Search

Since 1970, a United States prosecutor can prove perjury without specifying which statement is perjurious. A bold prosecutor could concede ignorance of which statement is false. A bolder prosecutor could further concede that the witness himself does not know. The boldest prosecutor could concede there is no specific lie. Instead of there being a statement that is intrinsically perjurious, the perjury is relational. Just as two statements can be inconsistent without either being inconsistent, two statements can be perjurious without either being perjurious. These consequences are reconciled with the generalisation that all perjury involves lying. Corollaries about culpability are drawn from the phenomenon of nonspecific perjury. The reasoning is generalised to other forms of illegal lying. [ABSTRACT FROM AUTHOR]

Brouwer's philosophy of mathematics is usually regarded as an intra-subjective, even solipsistic approach, an approach that also underlies his mathematical intuitionism, as he strived to create a mathematics that develops out of something inner and a-linguistic. Thus, points of connection between Brouwer's mathematical views and his views about and the social world seem improbable and are rarely mentioned in the literature. The current paper aims to challenge and change that. The paper employs a socially oriented prism to examine Brouwer's views on the construction, use, and practice of mathematics. It focuses on Brouwer's views on language, his social interactions, and the importance of group context as they appear in the significs dialogues. It does so by exploring the establishment and dissolution of the significs movement, focusing on Gerrit Mannoury's influence and relationship with Brouwer and analyzing several fragments from the significs dialogues while emphasizing the role Brouwer ascribed to groups in forming and sharing new ideas. The paper concludes by raising two questions that challenge common historical and philosophical readings of intuitionism. [ABSTRACT FROM AUTHOR]

The article is devoted to selected new trends in metaethics, ethics, political philosophy, and the practice of philosophy with the use of artificial intelligence. It consists of five parts. In the first part, I analyze the meaning of the question about new trends in analytical philosophy. In the second part, I present the latest trends in metaethics. In the third part, I briefly characterize the ethics of artificial intelligence. The fourth part, devoted to political philosophy, includes a presentation of the latest publications on meritocracy. The article concludes with considerations about practicing philosophy in the age of artificial intelligence. [ABSTRACT FROM AUTHOR]

My first aim in this paper is to use time diagrams in the style of Brentano to analyze constructions in Brouwer's separable mathematics more precisely. I argue that constructions must involve not only pairing and projecting as basic operations guaranteed by the intuition of twoity, as sometimes assumed in the literature, but also a recalling operation. My second aim is to argue that Brouwer's views on the intuition of twoity and arithmetic lead to an ontological explosion. Redeveloping the constructions of natural numbers and systems sketched in an appendix to Brouwer's Cambridge lectures, I observe that the only plausible way he can make some elementary arithmetic in his separable mathematics is by allowing for the same canonical number to be determined by multiple separable entities, resulting in an overabundant mathematical ontology. [ABSTRACT FROM AUTHOR]

In a recent paper, Nicolas Gisin suggests that by conducting physics with intuitionistic rather than classical mathematics, rich temporality—that is, passage and tense, and specifically the future's openness—can be incorporated into physics. Physics based on classical mathematics is tenseless and deterministic, and that, so he holds, renders it incongruent with experience. According to Gisin, physics ought to represent the indeterminate nature of reality, and he proposes that intuitionistic mathematics is the key to succeeding in doing so. While I share his insistence on the reality of passage and tense and on the future being real and open, I argue that the amendment he offers does not work. I show that, its attunement to time notwithstanding, intuitionistic mathematics is as tenseless as classical mathematics and that physics is bound to remain tenseless regardless of the math it employs. There is much to learn about tensed time, but the task belongs to phenomenology and not to physics. [ABSTRACT FROM AUTHOR]

We propose a model of physics that blends Rovelli’s relational quantum mechanics (RQM) interpretation with the language of finite information quantities (FIQs), defined by Gisin and Del Santo in the spirit of intuitionistic mathematics. We discuss deficiencies of using real numbers to model physical systems in general, and particularly under the RQM interpretation. With this motivation for an alternative mathematical language, we propose the use of FIQs to model the world under the RQM interpretation, wherein we view the propensities that make up a FIQ as quantifications of potential interaction. Under this model, the stable facts, relative facts, and shifting perspectives that make up the relational interpretation correspond to shifting digits and propensities of the FIQs. The model’s predictions agree with those of both classical and quantum physics, and it is indeterministic. We also propose explanations, with examples, for how the propensities of a FIQ are distributed, and how its digits become actualized. This is equivalent to the notion of the measurement problem, and the question of what causes wave function collapse. In short, by stepping through the “new door” opened by the language of FIQs, we attempt to describe the world under the relational interpretation. [ABSTRACT FROM AUTHOR]

