This paper defends the heretical view that, at least in some cases, we ought to assign legal liability based on purely statistical evidence. The argument draws on prominent civil law litigation concerning pharmaceutical negligence and asbestos-poisoning. The overall aim is to illustrate moral pitfalls that result from supposing that it is never appropriate to rely on bare statistics when settling a legal dispute.

Reliance on intuitive and emotive responses is widespread across many areas of bioethics, and the current debate on biotechnological human enhancement is particularly interesting in this respect. A strand of “bioconservatives” that has explicitly drawn connections to the modern conservative tradition, dating back to Edmund Burke, appeals explicitly to the alleged wisdom of our intuitions and emotions to ground opposition to some biotechnologies or their uses. So-called bioliberals, those who in principle do not oppose human bioenhancement, tend to rely on (...) rational arguments and to see intuitions and emotions mostly as sources of biases. This approach often translates into shifting the burden of proof onto bioconservatives and challenging them to provide arguments against the proposed enhancement to back what bioliberals perceive as merely intuitive, emotive, and irrational reactions. -/- In this article, I am going to show that the methodological divide between bioliberals and bioconservatives is less significant than at first glance it appears to be and less significant than it is often taken to be. I will do so by defending two theses. The first is that reliance on intuitions and emotions is not a prerogative of bioconservatives: bioliberals have their typical intuitions and emotive responses and are for this reason exposed to potential biases in the same way as bioconservatives are. The second thesis is that reliance on intuitions and emotions is not necessarily antithetic to reason and rationality. This latter thesis has been philosophically defended with particular reference to the debate on biotechnologies, while the former is perhaps more controversial and more difficult to accept—at least for bioliberals. In both cases, I will support the claims by drawing on resources from the field of moral psychology and the sciences of the mind and, particularly, by applying to some positions in the enhancement debate recent findings about the role of intuitions and emotions in human moral assessment. This new empirically informed perspective holds promises for solving the methodological controversy between bioconservatives and bioliberals, thus enabling proper dialogue and debate between the two sides. (shrink)

A proof P of a theorem T is transferable when a typical expert can become convinced of T solely on the basis of their prior knowledge and the information contained in P. Easwaran has argued that transferability is a constraint on acceptable proof. Meanwhile, a proof P is fixable when it’s possible for other experts to correct any mistakes P contains without having to develop significant new mathematics. Habgood-Coote and Tanswell have observed that some acceptable proofs are both fixable (...) and in need of fixing, in the sense that they contain nontrivial mistakes. The claim that acceptable proofs must be transferable seems quite plausible. The claim that some acceptable proofs need fixing seems plausible too. Unfortunately, these attractive suggestions stand in tension with one another. I argue that the transferability requirement is the problem. Acceptable proofs need only satisfy a weaker requirement I call “corrigibility”. I explain why, despite appearances, the corrigibility standard is preferable to stricter alternatives. (shrink)

The suggestion of something akin to a ‘relativist solution to the Mind-Body problem’ has recently been held by some scientists and philosophers; either explicitly (Galadí, 2023; Lahav & Neemeh, 2022; Ludwig, 2015) or in more implicit terms (Solms, 2018; Velmans, 2002, 2008). In this paper I provide an argument in favor of a relativist approach to the Mind-Body problem, more specifically, an argument for ‘1st/3rd person relativism’, the claim that ‘The truth value of some sentences or propositions is relative to (...) 1st and 3rd person perspectives’. The argument for 1st/3rd person relativism is close to a forma proof. It is shown that, just by assuming the 1st/3rd person distinction itself and using first order logic and set theory, ‘1st/3rd person relativism’ follows as a theorem. Some consequences of ‘1st/3rd person relativism’ to the Mind-Body Problem are evaluated. It is shown that ‘1st/3rd person relativism’ predicts the existence of an (apparent) Explanatory Gap; explains why the Explanatory Gap is just apparent (and the origins of such illusion); dissolves the Hard-Problem; provides a possible solution the problem of Mental Causation; explains why Mental Causation looks like a problem in the first place and accurately predicts the actual empirically found correlation and covariation between conscious experiences and brain states. This explanatory power of ‘1st/3rd person relativism’ is particularly impressive since it was not designed as a possible solution to the Mind-Body problem in the first place. (shrink)

The study aimed to determine if differentiated instruction effectively addresses learning gaps in mathematics. In particular, it explored how it can improve the student’s learning gaps concerning mathematical performance and confidence. The study employed a quasi-experimental design with 30 purposively-selected Grade 10 participants divided into differentiated (n = 15) and control groups (n = 15), ensuring the utmost ethical measures. The mean and standard deviation were used to describe the participants’ performance and confidence. Independent samples t-tests were used (...) to determine the significant differences in the performance and confidence between the two groups. In contrast, dependent samples t-tests were used to determine the significant differences in each group’s pre and posttest performance and confidence. Findings bared that the differentiated instruction successfully addressed students’ performance in mathematics even in a short period. It also increased the participants’ confidence when answering fundamental problems. Continuing differentiated instruction activities are recommended since it benefits students who struggle in mathematics, particularly in answering fundamental operations. Differentiated teaching activities in mathematics can boost academic achievement and engagement and prepare students for future success while fostering a positive and inclusive classroom culture that values individual learning needs and preferences. (shrink)

