Additive theories of rationality, as I use the term, are theories that hold that an account of our capacity to reflect on perceptually-given reasons for belief and desire-based reasons for action can begin with an account of what it is to perceive and desire, in terms that do not presuppose any connection to the capacity to reflect on reasons, and then can add an account of the capacity for rational reflection, conceived as an independent capacity to ‘monitor’ and ‘regulate’ our (...) believing-on-the-basis-of-perception and our acting-on-the-basis-of-desire. I show that a number of recent discussions of human rationality are committed to an additive approach, and I raise two difficulties for this approach, each analogous to a classic problem for Cartesian dualism. The interaction problem concerns how capacities conceived as intrinsically independent of the power of reason can interact with this power in what is intuitively the right way. The unity problem concerns how an additive theorist can explain a rational subject's entitlement to conceive of the animal whose perceptual and desiderative life he or she oversees as ‘I’ rather than ‘it’. I argue that these difficulties motivate a general skepticism about the additive approach, and I sketch an alternative, ‘transformative’ framework in which to think about the cognitive and practical capacities of a rational animal. (shrink)

In this paper I argue that de Finetti provided compelling reasons for rejecting countable additivity. It is ironical therefore that the main argument advanced by Bayesians against following his recommendation is based on the consistency criterion, coherence, he himself developed. I will show that this argument is mistaken. Nevertheless, there remain some counter-intuitive consequences of rejecting countable additivity, and one in particular has all the appearances of a full-blown paradox. I will end by arguing that in fact it is no (...) paradox, and that what it shows is that conditionalisation, often claimed to be integral to the Bayesian canon, has to be rejected as a general rule in a finitely additive environment. (shrink)

Let X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }$$\end{document} be a set of outcomes, and let I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{I }$$\end{document} be an infinite indexing set. This paper shows that any separable, permutation-invariant preference order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} on XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }^\mathcal{I }$$\end{document} admits an additive representation. That is: there exists a linearly ordered abelian group R\documentclass[12pt]{minimal} (...) \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R }$$\end{document} and a ‘utility function’ u:X⟶R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u:\mathcal{X }{{\longrightarrow }}\mathcal{R }$$\end{document} such that, for any x,y∈XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$\end{document} which differ in only finitely many coordinates, we have x≽y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x}\succcurlyeq \mathbf{y}$$\end{document} if and only if ∑i∈Iu-u≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i\in \mathcal{I }} \left[u-u\right]\ge 0$$\end{document}. Importantly, and unlike almost all previous work on additive representations, this result does not require any Archimedean or continuity condition. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} also satisfies a weak continuity condition, then the paper shows that, for anyx,y∈XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$\end{document}, we have x≽y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x}\succcurlyeq \mathbf{y}$$\end{document} if and only if ∗∑i∈Iu≥∗∑i∈Iu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\sum _{i\in \mathcal{I }} u\ge {}^*\!\sum _{i\in \mathcal{I }}u$$\end{document}. Here, ∗∑i∈Iu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\sum _{i\in \mathcal{I }} u$$\end{document} represents a nonstandard sum, taking values in a linearly ordered abelian group ∗R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\mathcal{R }$$\end{document}, which is an ultrapower extension of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R }$$\end{document}. The paper also discusses several applications of these results, including infinite-horizon intertemporal choice, choice under uncertainty, variable-population social choice and games with infinite strategy spaces. (shrink)

This paper addresses the issue of finite versus countable additivity in Bayesian probability and decision theory -- in particular, Savage's theory of subjective expected utility and personal probability. I show that Savage's reason for not requiring countable additivity in his theory is inconclusive. The assessment leads to an analysis of various highly idealised assumptions commonly adopted in Bayesian theory, where I argue that a healthy dose of, what I call, conceptual realism is often helpful in understanding the interpretational value of (...) sophisticated mathematical structures employed in applied sciences like decision theory. In the last part, I introduce countable additivity into Savage's theory and explore some technical properties in relation to other axioms of the system. (shrink)

