Cohen’s hostility to Rawls’ justification of the Difference Principle by social facts spawned Cohen’s general thesis that ultimate principles of justice and morality are fact-insensitive, but explain how any fact-sensitive principle is grounded in facts. The problem with this thesis, however, is that when facts F ground principle P, reformulating this relation as the "fact-insensitive" conditional “If F, then P” is trivial and thus explanatorily impotent. Explanatory, hence justificatory, force derives either from subsumption under more general principles, or precisely exhibiting (...) value in light of relevant (actual or hypothetical) facts. In examples where no subsumption occurs, actual facts trivially become hypothetical facts in "fact-insensitive" conditionals, an empty formalism. Indeed, Rawls’ grounding of principles of justice in “conditions of life” can easily be reformulated as a conditional principle “sensitive” only to hypothetical such conditions, and thus formally fact-insensitive in Cohen's sense, for all Cohen’s ire against Rawls’s grounding.Moreover, any plausible “ultimate fact-insensitive principle” must be intricately qualified, which tacit ceteris paribus clauses mask. Each qualification implies prioritisation of one principle over another in conceivable circumstances, and wherever the now qualified principle is given scope, that too implies prioritisation over competing principles in typical circumstances. Any principle is thus sensitive to conceivable circumstances of application, as recognised by more sophisticated intuitionisms. Non-trivial ultimate principles – luck egalitarianism, act utilitarianism, etc. - require defense, which inevitably involves showing how they best interpret and respond to facts about human needs, goals, and capacities in predictable circumstances. Finally, the substantive debate between Rawls and Cohen about which facts are relevant to the DP is only obscured by the doctrine of fact-insensitivity. (shrink)

The dominance of the modern concept of risk and calculability is challenged by and has to be distinguished from “manufactured uncertainties.” Typically today, conflict and controversy flare up around this particular type of new manufactured risk. Neither natural disasters – threats – coming from the outside and thus attributable to God or nature have this effect any longer. Nor do the specific calculable uncertainties – “risks” – which are determinable with actuarial precision interms of a probability calculus backed (...) up by insurance and monetary compensation fall in this category. At the centre of attention today, by contrast, are “manufactured uncertainties.” They are distinguished by the fact that they are dependent on human decisions, created by society itself, immanent to society and thus non-externalizable, collectively imposed and thus individually unavoidable. (shrink)

We describe a software system for the analysis of defined benefit actuarial plans. The system uses a recursive formulation of the actuarial stochastic processes to implement precise and efficient computations of individual and group cash flows.

This handbook with exercises reveals the mathematical beauty of formalisms hitherto mostly used for software and hardware design and verification.

The epsilon calculus is a logical formalism developed by David Hilbert in the service of his program in the foundations of mathematics. The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. Specifically, in the calculus, a term εx A denotes some x satisfying A(x), if there is one. In Hilbert's Program, the epsilon terms play the role of ideal elements; the aim of Hilbert's finitistic consistency proofs is to give a procedure which removes (...) such terms from a formal proof. The procedures by which this is to be carried out are based on Hilbert's epsilon substitution method. The epsilon calculus, however, has applications in other contexts as well. The first general application of the epsilon calculus was in Hilbert's epsilon theorems, which in turn provide the basis for the first correct proof of Herbrand's theorem. More recently, variants of the epsilon operator have been applied in linguistics and linguistic philosophy to deal with anaphoric pronouns. (shrink)

In actuarial parlance, the price of an insurance policy is considered fair if customers bearing the same risk are charged the same price. The estimate of this fair amount hinges on the expected value obtained by weighting the different claims by their probability. We argue that, historically, this concept of actuarial fairness originates in an Aristotelian principle of justice in exchange (equality in risk). We will examine how this principle was formalized in the 16th century and shaped in (...) life insurance during the following two hundred years, in two different interpretations. The Domatian account of actuarial fairness relied on subjective uncertainty: An agreement on risk was fair if both parties were equally ignorant about the chances of an uncertain event. The objectivist version grounded any agreement on an objective risk estimate drawn from a mortality table. We will show how the objectivist approach collapsed in the market for life annuities during the 18th century, leaving open the question of why we still speak of actuarial fairness as if it were an objective expected value. (shrink)

