If there is a good wellordering of the reals, then there is a wellfounded relation for which the comparison relation is not projective.

We show that it is relatively consistent with ZFC that there is a projective wellfounded relation with rank higher than all projective prewellorderings.

If there is a good $\Delta^1_3$ wellordering of the reals, then there is a $\Pi^1_1$ wellfounded relation for which the comparison relation is not projective.

In a previous article on cosmological arguments, I have put forward a few examples of complete infinite and circular explanations, and argued that complete non-wellfounded explanations such as these might explain the present state of the world better than their well-founded theistic counterparts (Billon, 2021). Although my aim was broader, the examples I gave there implied merely causal explanations. In this article, I would like to do three things: • Specify some general informative conditions for complete and incomplete non- (...) class='Hi'>wellfounded causal explanations that can be used to assess candidate explanations and to generate new examples of complete non-wellfounded explanations. • Show that these conditions, which concern chains of causal explanations, easily generalize to chains of metaphysical, grounding explanations and even to chains involving other “determination relations” such as supervenience. • Apply these general conditions to the recent debates against the existence of nonwellfounded chains of grounds and show, with a couple of precise examples, that the latter can be complete, and that just like in the case of causal explanations, non-wellfoundedness can in fact be an aset rather than a liability. (shrink)

We prove that if I is a partially ordered set in a countable transitive model M of ZFC then M can be extended by a generic sequence of reals a i , i ∈ I, such that ℵ M 1 is preserved and every a i is Sacks generic over $\mathfrak{M}[\langle \mathbf{a}_j: j . The structure of the degrees of M-constructibility of reals in the extension is investigated. As applications of the methods involved, we define a cardinal invariant to distinguish (...) product and iterated Sacks extensions, and give a short proof of a theorem (by Budinas) that in ω 2 -iterated Sacks extension of L the Burgess selection principle for analytic equivalence relations holds. (shrink)

Ross Cameron (2022) argues that metaphysical infinitists should reject the generally accepted idea that metaphysical determination relations back metaphysical explanations. Otherwise it won’t be possible for them to come up with successful explanations for the existence of dependent entities in non-wellfounded chains of dependence. I argue that his argument suffers from what he calls the finitist dogma, although indirectly so. However, there is a better way of motivating Cameron’s conclusion. Assuming Cameron’s principle of Essence, explanations for the existence of (...) dependent entities turn out to be circular if determination relations back explanations. This latter argument provides a stronger case as it puts the foundationalist under significant pressure, besides putting the infinitist under some pressure, to deny the idea that determination relations back explanations. (shrink)

The philosophies of education serve as the guide of the teachers in handling the teaching-learning process. However, a belief will remain as a belief unless it is practiced. This study aimed to find the relationship between the philosophical beliefs and practices of the 30 teachers of the schools in San Roque, Mabini, Bohol - San Roque Elementary School and San Roque National High School, S.Y. 2019-2020. The study utilized a quantitative method descriptive survey research design. The research instrument used was (...) a self-constructed questionnaire in which the reliability was tested by a Cronbach’s Alpha using Statistical Package for the Social Science (SPSS) software. Simple percentage and weighted mean were some of the statistical treatments used. Kendall’s tau_b, a non-parametric test, was used to determine the correlation between philosophical beliefs and their equivalent pedagogical practices. All of the statistical treatments were also run using the SPSS software. Further, the researcher assured the proper observation of the research ethics protocol. Based on the results, the philosophies of Progressivism (3.63) and Perennialism (3.31) were the most and least prevailing philosophical beliefs, respectively. On the other hand, the most and least prevailing philosophies practiced are Progressivism (3.69) and Essentialism (3.19), respectively. Moreover, research findings revealed that there was a significant relationship between the philosophical beliefs of teachers and their pedagogical practices. In conclusion, this study claims that teachers’ pedagogical practices are dependent to their philosophical beliefs; and as the teachers believed on a particular philosophy of education, they practice it accordingly to the extent of their adherence to the belief. (shrink)

