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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]

In this article, we will consider the application of intuitionistic tense logics to construct logical systems in which the thesis of logical determinism cannot be expressed.

The interests of the Chilean engineer and philosopher Manuel Atria Ramírez cover a variety of subjects and disciplines, but most of his published work focuses on philosophy of mathematics and physics. After a first phase of youth, of evident Thomistic inclination, his mature thought shows the influence of the great epistemological schools that thrive during the first half of the 20th century. Beyond illustrating the reception that was given in those years in the Latin American academic scene to the new European philosophical tendencies, we think that the concise expository style of the author and his deep understanding of mathematical physics and its history, have the merit to pose with freshness and originality the question referring to the ultimate foundation of mathematics, and its connection with reality. In this article we present and analyze the fundamental lines of Professor Atria`s philosophy of mathematics and physics, as well as those questions open for further research.

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]

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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]

TT□C is a generic family of effectful, extensional type theories with a forcing interpretation parameterized by modalities. This paper identifies a subclass of TT□C theories that internally realizes continuity principles through stateful computations, such as reference cells. The principle of continuity is a seminal property that holds for a number of intuitionistic theories such as System T. Roughly speaking, it states that functions on real numbers only need approximations of these numbers to compute. Generally, continuity principles have been justified using semantical arguments, but it is known that the modulus of continuity of functions can be computed using effectful computations such as exceptions or reference cells. In this paper, the modulus of continuity of the functionals on the Baire space is directly computed using the stateful computations enabled internally in the theory. [ABSTRACT FROM AUTHOR]

In 2017, Michael Nair-Collins formulated his Transitivity Argument which claimed that brain-dead patients are alive according to a concept that defines death in terms of the loss of moral status. This article challenges Nair-Collins' view in three steps. First, I elaborate on the concept of moral status, claiming that to understand this notion appropriately, one must grasp the distinction between direct and indirect duties. Second, I argue that his understanding of moral status implicit in the Transitivity Argument is faulty since it is not based on a distinction between direct and indirect duties. Third, I show how this flaw in Nair-Collins' argument is grounded in the more general problems between preference utilitarianism and desire fulfillment theory. Finally, I present the constructivist theory of moral status and the associated moral concept of death and explain how this concept challenges the Transitivity Argument. According to my view, brain death constitutes a valid criterion of death since brain death is incompatible with the preserved capacity to have affective attitudes and to value anything. [ABSTRACT FROM AUTHOR]

Research in the field of sports sciences undergoes substantial turmoil and instability, stemming from both quantitative and qualitative research methods. This turbulence prompts the expansion of theoretical and empirical knowledge in the domain. Qualitative research, which can be approached through positivist or critical lenses, necessitates researchers to articulate their research philosophy and epistemological stance upfront. Notably, the methodological approach, rooted in intuitionism for analyzing social phenomena, undergoes similar shifts in sociology and sports management. This approach not only invites critique but also spawns numerous theoretical underpinnings. To counter reductionist tendencies in theoretical frameworks, the Multi Grounded Theory (MGT) emerges, aiming to harmonize intuitive and reductionist analytical methods within the framework of Hegelian thesis and antithesis. Hence, authors employing MGT move beyond the pure inductive approach in MGT by explicitly incorporating external theories. By bridging the gap between theoretical frameworks and empirical data, MGT promises to offer a holistic understanding of sports-related phenomena, empowering researchers to cause innovative theories and drive the advancement of knowledge in the field. [ABSTRACT FROM AUTHOR]

Copyright of Veritas is the property of EDIPUCRS - Editora Universitaria da PUCRS and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

At the beginning of the twentieth century, among the foundational schools of mathematics appeared 'intuitionism' by Dutchman L. E. J. Brouwer, who based arithmetic on the intuition of time and all mental constructions that could be made out of it. His pupil Arend Heyting was the first populariser of intuitionism, and he repeatedly emphasised that no philosophy was required to practise intuitionism so that such mathematics could be shared by anyone. Still, stimulated by invitations to humanistic conferences, he wrote a series of notes, preserved in the State Archives, Haarlem, about solipsism. In them, he operated a series of theoretical reflections consisting of the stripping away of the patterns that are part of our consciousness and their subsequent progressive re‐introduction, in order to understand the formation of our Self as distinct from the natural world and other humans. In 1996, following a stroke, neuroscientist Jill Bolte Taylor experienced the deprivation of certain abilities in her brain and their successive regaining. In her experience there are remarkable similarities with what Heyting had hypothesised about the formation of the self. The purpose of this article is to highlight them and point out how Heyting's theoretical construct was found to be embodied in the brain. [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]

