical use of reason. It does so by investigating the role of hypotheses in Kant's philosophy of science. We first situate Kant’s account of hypotheses in the context of eighteenth-century German philosophy of science, focusing on the works of Wolff, Meier, and Crusius. We contrast different conceptions of hypotheses of these authors and elucidate the different theories of probability informing them. We then adopt a more systematic perspective to discuss (...) Kant's idea that scientific hypotheses must articulate real possibilities. We argue that Kant's views on the intelligibility of scientific hypotheses constitute a valuable perspective on scientific understanding and the constraints it imposes on scientific rationality. (shrink)

It is argued that logic, and in particular mathematical logic, should play a key role in the undergraduate curriculum for students in the computing fields, which include electrical engineering (EE), computer engineering (CE), and computer science (CS). This is based on 1) the history of the field of computing and its close ties with logic, 2) empirical results showing that students with better logical thinking skills perform better in tasks such as programming and mathematics, and 3) the skills students are (...) expected to have in the job market. Further, we believe teaching logic to students explicitly will improve student retention, especially involving underrepresented minorities. Though this work focuses specifically on the computing fields, these results demonstrate the importance of logic education to STEM (science, technology, engineering, and mathematics) as a whole. (shrink)

In this article, I define a cultural epistemology as a set of socially reinforced assumptions about how knowledge and truth are produced. Unlike a philosophical epistemology, a cultural epistemology is largely the product of culture and largely invisible. As products of culture, cultural epistemology are relatively unquestioned and, in many cases, philosophically unsophisticated. There are three common types of cultural epistemologies, influenced by who holds power in a given society: an epistemological monarchy, an epistemological oligarchy and an epistemological democracy. A (...) given cultural epistemology subsequently influences the legitimacy of culture and government itself. I argue argue that Americans tend to hold an "epistemological democracy" with two distinct facets: a trust in consensus, linked to living in a democratic republic, and a trust in their own self-interest, linked to market economics and capitalism more generally. These two facets combine, interact, and even oppose each other to legitimize and challenge American culture and its political and economic systems. (shrink)

This work deals with problems involving infinities and infinitesimals. It explores the ideas behind zero, its relationship to ontological nothingness, finititude (such as finite numbers and quantities), and the infinite. The idea of infinity and zero are closely related, despite what many perceive as an intuitive inverse relationship. The symbol 0 generally refers to nothingness, whereas the symbol infinity refers to ``so much'' that it cannot be quantified or captured. The notion of finititude rests somewhere between complete nothingness and something (...) having no end. -/- My concern is that many of the philosophers arguing for or against the ontology (or possibility) of an actual infinite set are unaware or unfamiliar with the mathematical literature attempting to clearly and rigorously define these terms. I believe it is a mistake to leave mathematicians out of this conversation, as analysts in particular have defined (and used) infinity in a way that is relevant to the ongoing debate between philosophers. -/- For this paper, I have selected examples from Principles of Analysis by Walter Rudin, which has become a standard text for Real Analysis classes taken by pure mathematicians and engineers. I will also restrict the scope of examples to Euclidean spaces for simplicity. Finally, I will be using addition, + and multiplication * in their usual way. (shrink)

In this article, logical concepts are defined using the internal syntactic and semantic structure of language. For a first-order language, it has been shown that its logical constants are connectives and a certain type of quantifiers for which the universal and existential quantifiers form a functionally complete set of quantifiers. Neither equality nor cardinal quantifiers belong to the logical constants of a first-order language.

In this paper, I wish to criticize the notion that consensus is an epistemology. While I have never seen it explicitly claimed that “consensus is an epistemology,” I have nonetheless seen it implied in many scholarly (and layperson) articles. This occurs whenever articles cite, “a majority of scholars agree that…” or “most scientists/researchers think…” In our democratic, individualistic society, we put a value on high value votes and the quantification of majority viewpoints, whether it be in political polls (due to (...) living in a democracy), online on websites with ranking or voting systems such as Reddit, YouTube, or Instagram. The danger here is we may defer too much to the consensus view without appropriate justification. I will give several reasons why we should be skeptical of appeals to consensus (including a consensus of experts), which include its non-obvious link to ontology, the fact that consensus changes, and a lack of specificity of who counts as an ``expert,'' the lack of specificity on how to handle consensus (if it does occur), and epistemic undercutting defeaters from evolutionary epistemology. While an appeal to consensus may not tell us much about whether a theory is true, they can give us practical or pragmatic reasons for trusting a position. (shrink)

The Russell-Myhill paradox puts pressure on the Russellian structured view of propositions by showing that it conflicts with certain prima facie attractive ontological and logical principles. I describe several versions of RMP and argue that structurists can appeal to natural assumptions about metaphysical grounding to provide independent reasons for rejecting the ontological principles used in these paradoxes. It remains a task for future work to extend this grounding-based approach to all variants of RMP.

