Stewart Shapiro

Early in Book 1 of Herodotus' Histories, Solon speaks to Croesus about the jealousy of the gods and the ephemeral nature of human happiness . Since Solon's speech is so prominently placed, and since it introduces themes that recur throughout the Histories, it has traditionally been seen as programmatic, i.e., as expressing Herodotus' own views about the gods and human happiness. Although the assumption that Solon speaks for Herodotus has long been the standard view, it has recently been challenged on the grounds that Herodotus himself never directly affirms his agreement with what Solon says. On the other hand, many scholars have noted that Herodotus does not always state his views directly: he often uses literary devices such as analogy, juxtaposition, and the repetition of narrative patterns to indicate his views. After isolating the three main principles of Solon's speech , this paper examines the direct and indirect evidence for Herodotus' acceptance of these principles. After examining the evidence in Books 1-9 of the Histories, the paper concludes that Herodotus does indeed agree with the views of his character Solon, and that he makes his agreement clear both explicitly and implicitly, through analogy, repetition, and juxtaposition. Moreover, the position of Solon's speech indicates that Herodotus meant it to be programmatic, setting forth basic assumptions about the nature of human life and its relation to the gods which could then provide a philosophical framework for the Histories as a whole

We have measured the momentum and energy dependence of the spin fluctuations in the low-temperature mixed-valent state and the high-temperature integral-valent state of YbInCu4, using flux-grown single crystals and a triple axis spectrometer. The magnetic scattering can be fit with a Lorentzian power spectrum, with positions and halfwidths equal E+0 = 2.3 meV and Γ + = 1.8 meV in the high-temperature state at 50 K and E-0 = 40.2 meV and Γ- = 12.3 meV in the low-temperature state at 20 K. Within the experimental uncertainties, the line shape as a function of energy transfer does not depend on momentum transfer Q. The static susceptibility χ is no more than 10% larger at the zone boundary than at zone center for the [1,0,0] direction. The results compare reasonably well to the predictions of the Anderson impurity model and to earlier measurements on polycrystalline samples.

As the hospice movement continues to grow, caregivers are increasingly required to interact with dying patients for longer periods and in more intimate and more meaningful ways. Practical models of competent and compassionate communication and understanding need to be developed to accommodate the changing environment of the patient and caregiver and their relationship. We therefore: examine current death education trends in hospice care and education; and describe the need for a more expansive and transpersonal view, and ways of fulfilling that need. The aware patient and the aware caregiver can learn much during the dying trajectory. We hope that someday the potential for expansiveness during the dying trajectory will be more commonly recognized and accommodated as a doorway to the transpersonal.

A typical interpreted formal language has (first‐order) variables that range over a collection of objects, sometimes called a domain‐of‐discourse. The domain is what the formal language is about. A language may also contain second‐order variables that range over properties, sets, or relations on the items in the domain‐of‐discourse, or over functions from the domain to itself. For example, the sentence ‘Alexander has all the qualities of a great leader’ would naturally be rendered with a second‐order variable ranging over qualities. Similarly, the sentence ‘there is a property that holds of all and only the prime numbers’ has a variable ranging over properties of natural numbers. Third‐order variables range over properties of properties, sets of sets, functions from properties to sets, etc. For example, according to some logicist accounts, the number 4 is the property shared by all properties that apply to exactly four objects in the domain. Accordingly, the number 4 is a third‐order item. Fourth‐order variables, and beyond, are characterized similarly. The phrase ‘higher‐order variable’ refers to the variables beyond first‐order.

The concept of truth is now a major research subject in analytic philosophy. At the same time, working in different areas, mathematical logicians have developed sophisticated theories of truth and its formal paradoxes. Recent developments of semantical paradoxes in logical theories are highly relevant for philosophical research on the notion of truth. And conversely, philosophical guidance is necessary for the development of logical theories of truth and the paradoxes. From this perspective, this volume intends to reflect and promote deeper interaction and collaboration between philosophers and logicians investigating the concept of truth than has existed so far. In addition to an extended introductory overview of recent work in the theory of truth, the volume includes articles by leading philosophers and logicians on subjects and debates that interface logical and philosophical theories of truth. The volume is intended for graduate students in philosophy and in logic who want an introduction to contemporary research in this area, as well as for professional philosophers and logicians. Topics covered include a defense of minimalism, metaworlds, the role of correspondence theory, and truth, provability and its criteria. The volume includes contributions by John P. Burgess, Paul Horwich, and Michael Sheard. The contributors are united in their belief that at present the connections between truth and intentional notions are poorly understood, and the entire area is in an unsatisfactory state. Controversy abounds: Is truth a logical-mathematic concept or a properly philosophical one? Does truth pertain to temporal events or possible worlds? Axiomatic theories do not seem to possess the naturalness and elegance intentional notions of truth. The editors intend this volume to offer semantical guidance to those who would construct interesting axiomatic systems concerning intentional notions. Volker Halbach is on the faculty of philosophy of the Universitt Konstanz in Germany. He has contributed articles for Mind, Notre Dame Journal of Formal Logic, Journal of Symbolic Logic, Synthese, and other leading publications in philosophy. Leon Horsten is in the Department of Philosophy, University of Leuven. His work on the subject of this book has appeared in the Journal of Philosophical Logic.