De Finetti would claim that we can make sense of a draw in which each positive integer has equal probability of winning. This requires a uniform probability distribution over the natural numbers, violating countable additivity. Countable additivity thus appears not to be a fundamental constraint on subjective probability. It does, however, seem mandated by Dutch Book arguments similar to those that support the other axioms of the probability calculus as compulsory for subjective interpretations. These two lines of reasoning can be (...) reconciled through a slight generalization of the Dutch Book framework. Countable additivity may indeed be abandoned for de Finetti's lottery, but this poses no serious threat to its adoption in most applications of subjective probability. Introduction The de Finetti lottery Two objections to equiprobability 3.1 The ‘No random mechanism’ argument 3.2 The Dutch Book argument Equiprobability and relative betting quotients The re-labelling paradox 5.1 The paradox 5.2 Resolution: from symmetry to relative probability Beyond the de Finetti lottery. (shrink)

The principle of mass additivity states that the mass of a composite object is the sum of the masses of its elementary components. Mass additivity is true in Newtonian mechanics but false in special relativity. Physicists have explained why mass additivity is true in Newtonian mechanics by reducing it to Newton’s microphysical laws. This reductive explanation does not fit well with deducibility theories of reductive explanation such as the modern Nagelian theory of reduction, and the a priori entailment theory of (...) reduction that is prominent in the philosophy of mind. Nonetheless, I argue that a reconstruction of the explanation that incorporates distinctively philosophical concepts in fact fits both theories. I discuss the implications of this result for both theories and for the reductive explanation of consciousness. (shrink)

In this article, we tell a story about incompleteness in modal logic. The story weaves together an article of van Benthem, “Syntactic aspects of modal incompleteness theorems,” and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, ${\cal V}$-baos. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s article resolves the open question (...) in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to ${\cal V}$-baos, namely the provability logic $GLB$. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is ${\cal V}$-complete. After these results, we generalize the Blok Dichotomy to degrees of ${\cal V}$-incompleteness. In the end, we return to van Benthem’s theme of syntactic aspects of modal incompleteness. (shrink)

The term ‘mutual constitution’ appears with regularity in scholarship on intersectionality, but what does it mean? We could not easily answer this question in the usual way – by reading books and articles about it – because the term has not received direct, widespread or sustained engagement in feminist theory. This led us to analyse a wide range of feminist scholarship – the entire set of 379 articles in women’s studies journals that consider both intersectionality and mutual constitution – to (...) determine whether there are patterns and commonalities in the ways this important theoretical term is used. Our analysis reveals that while there is widespread agreement that mutual constitution does not allow for an additive or binary approach, this is the only major point of shared understanding of this term. Scholars disagree over whether mutual constitution is, in fact, the same thing as intersectionality, and in practice, clusters of disciplines use the term with different norms and levels of precision. Because of the explanatory potential of this term in intersectional theory, we recommend on the basis of our analysis that social scientists reconsider the convention of asserting that entities such as race, class and gender are mutually constituted and borrow the methodological tools from feminist historians, literary critics and other humanists that would allow for a genuine determination and demonstration of when entities are mutually constituted. (shrink)

How can we resist the repugnant conclusion? James Griffin has plausibly suggested that part way through the sequence we may reach a world—let us call it “J”—in which the lives are lexically superior to those that follow. If it would be preferable to live a single life in J than through any number of lives in the next one, then it would be strange to judge K the better world. Instead, we may reasonably “suspend addition” and judge J superior, (...) as if aggregating the lives in the larger world intrapersonally. I argue that the addition of new people with separate preferences renders this inference illicit when comparing J+ and K. When one pairwise comparison suspends addition and the other does not, the result is an intransitive value judgement: J < J+ < K < J, producing the mere addition paradox. (shrink)

This paper explores lexicographically additive representations of multi-attribute preferences on finite sets. Lexicographic additivity combines a lexicographic feature with local value tradeoffs. Tradeoff structures are governed by either transitive or nontransitive additive conjoint measurement. Alternatives are locally traded off when they are close enough within threshold associated with a dominant subset of attributes.

This paper studies how the introduction of non-additive probabilities (capacities) affects the solvability of strategic games.

