There is widespread excitement in the literature about the method of arbitrary functions: many take it to show that it is from the dynamics of systems that the objectivity of probabilities emerge. In this paper, I differentiate three ways in which a probability function might be objective, and I argue that the method of arbitrary functions cannot help us show that dynamics objectivise probabilities in any of these senses.

A conception of probability that can be traced back to Johannes von Kries is introduced: the “Spielraum” or range conception. Its close connection to the so-called method of arbitrary functions is highlighted. Possible interpretations of it are discussed, and likewise its scope and its relation to certain current interpretations of probability. Taken together, these approaches form a class of interpretations of probability in its own right, but also with its own problems. These, too, are introduced, discussed, and proposals (...) in response to them are surveyed, some of which also go back to von Kries. (shrink)

In contemporary mathematical logic, the debate over the notion of arbitrary function is reflected in the problem of the interpretation of second-order quantifiers. The distinction between first- and second-order logic hangs on the range of the quantifiers: in first-order logic, quantifiers range uniquely on elements of the structure under study, while in second-order logic, quantifiers can range over the latter's subsets, sets of subsets, and so forth. The range of second-order quantifiers admits of various interpretations from more encompassing or (...) standard ones to more restrictive or non-standard ones. When Leon Henkin made this distinction explicitly in his paper on the ‘Completeness in the Theory of Types’, he considered only one such non-standard interpretation, with his general models, where the higher-order variables are subjected to closure conditions with respect to Boolean and projective operations. (shrink)

The paper argues that unlike Ramsey, Frege and Russell lacked the idea of an arbitrary function and this had important consequences for their foundational programs.

Hintikka and Sandu have recently claimed that Frege's notion of function was substantially narrower than that prevailing in real analysis today. In the present note, their textual evidence for this claim is examined in the light of relevant historical and biographical background and judged insufficient.

In his article The Foundations of Mathematics (1925) Ramsey was concerned with the nature of the statements of 'pure mathematics' and the way these statements differ from those in empirical sciences. He thought that the answer given to these questions by Hilbert and the formalist school according to which mathematical statements are meaningless formulas, is unsatisfactory for several reasons, which will not be discussed here. He also expressed serious doubts about the intuitionist program developed by Brouwer and Weyl. It is (...) the logicist school of Frege, Russell and Whitehead, which attempted to reduce mathematics to few logical concepts, that came closest to Ramsey’s views, although he was not completely satisfied with it, either. (shrink)

In this chapter I will look at the semantic analysis of indefinites in English, and their treatment in the framework of Dynamic Logic, choice functions (epsilon terms), and Kit Fine’s arbitrary objects. In the end I will make some comparative remarks about the latter and the account in terms of Skolem functions that Fine criticized, and propose an alternative framework (team semantics).

This paper explores the concept of a social evaluation functional in the case of an arbitrary set of alternatives. In the first part, a characterization of projective social evaluations functionals is shown whenever the common restricted domain is the set of all bounded utility functions equipped with the supremum norm topology. The result makes a crucial use, among others, of a continuity axiom. In the second part, a comparison meaningful property is introduced for a social evaluation functional which (...) allows us for obtaining a more general result with no continuity requirements. Finally, an impossibility theorem, which is reminiscent of that is obtained by Chichilnisky in (Q J Econ 97:337–352, 1982) but without using topological conditions, is offered. (shrink)

It is widely held that entities have moral status if they possess a status-conferring property to a sufficient degree. However, this means that for at least one degree to which an entity can possess the status-conferring property and that grounds moral status, there is some incrementally lower degree of possessing the property that does not ground moral status. Critics maintain that this renders any threshold for moral status arbitrary. In this paper, we reject common responses to this arbitrariness objection, (...) such as that moral status thresholds are not arbitrary but merely vague. Instead, we defend the moderate discontinuity view. This view holds that thresholds denote moderate rather than radical shifts in moral status and that significant shifts in the moral status of entities on opposite sides of the threshold are a function of their distance from the threshold rather than of the threshold itself. Crucially, it follows from this that there is no principled way to reconcile the commitment to the moral equality of persons with the commitment to the moral superiority of persons over nonhuman animals. (shrink)