Barbarism was by no means unique to the past 100 years, Jonathan Glover tells us, but ''it is still right that much of 20th-century history has been a very unpleasant surprise.'' This was the century of Passchendaele, Dresden, Nanking, Nagasaki and Rwanda; of the Final Solution, the gulag, the Great Leap Forward, Year Zero and ethnic cleansing -- names that stand for killings in the six and seven figures and for suffering beyond comprehension. The technological progress that inspired the optimism (...) of the Victorians turned out also to multiply the effects of old-fashioned evil and criminal stupidity. (shrink)

Presents the fundamental ideas of Indian thinkers that have shaped the mind of Indian from 1770 to the post-modern era in the middle of 20th century in India. Lists the most Indian influential figures in the field of philosophy, political theory, activicism such as Rabindranath Tagore, Ram Mohan Roy, Swami Vivekananda, and Mohandas Karamchand Gandhi.

The satisfiability problem for the two-variable fragment of first-order logic is investigated over finite and infinite linearly ordered, respectively wellordered domains, as well as over finite and infinite domains in which one or several designated binary predicates are interpreted as arbitrary wellfounded relations. It is shown that FO 2 over ordered, respectively wellordered, domains or in the presence of one well-founded relation, is decidable for satisfiability as well as for finite satisfiability. Actually the complexity of these decision problems (...) is essentially the same as for plain unconstrained FO 2 , namely non-deterministic exponential time. In contrast FO 2 becomes undecidable for satisfiability and for finite satisfiability, if a sufficiently large number of predicates are required to be interpreted as orderings, wellorderings, or as arbitrary wellfounded relations. This undecidability result also entails the undecidability of the natural common extension of FO 2 and computation tree logic CTL. (shrink)

We consider foundational questions related to the definition of functions by corecursion. This method is especially suited to functions into the greatest fixed point of some monotone operator, and it is most applicable in the context of non-wellfounded sets. We review the work on the Special Final Coalgebra Theorem of Aczel [1] and the Corecursion Theorem of Barwise and Moss [4]. We offer a condition weaker than Aczel's condition of uniformity on maps, and then we prove a result relating (...) the operators satisfying the new condition to the smooth operators of [4]. (shrink)

In [10] Friedman showed that is a conservative extension of <ε0for-sentences wherei= min, i.e.,i= 2, 3, 4 forn= 0, 1, 2 +m. Feferman [5], [7] and Tait [11], [12] reobtained this result forn= 0, 1 and even with instead of. Feferman and Sieg established in [9] the conservativeness of over <ε0for-sentences for alln. In each paper, different methods of proof have been used. In particular, Feferman and Sieg showed how to apply familiar proof-theoretical techniques by passing through languages with Skolem (...) functionals.In this paper we study the same choice principles in the presence of theBar Rule, which permits one to infer the scheme of transfinite induction on a primitive recursive relation ≺ when it has been proved that ≺ is wellfounded. The main result characterizes + as a conservative extension of a system of the autonomously iterated-comprehension axiom for-sentences. Forn= 0 this has been proved by Feferman in the form that + is a conservative extension of <Γ0; this was first done in [8] by use of the Gödel functional interpretation for the stronger systemZω+μ+ + and then more recently by the simpler methods of [9]. Jäger showed how the latter methods could also be used to obtain the general result of Theorem 1 below. (shrink)

Building on previous work by Mints, Buchholz and Schwichtenberg, a simplified version of continuous normalization for the untyped λ-calculus and Gödel’s is presented and analysed in the coalgebraic framework of non-wellfounded terms with so-called repetition constructors.The primitive recursive normalization function is uniformly continuous w.r.t. the natural metric on non-wellfounded terms. Furthermore, the number of necessary repetition constructors is locally related to the number of reduction steps needed to reach the normal form and its size.It is also shown how (...) continuous normal forms relate to derivations of strong normalizability in the typed λ-calculus and how this leads to new bounds for the sum of the height of the reduction tree and the size of the normal form.Finally, the methods are extended to an infinitary λ-calculus with ω-rule and permutative conversions and this is used to derive a strong form of normalization for an iterative version of Gödel’s system , leading to a value table semantics for number-theoretic functions. (shrink)

A foundational algebra ( , f, ) consists of a hemimorphism f on a Boolean algebra with a greatest solution to the condition f(x). The quasi-variety of foundational algebras has a decidable equational theory, and generates the same variety as the complex algebras of structures (X, R), where f is given by R-images and is the non-wellfounded part of binary relation R.The corresponding results hold for algebras satisfying =0, with respect to complex algebras of wellfounded binary relations. (...) These algebras, however, generate the variety of all ( ,f) with f a hemimorphism on ). (shrink)