Is the overall value of a world just the sum of values contributed by each value-bearing entity in that world? Additively separable axiologies (like total utilitarianism, prioritarianism, and critical level views) say 'yes', but non-additive axiologies (like average utilitarianism, rank-discounted utilitarianism, and variable value views) say 'no'. This distinction appears to be practically important: among other things, additive axiologies generally assign great importance to large changes in population size, and therefore tend to strongly prioritize the long-term survival of humanity over (...) the interests of the present generation. Non-additive axiologies, on the other hand, need not assign great importance to large changes in population size. We show, however, that when there is a large enough `background population' unaffected by our choices, a wide range of non-additive axiologies converge in their implications with additive axiologies—for instance, average utilitarianism converges with critical-level utilitarianism and various egalitarian theories converge with prioritarianism. We further argue that real-world background populations may be large enough to make these limit results practically significant. This means that arguments from the scale of potential future populations for the astronomical importance of avoiding existential catastrophe, and other arguments in practical ethics that seem to presuppose additive separability, may succeed in practice whether or not we accept additive separability as a basic axiological principle. (shrink)

By the Galvin–Mycielski–Solovay theorem, a subset X of the line has Borel’s strong measure zero if and only if $M+X\neq \mathbb {R}$ for each meager set M.A set $X\subseteq \mathbb {R}$ is meager-additive if $M+X$ is meager for each meager set M. Recently a theorem on meager-additive sets that perfectly parallels the Galvin–Mycielski–Solovay theorem was proven: A set $X\subseteq \mathbb {R}$ is meager-additive if and only if it has sharp measure zero, a notion akin to strong measure zero.We investigate the (...) validity of this result in Polish groups. We prove, e.g., that a set in a locally compact Polish group admitting an invariant metric is meager-additive if and only if it has sharp measure zero. We derive some consequences and calculate some cardinal invariants. (shrink)

This article develops an axiom system to justify an additive representation for a preference relation ${\succsim}$ on the product ${\prod_{i=1}^{n}A_{i}}$ of extensive structures. The axiom system is basically similar to the n-component (n ≥ 3) additive conjoint structure, but the independence axiom is weakened in the system. That is, the axiom exclusively requires the independence of the order for each of single factors from fixed levels of the other factors. The introduction of a concatenation operation on each factor A i (...) makes it possible to yield a special type of restricted solvability, i.e., additive solvability and the usual cancellation on ${\prod_{i=1}^{n}A_{i}}$ . In addition, the assumption of continuity and completeness for A i implies a stronger type of solvability on A i . The additive solvability, cancellation, and stronger solvability axioms allow the weakened independence to be effective enough in constructing the additive representation. (shrink)

We give a new approach to the failure of the Canonical Base Property (CBP) in the so far only known counterexample, produced by Hrushovski, Palacín and Pillay. For this purpose, we will give an alternative presentation of the counterexample as an additive cover of an algebraically closed field. We isolate two fundamental weakenings of the CBP, which already appeared in work of Chatzidakis and Moosa-Pillay and show that they do not hold in the counterexample. In order to do so, a (...) study of imaginaries in additive covers is developed. As a by-product of the presentation, we observe that a pure binding-group-theoretic account of the CBP is unlikely. (shrink)

We prove that the additive group of the rationals does not have an automatic presentation. The proof also applies to certain other abelian groups, for example, torsion-free groups that are p-divisible for infinitely many primes p, or groups of the form ⊕ p∈I Z(p ∞ ), where I is an infinite set of primes.

Provision of additional food supplements for the purpose of biological conservation has been widely researched both theoretically and experimentally. The study of these biosystems is usually done using predator–prey models. In this paper, we consider an additional food provided predator–prey system in the presence of the inhibitory effect of the prey. This model is analyzed in the control parameter space using the control parameters, quality and quantity of additional food. The findings suggest that with appropriate choice of additional food to (...) predators, the biosystem can be controlled and steered to a desirable state. It is also possible to eliminate either of the interacting species. The vital role of the quality and quantity of the additional food in the system dynamics cautions the eco manager on the choice of the additional food for realizing the goal in the biological conservation programme. (shrink)

This paper explores a scenario where a decision maker evaluates bundles by adding up the utility of the options that attract her attention. We introduce a novel attention rule called the “idempotent attention rule” and examine additive representations under this rule. With idempotent attention rules, we can narrow our focus to a subset of bundles to reveal attention rules and utility functions. As a generalization of attention filters, this rule sheds light on how alternatives interact in forming attention.