Drawing on calls by researchers to examine corporate scandals involving potential conflicts of interest or compromise to professional independence involving the actuarial profession, this article outlines one such case. The consulting actuaries – to a large Australian listed company, James Hardie Industries Limited – found themselves advising two parties in a corporate restructuring where the interests of each were sometimes competing and the interests of the public appeared to be ignored. The James Hardie case is instructive in a number (...) of ways: first, it demonstrates the subtlety with which conflicts of interest or pressures on professional independence can arise; second, it demonstrates how important professional issues can be obfuscated by more obvious and pressing financial and strategic issues; and finally it demonstrates that adherence to professional codes of conduct and the ease with which professional ethics can be compromised when those codes are vague and transgressions are rarely actionable. The James Hardie case highlights structural issues in the employment of consulting actuaries which presents risks to the profession. It demonstrates that the combination of an aggressive corporate management with a strategic agenda reliant on consulting actuaries that have a vested interest in promoting and maintaining valuable relationships, both financially and professionally, results in ethical challenges. (shrink)

Drawing on calls by researchers to examine corporate scandals involving potential conflicts of interest or compromise to professional independence involving the actuarial profession, this article outlines one such case. The consulting actuaries – to a large Australian listed company, James Hardie Industries Limited – found themselves advising two parties in a corporate restructuring where the interests of each were sometimes competing and the interests of the public appeared to be ignored. The James Hardie case is instructive in a number (...) of ways: first, it demonstrates the subtlety with which conflicts of interest or pressures on professional independence can arise; second, it demonstrates how important professional issues can be obfuscated by more obvious and pressing financial and strategic issues; and finally it demonstrates that adherence to professional codes of conduct and the ease with which professional ethics can be compromised when those codes are vague and transgressions are rarely actionable. The James Hardie case highlights structural issues in the employment of consulting actuaries which presents risks to the profession. It demonstrates that the combination of an aggressive corporate management with a strategic agenda reliant on consulting actuaries that have a vested interest in promoting and maintaining valuable relationships, both financially and professionally, results in ethical challenges. (shrink)

Insurance is pervasive in many social settings. As a cooperative device based on risk pooling, it serves to attenuate the adverse consequences of various risks by offering policyholders coverage against the losses implied by adverse events in exchange for the payment of premiums. In the insurance industry, the concept of actuarial fairness serves to establish what could be adequate, fair premiums. Accordingly, premiums paid by policyholders should match as closely as possible their risk exposure. Such premiums are the product (...) of the probabilities of losses and the expected losses. This article presents a discussion of the fairness of actuarial fairness through three steps: defining the concept based on its formulation within the insurance industry; determining in which sense it may be about fairness; and raising some objections to the actual fairness of actuarial fairness. The necessity of a normative evaluation of actuarial fairness is justified by the influence of the concept on the current reforms of public insurance systems and the fact that it highlights the question of the repartition of the gains and burdens of social cooperation. (shrink)

We defend, from a contractarian perspective, that the fair price of an insurance policy is the amount that the contracting parties agree when they are both equally uncertain about the insured event. Drawing on the approach developed by R. Sugden in _The Community of Advantage_, we answer two standard objections raised against contractarianism in the actuarial sciences: (1) people are not wise enough to assess their actuarial risks; (2) they are not rational enough to decide which insurance policy (...) suits them better. We show under which circumstances people can make fair actuarial agreements, without presupposing any objective risk or rationality benchmarks. (shrink)

