The following syllogism is considered: a string is not an expression unless it is tokenable; no one can utter, write, or in anyway token an infinite string; so no infinite string is an expression. The second premise is rejected. But the tokenability of an infinite sentence is not sufficient for it being an infinite expression. A further condition is that no finite sentence expresses that sentence’s truth-conditions. So it is an open question whether English contains (...) class='Hi'>infinite expressions. (shrink)

The infinitary languages studied in this book are those in which quantification of infinitely many variables simultaneously, and conjunctions or alternations of infinitely many are permitted. Infinitary concatenation and infinitary propositional logics are first discussed, and a completeness theorem is proved about the latter. The later chapters deal with infinitary predicate languages and Scott's proof of incompleteness is introduced. Throughout the discussion, unsolved problems are mentioned and areas undergoing current development are emphasized. A short bibliography lists most recent articles on (...) this new branch of logic.—P. J. M. (shrink)

Infinite in All Directions is a popularized science at its best. In Dyson's view, science and religion are two windows through which we can look out at the world around us. The book is a revised version of a series of the Gifford Lectures under the title "In Praise of Diversity" given at Aberdeen, Scotland. They allowed Dyson the license to express everything in the universe, which he divided into two parts in polished prose: focusing on the diversity of (...) the natural world as the first, and the diversity of human reactions as the second half. Chapter 1 is a brief explanation of Dyson's attitudes toward religion and science. Chapter 2 is a one-hour tour of the universe that emphasizes the diversity of viewpoints from which the universe can be encountered as well as the diversity of objects which it contains. Chapter 3 is concerned with the history of science and describes two contrasting styles in science: one welcoming diversity and the other deploring it. He uses the cities of Manchester and Athens as symbols of these two ways of approaching science. Chapter 4, concerned with the origin of life, describes the ideas of six illustrious scientists who have struggled to understand the nature of life from various points of view. Chapter 5 continues the discussion of the nature and evolution of life. The question of why life characteristically tends toward extremes of diversity remains central in all attempts to understand life's place in the universe. Chapter 6 is an exercise in eschatology, trying to define possible futures for life and for the universe, from here to infinity. In this chapter, Dyson crosses the border between science and science fiction and he frames his speculations in a slightly theological context. (shrink)

Después de haber analizado las razones que indujeron a las antiguas matemáticas griegas y de que Aristóteles sólo admitiera una débil forma de lo infinito, se explora una ampliación de este concepto más allá de sus referencias numéricas y geométricas. El infinito puede expresar la “inagotable” riqueza ontológica de los atributos de las entidades individuales o, en otro sentido, el infinito puede ser entendido como aquello “ilimitado”. En este segundo sentido la “negación” se presenta como una fuerza positiva en la (...) formación del sentido ontológico de lo infinito.After having analyzed the reasons that induced ancient Greek mathematics and Aristotle to admit only a weak form of the infinite a broadening of this notion beyond its numerical and geometrical references is explored. The infinite can express the “inexhaustible” ontological richness of the attributes of individual entities or, in another sense, the infinite can be understood as the “unlimited”. In this second sense the “negation” appears as a positive force in the shaping of the ontological meaning of the infinite. (shrink)

Attempts a faithful account of Bruno's thought as expressed in his writings and to give an analysis of his thought as he developed it in regard to the issue of the infinitive.

Attempts a faithful account of Bruno's thought as expressed in his writings and to give an analysis of his thought as he developed it in regard to the issue of the infinitive.

In quantum field theory divergent expressions are "discarded", leaving finite expressions which provide the best predictions anywhere in science. In fact, this "renormalization procedure" involves no mystery or illegitimate operations. This paper explains, in terms accessible to non-experts, how the procedure really works and explores some different ways in which physicists have suggested that one understand it.

We analyze recent criticisms of the use of hyperreal probabilities as expressed by Pruss, Easwaran, Parker, and Williamson. We show that the alleged arbitrariness of hyperreal fields can be avoided by working in the Kanovei–Shelah model or in saturated models. We argue that some of the objections to hyperreal probabilities arise from hidden biases that favor Archimedean models. We discuss the advantage of the hyperreals over transferless fields with infinitesimals. In Paper II we analyze two underdetermination theorems by Pruss and (...) show that they hinge upon parasitic external hyperreal-valued measures, whereas internal hyperfinite measures are not underdetermined. (shrink)

This paper develops some important observations from a recent article by Maria Rosa Antognazza published in The Leibniz Review 2015 under the title “The Hypercategorematic Infinite”, from which I take up the characterization of God, the most perfect Being, as infinite in a hypercategorematic sense, i.e., as a being beyond any determination. By contrast, creatures are determinate beings, and are thus limited and particular expressions of the divine essence. But since Leibniz takes both God and creatures to (...) be infinite, creatures are simultaneously infinite and limited. This leads to seeing creatures as infinite in kind, in distinction from the absolute and hypercategorematic infinity of God. I present three lines of argument to substantiate this point: (1) seeing creatures as entailing a particular sequence of perfections and imperfections; (2) seeing creatures under the rubric of an intermediate degree of infinity and perfection that Leibniz, in 1676, calls “maximum in kind”; and (3) observing that primitive force, a defining feature of created substance, may be seen as infinite in a metaphysical sense. This leads to viewing Leibniz’s use of infinity within a Neoplatonic framework of descending degrees of Being: from the hypercategorematic infinite, identified with the most perfect Being; to the intermediate degree of maximum in kind, identified with creatures; to the lowest degree of entia rationis (or beings of reason), identified with mathematical and abstract entities. (shrink)

