In 1936 Tarski sketched a rigorous definition of the concept of logical consequence which, he claimed, agreed quite well with common usage-or, as he also said, with the common concept of consequence. Commentators of Tarski's paper have usually been elusive as to what this common concept is. However, being clear on this issue is important to decide whether Tarski's definition failed (as Etchemendy has contended) or succeeded (as most commentators maintain). I argue that the common concept of consequence that Tarski (...) tried to characterize is not some general, all-purpose notion of consequence, but a rather precise one, namely the concept of consequence at play in axiomatics. I identify this concept and show that Tarski's definition is fully adequate to it. (shrink)

Because of its capacity to characterize mathematical concepts and structures?a capacity which first-order languages clearly lack?second-order languages recommend themselves as a convenient framework for much of mathematics, including set theory. This paper is about the credentials of second-order logic:the reasons for it to be considered logic, its relations with set theory, and especially the efficacy with which it performs its role of the underlying logic of set theory.

There is an apparent tension between the open-ended aspect of the ordinal sequence and the assumption that the set-theoretical universe is fully determinate. This tension is already present in Cantor, who stressed the incompletable character of the transfinite number sequence in Grundlagen and avowed the definiteness of the totality of sets and numbers in subsequent philosophical publications and in correspondence. The tension is particularly discernible in his late distinction between sets and inconsistent multiplicities. I discuss Cantor’s contrasting views, and I (...) conclude that his account of that distinction is only tenable if the definiteness of the set-theoretical universe is rejected. (shrink)

Dating and hookup apps (DHAs) are now widely used and may be transforming our intimate relationships. The apps are beneficial in fostering intimate connections among those who are lonely, who are members of minority or marginalized groups, or who live nomadic lifestyles because of work or recreational travel. However, the wider social and relational changes that DHAs portend are merely beginning to be seriously discussed by academics (Arias et al., 2017). In this chapter, we employ concepts from the philosophy of (...) technology that inform our analysis of these changes. We also collate some of the morally significant impacts of DHAs and present some of the elements of the architecture of DHAs that facilitate them. Although there are many reasons to be concerned about DHAs, we argue that there is room for cautious optimism that DHAs may be vehicles of positive social and moral change. (shrink)

This article identifies several critical problems with the last 30 years of research into hostile communication on the internet and offers suggestions about how scholars might address these problems and better respond to an emergent and increasingly dominant form of online discourse which I call ‘e-bile’. Although e-bile is new in terms of its prevalence, rhetorical noxiousness, and stark misogyny, prototypes of this discourse—most commonly referred to as ‘flaming’—have always circulated on the internet, and, as such, have been discussed by (...) scholars from a range of disciplines. Nevertheless, my review of this vast body of literature reveals that online hostility has historically posed a number of conceptual, methodological, and epistemological challenges due to which scholars have typically underplayed, overlooked, ignored, or otherwise marginalised its prevalence and serious ethical and material ramifications. Fortunately, lessons learned from my analysis suggests promising approaches for future research into this challenging form of new media discourse. (shrink)

George Boolos has described an interpretation of a fragment of ZFC in a consistent second-order theory whose only axiom is a modification of Frege's inconsistent Axiom V. We build on Boolos's interpretation and study the models of a variety of such theories obtained by amending Axiom V in the spirit of a limitation of size principle. After providing a complete structural description of all well-founded models, we turn to the non-well-founded ones. We show how to build models in which foundation (...) fails in prescribed ways. In particular, we obtain models in which every relation is isomorphic to the membership relation on some set as well as models of Aczel's anti-foundation axiom (AFA). We suggest that Fregean extensions provide a natural way to envisage non-well-founded membership. (shrink)

In this paper an attempt is made to present Skolem's argument, for the relativity of some set-theoretical notions as a sensible one. Skolem's critique of set theory is seen as part of a larger argument to the effect that no conclusive evidence has been given for the existence of uncountable sets. Some replies to Skolem are discussed and are shown not to affect Skolem's position, since they all presuppose the existence of uncountable sets. The paper ends with an assessment of (...) the assumptions on which Skolem's argument rests from a present-day perspective. (shrink)

In this paper, Tarskis notion of Logical Consequence is viewed as a special case of the more general notion of being a theorem of an axiomatic theory. As was recognized by Tarski, the material adequacy of his definition depends on having the distinction between logical and non logical constants right, but we find Tarskis analysis persuasive even if we dont agree on what constants are logical. This accords with the view put forward in this paper that Tarski indeed captures the (...) more inclusive notion of theoremhood in an axiomatic theory. The approach to logical consequence via axiomatic theories leads us to grant centrality to inference schemas rather than to full-fledged arguments and to view the logically valid schemas as a subclass of generally valid schemas. (shrink)

In this paper we discuss the way logical consequence depends on what sets there are. We try to find out what set-theoretical assumptions have to be made to determine a logic, i.e., to give a definite answer to whether any given argument is correct. Consideration of second order logic -which is left highly indetermined by the usual set-theoretical axioms- prompts us to suggest a slightly different but natural nation of logical consequence, which reduces second order logic indeterminacy without interfering with (...) first order logic. (shrink)

The efficacies of Western and Chinese medicine have been under debate for a long time, and the whole issue still raises questions for the contemporary world. The article emphasizes the humanistic approach as well as the scientific method of the early Protestant medical missionaries to China, so as to give a more comprehensive scope to understand their historical roles and practices in a cross-cultural context. The authors also wish to call for a global readership to further discuss this historical legacy (...) with regard to the reception of Christianity in contemporary China and other East Asian countries. (shrink)

Second-order languages, canonically understood, allow quantification over all sets of objects in the range of the first-order variables. In this chapter two arguments are given against the suitability of using second-order consequence as the consequence relation of axiomatic theories. According to the first argument, second-order languages are inadequate for axiomatizing set theory because of the strong set-theoretic content coded by second-order consequence. The second more general argument is directed against the determinacy of second-order consequence, that is, against the assumption that (...) this is a definite relation. Only taking a strong realist view of set theory can one maintain that it is. (shrink)