Sum rules are derived for the quantum wave functions of the Hadamard billiard in arbitrary dimensions. This billiard is a strongly chaotic (Anosov) system which consists of a point particle moving freely on a D-dimensional compact manifold (orbifold) of constant negative curvature. The sum rules express a general (two-point)correlation function of the quantum mechanical wave functions in terms of a sum over the orbits of the corresponding classical system. By taking the trace of the orbit sum rule (...) or pre-trace formula, one obtains the Selberg trace formula. The sum rules are applied in two dimensions to a compact Riemann surface of genus two, and in three dimensions to the only non-arithmetic tetrahedron existing in hyperbolic 3-space. It is shown that the quantum wave functions can be computed from classical orbits. Conversely, we demonstrate that the structure of classical orbits can be extracted from the quantum mechanical energy levels and wave functions (inverse quantum chaology). (shrink)

Arbitrariness is a distinctive feature of human language, and a growing body of comparative work is investigating its presence in animal communication. But what is arbitrariness, exactly? We propose to distinguish four notions of semiotic arbitrariness: a notion of opaque association between sign forms and semiotic functions, one of sign‐function mapping optionality, one of acquisition‐dependent sign‐function coupling, and one of lack of motivatedness. We characterize these notions, illustrate the benefits of keeping them apart, and describe two reactions to our (...) proposal: abandoning arbitrariness‐talk in favor of the newly introduced conceptual vocabulary, or feeding the distinctions back into the parent concept. (shrink)

This chapter presents the main existing ways of spelling out the function of social institutions: (a) historical social functionalism, (b) structural functionalism, (c) Pettit’s recent virtual social functionalism, and (d) intentionalist social functionalism. In each case, the chapter then presents, updates, and analyzes the most compelling objections to these traditional forms of social functionalism. The upshot of the chapter is the conclusion that a new approach to the topic of the function of social institutions is needed.

Arbitrariness is a distinctive feature of human language, and a growing body of comparative work is investigating its presence in animal communication. But what is arbitrariness, exactly? We propose to distinguish four notions of semiotic arbitrariness: a notion of opaque association between sign forms and semiotic functions, one of sign-function mapping optionality, one of acquisition-dependent sign-function coupling, and one of lack of motivatedness. We characterize these notions, illustrate the benefits of keeping them apart, and describe two reactions to our (...) proposal: abandoning arbitrariness-talk in favor of the newly introduced conceptual vocabulary, or feeding the distinctions back into the parent concept. (shrink)

In [1] the class of safe recursive functions over an arbitrary structure is defined. We prove that in this class, one cannot define a total pairing function independently of the structure.

Part I of Frege’s Grundgesetze is devoted to the “exposition [Darlegung]” of his formal system.

Jacques Monod (1971) argued that certain molecular processes rely critically on the property of chemical arbitrariness, which he claimed allows those processes to “transcend the laws of chemistry”. It seems natural, as some philosophers have done, to interpret this in modal terms: a biological relationship is chemically arbitrary if it is possible, within the constraints of chemical “law”, for that relationship to have been otherwise than it is. But while modality is certainly important for understanding chemical arbitrariness, understanding its (...) biological role also requires an account of the concrete causal-functional features that distinguish arbitrary from non-arbitrary phenomena. In this paper I elaborate on this under-emphasised aspect by offering a general account of these features: arbitrary relations are instantiated by mechanisms that involve molecular adapters, which causally couple two properties or processes which would otherwise be uncorrelated. Additionally, adapters work by acting as intermediate rather than cooperating causes. (shrink)

Using coding devices based on a theorem due to Zsigmondy, Birkhoff and Vandiver, we first define in terms of successor S and coprimeness predicate $\perp$ a full arithmetic over the set of powers of some fixed prime, then we define in the same terms a restriction of the exponentiation. Hence we prove the main result insuring that all arithmetical relations and functions over prime powers and their opposite are $\{S, \perp\}$ -definable over Z. Applications to definability over Z and (...) N are stated as corollaries of the main theorem. (shrink)