This paper extends the axiomatic treatment of intuitionistic mereology introduced in Maffezioli and Varzi (Synthese, 198(S18), 4277–4302 2021) by examining the behavior of constructive notions of overlap and disjointness. We consider both (i) various ways of defining such notions in terms of other intuitionistic mereological primitives, and (ii) the possibility of treating them as mereological primitives of their own. [ABSTRACT FROM AUTHOR]

Hegel interpreters commonly reject attempts to situate Hegel's logic in relation to modern movements. Appealing to his criticisms of the logic of Verstand or mere understanding with its fixed logical structure, Hegel's logic, it is pointed out, was a logic of Vernunft or reason—a logic more at home in the thought of Plato and Aristotle than in modern mathematical forms. Contesting this implied dichotomy, it is here argued that the ancient roots of Hegel's logic, especially as transmitted by late Neopythagorean/Neoplatonic thinkers such as Proclus, gave it many features similar to ones later found in the type of algebraic transformation of Aristotle, started first by Leibniz, reanimated by Boole in the mid-nineteenth century and then developed by others such as C. S. Peirce and Arend Heyting. In particular, the ancient mathematics upon which Hegel had drawn allowed him to anticipate an answer to the criticism that Frege would later aim at Boole, concerning his inability to unite opposed class and propositional calculi. Hegel's logic would be a hybrid, incorporating features found later in intuitionist and classical logic, but it could be so because of the way he had called upon the mathematics of the ancient Platonist tradition. [ABSTRACT FROM AUTHOR]

We provide a framework for understanding agnosticism. The framework accounts for the varieties of agnosticism while vindicating the unity of the phenomenon. This combination of unity and plurality is achieved by taking the varieties of agnosticism to be represented by several agnostic stances, all of which share a common core provided by what we call the minimal agnostic attitude. We illustrate the fruitfulness of the framework by showing how it can be applied to several philosophical debates. In particular, several philosophical positions can be aptly conceived of as instances of agnosticism whilst retaining their differences and distinguishing features. [ABSTRACT FROM AUTHOR]

Most people believe some moral propositions are true. Most people would say that they know that rape is wrong, torturing people is wrong, and so on. But despite decades of intense epistemological study, philosophers cannot even provide a rudimentary sketch of moral knowledge. In my view, the fact that we have very strong epistemic confidence in some fundamental moral propositions and the fact that it is extremely difficult for us to provide even the basics of an account of moral knowledge gives us an important clue. Both of these facts stem from the very nature of moral knowledge. In this paper I provide an intuitionist account of moral knowledge. I try to remove misunderstanding and add to our understanding of the contemporary account of self-evident moral propositions. For a theory of moral knowledge to be acceptable it must explain both the moral knowledge we have and why it is so very difficult for us to explain. My theory meets both requirements. [ABSTRACT FROM AUTHOR]

How Moore's open question argument works, insofar as it does, remains a matter of controversy. The author's purpose here is to construct an open question argument based on a novel interpretation of how Moore's argument might work. In order to sidestep exegetical questions, he does not claim to be offering Moore's own argument. Rather, the author offers a reconstruction, making use of important elements of Moore's methodology and assumptions that could be reasonable within a Moorean viewpoint. The crucial role within the argument is played by what the author calls the real thought move. He shows that the reconstructed argument is more defensible from some standard objections than the common construction. The author finishes by drawing attention to a neglected objective that would make sense within Moore's viewpoint, showing that it fits with a major commitment of Bonjour's moderate rationalism, and showing how the package might be important for the non-naturalist today. [ABSTRACT FROM AUTHOR]

One of the most important "folk" anti‐realist thoughts about certain areas of our thought and discourse—basic taste, for instance, or comedy—is that their lack of objectivity crystallises in the possibility of "faultless disagreements": situations where one party accepts P, another rejects P, and neither is guilty of any kind of mistake of substance or shortcoming of cognitive process. On close inspection, however, it proves challenging to make coherent sense of this idea, and a majority of theorists have come to reject it as incoherent. There are two significant exceptions in the contemporary literature: relativists often hold it up as something of a coup for their view that it can make straightforward sense of faultless disagreement; and the author of this paper has argued (Wright 2006) that making judicious intuitionistic revisions to classical logic can provide resources that suffice to stabilise the notion. The present paper argues that neither relativism nor intuitionism in fact provides a satisfactory account and indicates how an alethic pluralist framework enables us to do better. [ABSTRACT FROM AUTHOR]

In this paper I will make a review of the core elements of the recent natural law ethical theories in analytic philosophy. I will expose the elements they share, and then its more essential differences, especially those related to epistemology and metaphysics. Finally, I shall explore the possibilities of development for natural law theories, and its metaethical implications, especially those related to a metaphysical and epistemological grounding for natural law. [ABSTRACT FROM AUTHOR]

