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)

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)

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)

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.

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)

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)

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)

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)

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

“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)

The lambda calculus is a universal programming language. It can represent the computable functions, and such offers a formal counterpart to the point of view of functions as rules. Terms represent functions and this allows for the application of a term/function to any other term/function, including itself. The calculus can be seen as a formal theory with certain pre-established axioms and inference rules, which can be interpreted by models. Dana Scott proposed the first non-trivial model of the (...) extensional lambda calculus, known as $ D_{\infty }$, to represent the $\lambda $-terms as the typical functions of set theory, where it is not allowed to apply a function to itself. Here we propose a construction of an $\infty $-groupoid from any lambda model endowed with a topology. We apply this construction for the particular case $D_{\infty }$, and we see that the Scott topology does not provide enough information about the relationship between higher homotopies. This motivates a new line of research focused on the exploration of $\lambda $-models with the structure of a non-trivial $\infty $-groupoid to generalize the proofs of term conversion to higher-proofs in $\lambda $-calculus. (shrink)

We work with weakly precomplete equivalence relations introduced by Badaev. The weak precompleteness is a natural notion inspired by various fixed point theorems in computability theory. Let E be an equivalence relation on the set of natural numbers $$\omega $$, having at least two classes. A total function f is a diagonal function for E if for every x, the numbers x and f(x) are not E-equivalent. It is known that in the case of c.e. relations E, the weak precompleteness (...) of E is equivalent to the lack of computable diagonal functions for E. Here we prove that this result fails already for $$\Delta ^0_2$$ equivalence relations, starting with the $$\Pi ^{-1}_2$$ level. We focus on the Turing degrees of possible diagonal functions. We prove that for any noncomputable c.e. degree $${\textbf{d}}$$, there exists a weakly precomplete c.e. equivalence E admitting a $${\textbf{d}}$$ -computable diagonal function. We observe that a Turing degree $${\textbf{d}}$$ can compute a diagonal function for every $$\Delta ^0_2$$ equivalence relation E if and only if $${\textbf{d}}$$ computes $${\textbf{0}}'$$. On the other hand, every PA degree can compute a diagonal function for an arbitrary c.e. equivalence E. In addition, if $${\textbf{d}}$$ computes diagonal functions for all c.e. E, then $${\textbf{d}}$$ must be a DNC degree. (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.

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.

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)

The monadic second-order theory of trees allows quantification over elements and over arbitrary subsets. We classify the class of trees with respect to the question: does a tree T have definable Skolem functions? This continues [6] where the question was asked only with respect to choice functions. A natural subclass is defined and proved to be the class of trees with definable Skolem functions. Along the way we investigate the spectrum of definable well orderings of well (...) ordered chains. (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)

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.

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 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.

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)

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.

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)

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)

Drawing on Christopher Boorse's Biostatistical Theory (BST), Norman Daniels contends that a genuine health need is one which is necessary to restore normal functioning – a supposedly objective notion which he believes can be read from the natural world without reference to potentially controversial normative categories. But despite his claims to the contrary, this conception of health harbors arbitrary evaluative judgments which make room for intractable disagreement as to which conditions should count as genuine health needs and therefore which (...) needs should be met. I begin by offering a brief summary of Boorse's BST, the theory to which Daniels appeals for providing the conception of health as normal functioning upon which his overall distributive scheme rests. Next, I consider what I call practical objections to Daniels's use of Boorse's theory. Finally I recount Elseljin Kingma's theoretical objection to Boorse's BST and discuss its impact on Daniels's overall theory. Though I conclude that Boorse's view, so weakened, will no longer be able to sustain the judgments which Daniels's theory uses it to reach, in the end, I offer Daniels an olive branch by briefly sketching an alternative strategy for reaching suitably objective conclusions regarding the health and/or disease status of various conditions. (shrink)

The sum, difference, product and quotient of two functions with different domains are usually defined only on their common domain. This paper extends these definitions so that the sum and other operations are essentially defined anywhere that at least one of the components is defined. This idea is applied to propositions and events, expressed as indicator functions, to define conditional propositions and conditional events as three-valued indicator functions that are undefined when their condition is false. Extended operations (...) of "and", "or", "not" and "conditioning" are then defined on these conditional events with variable conditions. The probabilities of the disjunction and of the conjunction of two conditionals are expressed in terms of the conditional probabilities of the component conditionals. In a special case, these are shown to be weighted averages of the component conditional probabilities where the weights are the relative probabilities of the various conditions. Next, conditional random variables are defined to be random variables X whose domain has been restricted by a condition on a second random variable Y. The extended sum, difference, product and conditioning operations on functions are then applied to these conditional random variables. The expectation of a random variable and the conditional expectation of a conditional random variable are recounted. Theorem 1 generalizes the standard result that the conditional expectation of the sum of two conditional random variables with disjoint and exhaustive conditions is a weighted sum of the conditional expectations of the component conditional random variables. Because of the extended operations, the theorem is true for arbitrary conditions. Theorem 2 gives a formula for the expectation of the product of two conditional random variables. After the definition of independence of two random variables is extended to accommodate the extended operations, it is applied to the formula of Theorem 2 to simplify the expectation of a product of conditional random variables. Two examples end the paper. The first concerns a work force of n workers of different output levels and work shifts. The second example involves two radars with overlapping surveillance regions and different detection error rates. One radar's error rate is assumed to be sensitive to fog and the other radar's error rate is assumed to be sensitive to air traffic density. The combined error rate over the combined surveillance region given heavy fog and moderate air traffic is computed. (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)