We propose an indexed natural deduction system for the modal logic S5 , ideally following Wansing's previous work in the context of tableaux sequents. The system, given both in the classical and intuitionistic versions (called N 5 c and N 5 i respectively), is designed to match as much as possible the structure and properties of the standard system of natural deduction for first-order logic, exploiting the formal analogy between modalities and quantifiers. We study a (syntactical) normalization theorem for both N 5 c and N 5 i and its main consequences, the sub-formula principle and the consistency theorem. In particular, we propose an intuitionistic encoding of classical S5 (via a suitable extension of the Gödel translation for first-order classical logic). Moreover, via the BHK interpretation of intuitionistic proofs, we propose a suitable Curry–Howard isomorphism for N 5 i . By translation into the natural deduction system given by Galmiche and Salhi in [(2010b). Label-free proof systems for intuitionistic modal logic is5. In E. M. Clarke & A. Voronkov (Eds.), Logic for programming, artificial intelligence, and reasoning (pp. 255–271). Springer Berlin Heidelberg.], we prove the equivalence of N 5 i w.r.t. an Hilbert-style axiomatization of IS5 . However, when considering the sheer provability of labelled formulas, our system is comparable to the one presented by Simpson in [(1993). The proof theory and semantics of intuitionistic modal logic [PhD thesis], University of Edinburgh, UK.]. Nevertheless, it remains uncertain whether it is feasible to establish a translation between the corresponding derivations. [ABSTRACT FROM AUTHOR]

In one of his meetings with members of the Vienna Circle, Wittgenstein discusses Hermann Weyl’s brief conversion to intuitionism and criticizes his arguments against applying the law of the excluded middle to generalizations over the natural numbers. Like Weyl, however, Wittgenstein rejects the classical model theoretic conception of generality when it comes to infinite domains. Nonetheless, he disagrees with him about the reasons for doing so. This paper provides an account of Wittgenstein’s criticism of Weyl that is based on his differing understanding of what a general statement over infinite domains consists in. This difference in their conception of generality is argued to be central to the middle Wittgenstein’s overall stance on intuitionism as well. While Weyl (and other intuitionists) reject the law of the excluded middle on grounds of constructivity, Wittgenstein argues that general statements over infinite domains do not express propositions in the first place. The origin of this position as well as its consequences for contemporary debates on generality are further assessed. [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]

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We present four different approaches to prove the soundness theorem for variants with t-truth of functional interpretations. To showcase our different methods we focus on the intuitionistic nonstandard bounded functional interpretation of the nonstandard extensional Heyting arithmetic in all finite types because a version with t-truth for this interpretation has not been given before. Also, because it is a more involved interpretation than others since it includes both nonstandard principles and majorisability. This leads us to believe that if the approaches work for this more complicated functional interpretation, then they should also work for simpler functional interpretations (and realisabilities). [ABSTRACT FROM AUTHOR]

Brouwer’s intuitionistic program was an intriguing attempt to reform the foundations of mathematics that eventually did not prevail. The current paper offers a new perspective on the scientific community’s lack of reception to Brouwer’s intuitionism by considering it in light of Michael Friedman’s model of parallel transitions in philosophy and science, specifically focusing on Friedman’s story of Einstein’s theory of relativity. Such a juxtaposition raises onto the surface the differences between Brouwer’s and Einstein’s stories and suggests that contrary to Einstein’s story, the philosophical roots of Brouwer’s intuitionism cannot be traced to any previously established philosophical traditions. The paper concludes by showing how the intuitionistic inclinations of Hermann Weyl and Abraham Fraenkel serve as telling cases of how individuals are involved in setting in motion, adopting, and resisting framework transitions during periods of disagreement within a discipline. [ABSTRACT FROM AUTHOR]

The foundations of mathematics cover mathematical as well as philosophical problems. At the turn of the 20th century logicism, formalism and intuitionism, main foundational schools were developed. A natural problem arose, namely of how much the foundations of mathematics influence the real practice of mathematicians. Although mathematics was and still is declared to be independent of philosophy, various foundational controversies concerned some mathematical axioms, e.g. the axiom of choice, or methods of proof (particularly, non-constructive ones) and sometimes qualified them as admissible (or not) in mathematical practice, relatively to their philosophical (and foundational) content. Polish Mathematical School was established in the years 1915–1920. Its research program was outlined by Zygmunt Janiszewski (the Janiszewski program) and suggested that Polish mathematicians should concentrate on special branches of studies, including set theory, topology and mathematical logic. In this way, the foundations of mathematics became a legitimate part of mathematics. In particular, the foundational investigations should be conducted independently of philosophical assumptions and apply all mathematically accepted methods, finitary or not, and the same concerns other branches of mathematics. This scientific ideology contributed essentially to the remarkable development of logic, set theory and topology in Poland.

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]

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