While there are several arguments on either side, it is far from clear as to whether or not countable additivity is an acceptable axiom of subjective probability. I focus here on de Finetti's central argument against countable additivity and provide a new Dutch book proof of the principle, To argue that if we accept the Dutch book foundations of subjective probability, countable additivity is an unavoidable constraint.

A principal aim of the branch of ethics called ‘population theory’ or ‘population ethics’ is to find a plausible welfarist axiology, capable of comparing total outcomes with respect to value. This has proved an exceedingly difficult task. In this paper I shall state and discuss two ‘trilemmas’, or choices between three unappealing alternatives, which the population ethicist must face. The first trilemma is not new. It originates with Derek Parfit's well-known ‘Mere Addition Paradox’, and was first explicitly stated by (...) Yew-Kwang Ng. I shall argue that one horn of this trilemma is less unattractive than Parfit and others have claimed. The second trilemma, which is a kind of mirror image of the first, appears hitherto to have gone unnoticed. Apart from attempting to resolve the two trilemmas, I shall suggest certain features which I believe a plausible welfarist axiology should possess. The details of this projected axiology will, however, be left open. (shrink)

Ordinal addition and multiplication can be extended in a natural way to all sets. I survey the structure of the sets under these operations. In particular, the natural partial ordering associated with addition of sets is shown to be a tree. This allows us to prove that any set has a unique representation as a sum of additively irreducible sets, and that the non-empty elements of any model of set theory can be partitioned into infinitely many submodels, each (...) isomorphic to the original model. Also any model of set theory has an isomorphic extension in which the empty set of the original model is non-empty. Among other results, the relations between the arithmetical operations and the transitive closure and the adductive hierarchy are elucidated. (shrink)

"Additive variance in fitness" is an important concept in the formal apparatus of population genetics. Wimsatt and Lloyd have argued that this concept can also be used to decide the "unit of selection" in an evolutionary process. The paper argues that the proposed criteria of Wimsatt and Lloyd are ambiguous, and several interpretations of their views are presented. It is argued that none of these interpretations provide acceptable criteria for deciding units of selection. The reason is that additive variance in (...) fitness can be both a cause of evolution, but also a byproduct of selection at another level. (shrink)

It is sometimes alleged that arguments that probability functions should be countably additive show too much, and that they motivate uncountable additivity as well. I show this is false by giving two naturally motivated arguments for countable additivity that do not motivate uncountable additivity.

It is customary to identify three broad classes of grading particles: additive particles like also, exclusive particles like only, and scalar particles like even (cf. König (1991); in the examples, grave accent stands for the main, falling accent).

This chapter will amount to a detailed exposition and exploration of one of the most prominent arguments against the existence of an unsurpassable world: the argument from addition. Endorsed by a variety of thinkers such as St. Thomas Aquinas, Alvin Plantinga, and William Rowe, the argument from addition uses the possibility of adding good things to a candidate unsurpassable world to argue that every world is surpassable. While widely endorsed, the argument has come under recent criticism. By carefully (...) working through a targeted version of the argument, I set out to establish the following: (i) that a world can always contain more good things; (ii) that a suitably restricted additive aggregation principle allows us to say that adding more good things in a certain way is an improvement, and (iii) that objections to the argument from widespread value incomparability fail. (shrink)

This article analyzes rent seeking with multiple additive efforts for each of two players. Impact on rent seeking occurs even when a player exerts only one effort. This contrasts with models of multiplicative efforts with impact on rent seeking only when a player exerts all its available efforts. An analytical solution is developed when the contest intensities are below one, and equal to one for one effort. Then, additional efforts causing interior solutions give players higher expected utilities and lower rent (...) dissipation, which contrasts with earlier findings for multiplicative efforts. Players cut back on the effort with contest intensity equal to one, and exert alternative efforts instead. Accounting for solutions which have to be determined numerically, a Nash equilibrium selection method is provided. For illustration, an example with maximum two efforts for each player is provided. Equilibria are shown where both players choose both efforts, or one player withdraws from its most costly effort. Both players may collectively prefer to exclude one of their efforts, though in equilibrium, they may prefer both efforts. When all contest intensities are equal to one or larger than one, only the one most cost-effective effort is exerted, due to the logic of linear or convex production. Rent dissipation increases in the contest intensity, and is maximum when the players are equally advantaged determined by unit effort cost divided by impact. (shrink)