We investigated whether mathematicians typically agree about the qualities of mathematical proofs. Between-mathematician consensus in proof appraisals is an implicit assumption of many arguments made by philosophers of mathematics, but to our knowledge the issue has not previously been empirically investigated. We asked a group of mathematicians to assess a specific proof on four dimensions, using the framework identified by Inglis and Aberdein (2015). We found widespread disagreement between our participants about the aesthetics, intricacy, precision and utility of the (...) proof, suggesting that a priori assumptions about the consistency of mathematical proof appraisals are unreasonable. (shrink)

If there is an area of discourse in which disagreement is virtually absent, it is mathematics. After all, mathematicians justify their claims with deductive proofs: arguments that entail their conclusions. But is mathematics really exceptional in this respect? Looking at the history and practice of mathematics, we soon realize that it is not. First, deductive arguments must start somewhere. How should we choose the starting points (i.e., the axioms)? Second, mathematicians, like the rest of us, are fallible. Their ability (...) to recognize whether a putative proof is correct is not infallible. In most cases, disagreement over the correctness of a putative proof is, however, evanescent. Once an error is spotted and communicated, the disagreement disappears. But this is not always the case. Sometimes it is recalcitrant; that is, it persists over time. In order to zoom in on this type of disagreement and explain its very possibility, we focus on a single case study: a decades-long (1921-1949) controversy between Federigo Enriques and Francesco Severi, two prominent exponents of the Italian school of algebraic geometry. We suggest that the instability of the mathematical community to which they belonged can be explained by the gap between an abstract criterion of rigor and local criteria of acceptability. It is this instability that made the existence of recalcitrant disagreement over putative proofs possible. We do not condemn speculative mathematics but rather its pretense of being rigorous mathematics. In this respect, we show that the overly self-confident Severi and the more intuitive, visionary Enriques had a completely different attitude. (shrink)

We argue that the evil-god challenge is not an additional challenge for theists above and beyond the (much older) gap problem. One version of the evil-god challenge is merely a specific instance of the gap problem, and another is dependent on that specific instance of the gap problem. Therefore, the various solutions to the gap problem that theists have developed double as responses to the evil-god challenge, placing the evil-god challenge in a more vulnerable position than has been supposed.

Many seem to think that AI-induced responsibility gaps are morally bad and therefore ought to be avoided. We argue, by contrast, that there is at least a pro tanto reason to welcome responsibility gaps. The central reason is that it can be bad for people to be responsible for wrongdoing. This, we argue, gives us one reason to prefer automated decision-making over human decision-making, especially in contexts where the risks of wrongdoing are high. While we are not the (...) first to suggest that responsibility gaps should sometimes be welcomed, our argument is novel. Others have argued that responsibility gaps should sometimes be welcomed because they can reduce or eliminate the psychological burdens caused by tragic moral choice-situations. By contrast, our argument explains why responsibility gaps should sometimes be welcomed even in the absence of tragic moral choice-situations, and even in the absence of psychological burdens. (shrink)

Miranda Fricker’s insightful work on epistemic injustice discusses two forms of epistemic injustice—testimonial injustice and hermeneutical injustice. Hermeneutical injustice occurs when the victim lacks the interpretative resources to make sense of her experience, and this lacuna can be traced down to a structural injustice. In this paper, I provide one model of how to fill the conceptual gap in hermeneutical injustice. First, I argue that the victims possess conceptual resources to make sense of their experiences, namely phenomenal concepts. Second, I (...) show how one might work the way up in a two-step process from a subjective, phenomenal concept to a novel, public concept. Finally, I discuss the conditions that have to be met for this process to be successful. The resulting model shows a way how the victims might alleviate hermeneutical injustice by developing novel concepts, given that the dominant group does not care about their predicament. (shrink)

Abstract: This article mainly intends to identify barriers that cause the under-representation of women in top leadership positions in higher education in Tanzania. The study comprises the sample of 250 respondents with the use of a case study research design constructed on the application of mixed methods (qualitative and quantitative). Thus, non-probability sampling was applied to qualitative data collection while probability sampling was used in quantitative data. The findings show the fundamental relationship between individual, administrative and societal factors that block (...) women to reach the top leadership position in universities. Besides, the study has demonstrated the gender mainstreaming as the system which can be connected to adjust the hierarchical structures, methods and culture and bring the positive condition which advance gender equality. Nevertheless, as the well-known saying "Fix the holes in the pipe" goes, the variety of actions and activities are needed. Thus, in accordance with the proof from the study and individual encounters, the researcher, marked on Focused Gender Research, Capacity Development Programs, Parliamentary Charter, Setting Targets as well as Regular Monitoring and Reporting as interventions sustainable for bridging the prevailing gender leadership gap in higher education in Tanzania. (shrink)

Sparrow argues that military robots capable of making their own decisions would be independent enough to allow us denial for their actions, yet too unlike us to be the targets of meaningful blame or praise—thereby fostering what Matthias has dubbed “the responsibility gap.” We agree with Sparrow that someone must be held responsible for all actions taken in a military conflict. That said, we think Sparrow overlooks the possibility of what we term “blank check” responsibility: A person of sufficiently high (...) standing could accept responsibility for the actions of autonomous robotic devices—even if that person could not be causally linked to those actions besides this prior agreement. The basic intuition behind our proposal is that humans can impute relations even when no other form of contact can be established. The missed alternative we want to highlight, then, would consist in an exchange: Social prestige in the occupation of a given office would come at the price of signing away part of one's freedoms to a contingent and unpredictable future guided by another agency. (shrink)