This article focuses on the moral development ideal of William Whewell, one of the most important intellectual figures of the 19th century Britain. According to Whewell, morality is an ideal of development. This is a process that continues throughout human life in which there is true happiness at the end. In every moment we live, we have the opportunity to be a more virtuous and better person. According to him, the most important task of human beings is to realize this ideal of development. The motivation behind this ideal is to bring morality, science and religion together in a rational system in harmony with each other. The source of this motivation is the criticism of irrationality and inconsistency directed by utilitarian thinkers (especially, John Stuart Mill), the most important opponents of ethical intuitionism. As a way of making these three areas consistent with each other, Whewell bases morality and science on a theological basis. Accordingly, God, the creator of the universe and human beings, has laid down certain principles that will regulate both nature and human actions. The duty of morality and science is to discover these principles. Therefore, morality and science are simply a process of discovery. Whewell argues that the basic principles of morality can only be known intuitively. The main reason why Whewell is considered within the ethical intuitionist tradition is this view. But this process of knowing the basic principles does not in any way imply an irrationality. The basic power that intuitively perceives these principles is reason. Conscience is a faculty that approves the judgments of reason. However, in Whewell's philosophy, conscience corresponds to an aspect of reason rather than being a faculty separate from reason. Thus, Whewell is distinguished from the intuitive moral thinkers who regard conscience as a faculty separate from reason, which grasps the basic principles of morality. According to Whewell, conscience is the guide of a human being but the judgments of conscience can sometimes be mistaken. Therefore, conscience must be constantly controlled and developed by reason. This point is where Whewell bases his moral philosophy. Because one's duty is to continuously improve their conscience and make all their judgments in accordance with the highest rule. This corresponds to moral development. The concepts of the highest good, duty, and virtue are extremely important in moral development. The ultimate rule in Whewell's philosophy expresses the will of God. The obligation to obey God's commands corresponds to duty, and fulfilling the duty corresponds to virtue. Thus, any violation of duty is a violation of God's command. Failing to do the duty is to go against the will of God. In a religious context, this corresponds to sin. Whewell argues, however, that the ideal of moral progress cannot be fulfilled by reason alone. The main reason for this is the limited structure of reason. Therefore, reason needs the help of revelation in this process. Nevertheless, in order to preserve the rational ground of his moral philosophy Whewell prioritizes reason over revelation and natural religion over revealed religion. However, Whewell still argues the help of revelation necessary for the fulfillment of the ideal of progress. He even says that those who do not accept this help of revelation and do not believe in revelation should be condemned. Because, according to him, this ideal of development can be completed with the information beyond the reach of reason that can only be bestowed by revelation. Whewell goes so far as to argue that no one was able to complete their moral evolution before Jesus Christ descended on earth. He also shows that revelation did not occur at that time as a justification. The question of whether Whewell's claim has an impact on the rationality of his philosophy is the main problem of this article. Here I argue that Whewell's approach that the ideal of moral development cannot be completed without the help of revelation makes his philosophy controversial, even if it does not completely remove the rational ground of moral theory. [ABSTRACT FROM AUTHOR]

In 1928 Ivan Orlov published a remarkable paper which contains the first formulation of a relevant logic. The paper remained largely unknown to English-speakers until this discovery of relevant logic was shown by Kosta Do en in 1992. By that time the material on relevant logic was well known; however, there is more of interest in Orlov's paper than this: his remarks on intuitionism, the motivation for his construction, and its wider implications. This paper explains and reflects on these matters. [ABSTRACT FROM AUTHOR]

Argument: The paper explores Hermann Weyl's turn to intuitionism through a philosophical prism of normative framework transitions. It focuses on three central themes that occupied Weyl's thought: the notion of the continuum, logical existence, and the necessity of intuitionism, constructivism, and formalism to adequately address the foundational crisis of mathematics. The analysis of these themes reveals Weyl's continuous endeavor to deal with such fundamental problems and suggests a view that provides a different perspective concerning Weyl's wavering foundational positions. Building on a philosophical model of scientific framework transitions and the special role that normative indecision or ambivalence plays in the process, the paper examines Weyl's motives for considering such a radical shift in the first place. It concludes by showing that Weyl's shifting stances should be regarded as symptoms of a deep, convoluted intrapersonal process of self-deliberation induced by exposure to external criticism. [ABSTRACT FROM AUTHOR]

We continue the investigation of the first paper where we studied logics with various negations including empirical negation and co-negation. We established how such logics can be treated uniformly with R. Sylvan's CCω as the basis. In this paper we use this result to obtain cut-free labelled sequent calculi for the logics. [ABSTRACT FROM AUTHOR]