Pascal's wager has fascinated philosophers far in excess of its reputation as effective apologetics. Very few of the wager's defenders, in fact, have retained more than an academic interest in its power to persuade. Partly this is a matter of good manners. Pascal is supposed to have pitched his wager at folks who understand only self-interested motivations, and today it is no longer fashionable for defenders of theism to disparage the character of their opponents. But partly the low-key concern with (...) apologetics expresses a philosophical judgement. Pascal's defenders have found the question of the wager's audience to be less philosophically engaging than the logic of its argument. I believe that this assessment is mistaken. The most puzzling feature of Pascal's wager is its invocation of infinite utility. What are finite human beings, theists or otherwise, supposed to make of the idea of an infinitely desirable happiness? There are, I will argue, two sorts of response to this question, and depending on which sort Pascal had in mind, the logic of his wager comes out very differently. (shrink)

Pascal's wager has fascinated philosophers far in excess of its reputation as effective apologetics. Very few of the wager's defenders, in fact, have retained more than an academic interest in its power to persuade. Partly this is a matter of good manners. Pascal is supposed to have pitched his wager at folks who understand only self-interested motivations, and today it is no longer fashionable for defenders of theism to disparage the character of their opponents. But partly the low-key concern with (...) apologetics expresses a philosophical judgement. Pascal's defenders have found the question of the wager's audience to be less philosophically engaging than the logic of its argument. I believe that this assessment is mistaken. The most puzzling feature of Pascal's wager is its invocation of infinite utility. What are finite human beings, theists or otherwise, supposed to make of the idea of an infinitely desirable happiness? There are, I will argue, two sorts of response to this question, and depending on which sort Pascal had in mind, the logic of his wager comes out very differently. (shrink)

One of the main logical functions of the truth predicate is to enable us to express so-called ‘infinite conjunctions’. Several authors claim that the truth predicate can serve this function only if it is fully disquotational, which leads to triviality in classical logic. As a consequence, many have concluded that classical logic should be rejected. The purpose of this paper is threefold. First, we consider two accounts available in the literature of what it means to express infinite conjunctions (...) with a truth predicate and argue that they fail to support the necessity of transparency for that purpose. Second, we show that, with the aid of some regimentation, many expressive functions of the truth predicate can actually be performed using truth principles that are consistent in classical logic. Finally, we suggest a reconceptualisation of deflationism, according to which the principles that govern the use of the truth predicate in natural language are largely irrelevant for the question of what formal theory of truth we should adopt. Many philosophers think that the paradoxes pose a special problem for deflationists; we will argue, on the contrary, that deflationists are in a much better position to deal with the paradoxes than their opponents. (shrink)

According to the disquotationalist theory of truth, the Tarskian equivalences, conceived as axioms, yield all there is to say about truth. Several authors have claimed that the expression of infinite conjunctions and disjunctions is the only purpose of the disquotationalist truth predicate. The way in which infinite conjunctions can be expressed by an axiomatized truth predicate is explored and it is considered whether the disquotationalist truth predicate is adequate for this purpose.

The problem of finding a covariant expression for the distribution and conservation of gravitational energy-momentum dates to the 1910s. A suitably covariant infinite-component localization is displayed, reflecting Bergmann's realization that there are infinitely many gravitational energy-momenta. Initially use is made of a flat background metric (or rather, all of them) or connection, because the desired gauge invariance properties are obvious. Partial gauge-fixing then yields an appropriate covariant quantity without any background metric or connection; one version is the collection of (...) pseudotensors of a given type, such as the Einstein pseudotensor, in _every_ coordinate system. This solution to the gauge covariance problem is easily adapted to any pseudotensorial expression (Landau-Lifshitz, Goldberg, Papapetrou or the like) or to any tensorial expression built with a background metric or connection. Thus the specific functional form can be chosen on technical grounds such as relating to Noether's theorem and yielding expected values of conserved quantities in certain contexts and then rendered covariant using the procedure described here. The application to angular momentum localization is straightforward. Traditional objections to pseudotensors are based largely on the false assumption that there is only one gravitational energy rather than infinitely many. (shrink)

Traditional act utilitarianism judges an action permissible just in case it produces as much aggregate utility as any alternative. It is often supposed that utilitarianism faces a serious problem if the future is infinitely long. For in that case, actions may produce an infinite amount of utility. And if that is so for most actions, then utilitarianism, it appears, loses most of its power to discriminate among actions. For, if most actions produce an infinite amount of utility, then (...) few actions produce non-maximal utility, and so most actions are permissible.1 I will argue that potentially infinite futures create no major problems for utilitarianism. Utilitarianism has, I argue, the resources to distinguish among actions all of which produce infinite amounts of utility -- judging some permissible and some impermissible. For brevity of expression I will focus on act utilitarianism, but all the points apply equally well to many other traditional forms of utilitarianism. (shrink)

It is argued that English is finite. By this is meant that it contains only finitely many expressions. The conclusion is reached by arguing: (1) only finitely many expressions of English are tokenable; (2) if E is an expression of English, then E is tokenable.