Most mental health research largely ignores or minimises gender and age differences in depression. In ‘Don't mind the gap: Why do we not care about the gender gap in mental health?’, Patalay and Demkowicz identify a dearth of research on the causal factors of depression in young women. They attribute this to an over-reliance on biological accounts of gender differences in depression. Patalay and Demkowicz conclude that a person-centred approach that meaningfully engages with the reports of young women with depression (...) is more likely to expose the social drivers of depression that impact this group. This commentary focuses on Patalay and Demkowicz's call to examine the patient's lived experience. We argue that there is an urgent need to reflect upon the methodologies involved in examining lived experience and how they can be best utilised. Ultimately, we advocate for an approach known as ‘phenomenological psychopathology’, through a phenomenological investigation of depression in young women, we can go some way towards closing the gender gap. (shrink)

The paper addresses two difficulties that arise in Treatise 1.2.5. First, Hume appears to be inconsistent when he denies that we have an idea of a vacuum or empty space yet allows for the idea of an “invisible and intangible distance.” My solution to this difficulty is to develop the overlooked possibility that Hume does not take the invisible and intangible distance to be a distance at all. Second, although Hume denies that we have an idea of a vacuum, some (...) texts in Treatise 1.2.5 are taken by interpreters to suggest that Hume nonetheless believes that there are vacuums in nature. I discuss the relevant texts and defend the view that Hume does not in fact countenance belief in vacuums. I conclude by outlining an interpretation of Hume’s intention in the Treatise that allows us to understand his discussion of ideas as having implications for the sciences. (shrink)

Callender and Cohen have proposed to apply a “Gricean strategy” to the constitution problem of scientific representation, taking inspiration from Grice’s reduction of linguistic meaning to mental states. They suggest that scientific representation can be reduced to stipulation by epistemic agents. This account has been criticised for not making a distinction between symbolic and epistemic representation and not taking into account the communal aspects of scientific representation. I argue that these criticisms would not apply if Grice’s actual strategy were properly (...) employed. I present Grice’s account of linguistic meaning and transpose his actual strategy and method to epistemic representation. This results in a reduction of epistemic representation to mental states that does not fall prey to the same criticisms. The main novelty of the resulting account is a distinction between contextual representational use and general representational status, which I address using the notion of indexicality. (shrink)

Kant’s aim in the Refutation of Idealism is to show that the temporal determination of inner experience presupposes outer experience. Commentators have rightly noted the extraordinarily compressed character of Kant's argument, and numerous gaps in the argument have been pointed out. In this paper I focus on two of these gaps and provide a reconstruction of Kant's argument that closes them.

Kant’s aim in the Refutation of Idealism is to show that the temporal determination of inner experience presupposes outer experience. Commentators have rightly noted the extraordinarily compressed character of Kant's argument, and numerous gaps in the argument have been pointed out. In this paper I focus on two of these gaps and provide a reconstruction of Kant's argument that closes them.

While seeking novel food sources to feed the increasing population of the globe, several alternatives have been discussed, including algae, fungi or in vitro meat. The increasingly propagated usage of farmed insects for human nutrition raises issues regarding food safety, consumer information and animal protection. In line with law, insects like any other animals must not be reared or manipulated in a way that inflicts unnecessary pain, distress or harm on them. Currently, there is a great need for research in (...) the area of insect welfare, especially regarding species-specific needs, health, farming systems and humane methods of killing. Recent results from neurophysiological, neuroanatomical and behavioral sciences prompt caution when denying consciousness and therefore the likelihood of presence of pain and suffering or something closely related to it to insects. From an animal protection point of view, these issues should be satisfyingly solved before propagating and establishing intensive husbandry systems for insects as a new type of mini-livestock factory farming. (shrink)

Writing strategic documents is a major practice of many actors striving to see their educational ideas realised in the curriculum. In these documents, arguments are systematically developed to create the legitimacy of a new educational goal and competence to make claims about it. Through a qualitative analysis of the writing strategies used in these texts, I show how two of the main actors in the Czech educational discourse have developed a proof that a new educational goal is needed. I draw (...) on the connection of the relational approach in the sociology of education with Lyotard’s analytical semantics of instances in the event. The comparison of the writing strategies in the two documents reveals differences in the formation of a particular pattern of justification. In one case the texts function as a herald of pure reality, and in the other case as a messenger of other witnesses. This reveals different regimens of proof, although both of them were written as prescriptive directives – normative models of the educational world. (shrink)

Understanding academic gender gaps is difficult because gender-imbalanced fields differ across many features, limiting researchers’ ability to systematically study candidate causes. In the present preregistered research, we isolate two potential explanations—brilliance beliefs and fixed versus growth intelligence mindsets—by comparing two fields that have inverse gender gaps and historic and topical overlap: philosophy and psychology. Many more men than women study philosophy and vice versa in psychology, with disparities emerging during undergraduate studies. No prior work has examined the contributions (...) of both self-perceptions of brilliance and fixed versus growth mindsets on choice of major among undergraduate students. We assessed field-specific brilliance beliefs, brilliance beliefs about self, and mindsets, cross-sectionally in 467 undergraduates enrolled in philosophy and psychology classes at universities in the United States and Canada via both in-person and online questionnaires. We found support for the brilliance beliefs about the self, but not mindset, explanation. Brilliance beliefs about oneself predicted women’s but not men’s choice of major. Women who believed they were less brilliant were more likely to study psychology (perceived to require low brilliance) over philosophy (perceived to require high brilliance). Findings further indicated that fixed versus growth mindsets did not differ by gender and were not associated with major. Together, these results suggest that internalized essentialist beliefs about the gendered nature of brilliance are uniquely important to understanding why men and women pursue training in different academic fields. (shrink)