I argue that racism is a form of necro-being entrapped in necro-tragedy. Necro-being, as I present it, is a condition that kills and prevents persons from being born. I defend a conception of tragedy: absolute necrotragedy; absolute irredeemable suffering in a non-moral universe. Explanations of racism are commonly subject to anomalies, for example, volitional accounts offer special desiderata to account for institutional racism; conversely for institutional accounts. I offer a way to see racism, given the existence of a vast array (...) of kinds of racism: a descriptive actuarial approach. The account is intended to avoid the quandaries and anomalies of ‘explanation.’ I use rational-intentional explanations (including accounts by Jorge Garcia and Charles Mills) and social kind racial realism as inadequate explanations and examples of explanations with anomalies. These accounts also help demonstrate that logical systems of racism are inadequate. All such accounts explain racism as a coherent system and offer correlative reasons for its wrongness. I consider death, mortality, morbidity, and irredeemable misery as primary indicators of racism across an array of types of racism globally: racial necro-being. (shrink)

A mathematical model in science can be formulated as a counterfactual conditional, with the model’s assumptions in the antecedent and its predictions in the consequent. Interestingly, some of these models appear to have assumptions that are metaphysically impossible. Consider models in ecology that use differential equations to track the dynamics of some population of organisms. For the math to work, the model must assume that population size is a continuous quantity, despite that many organisms are necessarily discrete. This means our (...) counterfactual representation of the model can have an impossible antecedent, giving us a counterpossible. Analogous counterpossibles arise in other sciences, as we’ll see. According to a prominent view in counterfactual semantics, the vacuity thesis, all counterpossibles are vacuously true, that is, true merely because their antecedents are necessarily false. But some counterpossible formulations of differential equation models in science are not all vacuously true—some are non-vacuously true, and some are false. I go on to show how an alternative semantics, one that employs impossible worlds, can deliver this judgment. (shrink)

Because of the “all-or-none” character of nervous activity, neural events and the relations among them can be treated by means of propositional logic. It is found that the behavior of every net can be described in these terms, with the addition of more complicated logical means for nets containing circles; and that for any logical expression satisfying certain conditions, one can find a net behaving in the fashion it describes. It is shown that many particular choices among possible neurophysiological assumptions (...) are equivalent, in the sense that for every net behaving under one assumption, there exists another net which behaves under the other and gives the same results, although perhaps not in the same time. Various applications of the calculus are discussed. (shrink)

The ArgumentHistorians of science naturally tend to express interest in other forms of intellectual activity only when these intersect with science. This tendncy has produced a number of enlightening studies of what happens when science and (for instance) law or theology come into contact, but little by way of how science enters into the calculations and social status of such forms of knowledge after the conjuction has passed. Recent work in the sociology of professions, in contrast, has focused attention precisely (...) on those moments when the expert knowledge produced by different group doesnotoverlap. This has been the contribution of Andrew Abbortt's theory of “jurisdictional boundaries” between competing professions. The case of Victorian actuaries who worked hard to maintain unique intellectual claims in the competitive life insurance industry while maintainiing strong social connections and overlaps in knowledge with organized science, challenges the way both historians of science and sociologists of professions view contending knowledge claims. By observing what motivated actuaries to forge an alliance with men of science in the 1820s, then tracing their gradual recognition of a need to distance themselves from certain of the “scientific” values that had earlier informed their collective identity, it is possible to make sense of connections as well as disconnections between science and other forms of knowledge. A history of quantification from the actuaries' perspective, in other words, allows us to view science bothasandincontext. (shrink)

Although Aristotle (Metaphysics, Book IV, Chapter 2) was perhaps the first person to consider the part-whole relationship to be a proper subject matter for philosophic inquiry, the Polish logician Stanislow Lesniewski [15] is generally given credit for the first formal treatment of the subject matter in his Mereology.1 Woodger [30] and Tarski [24] made use of a specific adaptation of Lesniewski's work as a basis for a formal theory of physical things and their parts. The term 'calculus of individuals' (...) was introduced by Leonard and Goodman [14] in their presentation of a system very similar to Tarski's adaptation of Lesniewski's Mereology. Contemporaneously with Lesniewski's development of his Mereology, Whitehead [27] and [28] was developing a theory of extensive abstraction based on the two-place predicate, 'x extends over y\ which is the converse of 'x is a part of y\ This system, according to Russell [22], was to have been the fourth volume of their Pήncipia Mathematica, the never-published volume on geometry. Both Lesniewski [15] and Tarski [25] have recognized the similarities between Whitehead's early work and Lesniewski's Mereology. Between the publication of Whitehead's early work and the publication of Process and Reality [29], Theodore de Laguna [7] published a suggestive alternative basis for Whitehead's theory. This led Whitehead, in Process and Reality, to publish a revised form of his theory based on the two-place predicate, 'x is extensionally connected with y\ It is the purpose of this paper to present a calculus of individuals based on this new Whiteheadian primitive predicate. (shrink)