For years, the European world saw millions of swans, and all of them without exception were white. If inductive reasoning is valid, one may conclude that all swans are white. However, this would be incorrect: in 1667 Dutch explorer Willem de Vlamingh observed black swans in Australia, falsifying the hypothesis that all swans are white. While often used as a cautionary tale for the use of induction, such as with Popper’s falsification principle, I want to explore a slightly different idea: (...) does the existence of fabricated or otherwise failed accounts of Black swans give me a reason to ignore future arguments or doubt their existence? I argue that while faulty or fabricated reports do not give us a logical reason to reject a claim, they can give us a practical one for refusing to consider future arguments. This practical vs logical distinction gives an inductive reason for believing that future arguments will be invalid, but keeps one from being able to necessarily deny the claim in question without further argumentation. (shrink)

Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial interpretation of the corresponding language. It follows from the analysis that (i) mathematical objects do not exist in the external world: they are our internally imagined objects, some of which, at least approximately, we can realize or represent; (ii) mathematical truths are not (...) truths about the external world but specifications (formulations) of mathematical conceptions; (iii) mathematics is first and foremost our imagined tool by which, with certain assumptions about its applicability, we explore nature and synthesize our rational cognition of it. (shrink)

This article informally presents a solution to the paradoxes of truth and shows how the solution solves classical paradoxes (such as the original Liar) as well as the paradoxes that were invented as counterarguments for various proposed solutions (“the revenge of the Liar”). This solution complements the classical procedure of determining the truth values of sentences by its own failure and, when the procedure fails, through an appropriate semantic shift allows us to express the failure in a classical two-valued language. (...) Formally speaking, the solution is a language with one meaning of symbols and two valuations of the truth values of sentences. The primary valuation is a classical valuation that is partial in the presence of the truth predicate. It enables us to determine the classical truth value of a sentence or leads to the failure of that determination. The language with the primary valuation is precisely the largest intrinsic fixed point of the strong Kleene three-valued semantics (LIFPSK3). The semantic shift that allows us to express the failure of the primary valuation is precisely the classical closure of LIFPSK3: it extends LIFPSK3 to a classical language in parts where LIFPSK3 is undetermined. Thus, this article provides an argumentation, which has not been present in contemporary debates so far, for the choice of LIFPSK3 and its classical closure as the right model for the truth predicate. In the end, an erroneous critique of Kripke-Feferman axiomatic theory of truth, which is present in contemporary literature, is pointed out. (shrink)

Much of the modern philosophy of causation has been governed by two ideas: (i) causes make their effects inevitable; (ii) a cause is something that makes a difference to whether its effect occurs. I focus on explaining the origin of idea (ii) and its connection to (i). On my view, the frequent attempts to turn (ii) into an analysis of causation are wrongheaded. Patterns of difference-making aren't what makes causal claims true. They merely provide a useful test for causal claims. (...) Moreover, what justifies us in using them as a test is idea (i). That's how (i) and (ii) are connected. (shrink)

Aristotle’s aitiai are middle terms in Aristotelian syllogisms. I argue that stating the aitia of a thing therefore amounts to re-describing this same thing in an alternative and illuminating way. This, in turn, means that a thing and its aitiai really are one and the same thing under different descriptions. The purpose of this paper is to show that this view is implied by Aristotle’s account of explanation, and that it makes more sense than one might expect.

In this essay, I will explain why Methodological Naturalism (MN) fails as a demarcating criteria for science. I will argue that MN is not precise enough to be useful for demarcation, unable to follow the evidence where it leads, not theologically neutral (despite its stated goals as such), and difficult to justify (and currently unjustified) as an ontological or epistemic principle.