I present a new argument for the repugnant conclusion. The core of the argument is a risky, intrapersonal analogue of the mere addition paradox. The argument is important for three reasons. First, some solutions to Parfit’s original puzzle do not obviously generalize to the intrapersonal puzzle in a plausible way. Second, it raises independently important questions about how to make decisions under uncertainty for the sake of people whose existence might depend on what we do. And, third, it suggests (...) various difficulties for leading views about the value of a person’s life compared to her nonexistence. (shrink)

This paper presents a dilemma for the additive model of reasons. Either the model accommodates disjunctive cases in which one ought to perform some act $$\phi $$ just in case at least one of two factors obtains, or it accommodates conjunctive cases in which one ought to $$\phi $$ just in case both of two factors obtains. The dilemma also arises in a revised additive model that accommodates imprecisely weighted reasons. There exist disjunctive and conjunctive cases. Hence the additive model (...) is extensionally inadequate. The upshot of the dilemma is that one of the most influential accounts of how reasons accrue to determine what we ought to do is flawed. (shrink)

ABSTRACT According to an influential line of argument, the past must be finite because no infinite series can be formed by successive addition. The present paper pinpoints the non sequitur at the heart of this argument, disentangles the ambiguities that disguise it, and dismantles the misleading picture of ‘traversing the infinite’ that gives the argument so much of its allure. Finally, the paper critically explores the related argument that a beginningless series of past events is impossible because there could (...) be no explanation of its having been ‘completed’ at one time rather than another. (shrink)

In the longstanding foundational debate whether to require that probability is countably additive, in addition to being finitely additive, those who resist the added condition raise two concerns that we take up in this paper. (1) Existence: Settings where no countably additive probability exists though finitely additive probabilities do. (2) Complete Additivity: Where reasons for countable additivity don’t stop there. Those reasons entail complete additivity—the (measurable) union of probability 0 sets has probability 0, regardless the cardinality of that union. (...) Then probability distributions are discrete, not continuous. We use Easwaran’s (Easwaran, Thought 2:53–61, 2013) advocacy of the Comparative principle to illustrate these two concerns. Easwaran supports countable additivity, both for numerical probabilities and for finer, qualitative probabilities, by defending a condition he calls the Comparative principle [C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document}]. For numerical probabilities, principle C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document} contrasts pairs, P1 and P2, defined over a common partition ∏\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\prod$$\end{document} = {ai: i ∈ I} of measurable events. C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document} requires that no P1 may be pointwise dominated, i.e., no (finitely additive) probability P2 exists such that for each i ∈ I, P2(ai) > P1(ai). By design, the cardinality of ∏\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\prod$$\end{document} is not limited in C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document}, which Easwaran asserts is important when arguing that the principle does not require more, or less, than that probability is countably additive. We agree that a numerical probability P satisfies principle C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document} in all partitions just in case P is countably additive. However, we show that for numerical probabilities, by considering the size of the algebra of events to which probability is applied, principle C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document} is subject to each of the above concerns, (1) and (2). Also, Easwaran considers principle C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document} with non-numerical, qualitative probabilities, where a qualitative probability may be finer than an almost agreeing numerical probability P. A qualitative probability is regular if possible events are strictly more likely than impossible events. Easwaran motivates and illustrates regular qualitative probabilities using a continuous, almost agreeing quantitative probability that is uniform on the unit interval. We make explicit the conditions for applying principle C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document} with qualitative probabilities and show that C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document} restricts regular qualitative probabilities to those whose almost agreeing quantitative probabilities are completely additive. For instance, Easwaran’s motivating example of a regular qualitative probability is precluded by principle C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document}. (shrink)