The calculus of Natural Calculation is introduced as an extension of Natural Deduction by proper term rules. Such term rules provide the capacity of dealing directly with terms in the calculus instead of the usual reasoning based on equations, and therefore the capacity of a natural representation of informal mathematical calculations. Basic proof theoretic results are communicated, in particular completeness and soundness of the calculus; normalisation is briefly investigated. The philosophical impact on a proof theoretic account of (...) the notion of meaning is considered. (shrink)

We present our calculus of higher-level rules, extended with propositional quantification within rules. This makes it possible to present general schemas for introduction and elimination rules for arbitrary propositional operators and to define what it means that introductions and eliminations are in harmony with each other. This definition does not presuppose any logical system, but is formulated in terms of rules themselves. We therefore speak of a foundational account of proof-theoretic harmony. With every set of introduction rules a canonical (...) elimination rule, and with every set of elimination rules a canonical introduction rule is associated in such a way that the canonical rule is in harmony with the set of rules it is associated with. An example given by Hazen and Pelletier is used to demonstrate that there are significant connectives, which are characterized by their elimination rules, and whose introduction rule is the canonical introduction rule associated with these elimination rules. Due to the availabiliy of higher-level rules and propositional quantification, the means of expression of the framework developed are sufficient to ensure that the construction of canonical elimination or introduction rules is always possible and does not lead out of this framework. (shrink)

Testimony about the future dangerousness of a person has become a central staple of many judicial processes. In settings such as bail, sentencing, and parole decisions, in rulings about the civil confinement of the mentally ill, and in custody decisions in a context of domestic violence, the assessment of a person’s propensity towards physical or sexual violence is regarded as a deciding factor. These assessments can be based on two forms of expert testimony: actuarial or clinical. The purpose of (...) this paper is to examine the scientific and epistemological basis of both methods of prediction or risk assessment. My analysis will reveal that this kind of expert testimony is scientifically baseless. The problems I will discuss will generate a dilemma for factfinders: on the one hand, given the weak predictive abilities of the branches of science involved, they should not admit expert clinical or actuarial testimony as evidence; on the other hand, there is a very strong tradition and a vast jurisprudence that supports the continued use of this kind of expert testimony. It is a clear case of the not so uncommon conflict between science and legal tradition. (shrink)

A sequent calculus is given in which the management of weakening and contraction is organized as in natural deduction. The latter has no explicit weakening or contraction, but vacuous and multiple discharges in rules that discharge assumptions. A comparison to natural deduction is given through translation of derivations between the two systems. It is proved that if a cut formula is never principal in a derivation leading to the right premiss of cut, it is a subformula of the conclusion. (...) Therefore it is sufficient to eliminate those cuts that correspond to detour and permutation conversions in natural deduction. (shrink)

Traces the development of the integral and the differential calculus and related theories since ancient times.

In this paper we introduce a Gentzen calculus for (a functionally complete variant of) Belnap's logic in which establishing the provability of a sequent in general requires \emph{two} proof trees, one establishing that whenever all premises are true some conclusion is true and one that guarantees the falsity of at least one premise if all conclusions are false. The calculus can also be put to use in proving that one statement \emph{necessarily approximates} another, where necessary approximation is a (...) natural dual of entailment. The calculus, and its tableau variant, not only capture the classical connectives, but also the `information' connectives of four-valued Belnap logics. This answers a question by Avron. (shrink)

We study an expansion of the Distributive Non-associative Lambek Calculus with conjugates of the Lambek product operator and residuals of those conjugates. The resulting logic is well-motivated, under-investigated and difficult to tackle. We prove completeness for some of its fragments and establish that it is decidable. Completeness of the logic is an open problem; some difficulties with applying the usual proof method are discussed.