In this article, modern standards of early years mathematics education are criticized and a proposal for change is presented. Today's early years mathematics education standards rest on a view of mathematics that became obsolete already at the end of the 19th century while the spirit of children's mathematics is precisely the spirit of modern mathematics. The proposal for change is not a return to the “new mathematics” movement, but something different.

The essential role of language in rational cognition is analysed. The approach is functional: only the results of the connection between language, reality, and thinking are considered. Scientific language is analysed as an extension and improvement of everyday language. The analysis gives a uniform view of language and rational cognition. The consequences for the nature of ontology, truth, logic, thinking, scientific theories, and mathematics are derived.

The sample space of the chance distribution at a given time is a class of possible worlds. Thanks to this connection between chance and modality, one’s views about modal space can have significant consequences in the theory of chance and can be evaluated in part by how plausible these implications are. I apply this methodology to evaluate certain forms of modal contingentism, the thesis that some facts about what is possible are contingent. Any modal contingentist view that meets certain conditions (...) that I specify generates difficulties in the philosophy of chance, including a problem usually associated with Humeanism that is known as ‘the problem of undermining futures’. I consider two well-known versions of modal contingentism that face this difficulty. The first version, proposed by Hugh Chandler and Nathan Salmon, rests on an argument for the claim that many individuals have their modal features contingently. The second version is motivated by the thesis that the existence of a possible world depends on the existence of the contingent individuals inhabiting it, and that many worlds are therefore contingent existents. (shrink)

Many philosophers and non-philosophers who reflect on the causal antecedents of human action get the impression that no agent can have morally relevant freedom. Call this the ‘non-existence impression.’ The paper aims to understand the (often implicit) reasoning underlying this impression. On the most popular reconstructions, the reasoning relies on the assumption that either an action is the outcome of a chance process, or it is determined by factors that are beyond the agent’s control or which she did not bring (...) about. I argue that arguments based on this premise fail to apply to some possible agents for whom the non-existence impression arises. On the alternative reconstruction I offer, the impression rests on the assumption that free will requires being involved in the ultimate explanation of one’s actions in a novel sense in which nothing can be involved in the ultimate explanation of anything. (shrink)

This chapter argues that the distinction between ambitious and modest transcendental arguments, developed and deployed by various authors in the wake of Stroud’s influential critique of transcendental reasoning, may be pointless when applied to transcendental arguments from performative inconsistency that have moral statements as their conclusions. If moral truth is assertorically constrained, then any modest moral transcendental argument from performative inconsistency can be converted into an ambitious moral transcendental argument. The chapter provides an account of performative inconsistency and suggests an (...) alternative to the widespread reading of transcendental conditionals in terms of an ‘if-then’-sentences whose antecedents express a proposition to the effect that some x is possible and whose consequents express a statement to the effect that some y is actual. (shrink)

Eric Watkins argues that according to Kant, causation is not a relation between two events, but a relation between the “causality” of a substance and an event. It is shown that his arguments are partly based on a confusion between causation and interaction. Further, Watkins claims that for Kant, causes cannot be temporally determined. If this were true, it would follow that there can be no causal chains, and that all factors that determine the time when an effect occurs do (...) not belong to its cause. However, it is not true. In order to understand Kant, one must distinguish between causation, action, and interaction. When two substances interact, each of them does something (an event), which causes something to happen to the other one. (shrink)

In this article I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us, all directions are the same to us and all units of length we use to create geometric figures are the same to us. On the other hand, through the process of algebraic simplification, this system of axioms directly provides the Weyl’s (...) system of axioms for Euclidean geometry. The system of axioms, together with its a priori interpretation, offers new views to philosophy and pedagogy of mathematics: it supports the thesis that Euclidean geometry is a priori, it supports the thesis that in modern mathematics the Weyl’s system of axioms is dominant to the Euclid’s system because it reflects the a priori underlying symmetries, it gives a new and promising approach to learn geometry which, through the Weyl’s system of axioms, leads from the essential geometric symmetry principles of the mathematical nature directly to modern mathematics. (shrink)