Arbitrariness is a distinctive feature of human language, and a growing body of comparative work is investigating its presence in animal communication. But what is arbitrariness, exactly? We propose to distinguish four notions of semiotic arbitrariness: a notion of opaque association between sign forms and semiotic functions, one of sign-function mapping optionality, one of acquisition-dependent sign-function coupling, and one of lack of motivatedness. We characterize these notions, illustrate the benefits of keeping them apart, and describe two reactions to our (...) proposal: abandoning arbitrariness-talk in favor of the newly introduced conceptual vocabulary, or feeding the distinctions back into the parent concept. (shrink)

“Due process of law” is arguably the most controversial and frequently-litigated phrase in the American Constitution. Although the dominant originalist view has long been that Fifth and Fourteenth Amendment’s Due Process of Law Clauses are solely “process” guarantees and don’t constrain the “substance” of legislation at all, originalist scholars have in recent years made fresh inquiries into the historical evidence and concluded that there’s a weighty case for some form of substantive due process. In this Article, we review and critique (...) these findings employing our theory of good-faith originalist interpretation and construction. We begin by investigating the “letter” of the Due Process of Law Clauses — that is, the original meaning of their texts. Next, to develop doctrine by which this meaning can be implemented, we identify the clauses’ original function — their “spirit” — of barring arbitrary exercises of power over individuals that rest upon mere will rather than constitutionally proper reasons. We contend that the original letter and spirit of the “due process of law” in both clauses requires federal and state legislators to exercise their discretionary powers in good faith by enacting legislation that is actually calculated to achieve constitutionally proper ends and imposes a duty upon both state and federal judges to make a good-faith determination of whether legislation is calculated to achieve constitutionally proper ends. Finally, we confront hard questions concerning the scope of the states’ reserved powers, acknowledging the flaws in the “police-power” jurisprudence associated with the so-called “Lochner era” and we delineate an approach that will better safeguard all “person(s)” against arbitrary power. By so doing, we assist state and federal legislators by providing clarity concerning the constitutionally proper ends that federal and state legislators can pursue; aid state and federal judges by equipping them to review legislators’ pursuit of those ends; and help members of the public by enabling them to monitor the performance of their legislative and judicial agents. (shrink)

A very weak omniscience principle is formulated, related omniscience principlesare considered, and the theorem that a function of bounded variation is the difference of two increasing functions is shown to be equivalent to the omniscience principle WLPO. It is a so shown that an arbitrary function with located variation on an interval is the difference of two increasing functions.

Both Frege's Grundgesetze, and Lagrange's treatises on analytical functions pursue a foundational purpose. Still, the former's program is not only crucially different from the latter's. It also depends on a different idea of what foundation of mathematics should be like . Despite this contrast, the notion of function plays similar roles in their respective programs. The purpose of my paper is emphasising this similarity. In doing it, I hope to contribute to a better understanding of Frege's logicism, especially in (...) relation to its crucial differences with a set-theoretic foundational perspective. This should also spread some light on a question arisen by J. Hintikka and G. Sandu in a widely discussed paper, namely whether Frege should or should not be credited with the notion of arbitrary function underlying our standard interpretation of second-order logic. In section 1, I account for Lagrange's notion of function. In section 2, I advance some remarks on connected historical matters. This will provide an appropriate framework for discussing the role played by the notion of function in Frege's Grundgesetze. Section 3 is devoted to this. Some concluding remarks will close the paper. (shrink)

Phenomenal Conservatism is an internalist theory of justification that upholds the idea that justification arises from a distinct picture of epistemic warrant based on seemings. Recently, it has enjoyed an ever-growing number of defenders due to its advantages over traditional internalist perspectives. One of its main merits resides in its promise to avoid long-standing challenges to foundationalism. Phenomenal Conservatism supposedly holds the resources to reject an infinite regress of reasons while simultaneously avoiding making the subject’s beliefs arbitrary. This paper (...) aims to show that Phenomenal Conservatism fails to dispel the accusation of arbitrariness and that this failure ultimately undoes its viability as an internalist theory of justification. The arbitrariness objection here developed is lifted from Stewart Cohen’s bootstrapping problem. It argues that by being unable to handle bootstrapping inferences, seemings fail to function as non-arbitrary reasons to believe. Additionally, it is suggested that bootstrapping-related considerations provide better support to one of the main internalist alternatives to Phenomenal Conservatism. (shrink)