The scope of Platonism is extended by introducing the concept of a “Platonic computer” which is incorporated in metacomputics. The theoretical framework of metacomputics postulates that a Platonic computer exists in the realm of Forms and is made by, of, with, and from metaconsciousness. Metaconsciousness is defined as the “power to conceive, to perceive, and to be self-aware” and is the formless, con-tentless infinite potentiality. Metacomputics models how metaconsciousness generates the perceived actualities including abstract entities and physical and nonphysical realities. (...) It is postulated that this is achieved via digital computation using the Platonic computer. The introduction of a Platonic computer into the realm of Forms thus bridges the “inverse explanatory gap” and therefore solves the “inverse hard problem of consciousness” in the philosophy of mind. (shrink)

The use of evolutionary explanations to explain phenomena at the individual level has been described by various authors as an explanatory error, the so-called Phylogeny Fallacy. In this paper, this fallacy will be analyzed in the context of teleosemantics, a central project of the philosophy of mind whose main aim is to naturalize intentional systems by appealing to their biological teleofunctions. I will argue that those teleosemantics projects that invoke evolutionary functions generally commit the fallacy. First, I will point to (...) various arguments in the literature that point to this fallacy. However, a more general argument will also be made. To illustrate this puzzling scenario, I will present two desiderata that any teleosemantic project must fulfill. I will argue that naturalizing intentionality based on natural selection creates an explanatory gap between types and tokens. This gap prohibits an adequate explanation of the desiderata. To close the gap, teleosemantics invokes replicator biology, a view of inheritance that has already been identified to commit the fallacy. This leads teleosemantics into a complex situation: to close the gap, it must commit the fallacy. (shrink)

Recent events have revived questions about the circumstances that ought to trigger therapists' duty to warn or protect. There is extensive interstate variation in duty to warn or protect statutes enacted and rulings made in the wake of the California Tarasoff ruling. These duties may be codified in legislative statutes, established in common law through court rulings, or remain unspecified. Furthermore, the duty to warn or protect is not only variable between states but also has been dynamic across time. In (...) this article, we review the implications of this variability and dynamism, focusing on three sets of questions: first, what legal and ethics-related challenges do therapists in each of the three broad categories of states (states that mandate therapists to warn or protect, states that permit therapists to breach confidentiality for warnings but have no mandate, and states that give no guidance) face in handling threats of violence? Second, what training do therapists and other professionals involved in handling violent threats receive, and is this training adequate for the task that these professionals are charged with? Third, how have recent court cases changed the scope of the duty? We conclude by pointing to gaps in the empirical and conceptual scholarship surrounding the duty to warn or protect. (shrink)

This paper argues that we should reject G. E. Moore’s anti-skeptical argument as it is presented in “Proof of an External World.” However, the reason I offer is different from traditional objections. A proper understanding of Moore’s “proof” requires paying attention to an important distinction between two forms of skepticism. I call these Ontological Skepticism and Epistemic Skepticism. The former is skepticism about the ontological status of fundamental reality, while the latter is skepticism about our empirical knowledge. Philosophers often assume (...) that Moore’s response to “external world skepticism” deals exclusively with the former, not the latter. But this is a mistake. I shall argue that Moore’s anti-skeptical argument targets an ontological form of skepticism. Thus, the conclusion is an ontological claim about fundamental reality, while the premises are epistemic claims. If this is correct, then the conclusion outstrips the scope of its premises and proves too much. (shrink)

This research note argues that political theorists of refuge ought to consider the experiences of refugees after they have received asylum in the Global North. Currently, much of the literature concerning the duties of states towards refugees implicitly adopts a blanket approach, rather than considering how varied identities may affect the remedies available to displaced people. Given the prevalence of racism, xenophobia, and homophobia in the Global North, and the growing norm of dissident persecution in foreign territory, protection is not (...) guaranteed after either territorial or legal admission. This research note considers the case of LGBTQ refugees in order to demonstrate the analytical potential of more inclusive and diverse normative approaches. Taking the origin and extension of harm seriously requires a conceptualization of sanctuary after asylum that accurately reflects the experiences of the displaced. In doing so, questions arise regarding the nature and efficacy of territorial asylum. (shrink)

It is widely accepted that in fallible reasoning potential error necessarily increases with every additional step, whether inferences or premises, because it grows in the same way that the probability of a lengthening conjunction shrinks. As it stands, this is disappointing but, I will argue, not out of keeping with our experience. However, consulting an expert, proof-checking, constructing gap-free proofs, and gathering more evidence for a given conclusion also add more steps, and we think these actions have the potential (...) to improve our reliability or justifiedness. Thus, the received wisdom about the growth of error implies a skepticism about the possibility of improving our reliability and level of justification through effort. Paradoxically, and even more implausibly, taking steps to decrease your potential error necessarily increases it. I will argue that the self-help steps listed here are of a distinctive type, involving composition rather than conjunction. Error grows differently over composition than over conjunction, I argue, and this dissolves the apparent paradox. (shrink)