The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. (...) Next, the geometrical meaning of the double contradiction is considered as follows. A tangent at a point on a convex curve is iterated. Then, the slope of the tangent at the point is sandwiched by two kinds of lines. The first kind of line crosses the curve at the original point and a point to the right of it. The second kind of line crosses the curve at the original point and a point to the left of it. Then, the double contradiction can be applied, and the slope of the tangent is determined as a single value. Finally, the meaning of this method for the foundation of mathematics is considered. We reflect on Dehaene’s notion that the foundation of mathematics is based on the intuitions, which evolve independently. Hence, there may be gaps between intuitions. In fact, the Ancient Greeks identified inconsistency between arithmetic and geometry. However, Eudoxus developed the theory of proportion, which is equivalent to the Dedekind Cut. This allows the iteration of an irrational number by rational numbers as precisely as desired. Simultaneously, we can define the irrational number by the double contradiction, although its existence is not guaranteed. Further, an area of a curved figure is iterated and defined by rectilinear figures using the double contradiction. (shrink)

The revised edition contains a new chapter which provides an elegant description of the semantics. The various classes of lambda calculus models are described in a uniform manner. Some didactical improvements have been made to this edition. An example of a simple model is given and then the general theory (of categorical models) is developed. Indications are given of those parts of the book which can be used to form a coherent course.

It is a commonplace that in the development of modern logic towards its actual shape at least two directions or traditions have to be distinguished. These traditions may be called, following the mo...

The aim of this study is to look at the the syntactic calculus of Bar-Hillel and Lambek, including semantic interpretation, from the point of view of constructive type theory. The syntactic calculus is given a formalization that makes it possible to implement it in a type-theoretical proof editor. Such an implementation combines formal syntax and formal semantics, and makes the type-theoretical tools of automatic and interactive reasoning available in grammar.In the formalization, the use of the dependent types of (...) constructive type theory is essential. Dependent types are already needed in the semantics of ordinary Lambek calculus. But they also suggest some natural extensions of the calculus, which are applied to the treatment of morphosyntactic dependencies and to an analysis of selectional restrictions. Finally, directed dependent function types are introduced, corresponding to the types of constructive type theory. (shrink)

By way of a close reading of Boole and Frege’s solutions to the same logical problem, we highlight an underappreciated aspect of Boole’s work—and of its difference with Frege’s better-known approach—which we believe sheds light on the concepts of ‘calculus’ and ‘mechanization’ and on their history. Boole has a clear notion of a logical problem; for him, the whole point of a logical calculus is to enable systematic and goal-directed solution methods for such problems. Frege’s Begriffsschrift, on the (...) other hand, is a visual tool to scrutinize concepts and inferences, and is a calculus only in the thin sense that every possible transition between sentences is fully and unambiguously specified in advance. While Frege’s outlook has dominated much of philosophical thinking about logical symbolism, we believe there is value—particularly in light of recent interest in the role of notations in mathematics and logic—in reviving Boole’s idea of an intrinsic link between, as he put it, a ‘calculus’ and a ‘directive method’ to solve problems. (shrink)

Traditionally, “the operator calculus of Born and Wiener” has been considered one of the four formulations of quantum mechanics that existed in 1926. The present paper reviews the operator calculus as applied by Max Born and Norbert Wiener during the last months of 1925 and the early months of 1926 and its connections with the rise of the new quantum theory. Despite the relevance of this operator calculus, Born–Wiener’s joint contribution to the topic is generally bypassed in (...) historical accounts of quantum mechanics. In this study, we analyse the paper that epitomises the contribution, and we explain the main reasons for the apparent lack of interest in Born and Wiener’s work. We argue that they did not solve the main problem for which the tool was intended, that of linear motion, because of their reluctance to use Dirac delta functions. (shrink)