Additive particles are generally infelicitous in out-of-the-blue contexts. Kripke (in _Linguistic Inquiry_ 40(3):367–386, 2009 ) puts forward an influential idea that the additive presupposition they trigger has an anaphoric component that cannot be accommodated. Ruys (in _Linguistic Inquiry_ 46(2):343–361, 2015 ) proposes an alternative analysis that the observation can be explained in terms of conditions on deaccenting, claiming that in light of exceptional cases where additive particles can be used felicitously out of the blue, it is better to analyse the (...) additive presuppositions as existential presuppositions. However, such exceptional cases have not been given a systematic explanation so far. In this paper, we closely examine existing and novel examples where additive presuppositions are felicitously used out of the blue and argue that Ruys’s existential analysis as well as the Kripkean anaphoric approach face empirical issues. We propose a new analysis, according to which the _sui generis_ infelicity of additive particles in out-of-the-blue contexts is due to their focus sensitivity. Specifically, when an additive particle is used, two interpretive processes take place: (i) identification of a contextually relevant set of focus alternatives, and (ii) evaluation of the additive presupposition based on the focus alternatives so identified, which may result in accommodation. We claim that the felicity conditions on out-of-the-blue uses of additive particles emerge from the interplay between (i) and (ii). (shrink)

At a very fundamental level an individual (or a computer) can process only a finite amount of information in a finite time. We can therefore model the possibilities facing such an observer by a tree with only finitely many arcs leaving each node. There is a natural field of events associated with this tree, and we show that any finitely additive probability measure on this field will also be countably additive. Hence when considering the foundations of Bayesian statistics we may (...) as well assume countable additivity over a σ-field of events. (shrink)

The discussion of different principles of additivity for probability functions has been largely focused on the personalist interpretation of probability. Very little attention has been given to additivity principles for physical probabilities. The form of additivity for quantum probabilities is determined by the algebra of observables that characterize a physical system and the type of quantum state that is realizable and preparable for that system. We assess arguments designed to show that only normal quantum states are realizable and preparable and, (...) therefore, quantum probabilities satisfy the principle of complete additivity. We underscore the little remarked fact that unless the dimension of the Hilbert space is incredibly large, complete additivity in ordinary non-relativistic quantum mechanics reduces to countable additivity. We then turn to ways in which knowledge of quantum probabilities may constrain rational credence about quantum events and, thereby, constrain the additivity principle satisfied by rational credence functions. (shrink)

In this paper, we aim to account for the distribution and interpretation of a novel class of free choice items in Romanian, which we refer to as additive free choice items (ADD-FCIs). We show that the internal composition of ADD-FCIs, as well as their distribution, differs from that attested for other free choice paradigms discussed in the literature. Morphologically, ADD-FCIs are a more complex variant of regular universal FCIs, by virtue of an additional morpheme. This morpheme plays an additive role (...) when it functions as a stand-alone particle, and we propose that its role is similarly additive when it functions as an infix in ADD-FCIs. Couched in an exhaustification framework, we put forward a novel compositional account that can derive the interpretation of these ADD-FCIs. As for their distribution, these elements are only found in the presence of the conditional mood, as well as unconditional structures. We show how our analysis, coupled with the Viability Constraint used to explain the distribution of regular FCIs, can also account for their restricted distribution. In doing so, we further show how the pattern we investigate opens new perspectives regarding the licensing of free choice items in unconditionals. (shrink)

In addition to his large‐scale system of metaphysics, Schopenhauer produced many essays, and it was eventually these that made his name and drew attention to his philosophy. The biggest collection of them is called Parerga and Paralipomena. They are of help in understanding the philosophy, because they often contain bolder, more clear‐cut statements of the same points. They are written in an aphoristic style and are the source of many epigrams. For a long time they were more widely read (...) than the philosophy itself. (shrink)

"An Addition for the Catalogue des commentaires sur les « Parva Naturalia » (Bulletin 7, pp. 95-108)." Bulletin de Philosophie Médiévale, 8-9(), pp. 126–127.

We have provided a model-theoretic proof for the decidability of the additive structure of integers together with the function f mapping x to $$\lfloor \varphi x\rfloor $$ where $$\varphi $$ is the golden ratio.