Mathematicians distinguish between proofs that explain their results and those that merely prove. This paper explores the nature of explanatory proofs, their role in mathematical practice, and some of the reasons why philosophers should care about them. Among the questions addressed are the following: what kinds of proofs are generally explanatory (or not)? What makes a proof explanatory? Do all mathematical explanations involve proof in an essential way? Are there really such things as explanatory proofs, and (...) if so, how do they relate to the sorts of explanation encountered in philosophy of science and metaphysics? (shrink)

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order (...) logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics. (shrink)

In this paper, I propose that applying the methods of data science to “the problem of whether mathematical explanations occur within mathematics itself” (Mancosu 2018) might be a fruitful way to shed new light on the problem. By carefully selecting indicator words for explanation and justification, and then systematically searching for these indicators in databases of scholarly works in mathematics, we can get an idea of how mathematicians use these terms in mathematical practice and with what frequency. The results of (...) this empirical study suggest that mathematical explanations do occur in research articles published in mathematics journals, as indicated by the occurrence of explanation indicators. When compared with the use of justification indicators, however, the data suggest that justifications occur much more frequently than explanations in scholarly mathematical practice. The results also suggest that justificatory proofs occur much more frequently than explanatory proofs, thus suggesting that proof may be playing a larger justificatory role than an explanatory role in scholarly mathematical practice. (shrink)

A close look at Frege's proof in "Foundations of Arithmetic" that every number has a successor. The examination reveals a surprising gap in the proof, one that Frege would later fill in "Basic Laws of Arithmetic".

In this dissertation, we shall investigate whether Tennant's criterion for paradoxicality(TCP) can be a correct criterion for genuine paradoxes and whether the requirement of a normal derivation(RND) can be a proof-theoretic solution to the paradoxes. Tennant’s criterion has two types of counterexamples. The one is a case which raises the problem of overgeneration that TCP makes a paradoxical derivation non-paradoxical. The other is one which generates the problem of undergeneration that TCP renders a non-paradoxical derivation paradoxical. Chapter 2 deals with (...) the problem of undergeneration and Chapter 3 concerns the problem of overgeneration. Chapter 2 discusses that Tenant’s diagnosis of the counterexample which applies CR−rule and causes the undergeneration problem is not correct and presents a solution to the problem of undergeneration. Chapter 3 argues that Tennant’s diagnosis of the counterexample raising the overgeneration problem is wrong and provides a solution to the problem. Finally, Chapter 4 addresses what should be explicated in order for RND to be a proof-theoretic solution to the paradoxes. (shrink)

This paper presents proof that Buss's S22 can prove the consistency of a fragment of Cook and Urquhart's PV from which induction has been removed but substitution has been retained. This result improves Beckmann's result, which proves the consistency of such a system without substitution in bounded arithmetic S12. Our proof relies on the notion of "computation" of the terms of PV. In our work, we first prove that, in the system under consideration, if an equation is proved and either (...) its left- or right-hand side is computed, then there is a corresponding computation for its right- or left-hand side, respectively. By carefully computing the bound of the size of the computation, the proof of this theorem inside a bounded arithmetic is obtained, from which the consistency of the system is readily proven. This result apparently implies the separation of bounded arithmetic because Buss and Ignjatović stated that it is not possible to prove the consistency of a fragment of PV without induction but with substitution in Buss's S12. However, their proof actually shows that it is not possible to prove the consistency of the system, which is obtained by the addition of propositional logic and other axioms to a system such as ours. On the other hand, the system that we have considered is strictly equational, which is a property on which our proof relies. (shrink)

One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theorem that all consistent formal systems containing arithmetic include sentences that can neither be proved nor disproved within that system. However, such undecidable Gödel sentences can be established to (...) be true once we expand the formal system with Alfred Tarski s semantical theory of truth, as shown by Stewart Shapiro and Jeffrey Ketland in their semantical arguments for the substantiality of truth. According to them, in Gödel sentences we have an explicit case of true but unprovable sentences, and hence deflationism is refuted. -/- Against that, Neil Tennant has shown that instead of Tarskian truth we can expand the formal system with a soundness principle, according to which all provable sentences are assertable, and the assertability of Gödel sentences follows. This way, the relevant question is not whether we can establish the truth of Gödel sentences, but whether Tarskian truth is a more plausible expansion than a soundness principle. In this work I will argue that this problem is best approached once we think of mathematics as the full human phenomenon, and not just consisting of formal systems. When pre-formal mathematical thinking is included in our account, we see that Tarskian truth is in fact not an expansion at all. I claim that what proof is to formal mathematics, truth is to pre-formal thinking, and the Tarskian account of semantical truth mirrors this relation accurately. -/- However, the introduction of pre-formal mathematics is vulnerable to the deflationist counterargument that while existing in practice, pre-formal thinking could still be philosophically superfluous if it does not refer to anything objective. Against this, I argue that all truly deflationist philosophical theories lead to arbitrariness of mathematics. In all other philosophical accounts of mathematics there is room for a reference of the pre-formal mathematics, and the expansion of Tarkian truth can be made naturally. Hence, if we reject the arbitrariness of mathematics, I argue in this work, we must accept the substantiality of truth. Related subjects such as neo-Fregeanism will also be covered, and shown not to change the need for Tarskian truth. -/- The only remaining route for the deflationist is to change the underlying logic so that our formal languages can include their own truth predicates, which Tarski showed to be impossible for classical first-order languages. With such logics we would have no need to expand the formal systems, and the above argument would fail. From the alternative approaches, in this work I focus mostly on the Independence Friendly (IF) logic of Jaakko Hintikka and Gabriel Sandu. Hintikka has claimed that an IF language can include its own adequate truth predicate. I argue that while this is indeed the case, we cannot recognize the truth predicate as such within the same IF language, and the need for Tarskian truth remains. In addition to IF logic, also second-order logic and Saul Kripke s approach using Kleenean logic will be shown to fail in a similar fashion. (shrink)

