In connection with his interest in selfdistributive algebra, Richard Laver established two deep results with potential applications in low-dimen\-sional topology, namely the existence of what is now known as the Laver tables and the well-foundedness of the standard ordering of positive braids. Here we present these results and discuss the way they could be used in topological applications.

We construct a family $\fancyscript{F}$ , indexed by five integer parameters, of finite monogenerated left-distributive (LD) groupoids with the property that every finite monogenerated LD groupoid is a quotient of a member of $\fancyscript{F}$ . The combinatorial abundance of finite monogenerated LD groupoids is encoded in the congruence lattices of the groupoids $\fancyscript{F}$ , which we show to be extremely large.

Communication is an integral part of nursing practice—with patients and their relatives, other nurses and members of the healthcare team, and ancillary staff. Through interaction with the ‘other’, language and silence creates and recreates social realities. Acceptance, rejection or modification of social realities depends on what is expressed and by whom. Narratives that are offered can tell of some experiences and not others. Some nurses choose to be silent while others are silenced. In nursing situations recognising and allowing silence to (...) speak is a challenging but uniquely personal experience that embraces reflection in and on experiences, practice and self as a person and a professional. If enabled and truly heard, silence can speak more loudly than the hubbub of daily practice, allowing us to collectively question and challenge inherent assumptions and biases as professionals, and as a profession. Through a microcosm of Newly Graduated Nurses' lived experiences of nursing situations and expressions of silence individuals' discomfort and private efforts to ascribe meaning to experiences are reflected on. Returning to silence is to return to a constant process of professional transformation that can enable ways of knowing and being that can reform our profession from within and enable us to cast off shackles that bind us to a shameful cultural underbelly. (shrink)

The large cardinal axioms of the title assert, respectively, the existence of a nontrivial elementary embedding j:Vλ→Vλ, the existence of such a j which is moreover , and the existence of such a j which extends to an elementary j:Vλ+1→Vλ+1. It is known that these axioms are preserved in passing from a ground model to a small forcing extension. In this paper the reverse directions of these preservations are proved. Also the following is shown : if V is a model (...) of ZFC and V[G] is a -generic forcing extension of V, then in V[G], V is definable using the parameter Vδ+1, where. (shrink)

The rank-into-rank and stronger large cardinal axioms assert the existence of certain elementary embeddings. By the preservation of the large cardinal properties of the embeddings under certain operations, strong implications between various of these axioms are derived.

Quine’s ‘Two Dogmas of Empiricism’ is generally seen as overturning Carnap’s epistemological picture of mathematics and the sciences. However, I wish to stress how this paper grew out of arguments not having anything to do with large-scale epistemological concerns, but ones originally presented against quantified modal logic. Quine thought he could demonstrate the impossibility of adding anything like ordinary quantification to modal logic, but Barcan Marcus did exactly this. In fact, as I will argue, ‘Two Dogmas ...’ can be seen (...) as growing out of a 1947 paper of Quine’s where Barcan Marcus is the explicit target. Later, in certain exchanges with Quine, Barcan Marcus puts forward and defends a consistent and interesting position on matters of ontology and the philosophy of language. This chapter will examine Barcan Marcus’s role in both the debate over analyticity and the debate on (meta-)ontology. (shrink)

Carnap's position on mathematical truth in The Logical Syntax of Language has been attacked from two sides: Kreisel argues that it is formalistic but should not be, and Friedman argues that it is not formalistic but needs to be. In this paper I argue that the Carnap of Syntax does not eliminate our ordinary notion of mathematical truth in favour of a formal analogue; so Carnap's notion of mathematical truth is not formalistic. I further argue that there is no conflict (...) between Carnap's use of informal notions and his principle of tolerance; so Carnap's definition of mathematical truth need not be formalistic. CiteULike Connotea Del.icio.us What's this? (shrink)