This paper establishes a conjecture of Steel [7] regarding the structure of elementary embeddings from a level of the cumulative hierarchy into itself. Steel’s question is related to the Mitchell order on these embeddings, studied in [5] and [7]. Although this order is known to be illfounded, Steel conjectured that it has certain large wellfounded suborders, which is what we establish. The proof relies on a simple and general analysis of the much broader class of extender embeddings and a (...) variant of the Mitchell order called the internal relation. (shrink)

This paper is a systematic exploration of non-wellfounded mereology. Motivations and applications suggested in the literature are considered. Some are exotic like Borges’ Aleph, and the Trinity; other examples are less so, like time traveling bricks, and even Geach’s Tibbles the Cat. The authors point out that the transitivity of non-wellfounded parthood is inconsistent with extensionality. A non-wellfounded mereology is developed with careful consideration paid to rival notions of supplementation and fusion. Two equivalent axiomatizations are given, and (...) are compared to classical mereology. We provide a class of models with respect to which the non-wellfounded mereology is sound and complete. (shrink)

In this paper we present and study a categorical formulation of the W-types of Martin-Löf. These are essentially free term algebras where the operations may have finite or infinite arity. It is shown that W-types are preserved under the construction of sheaves and Artin gluing. In the proofs we avoid using impredicative or nonconstructive principles.

We study the non-wellfounded sets as fixed points of substitution. For example, we show that ZFA implies that every function has a fixed point. As a corollary we determine for which functions f there is a function g such that . We also present a classification of non-wellfounded sets according to their branching structure.

The paper explores the proof theory of non-wellfounded mereology with binary fusions and provides a cut-free sequent calculus equivalent to the standard axiomatic system.

In Rabinowicz (2008), I considered how value relations can best be analysed in terms of fitting pro-attitudes. In the formal model of that paper, fitting pro-attitudes are represented by the class of permissible preference orderings on a domain of items that are being compared. As it turns out, this approach opens up for a multiplicity of different types of value relationships, along with the standard relations of ‘better’, ‘worse’, ‘equally as good as’ and ‘incomparable in value’. Unfortunately, the approach is (...) vulnerable to a number of objections. I believe these objections can be avoided if one re-interprets the underlying notion of preference: instead of treating preference as a ‘dyadic’ attitude directed towards a pair of items, we can think of it as a difference of degree between ‘monadic’ attitudes of favouring. Each such monadic attitude has just one item as its object. Given this re-interpretation, permissible preferences can be modelled by the class of permissible assignments of degrees of favouring to items in the domain. From this construction, we can then recover the old modelling in terms of the class of permissible preference orderings, but the previous objections to that model no longer apply. (shrink)

In this article, I question the plausibility of Metz’s African moral theory from an oft-neglected moral topic of partiality. Metz defends an Afro-communitarian moral theory that posits that the rightness of actions is entirely definable by relationships of identity and solidarity (or, friendship). I offer two objections to this relational moral theory. First, I argue that justifying partiality strictly by invoking relationships (of friendship) ultimately fails to properly value the individual for her own sake – this is called the ‘focus (...) problem’ in the literature. Second, I argue that a relationship-based theory cannot accommodate the agent-related partiality since it posits some relationship to be morally fundamental. My critique ultimately reveals the inadequacy of a relationship-based moral theory insofar as it overlooks some crucial moral considerations grounded on the individual herself in her own right. (shrink)

ABSTRACT In the present study, we examined how the perceived attainability and relatability of moral exemplars predicted moral elevation and pleasantness among both adult and college student participants. Data collected from two experiments were analyzed with Bayesian multilevel modeling to explore which factors significantly predicted outcome variables at the story level. The analysis results demonstrated that the main effect of perceived relatability and the interaction effect between attainability and relatability shall be included in the best prediction model, and thus, were (...) deemed to predict the outcome variables significantly. The main effect of relatability as well as its interaction with attainability positively predicted elevation and pleasantness. We discussed educational implications of the findings in terms of how relatability may be the first point of emphasis for moral educators to focus on and attainability can then bolster the effectiveness. These relatable and attainable moral exemplars can be sources for moral elevation and pleasantness, which promote motivation to emulate moral behavior presented by the exemplars. (shrink)