Chapter 1 constitutes an introduction to Gentzen calculi from two perspectives, logical and philosophical. It introduces the notion of generalisations of Gentzen sequent calculus and the discussion on properties that characterize good inferential systems. Among the variety of Gentzen-style sequent calculi, I divide them in two groups: syntactic and semantic generalisations. In the context of such a discussion, the inferentialist philosophy of the meaning of logical constants is introduced, and some potential objections – mainly concerning the choice of working with (...) semantic generalizations – are addressed. Finally, I’ll introduce the case studies that I’ll be dealing with in part II. Chapter 2 is concerned with the origins and development of Jaśkowski’s discussive logic. The main idea of this chapter is to systematize the various stages of the development of discussive logic related researches from two different angles, i.e., its connections to modal logics and its proof theory, by highlighting virtues and vices. Chapter 3 focuses on the Gentzen-style proof theory of discussive logic, by providing a characterization of it in terms of labelled sequent calculi. Chapter 4 deals with the Gentzen-style proof theory of relevant logics, again by employing the methodology of labelled sequent calculi. This time, instead of working with a single logic, I’ll deal with a whole family of them. More precisely, I’ll study in terms of proof systems those relevant logics that can be characterised, at the semantic level, by reduced Routley-Meyer models, i.e., relational structures with a ternary relation between states and a unique base element. Chapter 5 investigates the proof theory of a modal expansion of intuitionistic propositional logic obtained by adding an ‘actuality’ operator to the connectives. This logic was introduced also using Gentzen sequents. Unfortunately, the original proof system is not cut-free. This chapter shows how to solve this problem by moving to hypersequents. Chapter 6 concludes the investigations and discusses the future of the research presented throughout the dissertation. (shrink)

The lack of gender parity in philosophy has garnered serious attention recently. Previous empirical work that aims to quantify what has come to be called “the gender gap” in philosophy focuses mainly on the absence of women in philosophy faculty and graduate programs. Our study looks at gender representation in philosophy among undergraduate students, undergraduate majors, graduate students, and faculty. Our findings are consistent with what other studies have found about women faculty in philosophy, but we were able to add (...) two pieces of new information. First, the biggest drop in the proportion of women in philosophy occurs between students enrolled in introductory philosophy classes and philosophy majors. Second, this drop is mitigated by the presence of more women philosophy faculty. (shrink)

Proofs of God in Early Modern Europe offers a fascinating window into early modern efforts to prove God’s existence. Assembled here are twenty-two key texts, many translated into English for the first time, which illustrate the variety of arguments that philosophers of the seventeenth and eighteenth centuries offered for God. These selections feature traditional proofs—such as various ontological, cosmological, and design arguments—but also introduce more exotic proofs, such as the argument from eternal truths, the argument from universal (...) aseity, and the argument ex consensu gentium. Drawn from the work of eighteen philosophers, this book includes both canonical figures (such as Descartes, Spinoza, Newton, Leibniz, Locke, and Berkeley) and noncanonical thinkers (such as Norris, Fontenelle, Voltaire, Wolff, Du Châtelet, and Maupertuis). -/- Lloyd Strickland provides fresh translations of all selections not originally written in English and updates the spelling and grammar of those that were. Each selection is prefaced by a lengthy headnote, giving a biographical account of its author, an analysis of the main argument(s), and important details about the historical context. Strickland’s introductory essay provides further context, focusing on the various reasons that led so many thinkers of early modernity to develop proofs of God’s existence. -/- Proofs of God is perfect for both students and scholars of early modern philosophy and philosophy of religion. (shrink)

A textbook on proof in mathematics, inspired by an Aristotelian point of view on mathematics and proof. The book expounds the traditional view of proof as deduction of theorems from evident premises via obviously valid steps. It deals with the proof of "all" statements, "some" statements, multiple quantifiers and mathematical induction.

By the middle of the seventeenth century we that find that algebra is able to offer proofs in its own right. That is, by that time algebraic argument had achieved the status of proof. How did this transformation come about?