In this paper I examine the fundamental views on the nature of logical and mathematical truth of both Frege and Carnap. I argue that their positions are much closer than is standardly assumed. I attempt to establish this point on two fronts. First, I argue that Frege is not the metaphysical realist that he is standardly taken to be. Second, I argue that Carnap, where he does differ from Frege, can be seen to do so because of mathematical results proved (...) in the early twentieth century. The differences in their views are, then, not primarily philosophical differences. Also, it might be thought that Frege was interested in analyzing our ordinary mathematical notions, while Carnap was interested in the construction of arbitrary systems. I argue that this is not the case: our ordinary notions play, in a sense, an even more important role in Carnap’s philosophy of mathematics than they do in Frege’s. Finally, I address Tyler Burge’s interpretation of Frege which is in opposition to any Carnapian reading of Frege. (shrink)

1 Where to begin? 'Life and times' Roland Barthes is generally acknowledged, even by those not conversant with his books, as one of the leading figures of ...

In 2020, it is clear that racism must be a moral priority for white American Catholics, as for all white Americans. To face racism maturely, our consciences need more robust formation than we received as children preparing for First Reconciliation—or as adolescents at Confirmation, when we were instructed that we must be prepared to act boldly to defend our commitment to Christ. One way to build a racially-mature conscience is to seek feedback from anti-racism accountability partners, as the author did. (...) Another is to explore the distinction in Catholic moral theology between invincible and vincible ignorance—what we can know and what we cannot—in order to discern whether an action or inaction is moral. Much of what liberal whites claim we can’t know without input from Blacks is readily available if we have ears to hear. (shrink)

This paper puts forward and defends an account of mathematical truth, and in particular an account of the truth of mathematical axioms. The proposal attempts to be completely nonrevisionist. In this connection, it seeks to satisfy simultaneously both horns of Benacerrafs work on informal rigour. Kreisel defends the view that axioms are arrived at by a rigorous examination of our informal notions, as opposed to being stipulated or arrived at by trial and error. This view is then supplemented by a (...) Fregean account of the objectivity and our knowledge of abstract objects. It is then argued that the resulting view faces no insurmountable metaphysical or epistemic obstacles. (shrink)

This collection of new essays addresses emotion in relation to the arts. The essays consider such topics as the paradox of fiction, emotion in the pure and abstract arts, and the rationality and ethics of emotional responses to art.

This paper argues that Carnap both did not view and should not have viewed Frege's project in the foundations of mathematics as misguided metaphysics. The reason for this is that Frege's project was to give an explication of number in a very Carnapian sense — something that was not lost on Carnap. Furthermore, Frege gives pragmatic justification for the basic features of his system, especially where there are ontological considerations. It will be argued that even on the question of the (...) independent existence of abstract objects, Frege and Carnap held remarkably similar views. I close with a discussion of why, despite all this, Frege would not accept the principle of tolerance. (shrink)

From 1953 to 1959 Gödel worked on a response to Carnap’s philosophy of mathematics. The drafts display Gödel’s familiarity with Carnap’s position from The Logical Syntax of Language, but they received a dismissive reaction on their eventual, posthumous, publication. Gödel’s two principal points, however, will here be defended. Gödel, though, had wished simply to append a few paragraphs to show that the same arguments apply to Carnap’s later views. Carnap’s position, however, had changed significantly in the intervening years, and to (...) a significant extent, this mature position realizes much of what Gödel himself wanted out of a philosophy of mathematics. (shrink)

Carnap’s ‘Empiricism, Semantics, and Ontology’ (Carnap (1950a), ESO hereafter) is certainly a classic of twentieth century analytic philosophy. For decades now, most undergraduates are expected to read it at some point in their studies. Lately, it is being seen as the inspiration for a host of positions in the field of metaontology. Despite the widespread agreement on the importance of the paper, there is a lack of agreement on what Carnap attempts to do in the paper. My main aim in (...) this chapter is to defend an interpretation of ESO by displaying its relation to Carnap’s earlier works. My secondary goal will be to draw a lesson from this discussion for Carnap’s later work on unobservable entities in science. (shrink)