‘What is characteristic of every mental activity’, according to Brentano, is ‘the reference to something as an object. In this respect every mental activity seems to be something relational.’ But what sort of a relation, if any, is our cognitive access to the world? This question – which we shall call Brentano’s question – throws a new light on many of the traditional problems of epistemology. The paper defends a view of perceptual acts as real relations of a subject (...) to an object. To make this view coherent, a theory of different types of relations is developed, resting on ideas on formal ontology put forward by Husserl in his Logical Investigations and on the theory of relations sketched in Smith's "Acta cum fundamentis in re". The theory is applied to the notion of a Cambridge change, which proves to have an unforeseen relevance to our understanding of perception. (shrink)

The discussion of collective agency involves the reduction problem of the concept of a collective. Individualism and Cartesian internalism have long restricted orthodox theories and made them face the tension between an irreducible concept of a collective and ontological reductionism. Heterodox theories as functionalism and interpretationism reinterpret the concept of agency and accept it as realized on the level of a collective. In order to adequately explain social phenomena that have relations as their essence, in this paper we propose a (...) relational, holistic account of collective agency and argue that functionalism and interpretationism can be integrated into such an account. (shrink)

Agents are often assumed to have degrees of belief (“credences”) and also binary beliefs (“beliefs simpliciter”). How are these related to each other? A much-discussed answer asserts that it is rational to believe a proposition if and only if one has a high enough degree of belief in it. But this answer runs into the “lottery paradox”: the set of believed propositions may violate the key rationality conditions of consistency and deductive closure. In earlier work, we showed that this problem (...) generalizes: there exists no local function from degrees of belief to binary beliefs that satisfies some minimal conditions of rationality and non-triviality. “Locality” means that the binary belief in each proposition depends only on the degree of belief in that proposition, not on the degrees of belief in others. One might think that the impossibility can be avoided by dropping the assumption that binary beliefs are a function of degrees of belief. We prove that, even if we drop the “functionality” restriction, there still exists no local relation between degrees of belief and binary beliefs that satisfies some minimal conditions. Thus functionality is not the source of the impossibility; its source is the condition of locality. If there is any non-trivial relation between degrees of belief and binary beliefs at all, it must be a “holistic” one. We explore several concrete forms this “holistic” relation could take. (shrink)

I distinguish between two senses in which feminists have argued that the knower is social: 1. situated or socially positioned and 2. interdependent. I argue that these two aspects of the knower work in cooperation with each other in a way that can produce willful hermeneutical ignorance, a type of epistemic injustice absent from Miranda Fricker's Epistemic Injustice. Analyzing the limitations of Fricker's analysis of the trial of Tom Robinson in Harper Lee's To Kill a Mockingbird with attention to the (...) way in which situatedness and interdependence work in tandem, I develop an understanding of willful hermeneutical ignorance, which occurs when dominantly situated knowers refuse to acknowledge epistemic tools developed from the experienced world of those situated marginally. Such refusals allow dominantly situated knowers to misunderstand, misinterpret, and/or ignore whole parts of the world. (shrink)

The standard account of modal expressions in natural language analyzes them as quantifiers over a set of possible worlds determined by the evaluation world and an accessibility relation. A number of authors have recently argued for an alternative account according to which modals are analyzed as quantifying over a domain of possible worlds that is specified directly in the points of evaluation. But the new approach only handles the data motivating it if it is supplemented with a non-standard account (...) of attitude verbs and conditionals. It can be shown the the relational account handles the same data equally well if it too is supplemented with a non-standard account of such expressions. (shrink)

This paper asks whether phenomenal intentionality (intentionality that arises from phenomenal consciousness alone) has a relational structure of the sort envisaged in Russell’s theory of acquaintance. I put forward three arguments in favor of a relation view: one phenomenological, one linguistic, and one based on the view’s ability to account for the truth conditions of phenomenally intentional states. I then consider several objections to the relation view. The chief objection to the relation view takes the form of (...) a dilemma between Platonic and Aristotelian conceptions of the properties constitutive of the contents of phenomenally intentional states on this view: the Aristotelian view seems unable to account for all the apparent contents of phenomenally intentional states, but the Platonic view seems to be ontologically unacceptable. I also consider other objections from physicalism, phenomenology, and epistemology. (shrink)