I propose a counterfactual theory of infinite regress arguments. Most theories of infinite regress arguments present infinite regresses in terms of indicative conditionals. These theories direct us to seek conditions under which an infinite regress generates an infinite inadmissible set. Since in ordinary language infinite regresses are usually expressed by means of infinite sequences of counterfactuals, it is natural to expect that an analysis of infinite regress arguments should be based on a (...) theory of counterfactuals. The Stalnaker–Lewis theory of counterfactuals, augmented with some fundamental notions from metric-spaces, provides a basis for such an analysis of infinite regress arguments. Since the technique involved in the analysis is easily adaptable to various analyses, it facilitates a rigorous comparison among conflicting philosophical analyses of any given infinite regress. (shrink)

We present an abstract social aggregation theorem. Society, and each individual, has a preorder that may be interpreted as expressing values or beliefs. The preorders are allowed to violate both completeness and continuity, and the population is allowed to be infinite. The preorders are only assumed to be represented by functions with values in partially ordered vector spaces, and whose product has convex range. This includes all preorders that satisfy strong independence. Any Pareto indifferent social preorder is then shown (...) to be represented by a linear transformation of the representations of the individual preorders. Further Pareto conditions on the social preorder correspond to positivity conditions on the transformation. When all the Pareto conditions hold and the population is finite, the social preorder is represented by a sum of individual preorder representations. We provide two applications. The first yields an extremely general version of Harsanyi's social aggregation theorem. The second generalizes a classic result about linear opinion pooling. (shrink)

It is well known that Leibniz advocated the actual infinite, but that he did not admit infinite collections or infinite numbers. But his assimilation of this account to the scholastic notion of the syncategorematic infinite has given rise to controversy. A common interpretation is that in mathematics Leibniz’s syncategorematic infinite is identical with the Aristotelian potential infinite, so that it applies only to ideal entities, and is therefore distinct from the actual infinite that (...) applies to the actual world. Against this, I argue in this paper that Leibniz’s actual infinite, understood syncategorematically, applies to any entities that are actually infinite in multitude, whether numbers, actual parts of matter, or monads. It signifies that there are more of them than can be assigned a number, but that there is no infinite number or collection of them, which notion involves a contradiction. Similarly, to say that a magnitude is actually infinitely small in the syncategorematic sense is to say that no matter how small a magnitude one takes, there is a smaller, but there are no actual infinitesimals. In geometry one may calculate with expressions apparently denoting such entities, on the understanding that they are fictions, standing for variable magnitudes that can be made arbitrarily small, so as to produce demonstrations that there is no error in the resulting expressions. (shrink)

Introduction. This paper is a sequel to our paper [3]. In that paper we introduced the notion of a finite approximation to an infinitely long formula, in a language L with infinitely long expressions of the type considered by Henkin in [2]. The results of the paper [3] show relationships between the models of an infinitely long sentence and the models of its finite approximations. In the present paper we shall apply the main result of [3] to prove a (...) number of theorems about ordinary finitely long sentences; these theorems are of a type which one might call “preservation theorems”. The following two known results are typical preservation theorems: A sentence φ is preserved under substructures if and only if φ is equivalent to some universal sentence ; φ is preserved under homomorphic images if and only if φ is equivalent to some positive sentence. An expository account of preservation theorems may be found in Lyndon [9].We shall use freely all of the notation introduced in [3], and shall not attempt to make this paper selfcontained. However, the reader should have no difficulty following this paper after he has read [3]. In § 1 we cover some preliminary topics. In § 2 and § 3 we shall illustrate our method in familiar situations by proving anew the two preservation theorems mentioned above. We shall then obtain four new preservation theorems in §§ 4, 5, and 6, involving respectively direct powers and roots, strong homomorphisms and retracts, and direct factors. The result in § 6 was stated in the abstract [4], while the results in § 5 were stated in the abstract [5]. (shrink)

The rationale clause infinitive is a modifier of the verb which expresses an agent's intention in acting as they do. The rationale clause is related to the matrix in two important ways: the null subject of the infinitive (PRO) is usually coreferential with a phrase in the matrix (control), and the intention it expresses is usually assigned to a phrase in the matrix (predication). Given that the controller and argument of the rationale clause are usually clausemates of the infinitive itself, (...) the strongest hypothesis is that both control and predication are strictly local. However, there has been much debate in the literature concerning the viability of such a locality constraint. In this article, I argue in favour of a locality-based account of control and predication in the rationale clause, comparing it with a discourse-semantic constraint approach and a logophoricity constraint approach. Data from two related modifier infinitives (the purpose clause and the telic clause) will be used to sharpen the discussion. (shrink)