This paper is part of a larger project concerning potentiality in mathematics. The first and simplest case is the traditional Aristotelian notion of potential infinity. An issue much like that of truthmaking arises in our explication of one of the options for potential infinity, namely how to make sense of generalizations from that perspective.We use the traditional intuitionistic notion of realizability to resolve the issue, and to help settle the correct logic for one kind of potential infinity.

Stewart Shapiro presents a distinctive original view of the foundations of mathematics, arguing that second-order logic has a central role to play in laying these foundations. He gives an accessible account of second-order and higher-order logic, paying special attention to philosophical and historical issues. Foundations without Foundationalism is a key contribution both to philosophy of mathematics and to mathematical logic. 'In this excellent treatise Shapiro defends the use of second-order languages and logic as frameworks for mathematics. His coverage of the wide range of logical and philosophical... is thorough, clear, and persuasive.' Michael D. Resnik, History and Philosophy of Logic.

It has long been known that (classical) Peano arithmetic is, in some strong sense, “equivalent” to the variant of (classical) Zermelo–Fraenkel set theory (including choice) in which the axiom of infinity is replaced by its negation. The intended model of the latter is the set of hereditarily finite sets. The connection between the theories is so tight that they may be taken as notational variants of each other. Our purpose here is to develop and establish a constructive version of this. We present an intuitionistic theory of the hereditarily finite sets, and show that it is definitionally equivalent to Heyting Arithmetic HA, in a sense to be made precise. Our main target theory, the intuitionistic small set theory SST is remarkably simple, and intuitive. It has just one non-logical primitive, for membership, and three straightforward axioms plus one axiom scheme. We locate our theory within intuitionistic mathematics generally.

According to singularism, ‘the students’ refers to a single collective entity, e.g. a sum or set. In contrast, according to pluralism, ‘the students’ plurally refers to multiple students at once, through the primitive relation of plural reference. Although it was originally designed exclusively for plural nouns, this paper addresses whether plural reference can be extended so as to provide an empirically adequate semantics for mass nouns, such as ‘the furniture’, as well, as certain pluralists have maintained. Our contention is that extant pluralist analyses of mass nouns suffer from two major setbacks. First, while they may or may not explain the semantic commonalities between plural and mass nouns, they do not explain their semantic differences. Secondly, pluralism is inconsistent with the possibility that mass nouns are gunky, i.e. fail to have atomic parts. However, we provide three kinds of semantic arguments showing that the gunkiness of at least some mass nouns is not only possible, but plausible.

The is an old question over whether there is a substantial disagreement between advocates of different logics, as they simply attach different meanings to the crucial logical terminology. The purpose of this article is to revisit this old question in light a pluralism/relativism that regards the various logics as equally legitimate, in their own contexts. We thereby address the vexed notion of translation, as it occurs between mathematical theories. We articulate and defend a thesis that the notion of “same meaning” is itself context-sensitive, depending on the purposes of a given conversation.

We consider a handful of solutions to the liar paradox which admit a naive truth predicate and employ a non-classical logic, and which include a proposal for classical recapture. Classical recapture is essentially the property that the paradox solvent (in this case, the non-classical interpretation of the connectives) only affects the portion of the language including the truth predicate—so that the connectives can be interpreted classically in sentences in which the truth predicate does not occur. We consider a variation on this theme where the logic to be recaptured is not classical but rather intuitionist logic, and consider the extent to which these handful of solutions to the liar admit of intuitionist recapture by sketching potential ways of altering their various methods for classical recapture to suit an intuitionist framework.