The concept of inertial frame of reference in classical physics and special theory of relativity is analysed. It has been shown that this fundamental concept of physics is not clear enough. A definition of inertial frame of reference is proposed which expresses its key inherent property. The definition is operational and powerful. Many other properties of inertial frames follow from the definition, or it makes them plausible. In particular, the definition shows why physical laws obey space and time symmetries and (...) the principle of relativity, it resolves the problem of clock synchronization and the role of light in it, as well as the problem of the geometry of inertial frames. (shrink)

A simple interpretation of quantity calculus is given. Quantities are described as two-place functions from objects, states or processes (or some combination of them) into numbers that satisfy the mutual measurability property. Quantity calculus is based on a notational simplification of the concept of quantity. A key element of the simplification is that we consider units to be intentionally unspecified numbers that are measures of exactly specified objects, states or processes. This interpretation of quantity calculus combines all the advantages of (...) calculating with numerical values (since the values of quantities are numbers, we can do with them everything we do with numbers) and all the advantages of calculating with standardly conceived quantities (calculus is invariant to the choice of units and has built-in dimensional analysis). This also shows that the standard metaphysics and mathematics of quantities and their magnitudes is not needed for quantity calculus. At the end of the article, arguments are given that the concept of quantity as defined here is a pivotal concept in understanding the quantitative approach to nature. As an application of this interpretation of quantity calculus, an easy proof of dimensional homogeneity of physical laws is given. (shrink)

Wenn Tugenden Praxisformen sind, dann kann man einiges über Praxisformen lernen, indem man nachsieht, was Tugenden sind. Ich werde dies im Folgenden partiell und indirekt tun, indem ich die sprachliche Form untersuche, in der Aristoteles die ethischen Tugenden beschreibt. Er tut dies im Wesentlichen dadurch, dass er den derart Tugendhaften in der dritten Person Singular beschreibt. Dann werde ich kurz auf die Grenzen von Aristoteles’ Verfahren zu sprechen kommen, indem ich auf eine den Tugenden und anderen Praxisformen inhärente Pluralität hinweise. (...) Eine vergleichbare Untersuchung der sprachlichen Form in der Nikomachischen Ethik ist mir nicht bekannt; die Hinweise auf die Pluralität der Tugend dagegen beanspruchen keine besondere Originalität. (shrink)

In recent years the term ‘recognition’ has been used in ever more variegated theoretical contexts. This article contributes to the discussion of how the concept(s) expressed by this term in different debates should be explicated and understood. For the most part it takes the concept itself as its topic rather than making theoretical use of it. Drawing on important work by Ikäheimo and Laitinen and taking Honneth’s tripartite distinction of recognition into love, respect, and esteem as a starting point it (...) introduces the conceptual distinction between recognitive attitudes, recognitive relations, and recognitive acts, discusses Brandom’s attempt at explaining self-consciousness in terms of reflexive recognition mediated by intersubjective recognitive relations and suggests some critical points on how Butler puts the concept of recognition to work in her conception of ethics. (shrink)

An argument is given that Euclidean geometry is a priori in the same way that numbers are a priori, the result of modeling, not the world, but our activities in the world.

According to Cantor (Mathematische Annalen 21:545–586, 1883 ; Cantor’s letter to Dedekind, 1899 ) a set is any multitude which can be thought of as one (“jedes Viele, welches sich als Eines denken läßt”) without contradiction—a consistent multitude. Other multitudes are inconsistent or paradoxical. Set theoretical paradoxes have common root—lack of understanding why some multitudes are not sets. Why some multitudes of objects of thought cannot themselves be objects of thought? Moreover, it is a logical truth that such multitudes do (...) exist. However we do not understand this logical truth so well as we understand, for example, the logical truth $${\forall x \, x = x}$$ . In this paper we formulate a logical truth which we call the productivity principle. Rusell (Proc Lond Math Soc 4(2):29–53, 1906 ) was the first one to formulate this principle, but in a restricted form and with a different purpose. The principle explicates a logical mechanism that lies behind paradoxical multitudes, and is understandable as well as any simple logical truth. However, it does not explain the concept of set. It only sets logical bounds of the concept within the framework of the classical two valued $${\in}$$ -language. The principle behaves as a logical regulator of any theory we formulate to explain and describe sets. It provides tools to identify paradoxical classes inside the theory. We show how the known paradoxical classes follow from the productivity principle and how the principle gives us a uniform way to generate new paradoxical classes. In the case of ZFC set theory the productivity principle shows that the limitation of size principles are of a restrictive nature and that they do not explain which classes are sets. The productivity principle, as a logical regulator, can have a definite heuristic role in the development of a consistent set theory. We sketch such a theory—the cumulative cardinal theory of sets. The theory is based on the idea of cardinality of collecting objects into sets. Its development is guided by means of the productivity principle in such a way that its consistency seems plausible. Moreover, the theory inherits good properties from cardinal conception and from cumulative conception of sets. Because of the cardinality principle it can easily justify the replacement axiom, and because of the cumulative property it can easily justify the power set axiom and the union axiom. It would be possible to prove that the cumulative cardinal theory of sets is equivalent to the Morse–Kelley set theory. In this way we provide a natural and plausibly consistent axiomatization for the Morse–Kelley set theory. (shrink)