We investigate the relationship between M. De's empirical negation in Kripke and Beth Semantics. It turns out empirical negation, as well as co-negation, corresponds to different logics under different semantics. We then establish the relationship between logics related to these negations under unified syntax and semantics based on R. Sylvan's CCw. [ABSTRACT FROM AUTHOR]

On a bilateralist reading, sequents are interpreted as statements to the effect that, given the assertion of the antecedent it is incoherent to deny the succedent. This interpretation goes against its own ecumenical ambitions, endowing Cut with a meaning very close to that of tertium non datur and thus rendering it intuitionistically unpalatable. This paper explores a top-down route for arguing that, even intuitionistically, a prohibition to deny is as strong as a licence to assert. [ABSTRACT FROM AUTHOR]

Abstract Recently, a number of formal systems for Nonstandard Analysis restricted to the language of finite types, i.e. nonstandard arithmetic , have been proposed. We single out one particular system by Dinis–Gaspar, which is categorised by the authors as being part of intuitionistic nonstandard arithmetic. Their system is indeed in consistent with the Transfer axiom of Nonstandard Analysis, and the latter axiom is classical in nature as it implies (higher-order) comprehension. Inspired by this observation, the main aim of this paper is to provide answers to the following questions: (Q1) In the spirit of Reverse Mathematics , what is the minimal fragment of Transfer that is inconsistent with the Dinis–Gaspar system? (Q2) What other axioms are inconsistent with the Dinis–Gaspar system? Our answer to the first question suggests that the aforementioned inconsistency actually derives from the axiom of extensionality relative to the standard world, and that other (much stronger) consequences of Transfer are actually harmless. Perhaps surprisingly, our answer to the second question shows that the Dinis–Gaspar system is inconsistent with a number of (non-classical) continuity theorems which one would – in our opinion – categorise as intuitionistic in the sense of Brouwer. Finally, we show that the Dinis–Gaspar system involves a standard part map , suggesting this system also pushes the boundary of what still counts as 'Nonstandard Analysis' or 'internal set theory'. [ABSTRACT FROM AUTHOR]

The field of sociology has long been subject to critique for alleged ideological bias and left-wing groupthink linked to its social justice mission. Critics contend that the construction of “sacred victims” by progressive intellectuals hinders their ability to objectively appraise the circumstances of such vulnerable groups. To address this criticism, we survey 479 sociologists in national universities and colleges in the U.S. regarding three sensitive controversies: urban poverty in the black community; gendered differences in occupational choices; and immigration. We find significant patterns in the data. Commitment to the field’s “moral mission,” preferred research paradigm, gender, and especially political orientation are all significant predictors of sociologists’ views. The results, we suggest, can be understood by conceptualizing the field of sociology as an “emotive community.” In doing so, we draw upon current social psychological research on moral foundations theory developed by Jonathan Haidt and colleagues. [ABSTRACT FROM AUTHOR]

In a recent paper, Wigglesworth claims that syntactic criteria of theoretical equivalence are not appropriate for settling questions of equivalence between logical theories, since such criteria judge classical and intuitionistic logic to be equivalent; he concludes that logicians should use semantic criteria instead. However, this is an artefact of the particular syntactic criterion chosen, which is an implausible criterion of theoretical equivalence (even in the non-logical case). Correspondingly, there is nothing to suggest that a more plausible syntactic criterion should not be used to settle questions of equivalence between different logical theories; such a criterion (which may already be found in the literature) is exhibited and shown to judge classical and intuitionistic logic to be inequivalent. [ABSTRACT FROM AUTHOR]

The conception of the paper is connected with bringing forward the reflection of Leon Petrażycki on intuitive law. For this purpose I analyze the genesis and dynamics of this phenomenon on the cultural-historical level, as well as with reference to issues belonging to the scope of positive law. In addition, I broaden the research field with the range of problems touching on intuitionism, morality, and also independent ethics of Janusz Kotarbinski. The starting point of the methodological optics I assume is constituted by the multi-aspectual transformations surrounding us in the sphere of axiology. Hence, if the pedagogical aspects are taken into account, it seems to me justified to undertake some actions in order to search for the logically consistent, sensible and universal solutions, which can become an ethical guide-post for the contemporary human being. [ABSTRACT FROM AUTHOR]

In The Boundary Stones of Thought (2015), Rumfitt defends classical logic against challenges from intuitionistic mathematics and vagueness, using a semantics of pre-topologies on possibilities, and a topological semantics on predicates, respectively. These semantics are suggestive but the characterizations of negation face difficulties that may undermine their usefulness in Rumfitt’s project. [ABSTRACT FROM AUTHOR]