‘Relational value’ (RV) has intuitive credibility due to the shortcomings of existing axiological categories regarding recognizing the ethical relevance of people’s relations to nature. But RV is justified by arguments and analogies that do not hold up to closer scrutiny, which strengthens the assumption that RV is redundant. While RV may provide reasons for ethically considering some relations, much work remains to show that RV is a concept that does something existing axiological concepts cannot, beyond empirically describing relations people have (...) to environmental areas and places. (shrink)

In this paper I examine the role of dispositional properties in the most frequently discussed interpretations of non-relativistic quantum mechanics. After offering some motivation for this project, I briefly characterize the distinction between non-dispositional and dispositional properties in the context of quantum mechanics by suggesting a necessary condition for dispositionality – namely contextuality – and, consequently, a sufficient condition for non-dispositionality, namely non-contextuality. Having made sure that the distinction is conceptually sound, I then analyze the plausibility of the widespread, monistic (...) ontological thesis about the reducibility of dispositional properties to categorical properties in the context of the philosophy of quantum mechanics. I conclude that with the exception of Bohmian mechanics, the other “minimally realist” views of quantum mechanics require essential dispositions, i.e., dispositions of a non-reducible kind. Interestingly, seen behind the lenses of dispositionalism, Bohr’s and Bohm’s interpretations of quantum mechanics are much closer than it is usually recognized, a fact that could teach us something about the way the quantum world is. (shrink)

Relating Narratives is a major new work by the philosopher and feminist thinker Adriana Cavarero. First published in Italian to widespread acclaim, Relating Narratives is a fascinating and challenging new account of the relationship between selfhood and narration. Drawing a diverse array of thinkers from both the philosophical and the literary tradition, from Sophocles and Homer to Hannah Arendt, Karen Blixen, Walter Benjamin and Borges, Adriana Cadarero's theory of the `narratable self' shows how narrative models in philosophy and literature can (...) open new ways of thinking about formation of human identities. By showing how each human being has a unique story that can be told about them, Adriana Cavarero inaugurates an important shift in thinking about subjectivity and identity which relies not upon categorical or discursive norms, but rather seeks to account for `who' each one of us uniquely is. (shrink)

It is a metaphysical orthodoxy that interesting non-symmetric relations cannot be reduced to symmetric ones. This orthodoxy is wrong. I show this by exploring the expressive power of symmetric theories, i.e. theories which use only symmetric predicates. Such theories are powerful enough to raise the possibility of Pythagrapheanism, i.e. the possibility that the world is just a vast, unlabelled, undirected graph.

Let ZFB be ZF + "every set is the same size as a wellfounded set". Then the following are true. Every sentence true in every (Rieger-Bernays) permutation model of a model of ZF is a theorem of ZFB. (i.e.. ZFB is the theory of Rieger-Bernays permutation models of models of ZF) ZF and ZFAFA are both extensions of ZFB conservative for stratified formulæ. The class of models of ZFB is closed under creation of Rieger-Bernays permutation models.

Conception and causation are fundamental notions in Spinoza's metaphysics. I argue against the orthodox view that, due to the causal axiom, if one thing is conceived through another thing, then the second thing causes the first thing. My conclusion forces us to rethink Spinoza's entitlement to some of his core commitments, including the principle of sufficient reason, the parallelism doctrine and the conatus doctrine.

The orthodox monadic determination thesis holds that we represent colour relations by virtue of representing colours. Against this orthodoxy, I argue that it is possible to represent colour relations without representing any colours. I present a model of iconic perceptual content that allows for such primitive relational colour representation, and provide four empirical arguments in its support. I close by surveying alternative views of the relationship between monadic and relational colour representation.

Relational egalitarianism is a theory of justice according to which justice requires that people relate as equals. According to some relational egalitarians, X and Y relate as equals if, and only if, they (1) regard each other as equals; and (2) treat each other as equals. In this paper, we argue that relational egalitarians must give up 1.