Functionalism is the doctrine that mental states, by their nature, just are certain causal-dispositional states: all that it is to be some arbitrary mental state is to be a state that is disposed to have certain causes and effects. In a series of papers, Nick Zangwill has argued that functionalism faces a problem accommodating the possibility of systematic irrationality – that is, the fact that thinkers can be disposed to be irrational. In brief: according to Zangwill, the functionalist theory (...) of the mind implies that thinkers, by their nature, are necessarily disposed to be rational. But this result is flatly inconsistent with the possibility of systematic irrationality. Here I show how the functionalist can reconcile the possibility of systematic irrationality with the ultimately causal-dispositional nature of mental states by endorsing the doctrine that the mind admits of divisions within itself. If I am correct, the functionalist can accommodate the possibility of systematic irrationality by holding that mental states are defined by the causal relations that they would be disposed to have if they ever ‘came into direct causal contact’ with one another. (shrink)

Quasi-probability distribution functions fj WW, fj MM for quantum spin-j systems are derived based on the Wigner-Weyl, Margenau-Hill approaches. A probability distribution fj sph which is nonzero only on the surface of the sphere of radius √j(j+1) is obtained by expressing the characteristic function in terms of the spherical moments. It is shown that the Wigner-Weyl distribution function turns out to be a distribution over the sphere in the classical limit.

My focus in this essay is on 'institutional' or 'procedural' criticisms of the death penalty. These criticisms take aim at the death penalty as it is carried out in practice. They begin with empirical observations about the imperfect functioning of the various institutions involved in death penalty administration, such as courts and the police. These institutional imperfections, it is claimed, result in the death penalty being imposed arbitrarily or capriciously; skews death penalty verdicts by various forms of deprivation and discrimination (...) (direct and indirect); and deliver an unacceptable rate of wrongful convictions. These facts are then taken as reasons to abolish or at least establish a moratorium on the death penalty until the imperfections are ironed out to an acceptable level. My aims in this essay are three-fold. First, I aim to unpack the term “arbitrary” as a term of criticism against the criminal justice system. I argue that claims about arbitrariness refer to a range of more specific forms of injustice: one should speak not of arbitrariness as such but rather of an arbitrariness basket which is home to many kinds of injustice. Second, I wish to defend what I call "modest abolitionism," which relies upon a general duty to mitigate severe injustices when their rectification is particularly difficult or resource-intensive. Finally, I respond to some important criticisms of abolitionist arguments which take the arbitrariness of death penalty administration as their core. (shrink)

The separation-of-powers literature has entertained the possibility of differentiating governmental functions at a conceptual, pre-institutional level, as a way of defining the separation of powers. However, it can be objected that attempts at differentiating functions at this level cannot escape a problem of arbitrariness. In this article, I develop an account of the separation of powers which addresses this problem. On my account, the legislative function is defined by the creation of validity claims, understood as claims making it (...) a matter of normative common-knowledge that such and such normative attitudes hold for the interaction between agents. The executive-judicial function is defined by the specification of rights and other jural positions out of the validity claims. This account of the separation of powers has the double advantage that it works at a conceptual, pre-institutional level, and that it is able to distinguish between the functions in a non-arbitrary manner. In developing this account, I also consider its practical implications. I argue that the conceptual distinction between the legislative and the executive-judicial functions is expressive of rule-of-law values, and that this might render it a sufficient but not necessary condition for the institutional separation of powers. (shrink)

The existence of dysfunctions precludes the possibility of identifying the function to do F with the capacity to do F. Nevertheless, we continuously infer capacities from functions. For this and other reasons stated in the first part of this article, I propose a new theory of functions (of the etiological sort), applying to organisms as well as to artefacts, in which to have some determinate probability P to do F (i.e. a probabilistic capacity to do F) is a (...) necessary condition for having the function to do F. The main objective of this paper is to justify the legitimacy of this condition when considering artefacts. I begin by distinguishing “perspectival probabilities”, which reflect a pragmatic interest or an arbitrary state of knowledge, from “objective probabilities”, which depend on some objective feature of the envisageditems. I show that objective probabilities are not necessarily based on physical constitution. I then explain why we should distinguish between considering an object as a physical body and considering it as an artefact, and why the probability of dysfunction to be taken into account is one relative to the object as member of an artefact category. After clarifying how an artefact category can be defined if it is not defined in physical terms, I establish the objectivity of the probability of dysfunction under consideration by showing how it is causally determined by objective factors regulating the production of items of a definite artefact type. Ifocus on the case of industrially produced artefacts where the objective factors determining the probability of dysfunction can be best seen. (shrink)