This short article is based on my special lecture entitled "Aristotle and the Philosophy of Education" at Tamagawa University Research Institute in Tokyo on September 19, 2015, through a recording of the spoken language transcribed in written form with some corrections. The lecture delivered on that day consists of two parts: referring to historical research and a statistical survey, the first half focuses on uncovering the fact that the philosophy of education has been slighted both in Japanese and Western academia (...) and that this fact is valid for both Aristotle's and contemporary studies on the philosophy of education; the second half endeavors to describe two possible research projects - namely, a classical-philosophical and interpretative study of Aristotle's philosophy of education and a contemporary-philosophical and Aristotelian study of the philosophy education. Some of these topics have been published but others have not. This short article concerns the first half of the lecture and measures the gap between philosophy and the philosophy of education in Japan through a statistical survey of the largest competitive research funding database in Japan: KAKEN. (shrink)

The aim of this doctoral thesis is to bridge the gap between theoretical ideals of authenticity and practical authenticity-related problems in healthcare. In this context, authenticity means being "genuine," "real," "true to oneself," or similar, and is assumed to be closely connected to the autonomy of persons. The thesis includes an introduction and four articles related to authenticity. The first article collects various theories intended to explain the distinction between authenticity and inauthenticity in a taxonomy that enables oversight and analysis. (...) It is argued that (in-)authenticity is difficult to observe in others. The second article offers a solution to this difficulty in one theory of authenticity. It is proposed that under certain circumstances, it is morally justified to judge that the desires underlying a person's decisions are inauthentic. The third article incorporates this proposition into an already established theory of personal autonomy. It is argued that the resulting conceptualization of autonomy is fruitful for action-guidance in authenticity-related problems in healthcare. The fourth article collects nine cases of possible authenticity-related problems in healthcare. The theory developed in the third article is applied to the problems, when this is allowed by the case-description, to provide guidance with regard to them. It is argued that there is not one universal authenticity-related problem but many different problems, and that there is thus likely not one universal solution to such problems but various particular solutions. (shrink)

Kevin Elliott and others separate two common arguments for the legitimacy of societal values in scientific reasoning as the gap and the error arguments. This article poses two questions: How are these two arguments related, and what can we learn from their interrelation? I contend that we can better understand the error argument as nested within the gap because the error is a limited case of the gap with narrower features. Furthermore, this nestedness provides philosophers with conceptual tools for analyzing (...) more robustly how values pervade science. (shrink)

This study employed an Interpretative Phenomenological Analysis (IPA) which aimed to explore the lived experiences, challenges, and coping mechanisms of teachers in the public school both in elementary and secondary schools in Malolos, Bulacan. The findings of this study revealed that most teachers are significantly challenged with the poor internet connection, multitasking and multitudes of paperwork to be submitted, communication with the parents and teachers and the different modalities of learning which are cited as the contributing factors of stress and (...) anxiety. As to the experiences of the teachers, they became a module writer, online teacher, reporter while doing the online classes and juggling with the work at home as a homemaker. However, the good experiences of this pandemic brought realization to the teachers that they were able to unleash their potential and utilized the skills needed to survive in this difficult time. Moreover, teachers coped up with the situations through prayers, listening and watching motivational videos, yoga and others became farmers too just to avoid stress. Finally, the teachers also gave their suggestions to further help other teachers also who are struggling during this pandemic. Majority of the participants gave emphasis to focus on stabilizing mental health. Others also highlighted the need to seek the assistance of other people who are knowledgeable in terms of ICT. Continuing innovation and researching to update oneself was also one of the suggested ways to deal with the current paradigm shift in education. (shrink)

Are there deep differences between these cultures in their ways of thinking? How can they be described? There is no neutral language for doing so. One can doubt all claims to deep essence as being metaphysical illusions and figments. However, the differences are certainly experienced. They can be characterized negatively. This is where Chinese and Western viewpoints meet. Whereas Jullien finds the cultural Other enabling him to think otherwise and effectively to keep the recursive self-negating aspect of discourse active and (...) alive in his own discourse, I attempt to do this by reference to the apophatic current underlying Western philosophical reflection in all its most radically critical and especially self-critical manifestations. Especially from consideration of Eastern approaches to universality, the apophatic wisdom that can be gathered here can serve to put us on guard against any universality that can be thought. Universality is not what it is thought to be by the universals that we can think. The naïve faith in thought and education as per se emancipatory is belied by history. The universal remains an ideal rather than a manifest fact in the course of history. It cannot be approached except always by way of the negative. (shrink)

Considered in light of the reader’s expectation of a thoroughgoing criticism of the pretensions of the rational psychologist, and of the wealth of discussions available in the broader 18th century context, which includes a variety of proofs that do not explicitly turn on the identification of the soul as a simple substance, Kant’s discussion of immortality in the Paralogisms falls lamentably short. However, outside of the Paralogisms (and the published works generally), Kant had much more to say about the (...) arguments for the soul’s immortality as he devoted considerable time to the topic throughout his career in his lectures on metaphysics. In fact, as I show in this paper, the student lecture notes prove to be an indispensable supplement to the treatment in the Paralogisms, not only for illuminating Kant’s criticism of the rational psychologist’s views on the immortality of the soul, but also in reconciling this criticism with Kant’s own positive claims regarding certain theoretical proofs of immortality. (shrink)

Gaisi Takeuti (1926–2017) is one of the most distinguished logicians in proof theory after Hilbert and Gentzen. He extensively extended Hilbert's program in the sense that he formulated Gentzen's sequent calculus, conjectured that cut-elimination holds for it (Takeuti's conjecture), and obtained several stunning results in the 1950–60s towards the solution of his conjecture. Though he has been known chiefly as a great mathematician, he wrote many papers in English and Japanese where he expressed his philosophical thoughts. In particular, he used (...) several keywords such as "active intuition" and "self-reflection" from Nishida's philosophy. In this paper, we aim to describe a general outline of our project to investigate Takeuti's philosophy of mathematics. In particular, after reviewing Takeuti's proof-theoretic results briefly, we describe some key elements in Takeuti's texts. By explaining these texts, we point out the connection between Takeuti's proof theory and Nishida's philosophy and explain the future goals of our project. (shrink)