We introduce a general approach to learn an additively decomposable multiattribute utility function from preference information provided by a Decision Maker. The decompositions under considerations here involve several non-necessarily disjoint factors grouping subsets of attributes in interaction. Such decompositions known as Generalized Additive Independence (GAI) utility functions allow interactions between attributes while preserving some additive decomposability of the evaluation model. More precisely, we focus in this paper on the determination of sparse ANOVA-like decompositions of the utility function that best fit (...) preference data and have good generalizing performance. Our learning approach aims to identify the factors of interacting attributes and the utility functions defined on these factors. The proposed approach applies both to continuous and discrete attributes. We present the computational models of this learning approach and then perform numerical tests on both synthetic and real data to demonstrate its practical effectiveness. (shrink)

We consider the structures $$, $$, $$, and $$ where $\mathbb {Z}$ is the additive group of integers, $\mathrm {SF}^{\mathbb {Z}}$ is the set of $a \in \mathbb {Z}$ such that $v_{p} < 2$ for every prime p and corresponding p-adic valuation $v_{p}$, $\mathbb {Q}$ and $\mathrm {SF}^{\mathbb {Q}}$ are defined likewise for rational numbers, and $<$ denotes the natural ordering on each of these domains. We prove that the second structure is model-theoretically wild while the other three structures are (...) model-theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences. (shrink)

The quantitative argument against the notion of a best possible world claims that, no matter how many worthwhile lives a world contains, another world contains more and is, other things being equal, better. Parfit’s ‘ Mere Addition Paradox ’ suggests that defenders of this argument must accept his ‘ Repugnant Conclusion ’ : that outcomes containing billions upon billions of lives barely worth living are better than outcomes containing fewer lives of higher quality. Several responses to the Paradox are (...) discussed and rejected as either inadequate or unavailable in a theistic context. The quantitative argument fails if some world is such that addition to it is not possible, i.e., if it is intensively infinite, as Liebniz claimed. If the notion of such a world is incoherent, then no world is quantitatively best and the quantitative argument succeeds, but only at the cost of embracing the Repugnant Conclusion. (shrink)

In this paper I argue that de Finetti provided compelling reasons for rejecting countable additivity. It is ironical therefore that the main argument advanced by Bayesians against following his recommendation is based on the consistency criterion, coherence, he himself developed. I will show that this argument is mistaken. Nevertheless, there remain some counter-intuitive consequences of rejecting countable additivity, and one in particular has all the appearances of a full-blown paradox. I will end by arguing that in fact it is no (...) paradox, and that what it shows is that conditionalisation, often claimed to be integral to the Bayesian canon, has to be rejected as a general rule in a finitely additive environment. (shrink)

The phenomenal overflow debate is a debate about the richness of phenomenal consciousness. There are two candidate views: the rich view and the sparse view. The rich view says phenomenal consciousness outstrips access consciousness and the contents of working memory. The sparse view denies this. Moreover, according to some conceptions of the sparse view, the subjective impression of richness depends on scene statistics and the refrigerator-light illusion. The purpose of this paper is to show there are additional resources available to (...) the sparse view in accommodating intuitions of richness that have yet to be fully appreciated by participants in the debate. To this end, research pertaining to feature binding and activity-silent working memory will be discussed. (shrink)

This study evaluated the effects of an exergame program combined with traditional tennis training on autonomic regulation, tennis technique, gross motor skills, clinical reaction time, and cognitive inhibitory control in children. Sixty-three children were randomized into four groups and compared at baseline, 6-month immediately post intervention and at 1-year follow-up post intervention. At 6-month post intervention the combined exergame and regular training sessions revealed: higher breathing frequency, heart rate and lower skin conductance levels during exergaming; additional benefits in the point (...) of contact and kinetic chain elements of the tennis forehand and backhand technique ; negative impact on the shot preparation and the follow-through elements ; higher ball skills ; higher percentages of clinical reaction time improvement and cognitive inhibitory control improvement in both congruent and incongruent trials. The 1-year follow-up test showed no differences in the tennis technique, clinical reaction time and cognitive inhibitory control improvement between groups with the same number of trainings per week. The findings support exergaming as an additional training tool, aimed to improve important cognitive-motor tennis skills by adding dynamics to the standardized training process. Caution should be placed to planning this training, e.g., in a mesocycle, since exergaming might decrease the improvement of specific tennis technique parts of the trainees.. (shrink)