Alethic functionalism is a sophisticated version of alethic pluralism according to which truth per se is a functional property supervening on lower level truth properties. After presenting alethic functionalism, I discuss four objections to it. I raise a new objection to alethic functionalism that if Objectivity, Norm of Belief, and End of Inquiry are the three truisms, correspondence is necessary and sufficient to satisfy the truisms, so that alethic functionalism capitulates to a correspondence theory of truth. Second, I present a (...) new argument that on alethic functionalism, truth per se is explanatorily epiphenomenal. Third, examining the case of mixed compound propositions shows that the principle which alethic functionalism adopts for determining which lower level truth property is required for a mixed compound proposition to be true has counterintuitive results and is arbitrary. Finally, I argue that alethic functionalism is unable to account for the intuition that a true truth attribution is true because of the proposition(s) to which truth is attributed. (shrink)

The significance of the concept of a mathematical transformation is explained. In particular, it is explained how to construct true statements concerning n-dimensional spaces, for arbitrary n, on the basis of true statements concerning two-dimensional spaces.

We extend the notion of limitwise monotonic functions to include arbitrary computable domains. We then study which sets and degrees are support increasing limitwise monotonic on various computable domains. As applications, we provide a characterization of the sets S with computable increasing η-representations using support increasing limitwise monotonic sets on ℚ and note relationships between the class of order-computable sets and the class of support increasing limitwise monotonic sets on certain domains.

This paper, a contribution to “micro set theory”, is the study promised by the first author in [M4], as improved and extended by work of the second. We use the rudimentarily recursive functions and the slightly larger collection of gentle functions to initiate the study of provident sets, which are transitive models of $\mathsf{PROVI}$, a subsystem of $\mathsf{KP}$ whose minimal model is Jensen’s $J_{\omega}$. $\mathsf{PROVI}$ supports familiar definitions, such as rank, transitive closure and ordinal addition—though not ordinal multiplication—and (...) Shoenfield’s unramified forcing. Providence is preserved under directed unions. An arbitrary set has a provident closure, and the extension of a provident $M$ by a set-generic $\mathcal{G}$ is the provident closure of $M\cup\{\mathcal{G}\}$. The improvidence of many models of $\mathsf{Z}$ is shown. The final section uses similar but simpler recursions to show, in the weak system $\mathsf{MW}$, that the truth predicate for $\dot{\varDelta}_{0}$ formulæ is $\Delta_{1}$. (shrink)

Duparc introduced a two-player game for a function f between zero-dimensional Polish spaces in which Player II has a winning strategy iff f is of Baire class 1. We generalize this result by defining a game for an arbitrary function f : X → Y between arbitrary Polish spaces such that Player II has a winning strategy in this game iff f is of Baire class 1. Using the strategy of Player II, we reprove a result concerning first (...) return recoverable functions. (shrink)

One of Clayton’s major contributions to our understanding of animal minds has been her work on episodic-like memory. A central reason for the success of this work was its focus on ecological validity: rather than looking for episodic memory for arbitrary stimuli in artificial contexts, focussing on contexts in which episodic memory would serve a biological function such as food caching. This review aims to deepen this insight by surveying the numerous functions that have been proposed for episodic (...) memory, articulating a philosophically grounded framework for understanding what exactly functions are, and drawing on these to make suggestions for future directions in the comparative cognitive psychology of episodic memory. Our review suggests four key insights. First, episodic memory may have more than one function and may have different functions in different species. Second, cross-disciplinary work is key to developing a functional account of episodic memory. Third, there is scope for further theoretical elaboration of proposals relating episodic memory to food caching and, in particular, future-oriented cognition. Finally, learning-related functions suggested by AI (artificial intelligence)-based models are a fruitful avenue for future behavioural research. (shrink)

LetB be the set of bounded observables on a quantum logic. A mapJ: B →R is called an expectation functional ifJ is normalized, positive, continuous, and compatibly linear. Two questions are considered. IsJ linear, and isJ an expectation relative to some state? It is shown that the answers are affirmative for hidden variable logics and most Hilbert space logics. An example is given which shows thatJ can be nonlinear on an arbitrary quantum logic.