Frege writes in Numbers and Arithmetic about kindergarten-numbers and "an a priori mode of cognition" that they may have "a geometrical source." This resembles recent findings on arithmetical cognition. In my paper, I explore this resemblance between Gottlob Frege's later position concerning the geometrical source of arithmetical knowledge, and some current positions in the literature dedicated to arithmetical cognition, especially that of Stanislas Dehaene. In my analysis, I shall try to mainly see to what extent (Frege's) logicism is compatible with (Dehaene's) intuitionism. [ABSTRACT FROM AUTHOR]

We present a bounded modified realisability and a bounded functional interpretation of intuitionistic nonstandard arithmetic with nonstandard principles. The functional interpretation is the intuitionistic counterpart of Ferreira and Gaspar's functional interpretation and has similarities with Van den Berg, Briseid and Safarik's functional interpretation but replacing finiteness by majorisability. We give a threefold contribution: constructive content and proof-theoretical properties of nonstandard arithmetic; filling a gap in the literature; being in line with nonstandard methods to analyse compactness arguments. [ABSTRACT FROM AUTHOR]

In § 7 of the Critique of Practical Reason Kant presents his first definition of the universal law of morality--called the Categorical Imperative. In the same place, he affirms that being conscious of the law of morality is a Faktum of reason. The ambiguity of this statement has prompted some authors to argue that Kant does not deduce the moral law satisfactorily nor does he clarify how we become conscious of it. Nevertheless, they all forget in their analysis a relevant aspect of Kantian moral philosophy that turns out to be of great importance in contemporary studies of the philosophy of action--the self-constituting character of the rational subject and the role it plays within the foundation of morality. I claim that the Faktum of reason must be understood fundamentally as a self-constituting act. [ABSTRACT FROM AUTHOR]

In this paper we draw a proposal to develop a logic of fictions in the game-theoretical approach of dialogical pragmatism. From one of the main criticisms that point to classical logic: the structural schizophrenia of its semantics (Lambert, 2004: 142-143; 160), we review the ontological commitments of the two main traditions of logic (Aristotle and Frege) to highlight their limits concerning the analysis of fictional discourse, and the overcoming from a pragmatic game perspective. [ABSTRACT FROM AUTHOR]

Jonathan Haidt proposed one of the most well-known versions of moral nativism, in which he defends the existence of certain moral foundations. In this paper, I examine the main criticisms of this theory, especially a) the claim that it implies a questionable thesis about the human mind; and b) that Haidt's description is incomplete inasmuch as it does not provide an explanation for the mental steps prior to the triggering of intuitions. As for the first criticism, in accepting the thesis of massive modularity, Haidt commits himself to a view of the human mind that is not consilient with the results of neuroscientific research. Nevertheless, since he is aware of this objection, he proposes a milder version of his theory to counter the critics of massive modularity. Regarding the second criticism, Haidt's theory is, in fact, incomplete; for it ignores that it is only possible that an action provokes an intuition after the observer has formed some kind of representation. From the analysis of these criticisms, I conclude that the Moral Foundations Theory, in its mild version, offers an apparently plausible explanation for moral judgments. Notwithstanding, Haidt's theory fails in some measure. It does not explain the role that assumptions about the mental states of the author of an action play in triggering moral intuitions. [ABSTRACT FROM AUTHOR]

My goal for this paper can be presented as follows: I will attempt to show that objections to intuitionism, although they are serious, do not undermine entirely its fertility for knowledge and moral reasoning. This is probably the perception of contemporary philosophers like David Enoch, Robert Audi, Russ Shafer-Landau or John McDowell. In order to fulfill the objective mentioned above, I will do the following. First, I will outline broadly two of the paradigmatic features of moral intuitionism in order to identify it as a particular metaethics doctrine. Secondly, I will summarize some of the main objections that have been raised in order to discredit the value of moral intuitionism as a source both of moral knowledge and of valid support for moral reasoning. In third place, I will try, also briefly, to explain some of the possible (not all of course) answers to the objections previously mentioned in the paper. Fourth, I will recapitulate the more fruitful aspects of intuitionism, especially in regard to moral reasoning. [ABSTRACT FROM AUTHOR]

In this paper, from a critical point of view, I underlined the fact that, in spite of the success of their intuitive reconstruction of mathematics, both Kantianism and Neo-intuitionism (not only the Brouwerian one!) do not represent a solid philosophical ground of them (mathematics). Besides, even Heyting says that for the better, to achieve the understanding in mathematical intuitionism it is not necessary to address its philosophical thesis or some psychological ones. Nevertheless, this does not mean that, in general, for a proper grounding of mathematical intuitionism, a suitable philosophical view is useless, such as a correlative theory of intuition, which (intuition), by the way, in every intuitionist perspective is assumed as a fundamental item of mathematical thinking. [ABSTRACT FROM AUTHOR]

In this note I am re ecting on interrelations between three concepts of truth: (1) that employed by Hilbert arguing his formalist view on the nature of math- ematics, (2) Freges idea of truth supported by mathematical intuition, and (3) known as Aristotelian correspondence idea of truth concerning any propositions not merely mathematical. [ABSTRACT FROM AUTHOR]