This paper explores the role of moral uncertainty in explaining the morally disruptive character of new technologies. We argue that existing accounts of technomoral change do not fully explain its disruptiveness. This explanatory gap can be bridged by examining the epistemic dimensions of technomoral change, focusing on moral uncertainty and inquiry. To develop this account, we examine three historical cases: the introduction of the early pregnancy test, the contraception pill, and brain death. The resulting account highlights what we call “differential (...) disruption” and provides a resource for fields such as technology assessment, ethics of technology, and responsible innovation. (shrink)

This paper delves into the use of machine learning-based systems in decision-making processes and its implications for moral responsibility as traditionally defined. It focuses on the emergence of responsibility gaps and examines proposed strategies to address them. The paper aims to provide an introductory and comprehensive overview of the ongoing debate surrounding moral responsibility in automated decision-making. By thoroughly examining these issues, we seek to contribute to a deeper understanding of the implications of AI integration in society.

The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional “dimension” allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the “distance between finiteness and infinity”, particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special “flat” case of Hilbert (...) mathematics. The following four essential problems are considered for the idea to be elucidated: Fermat’s last theorem proved by Andrew Wiles; Poincaré’s conjecture proved by Grigori Perelman and the only resolved from the seven Millennium problems offered by the Clay Mathematics Institute (CMI); the four-color theorem proved “machine-likely” by enumerating all cases and the crucial software assistance; the Yang-Mills existence and mass gap problem also suggested by CMI and yet unresolved. They are intentionally chosen to belong to quite different mathematical areas (number theory, topology, mathematical physics) just to demonstrate the power of the approach able to unite and even unify them from the viewpoint of Hilbert mathematics. Also, specific ideas relevant to each of them are considered. Fermat’s last theorem is shown as a Gödel insoluble statement by means of Yablo’s paradox. Thus, Wiles’s proof as a corollary from the modularity theorem and thus needing both arithmetic and set theory involves necessarily an inverse Grothendieck universe. On the contrary, its proof in “Fermat arithmetic” introduced by “epoché to infinity” (following the pattern of Husserl’s original “epoché to reality”) can be suggested by Hilbert arithmetic relevant to Hilbert mathematics, the mediation of which can be removed in the final analysis as a “Wittgenstein ladder”. Poincaré’s conjecture can be reinterpreted physically by Minkowski space and thus reduced to the “nonstandard homeomorphism” of a bit of information mathematically. Perelman’s proof can be accordingly reinterpreted. However, it is valid in Gödel (or Gödelian) mathematics, but not in Hilbert mathematics in general, where the question of whether it holds remains open. The four-color theorem can be also deduced from the nonstandard homeomorphism at issue, but the available proof by enumerating a finite set of all possible cases is more general and relevant to Hilbert mathematics as well, therefore being an indirect argument in favor of the validity of Poincaré’s conjecture in Hilbert mathematics. The Yang-Mills existence and mass gap problem furthermore suggests the most general viewpoint to the relation of Hilbert and Gödel mathematics justifying the qubit Hilbert space as the dual counterpart of Hilbert arithmetic in a narrow sense, in turn being inferable from Hilbert arithmetic in a wide sense. The conjecture that many if not almost all great problems in contemporary mathematics rely on (or at least relate to) the Gödel incompleteness is suggested. It implies that Hilbert mathematics is the natural medium for their discussion or eventual solutions. (shrink)

Faith is the highest truth that ensures the happiness and salvation of man in the world and in the Hereafter. But the essence of superstitious is invalid and wrong. The realization of this happiness and salvation is possible by having a true faith. Another consequence of the true faith is the ability to recognize that this belief is right. Believing in true faith, ensures rightness and makes possible to prove and disclose this truth. It is important to have true faith (...) and accurate affirmation. This certainty requires certain criteria for accuracy and precision. In addition, this situation involves advocating the faith and delivering it to other people. This is closely related to the recognition and proof of these truths. It also makes it necessary to explore the distinction between presumption and believing as well as knowledge and ignorance. It requires examination of the concepts of certainty, doubt, evidence, persuasion and proof and what they have implied in terms of faith. The examination of the proofs of faith in terms of persuasion and proof also increases the understanding of the certainty of a belief. (shrink)

The standard of proof in criminal trials in many liberal democracies is proof beyond a reasonable doubt, the BARD standard. It is customary to describe it, when putting a number on it, as requiring that the fact finder be at least 90% certain, after considering the evidence, that the defendant is guilty. Strikingly, no good reason has yet been offered in defense of using that standard. A number of non-consequentialist justifications that aim to support an even higher standard have been (...) offered; all are morally unsound. Meanwhile, consequentialist arguments plausibly support a substantially lower standard — in some cases so low as to undermine the idea that punishment is what is at stake. In this paper, I offer a new retributive justification that supports excluding the instrumental benefits of punishment from the balance that sets the standard. The resulting balance supports a standard arguably in the ballpark of the customary understanding of BARD: a standard requiring that the fact finder have a high, though not maximally high, degree of confidence that the defendant is guilty. (shrink)