In this paper, I discuss the aetiological account of biological interests, developed by Varner, in the context of artefactual organisms envisioned by current research in synthetic biology. In “Sections 2–5”, I present Varner's theory and criticise it for being incapable of ascribing non-derivative interests to artefactual organisms due to their lack of a history of natural selection. In “Sections 6–7”, I develop a new alternative to Varner's account, building on the organisational theory of biological teleology and function. I argue that (...) the organisational account of biological interest is superior to Varner's aetiological account because it can accommodate both artefactual and naturally evolved organisms, provides a non-arbitrary and practical way of determining biological interests, supports the claim that organisms have interests in a sense in which artefacts do not, and avoids the possibility of there being a conflict between what an organismic part is supposed to do and what is in the interest of the organism. (shrink)

I present an account of deterministic chance which builds upon the physico-mathematical approach to theorizing about deterministic chance known as 'the method of arbitrary functions'. This approach promisingly yields deterministic probabilities which align with what we take the chances to be---it tells us that there is approximately a 1/2 probability of a spun roulette wheel stopping on black, and approximately a 1/2 probability of a flipped coin landing heads up---but it requires some probabilistic materials to work with. I (...) contend that the right probabilistic materials are found in reasonable initial credence distributions. I note that, with some normative assumptions, the resulting account entails that deterministic chances obey a variant of Lewis's 'principal principle'. I additionally argue that deterministic chances, so understood, are capable of explaining long-run frequencies. (shrink)

This paper introduces and describes new protocols for proving knowledge of secrets without giving them away: if the verifier does not know the secret, he does not learn it. This can all be done while only using one-way hash functions. If also the use of encryption is allowed, these goals can be reached in a more efficient way. We extend and use the GNY authentication logic to prove correctness of these protocols.

We critique the organizational account of biological functions by showing how its basis in the closure of constraints fails to be objective. While the account treats constraints as objective features of physical systems, the number and relationship of potential constraints are subject to potentially arbitrary redescription by investigators. For example, we show that self-maintaining systems such as candle flames can realize closure on a more thorough analysis of the case, contradicting the claim that these “simple” systems lack functional (...) organization. This also raises problems for Moreno and Mossio’s associated theory of biological autonomy, which asserts that living beings are distinguished by their possession of a closed system of constraints that channel and regulate their metabolic processes. (shrink)

We present a model of computation for string functions over single-sorted, total algebraic structures and study some basic features of a general theory of computability within this framework. Our concept generalizes the Blum-Shub-Smale setting of computability over the reals and other rings. By dealing with strings of arbitrary length instead of tuples of fixed length, some suppositions of deeper results within former approaches to generalized recursion theory become superfluous. Moreover, this gives the basis for introducing computational complexity in (...) a BSS-like manner. Relationships both to classical computability and to Friedman's concept of eds computability are established. Two kinds of nondeterminism as well as several variants of recognizability are investigated with respect to interdependencies on each other and on properties of the underlying structures. For structures of finite signatures, there are universal programs with the usual characteristics. For the general case of not necessarily finite signature, this subject will be studied in a separate, forthcoming paper. (shrink)

Two-classes of phase-space spin quasi-distribution functions are introduced and discussed. The first class of these distributions is based on the delta function construction. It is shown that such a construction can be carried out for an arbitrary spin s and an arbitrary ordering of the spin operators. The second class of the spin distributions is constructed with the help of the spin coherent states. The connection of the spin coherent states to the Stratonovich formalism is established and (...) discussed. It is shown that the c-number phase-space description of quantum fluctuations provides a simple statistical picture of quantum fluctuations of spinoperators in terms of random directions on a unit sphere. For quantum states of the spin system the statistics of these random orientations is given by non-positive spin quasi-distribution functions. It is shown that the application of these spin quasi-distribution functions to the Einstein-Podolsky-Rosen correlations provide an insight into the quantum theory of measurement. (shrink)