The aim of the article. The purpose of this article is to comprehensively reconstruct and present the theory of knowing of values according to Henryk Elzenberg. Epistemology of values is one of the parts of the of Elzenberg's formal axiology. The reconstruction method and conceptual analysis were used for realisation of the proposal. Discussed concepts. The reconstruction begins with the presentation of the main elements of Elzenber's philosophy of values. First, the concepts of utilitarian and perfectionist value are distinguished, and later perfectionist value is defined as the 'feature' of an object, which is as it should be. Then, Elzenberg's notion of intuition as individual non-sensual representation is explained. Elzenberg's deliberations on the nature of primary judgments are reconstructed. Reflection leads to the conclusion that the fundamental judgments in axiology are individual judgments about features which endow values. Later, Elzenberg's critique of the positions negating the cognitive meaning of the evaluations are presented. The main point of criticism was to demonstrate that the presence of the notion of perfectionist value shows the inability to reduce values to subjective states of consciousness, and that the influence of emotions in the recognition of values can be limited. This leads to the question of a methodology of knowing values, which recommends asceticism and limiting emotions that may side by side determine the value judgment. Cognitive value of the text. The text synthesizes various threads of Elzenberg's epistemology of values. The fundamental to Elzenberg's link between the theory of knowing values and the rest of general axiological considerations is visible. Using the materials contained in unpublished manuscripts, there is some important information on the Elzenberg's concept of intuition. The presented concept shows various arguments that can be used against positions that negate the autonomous and objective character of values, especially against positions that deny the possibility of a competent adjudication of presence of values. [ABSTRACT FROM AUTHOR]

The translation into English of Aleksandr Yakovlevich Khinchin's (1894–1959) 1926 paper entitled ‘Ideas of intuitionism and the struggle for a subject matter in contemporary mathematics’ is made available for the first time. Here, Khinchin presented the famous foundational debate between L.E.J. Brouwer and David Hilbert of the 1920s in terms of a search for a mathematics with content. His main aim seems to have been to make intuitionism ideologically acceptable to his audience at the Communist Academy by means of the claim that insofar as Brouwer's intuitionism had a clear ‘subject matter’ and Hilbert's new program was a concession to intuitionism, the alleged victory of intuitionism not only implied the defeat of ‘empty’ formalism, but also showed the compatibility and affinity of Marxism with the newest developments in modern mathematics. This introduction provides a tentative exploration of the issue of what was tactical (or due to ideological pressure) and what was real scientific interest (or due to ignorance) (or what was both) in Khinchin's 1926 paper in the form of a detailed commentary, especially, on the tactical side of his presentation of the positions of Brouwer and Hilbert. [ABSTRACT FROM AUTHOR]

With the recent revival of moral intuitionism, the work of W. D. Ross has grown in stature. But if we look at some recent well-regarded histories, anthologies and companions of analytic philosophy, Ross is noticeably absent. This discrepancy of assessments raises the question of Ross’s place in the history of analytic philosophy. Hans-Johann Glock has recently claimed that Ross is not an analytic philosopher at all, but is instead a ‘traditional philosopher’. In this article, I will identify several undeniable features of analytic philosophy that Ross’s work bears: a focus on linguistic analysis, great respect for pre-theoretical thoughts, the conviction that philosophy is a collaborative, piecemeal enterprise and so on. Such an investigation, I claim, reveals two historically significant results: Ross was the first ethicist to fully draw from commonsense beliefs about morality in light of characteristic analytic considerations to secure his theory. Two, concerning the matter of whether the notions ‘right’ and ‘good’ are reducible to other notions, Ross appears to have been right: ‘right’ and ‘good’ are irreducible notions. The classical analytic metaethicists, who based their entire research programme on the promise of finding suitable reductive semantic analyses of ‘right’ and ‘good’, were wrong. These results, I believe, suffice to secure W. D. Ross a high place in the history of analytic philosophy. [ABSTRACT FROM PUBLISHER]

Accounts of non-naturalist moral perception have been advertised as an empiricist-friendly epistemological alternative to moral rationalism. I argue that these accounts of moral perception conceal a core commitment of rationalism--to substantive a priori justification--and embody its most objectionable feature--namely, "mysteriousness." Thus, accounts of non-naturalist moral perception do not amount to an interesting alternative to moral rationalism. [ABSTRACT FROM AUTHOR]