This paper will deal with three questions regarding Carnap's transition from the position he held at the time of writing Syntax to the doctrines he held during his semantic phase: (1) What was Carnap's attitude towards truth at the time of writing Syntax? (2) What was Carnap's position regarding questions of reference and ontology at the time of writing Syntax? (3) Was Carnap's acceptance of Tarski's analysis of truth and reference detrimental to his philosophical project? Section 1 of this paper (...) will deal with the first of these questions. Special attention will be paid to identifying what it was that prevented Carnap from defining a truth predicate for descriptive languages in Syntax. Section 2 of this paper will deal with the question of Carnap's attitude towards reference and ontology in Syntax. It will be shown that the attempt in Syntax to address ontological questions is seriously defective. Section 3 of this paper addresses the last of the questions posed above. It is argued that in the light of what is established in Sections 1 and 2, Carnap could not have retained the position of Syntax with respect to truth and reference. (shrink)

This paper investigates the relation between Carnap and Quine’s views on analyticity on the one hand, and their views on philosophical analysis or explication on the other. I argue that the stance each takes on what constitutes a successful explication largely dictates the view they take on analyticity. I show that although acknowledged by neither party (in fact Quine frequently expressed his agreement with Carnap on this subject) their views on explication are substantially different. I argue that this difference not (...) only explains their differences on the question of analyticity, but points to a Quinean way to answer a challenge that Quine posed to Carnap. The answer to this challenge leads to a Quinean view of analyticity such that arithmetical truths are analytic, according to Quine’s own remarks, and set theory is at least defensibly analytic. (shrink)

Say that the property Φ of a cardinal λ strongly implies the property Ψ. If and only if for every λ,Φ implies that Ψ and that for some λ′<λ,Ψ. Frequently in the hierarchy of large cardinal axioms, stronger axioms strongly imply weaker ones. Some strong implications are proved between axioms of the form “there is an elementary embedding j:Lα[Vλ+1]→Lα[Vλ+1] with ”.

Gregory Lavers gives us a timeline of Waismann’s career, an overview of Waismann’s most significant publications in this later period and a detailed walkthrough from the first to the last paper of Waismann’s series on analyticity, “Analytic - Synthetic”. Lavers closes his paper with comparisons of Waismann and Quine as well as Waismann and Carnap. Both Waismann and Quine argue that the concept of analyticity is vague and both reject reductionism. However, behind these superficial similarities we find fundamentally different epistemologies. (...) According to Lavers, the web of inferential relations, spanning from every experience to any item of scientific knowledge, that supports the outlook suggested by Quine, is rejected as manifestly wrong by Waismann. Conversely, Lavers shows that despite superficial contrasts between Waismann and Carnap—Waismann being interested in the subtleties of natural language, Carnap in replacing these through explication—the two do not really oppose each other’s’ views on analyticity and necessary truth. (shrink)

Growing nursing workforce maldistributions impede rural healthcare access globally. In‐depth exploration of underlying philosophical ideas about rural health in nursing curricular could support recruitment and retention of nurses who are well positioned to support and advocated for health care and services relevant to their communities. Through a lens of positionality, the purpose of this paper is to explore rural health and nursing within the United States and Australia from the perspective of undergraduate students. Recognizing that both countries have ‘first world’ (...) health services, issues of access to services associated with systemic and structural biases were significant features identified by participants. Their perceptions were supported by discussion with attendees of the 26th International Nursing Philosophy Conference from several countries, during a presentation titled ‘What is different about rural health nursing: A philosophical exploration.’ This international consistency suggests that systemic and structural biases are global phenomena. While exposure to rural health and rural nursing may be beneficial for recruitment, preparing nurses for rural practice requires more than educating students to complete clinical tasks. Uncovering collaboration, advocacy and policy development opportunities for nurses in rural communities could move the dial from recruiting rural nurses to fulfill immediate needs to retention of well‐educated and highly skilled nurses. Building a stronger philosophical base for rural healthcare inclusive of all people, honouring their distinguishing characteristics such as age, gender, ethnicity, is long overdue. Positionality can inform how nurses across the globe relate to other countries' governments, cultures, norms, values, expectations, etc., which can influence the advancement of the profession and address social inequities among rural populations. (shrink)

