Chronological table -- Introduction -- The dictionary.

This article argues that what was understood as democracy in the eighteenth century differs fundamentally from modern democracy. While modern democratic states take locally born or naturalized personhood as the criterion of citizenship, eighteenth-century advocates of democracy demanded proof of political competence to allow participation in politics. While the requirement of competence to engage in any activity is not unreasonable, if defined, as it was by most Enlightenment thinkers, as a combination of independence, cultural standing and wealth, it is clearly (...) elitist. Enlightenment criticisms of birth and status as criteria of political standing should not be mistaken for demands for broadly inclusive, modern conceptions of equality and democracy. Rather, most Enlightenment thinkers and publicists, fully aware of the implications of widespread and debilitating poverty, expressed reservations about granting the ‘people’ an effective role in politics. Far from being ‘radical’, Enlightenment political theory called for the replacement of an elitism based on status and privilege with an elitism of ability and wealth. (shrink)

The postmodern critique of the Enlightenment is much concerned with what it regards as the unwillingness of progressive thinkers of the eighteenth century to accept the legitimacy of national or cultural groups that differed significantly from norms in Western Europe. My aim is to examine how eighteenth-century thinkers, including Hume, Montesquieu, Voltaire, Condorcet, and the Abbé Grégoire, perceived prototypical “others” such as Blacks and Jews, by looking at the sources—from contemporary medical science to travel literature, proto-anthropology, history, biblical scholarship and (...) reformist projects—on which these views were based. Perceptions of Blacks cannot easily be separated from the issue of slavery, nor that of the Jews from biblical history and theology. I argue that those who wanted to exclude these groups from mainstream society generally based their arguments on a one-dimensional, self-referential empirical methodology, while those who argued for their eventual inclusion usually posite.. (shrink)

From Carl Becker’s The Heavenly City of the Eighteenth-Century Philosophers to recent work on religion in the Enlightenment, it has been argued that the Enlightenment has significant religious elem...

This article addresses a number of issues relevant to the interpretation of the Enlightenment raised by Jonathan Israel in his recent book, Enlightenment Contested. After a brief summary of the main points of the book it considers whether, as Israel claims, the core of the Enlightenment is a materialist monist metaphysic first fully articulated by Spinoza, and whether it is convincing to make materialism and atheism the main criteria of Enlightenment thought. The argument that Spinoza and Pierre Bayle should be (...) seen as co-founders of the Enlightenment is also examined. The article further questions the cogency of the sharp distinction drawn between the moderate and radical wings of the movement, and seeks to determine whether the model of radicalism used by Israel has the consistency ascribed to it, whether it was as widely disseminated as claimed, and whether thinkers described as radical argued and wrote as Israel's model of radicalism would lead us to expect. (shrink)

This dictionary offers a balanced overview and helps readers understand and appreciate the Enlightenment Movement. Cross-referenced dictionary entries cover the significant persons, places, events, institutions, and literary works of the movement, and a chronological table charts the progression of the movement by indicating the date, the main figures involved, the political or society events, and the science, arts, or letters that resulted.

There have been many interpretations of the French Revolution. For reactionary theorists of the late eighteenth and early nineteenth century, it was a conspiracy of freethinkers, Protestants, freem...

Driven by interests in workforce planning and patient safety, a growing body of literature has begun to identify the reality and the prevalence of missed nursing care, also specified as care left undone, rationed care or unfinished care. Empirical studies and conceptual considerations have focused on structural issues such as staffing, as well as on outcome issues – missed care/unfinished care. Philosophical and ethical aspects of unfinished care are largely unexplored. Thus, while internationally studies highlight instances of covert rationing/missed care/care (...) left undone – suggesting that nurses, in certain contexts, are actively engaged in rationing care – in terms of the nursing and nursing ethics literature, there appears to be a dearth of explicit decision-making frameworks within which to consider rationing of nursing care. In reality, the assumption of policy makers and health service managers is that nurses will continue to provide full care – despite reducing staffing levels and increased patient turnover, dependency and complexity of care. Often, it would appear that rationing/missed care/nursing care left undone is a direct response to overwhelming demands on the nursing resource in specific contexts. A discussion of resource allocation and rationing in nursing therefore seems timely. The aim of this discussion paper is to consider the ethical dimension of issues of resource allocation and rationing as they relate to nursing care and the distribution of the nursing resource. (shrink)

Two countable well orderings are weakly comparable if there is an order preserving injection of one into the other. We say the well orderings are strongly comparable if the injection is an isomorphism between one ordering and an initial segment of the other. In [5], Friedman announced that the statement “any two countable well orderings are strongly comparable” is equivalent to ATR 0 . Simpson provides a detailed proof of this result in Chapter 5 of [13]. More recently, Friedman has (...) proved that the statement “any two countable well orderings are weakly comparable” is equivalent to ATR 0 . The main goal of this paper is to give a detailed exposition of this result. (shrink)