Intuitionism, as well as logicism and formalism, has been one of the main approaches in the foundations of mathematics. It also constitutes an approach on mathematics and mostly on the progress of mathematics. The formalist direction in the development of the intuitionist mathematics gains more and more advocates, which does not mean that the other approaches mentioned above (including the proper intuitionist one) were forsaken. Which of the two opposed approaches on neointuitionist formalist mathematics is the right one is no longer an issue regarding the way the development of mathematics is conceived, but the solution of the issue depends on the very course of this development. [ABSTRACT FROM AUTHOR]

In this paper we study a model of intuitionistic higher-order logic which we call the Muchnik topos. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, the Muchnik reals, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a choice principle (∀x ∃yA(x, y)) ⇒ ∃w ∀xA(x,wx) and a bounding principle (∀x ∃yA(x, y)) ⇒ ∃z ∀x ∃y(y ⩽T (x, z) ∧ A(x, y)) where x, y, z range over Muchnik reals, w ranges over functions from Muchnik reals to Muchnik reals, and A(x, y) is a formula not containing w or z. For the convenience of the reader, we explain all of the essential background material on intuitionism, sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems, Muchnik degrees, and Kolmogorov's calculus of problems. In a separate document we provide an English translation of Muchnik's 1963 paper on Muchnik degrees. [ABSTRACT FROM AUTHOR]

This paper reviews the existing definitions and formulas of entropy for interval-valued intuitionistic fuzzy sets (IVIFSs) and demonstrates that they cannot fully capture the uncertainty of IVIFSs. Then considering both fuzziness and intuitionism of IVIFSs, we introduce a novel axiomatic definition of entropy for IVIFSs and develop several entropy formulas. Example analyses show that the developed entropy formulas can fully reflect both fuzziness and intuitionism of IVIFSs. Furthermore, based on the entropy formulas of IVIFSs, a method is proposed to solve multi-attribute decision making problems with IVIFSs. Additionally, an investment alternative selection example is provided to validate the practicality and effectiveness of the method. [ABSTRACT FROM AUTHOR]

I address Sinnott-Armstrong's argument that evidence of framing effects in moral psychology shows that moral intuitions are unreliable and therefore not noninferentially justified. I begin by discussing what it is to be epistemically unreliable and clarify how framing effects render moral intuitions unreliable. This analysis calls for a modification of Sinnott-Armstrong's argument if it is to remain valid. In particular, he must claim that framing is sufficiently likely to determine the content of moral intuitions. I then re-examine the evidence which is supposed to support this claim. In doing so, I provide a novel suggestion for how to analyze the reliability of intuitions in empirical studies. Analysis of the evidence suggests that moral intuitions subject to framing effects are in fact much more reliable than perhaps was thought, and that Sinnott-Armstrong has not succeeded in showing that noninferential justification has been defeated. [ABSTRACT FROM AUTHOR]

Ramsey's Theorem for pairs is a fundamental result in combinatorics which cannot be intuitionistically proved. In this paper we present a new form of Ramsey's Theorem for pairs we call the H -closure Theorem, where H stands for “homogeneous”. The H -closure Theorem is a property of well-founded relations, intuitionistically provable, informative, and simple to use in intuitionistic proofs. Using our intuitionistic version of Ramsey's Theorem we intuitionistically prove the Termination Theorem by Podelski and Rybalchenko [25] . The Termination Theorem concerns an algorithm inferring termination for while-programs, and was originally proved from the classical Ramsey Theorem. Vytiniotis, Coquand, and Wahlstedt provided an intuitionistic proof of the Termination Theorem [29] , using the Almost Full Theorem [11] , an intuitionistic version of Ramsey's Theorem different from the H -closure Theorem. We provide a second intuitionistic proof of the Termination Theorem using the H -closure Theorem. In another paper, we use our proof to extract bounds for the Termination Theorem [5] . [ABSTRACT FROM AUTHOR]

After introducing semantic anti-realism and the paradox of knowability, the paper offers a reconstruction of the anti-realist argument from understanding. The proposed reconstruction validates an unrestricted principle to the effect that truth requires the existence of a certain kind of "demonstration". The paper shows that that principle fails to imply the problematic instances of the original unrestricted feasible-knowability principle but that the overall view underlying the new principle still has unrestricted epistemic consequences. Appealing precisely to the paradox of knowability, the paper also argues, against the BHK semantics, for the non-constructive character of the demonstrations envisaged by semantic anti-realism, and contends that, in such setting, one of the most natural arguments for a broadly intuitionist revision of classical logic loses all its force. [ABSTRACT FROM AUTHOR]

This article shows that an apparent puzzle in finance and accounting is resolved by changing from classical to constructive (more specifically, intuitionistic) mathematics. Our position is that it is unproblematic if real-world actors behave inconsistently with nonconstructive mathematical results. Thus the solution to our puzzle lies in a deeper challenge to the standard view of how rational agents reason. The alternative we propose gives refutable predictions that find support in the empirical capital markets literature. Our main tool, intuitionistic mathematics, plays a similar role in the mathematical philosophy literature to that of ambiguity in decision theoretic work. [ABSTRACT FROM AUTHOR]