Abstract At the time of The Logical Syntax of Language (Syntax), Quine was, in his own words, a disciple of Carnap’s who read this work page by page as it issued from Ina Carnap’s typewriter. The present paper will show that there were serious problems with how Syntax dealt with ontological claims. These problems were especially pronounced when Carnap attempted to deal with higher order quantification. Carnap, at the time, viewed all talk of reference as being part of the misleading (...) material mode of speech, and as such dismissed, rather than addressed, ontological problems. Central to the analysis in the present paper is the concept of an explication, which was seen by both Carnap and Quine as being of great philosophical importance. It will be shown that the concept of explication played a significant role in how each formulated their mature position on ontology. Both these final positions on ontology can also be seen as a evolving in reaction to Carnap’s flawed handling of ontological matters at the time of Syntax. Carnap, influenced by Tarski’s work on semantics, comes to believe that the concept of reference can be given an acceptable explication, and that by doing so we can see reference to abstract objects as unobjectionable. As a result, Carnap develops a position very different from the one presented in Syntax. Quine strongly rejected the instrumentalism of Syntax, and sought to give an explication of ontological questions that was language independent. This paper closes with a discussion of each’s understanding of the other’s position. (shrink)

This paper argues that Frege is not the metaphysical platonist about mathematics that he is standardly taken to be. It is shown that Frege’s project has two distinct stages: the identification of what is true of our ordinary notions, and then the provision of a systematic account that shares the identified features. Neither of these stages involves much metaphysics. The paper criticizes in detail Dummett’s interpretation of §§55-61 of Grundlagen. These sections fall under the heading ‘Every number is a self-subsistent (...) object’ and are described by Dummett as containing the worst arguments put forward by Frege. It is argued that essentially all of Dummett’s interpretive points are mistaken. Finally, I show that Frege’s claims about the independence of mathematics from humans and their activities does not commit him to any particularly metaphysical position either. (shrink)

Of Zeno’s book of forty paradoxes, it was the first that attracted Socrates’ attention. This is the paradox of the like and the unlike. On contemporary assessments, this paradox is largely considered to be Zeno’s weakest surviving paradox. All of these assessments, however, rely heavily on reconstructions of the paradox. It is only relative to these reconstructions that there is nothing paradoxical involved, or that there is some rather obvious mistake being made. This paper puts forward and defends a novel (...) interpretation of this paradox, according to which the concept of a unit plays a central role. There is every reason to think the paradox turns on the concept of a unit: after the presentation of the paradox the text of the Parmenides immediately turns to a discussion of units, and the concept of a unit is also central to the Greek conception of a plurality. If this interpretation is correct then the paradox that Zeno presented was the same as one discussed and solved in Frege’s Grundlagen. (shrink)

Pathologies of Rational Choice Theory: A Critique of Applications on Political Science, Donald P. Green and Ian ShapiroThe Rational Choice Controversy: Economic Models of Politics Reconsidered. Jeffrey Friedman.

This book has two main goals: first, to show that Quine's New Foundations set theory is better motivated than often assumed; and second, to defend Quine's philosophy of set theory. It is divided into three parts. The first concerns the history of set theory and argues against readings that see the iterative conception of set being the dominant notion of set from the very beginning. The second part concerns Quine's philosophy of set theory. Part 3 is a contemporary assessment of (...) the philosophical status of NF. Here one of the central targets is Boolos's defense of the iterative conception of set and his dismissal of NF as completely unnatural.As this is a very short review, I will avoid specific points and... (shrink)