Based on the relevant logic R, the system R# was proposed as a relevant Peano arithmetic. R# has many nice properties: the most conspicuous theorems of classical Peano arithmetic PA are readily provable therein; it is readily and effectively shown to be nontrivial; it incorporates both intuitionist and classical proof methods. But it is shown here that R# is properly weaker than PA, in the sense that there is a strictly positive theorem QRF of PA which is unprovable in R#. (...) The reason is interesting: if PA is slightly weakened to a subtheory P+, it admits the complex ring C as a model; thus QRF is chosen to be a theorem of PA but false in C. Inasmuch as all strictly positive theorems of R# are already theorems of P+, this nonconservativity result shows that QRF is also a nontheorem of R#. As a consequence, Ackermann's rule γ is inadmissible in R#. Accordingly, an extension of R# which retains its good features is desired. The system R##, got by adding an omega-rule, is such an extension. Central question: is there an effectively axiomatizable system intermediate between R# and R##, which does formalize arithmetic on relevant principles, but also admits γ in a natural way? (shrink)

Reflection, in the sense of [Fr03a] and [Fr03b], is based on the idea that a category of classes has a subclass that is “similar” to the category. Here we present axiomatizations based on the idea that a category of classes that does not form a class has extensionally different subclasses that are “similar”. We present two such similarity principles, which are shown to interpret and be interpretable in certain set theories with large cardinal axioms.

We study the formalization within sybsystems of second-order arithmetic of theorems concerning periodic points in dynamical systems on the real line. We show that Sharkovsky's theorem is provable in WKL0. We show that, with an additional assumption, Sharkovsky's theorem is provable in RCA0. We show that the existence for all n of n-fold iterates of continuous mappings of the closed unit interval into itself is equivalent to the disjunction of Σ02 induction and weak König's lemma.

We now fix A ⊆ Q. We study a fundamental class of digraphs associated with A, which we call the A-digraphs. An A,kdigraph is a digraph (Ak,E), where E is an order invariant subset of A2k in the following sense. For all x,y ∈ A2k, if x,y have the same order type then x ∈ E ↔ y ∈ E.

The subtle, almost ineffable, and ineffable cardinals were introduced in an unpublished 1971 manuscript of R. Jensen and K. Kunen. The concepts were extended to that of k-subtle, k-almost ineffable, and k-ineffable cardinals in 1975 by J. Baumgartner. In this paper we give a self contained treatment of the basic facts about this level of the large cardinal hierarchy, which were established by J. Baumgartner. In particular, we give a proof that the k-subtle, k-almost ineffable, and k-ineffable cardinals define three (...) properly intertwined hierarchies with the same limit, lying strictly above “total indescribability” and strictly below “arrowing ω”. The innovation here is presented in Section 2, where we take a distinctly minimalist approach. Here the subtle cardinal hierarchy is characterized by very elementary properties that do not mention closed unbounded or stationary sets. This development culminates in a characterization of the hierarchy by means of a striking universal second-order property of linear orderings. (shrink)

We consider a number of statements involving restrictions and extensions of algebras, and derive connections with large cardinal axioms.

The kind of unknowability I will discuss concerns the count of certain natural finite sets of objects. Even the situation with regard to our present strong formal systems is rather unclear. One can just profitably focus on that, putting aside issues of general unknowability.

Let R Õ [1,n]3k ¥ [1,n]k. We define R = {y Œ [1,n]k:($xŒA3)(R(x,y))}. We say that R is strictly dominating if and only if for all x,yŒ[1,n]k, if R(x,y) then max(x) < max(y).

Most of my intellectual efforts have focused around a single general question in the foundations of mathematics (f.o.m.). I became keenly aware of this question as a student at MIT around 40 years ago, and readily adopted it as the principal driving force behind my research.

We present some examples of mathematically natural nonrecursive sets of integers and relations on integers by combining results from Part 1, recursion theory, and from the negative solution to Hilbert’s 10th Problem ([3], [1], and [2]).

An unprovable theorem is a mathematical result that can-not be proved using the com-monly accepted axioms for mathematics (Zermelo-Frankel plus the axiom of choice), but can be proved by using the higher infinities known as large cardinals. Large car-dinal axioms have been the main proposal for new axioms originating with Gödel.

An extreme kind of logic skeptic claims that "the present formal systems used for the foundations of mathematics are artificially strong, thereby causing unnecessary headaches such as the Gödel incompleteness phenomena". The skeptic continues by claiming that "logician's systems always contain overly general assertions, and/or assertions about overly general notions, that are not used in any significant way in normal mathematics. For example, induction for all statements, or even all statements of certain restricted forms, is far too general - mathematicians (...) only use induction for natural statements that actually arise. If logicians would tailor their formal systems to conform to the naturalness of normal mathematics, then various logical difficulties would disappear, and the story of the foundations of mathematics would look radically different than it does today. In particular, it should be possible to give a convincing model of actual mathematical practice that can be proved to be free of contradiction using methods that lie within what Hilbert had in mind in connection with his program”. Here we present some specific results in the direction of refuting this point of view, and introduce the Strict Reverse Mathematics (SRM) program. (shrink)

It has been accepted since the early part of the Century that there is no problem formalizing mathematics in standard formal systems of axiomatic set theory. Most people feel that they know as much as they ever want to know about how one can reduce natural numbers, integers, rationals, reals, and complex numbers to sets, and prove all of their basic properties. Furthermore, that this can continue through more and more complicated material, and that there is never a real problem.

In a modal system of arithmetic, a theory S has the modal disjunction property if whenever $S \vdash \square\varphi \vee \square\psi$ , either $S \vdash \square\varphi$ or $S \vdash \square\psi. S$ has the modal numerical existence property if whenever $S \vdash \exists x\square\varphi(x)$ , there is some natural number n such that $S \vdash \square\varphi(\mathbf{n})$ . Under certain broadly applicable assumptions, these two properties are equivalent.