The concept of truth has many aims but only one source. The article describes the primary concept of truth, here called the synthetic concept of truth, according to which truth is the objective result of the synthesis of us and nature in the process of rational cognition. It is shown how various aspects of the concept of truth -- logical, scientific, and mathematical aspect -- arise from the synthetic concept of truth. Also, it is shown how the paradoxes of truth (...) arise. (shrink)

In this chapter, we assess the current state-of-the-art of the Hungarian sharing economy sector relying on statistics, previous surveys, and expert interviews around case examples. Although we record a fast emergence of an increasing number and a widening variety of multinational and home-grown initiatives, we also contend that in Hungary, the innovation ecosystem of the collaborative economy is still relatively feeble. The linkages that are created through these initiatives are controversial sociologically. The main end-users are highly educated young urbanites. In (...) contrast, on the service provider front, the non-formal workforce is quite vulnerable as the current regulations hardly provide any protection to platform workers. The motivations of the key players in the sector are varied, as only a few locally based initiatives are transformative. In contrast, most examples are solely linked to finding new business opportunities in a shrinking economy. (shrink)

According to a standard interpretation of the term, ‘gender’ denotes sets of social roles and expectations conventionally associated with the sexual physiology of human beings. Originally introduced in psychology, the term is now widely used in the social sciences and humanities, as well as in the biological sciences. In this article we introduce and discuss the central themes of contemporary philosophical debates on gender. Particular attention is paid to recent feminist arguments concerning the distinction between sex and gender, and to (...) how feminist debates today intersect with neuroscience research on sex-related differences between human brains. (shrink)

This paper examines two contemporary answers to the question of whether moral values and norms are apt for rational criticism and justification: Richard Rorty’s radically contextualist approach—which is centered around the notion of contingency and is characterized by a dismissal of all claims to philosophical justification—and Karl-Otto Apel’s transcendental-pragmatic version of discourse ethics—which encompasses highly ambitious claims to justification and universal validity. Contrasting the key theses of Rorty’s contextualism with those of Apel’s universalist discourse ethics and reconstructing their respective conceptions (...) of moral progress we argue that neither Rorty’s nor Apel’s position is convincing. (shrink)

Contingentism is the view that it is possible for there to be contingent existents. Timothy Williamson has argued that contingentists cannot provide a satisfactory interpretation of the possible-world semantics for modal logic. This paper aims to provide such an interpretation on behalf of contingentists.

On the basis of elementary thinking about language functioning, a solution of truth paradoxes is given and a corresponding semantics of a truth predicate is founded. It is shown that it is precisely the two-valued description of the maximal intrinsic fixed point of the strong Kleene three-valued semantics.

Complex logic is a novel logical framework, which formalizes the semantics of the categories of matter, space, and time in a system of logic that operates with complex logical objects. A complex logical object represents a superposition of a logical statement and its logical negation positioning any statement co-relatively to its logical negation. In the system of logical notations, where S is a logical statement and Not S is its logical negation, complex logic includes co-relative logical positions of S and (...) Not S with the probabilities of their truth within the scale 0... 1, excluding the boundary values of 0 and 1, into a single logical superposition. Thus, the logical superposition, summing up the probabilistic positions of S and Not S, has an invariant truth value equal to 1. In this context, complex logic proves to be invariant to reality, offering a unified logical interpretation of reality, spanning from quantum to cosmological scales. By defining the nature of matter and reality as a complex two-format processual essence with corresponding logical consequences, complex logic enhances our concept and understanding of reality. (shrink)