I propose a new semantics for intuitionistic logic, which is a cross between the construction-oriented semantics of Brouwer-Heyting-Kolmogorov and the condition-oriented semantics of Kripke. The new semantics shows how there might be a common semantical underpinning for intuitionistic and classical logic and how intuitionistic logic might thereby be tied to a realist conception of the relationship between language and the world. [ABSTRACT FROM AUTHOR]

In moral epistemology, the method of reflective equilibrium is often characterized in terms of intuitions or understood as a method for justifying intuitions. An analysis of reflective equilibrium and current theories of moral intuitions reveals that this picture is problematic. Reflective equilibrium cannot be adequately characterized in terms of intuitions. Although the method presupposes that we have initially credible commitments, it does not presuppose that they are intuitions. Nonetheless, intuitions can enter the process of developing a reflective equilibrium and, if the process is successful, be justified. Since the method of reflective equilibrium does not essentially involve intuitions, it does not constitute a form of intuitionism in any substantial sense. It may be classified as intuitionist only in the minimal sense of not reducing justification to a matter of inference relations alone. [ABSTRACT FROM AUTHOR]

Derek Parfit’s long-awaited work On What Matters is a very ambitious, very strange production seeking to defend both a nonreductive and nonnaturalistic but nonmetaphysical and nonontological form of cognitive intuitionism or rationalism and an ethical theory (the Triple Theory) reflecting the convergence of Kantian universalizability, Scanlonian contractualism, and rule utilitarianism. Critics have already countered that Parfit’s metaethics is unbelievable and his convergence thesis unconvincing, but On What Matters is a truly Sidgwickian work, the implications of which largely remain to be worked out. Parfit does not go far enough in spelling out exactly what matters and why, what normative reasons we actually have, and where we should go from here, if we take him seriously. [ABSTRACT FROM PUBLISHER]

Most empirical accounts of design suggest that designing is an activity where objects and representations are progressively constructed. Despite this fact, whether design is a constructive process or not is not a question directly addressed in the current design research. By contrast, in other fields such as Mathematics or Psychology, the notion of constructivism is seen as a foundational issue. The present paper defends the point of view that forms of constructivism in design need to be identified and integrated as a foundational element in design research as well. In fact, a look at the literature reveals at least two types of constructive processes that are well embedded in design research: first, an interactive constructivism, where a designer engages a conversation with media, that allows changing the course of the activity as a result of this interaction; second, a social constructivism, where designers need to handle communication and negotiation aspects, that allows integrating individuals' expertise into the global design process. A key feature lacking these well-established paradigms is the explicit consideration of creativity as a central issue of design. To explore how creative and constructivist aspects of design can be taken into account conjointly, the present paper pursues a theoretical approach. We consider the roots of constructivism in mathematics, namely the Intuitionist Mathematics, in order to shed light on the original insights that led to the development of a notion of constructivism. Intuitionists describe mathematics as the process of mental mathematical constructions realized by a creative subject over time. One of the most original features of intuitionist constructivism is the introduction of incomplete objects into the heart of mathematics by means of lawless sequences and free choices. This allows the possibility to formulate undecided propositions and the consideration of creative acts within a formal constructive process. We provide an in-depth analysis of intuitionism from a design standpoint showing that the original notion is more than a pure constructivism where new objects are a mere bottom-up combination of already known objects. Rather, intuitionism describes an imaginative constructivist process that allows combining bottom-up and top-down processes and the expansion of both propositions and objects with free choices of a creative subject. We suggest that this new form of constructivism we identify is also relevant in interpreting conventional design processes and discuss its status with respect to other forms of constructivism in design. [ABSTRACT FROM AUTHOR]

We describe a variety of sets internal to models of intuitionistic set theory that (1) manifest some of the crucial behaviors of potentially infinite sets as described in the foundational literature going back to Aristotle, and (2) provide models for systems of predicative arithmetic. We close with a brief discussion of Church's Thesis for predicative arithmetic. [ABSTRACT FROM AUTHOR]

This article examines the rehabilitation of ethical intuitionism mainly concerning the present discussion about the foundations of bioethics according to the normative theory. In addition to a brief review of the emergence and development of intuitionism, the first part of the work provides a critical analysis of a proposal to reformulate this ethical line on the basis of Wittgenstein's philosophical thought. In order to show that there is little reason to believe Wittgenstein was an intuitionist in the classical sense, the article presents, in the third part, an alternative called "practical cognitivism" which enables to solve both meta-ethical problems and those inherent to normative intuitionism. The plausibility of this project will be evaluated discussing the fundamental principles of bioethics. [ABSTRACT FROM AUTHOR]