In the year 1714, Johann Christian Lange published a baroque textbook about a logic machine, supposed to simulate human cognitive abilities such as perception, judgement, and reasoning. From today’s perspective, it can be argued that this blueprint is based on an inference engine applied to a strict ontology which serves as a knowledge base. In this paper, I will first introduce Lange’s approach in the period of baroque logic and then present a diagrammatic modernization of Lange’s principles, entitled Calculus (...) CL. Finally, I would like to discuss the possibilities of how to apply CL to modern cases of concern by using an example from epidemiology. (shrink)

The actuarial profession has a long history of providing critical expertise to society. The services delivered are some of the most complex and mysterious to outsiders of all professions but little has been written about the professional responsibilities of actuaries in the academic literature beyond that of the profession itself. This paper makes the case that the issues surrounding professional independence of actuaries are, in principle, similar to those that faced the audit profession before the scandals and resultant regulatory (...) changes early this century. It is argued that, despite the position taken by the actuarial profession and management, the status quo raises genuine concerns about conflicts of interest and independence and that the risks that arise are of sufficient magnitude that they should at least be the subject of a full debate. (shrink)

We attempt to extend the nominalistic project initiated in Hartry Field's Science Without Numbers to modern physical theories based in differential geometry.

Provides computer science students and researchers with a firm background in lambda-calculus and combinators.

Say that space is ‘gunky’ if every part of space has a proper part. Traditional theories of gunk, dating back to the work of Whitehead in the early part of last century, modeled space in the Boolean algebra of regular closed subsets of Euclidean space. More recently a complaint was brought against that tradition in Arntzenius and Russell : Lebesgue measure is not even finitely additive over the algebra, and there is no countably additive measure on the algebra. Arntzenius advocated (...) modeling gunk in measure algebras instead—in particular, in the algebra of Borel subsets of Euclidean space, modulo sets of Lebesgue measure zero. But while this algebra carries a natural, countably additive measure, it has some unattractive topological features. In this paper, we show how to construct a model of gunk that has both nice rudimentary measure-theoretic and topological properties. We then show that in modeling gunk in this way we can distinguish between finite dimensions, and that nothing in lost in terms of our ability to identify points as locations in space. (shrink)

This paper presents a sound and complete proof system for the first order fragment of Discourse Representation Theory. Since the inferences that human language users draw from the verbal input they receive for the most transcend the capacities of such a system, it can be no more than a basis on which more powerful systems, which are capable of producing those inferences, may then be built. Nevertheless, even within the general setting of first order logic the structure of the formulas (...) of DRS-languages, i.e. of the Discourse Representation Structures suggest for the components of such a system inference rules that differ somewhat from those usually found in proof systems for the first order predicate calculus and which are, we believe, more in keeping with inference patterns that are actually employed in common sense reasoning.This is why we have decided to publish the present exercise, in spite of the fact that it is not one for which a great deal of originality could be claimed. In fact, it could be argued that the problem addressed in this paper was solved when Gödel first established the completeness of the system of Principia Mathematica for first order logic. For the DRS-languages we consider here are straightforwardly intertranslatable with standard formulations of the predicate calculus; in fact the translations are so straightforward that any sound and complete proof system for first order logic can be used as a sound and complete proof system for DRSs: simply translate the DRSs into formulas of predicate logic and then proceed as usual. As a matter of fact, this is how one has chosen to proceed in some implementations of DRT, which involve inferencing as well as semantic representation; an example is the Lex system developed jointly by IBM and the University of Tübingen ). (shrink)

A sequent calculus is given in which the management of weakening and contraction is organized as in natural deduction. The latter has no explicit weakening or contraction, but vacuous and multiple discharges in rules that discharge assumptions. A comparison to natural deduction is given through translation of derivations between the two systems. It is proved that if a cut formula is never principal in a derivation leading to the right premiss of cut, it is a subformula of the conclusion. (...) Therefore it is sufficient to eliminate those cuts that correspond to detour and permutation conversions in natural deduction. (shrink)