This essay argues that "the family" should be understood in functional terms:whatever functions as a family should have the legal status of a family. Theauthor's argument thus avoids two extreme positions. The first is the position ofthe hard-line "platonic" essentialists who, on grounds of nature, supernature, orcultural history, argue that a family unit must comprise heterosexual partners.The second is the position of the radical relativist, who argues that there are noessences whatsoever or that essences are purely arbitrary. Treating (...) the family infunctionalist terms, the author argues, would have positive consequences thatwould strengthen the social fabric. (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)

International human rights law has struggled to define a standard for determining the extraterritorial applicability of its norms that would reconcile the ethos of universal entitlement, on the one hand, with the centrality of borders in delineating state powers and responsibilities under international law, on the other hand. The case law of the UN Human Rights Committee and the European Court of Human Rights favors barring states from engaging in conduct outside their borders that would be impermissible if undertaken inside (...) their borders. Still, attempts to demarcate the precise scope of extraterritorial application through allusion to degrees of control over individuals or areas, or by the nature of the obligation itself – have led to unsatisfactory, if not arbitrary results. This article opines a move to functionalism as the basis for extraterritorial applicability – requiring states to protect IHRL in situations they can do so. Under this approach, which takes universality seriously, borders lose much their normative significance. I suggest limiting the functional approach to extraterritorial applicability in accordance with two key notions: the intensity of power relations – factual relations of power entailing direct, significant and foreseeable potential impact – should result in the application of IHRL obligations; or, alternatively, special legal relations – relations of power that put the state in a unique legal position to afford IHRL protection would also justify the imposition of extraterritorial obligations. (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)

Drawing on Christopher Boorse's Biostatistical Theory (BST), Norman Daniels contends that a genuine health need is one which is necessary to restore normal functioning – a supposedly objective notion which he believes can be read from the natural world without reference to potentially controversial normative categories. But despite his claims to the contrary, this conception of health harbors arbitrary evaluative judgments which make room for intractable disagreement as to which conditions should count as genuine health needs and therefore which (...) needs should be met. I begin by offering a brief summary of Boorse's BST, the theory to which Daniels appeals for providing the conception of health as normal functioning upon which his overall distributive scheme rests. Next, I consider what I call practical objections to Daniels's use of Boorse's theory. Finally I recount Elseljin Kingma's theoretical objection to Boorse's BST and discuss its impact on Daniels's overall theory. Though I conclude that Boorse's view, so weakened, will no longer be able to sustain the judgments which Daniels's theory uses it to reach, in the end, I offer Daniels an olive branch by briefly sketching an alternative strategy for reaching suitably objective conclusions regarding the health and/or disease status of various conditions. (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.

The monadic second-order theory of trees allows quantification over elements and over arbitrary subsets. We classify the class of trees with respect to the question: does a tree T have a definable choice function (by a monadic formula with parameters)? A natural dichotomy arises where the trees that fall in the first class don't have a definable choice function and the trees in the second class have even a definable well ordering of their elements. This has a close connection (...) to the uniformization problem. (shrink)

In this paper, we develop an organizational account that defines biological functions as causal relations subject to closure in living systems, interpreted as the most typical example of organizationally closed and differentiated self-maintaining systems. We argue that this account adequately grounds the teleological and normative dimensions of functions in the current organization of a system, insofar as it provides an explanation for the existence of the function bearer and, at the same time, identifies in a non-arbitrary way (...) the norms that functions are supposed to obey. Accordingly, we suggest that the organizational account combines the etiological and dispositional perspectives in an integrated theoretical framework. IntroductionDispositional ApproachesEtiological TheoriesBiological Self-maintenance Closure, teleology, and normativityOrganizational differentiationFunctions C1: Contributing to the maintenance of the organization C2: Producing the functional trait Implications and Objections Functional versus useful Dysfunctions, side effects, and accidental contributionsProper functions and selected effectsReproductionRelation with other ‘unitarian’ approachesConclusions. (shrink)

Extensive measurement theory is developed in terms of theratio of two elements of an arbitrary (not necessarily Archimedean) extensive structure; thisextensive ratio space is a special case of a more general structure called aratio space. Ratio spaces possess a natural family of numerical scales (r-scales) which are definable in non-representational terms; ther-scales for an extensive ratio space thus constitute a family of numerical scales (extensive r-scales) for extensive structures which are defined in a non-representational manner. This is interpreted as (...) involving arelational theory of quantity which contrasts in certain respects with thequalitative theory of quantity implicit in standard representational extensive measurement theory. The representational properties of extensiver-scales are investigated, and found to coincide withweak extensive measurement in the sense of Holman. This provides support for the thesis (developed in a separate paper) that weak extensive measurement is a more natural model of actual physical extensive scales than is the standard model using strong extensive measurement. Finally, the present apparatus is applied to slightly simplify the existing necessary and sufficient conditions for strong extensive measurement. (shrink)