It is quite well-known from Kurt G¨odel’s (1931) ground-breaking Incompleteness Theorem that rudimentary relations (i.e., those definable by bounded formulae) are primitive recursive, and that primitive recursive functions are representable in sufficiently strong arithmetical theories. It is also known, though perhaps not as well-known as the former one, that some primitive recursive relations are not rudimentary. We present a simple and elementary proof of this fact in the first part of the paper. In the second part, we review some possible (...) notions of representability of functions studied in the literature, and give a new proof of the equivalence of the weak representability with the (strong) representability of functions in sufficiently strong arithmetical theories. (shrink)

We model infinite regress structures -not arguments- by means of ungrounded recursively defined functions in order to show that no such structure can perform the task of providing determination to the items composing it, that is, that no determination process containing an infinite regress structure is successful.

The classification of preserved hypothalamic activity in brain death and brainstem death as functional or non-functional has become a subject of debate. While proponents of the neurological criterion claim that these activities lack functional significance (Shemie et al. 2014), Nair-Collins and Joffe (2023) argue for their functional physiological role. However, the interpretation of the term "function" within the medico-legal framework, where death is characterized by the irreversible cessation of all brain functions, remains unclear. -/- My intention here is not (...) to engage in a debate over the validity of different viewpoints regarding the functional significance of preserved activities in the hypothalamus or the heart. Rather, I aim to highlight the critical importance of exploring and defining the concept of function in medicine, which currently lacks clarity. My point is that, whatever our position in the brain death debate, reaching a consensus on this conceptual matter would minimize misunderstandings and establish a robust foundation for further insightful analysis. (shrink)

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

Inductively defined structures are ubiquitous in mathematics; their specification is unambiguous and their properties are powerful. All fields of mathematical logic feature these structures prominently: the formula of a language, the set of theorems, the natural numbers, the primitive recursive functions, the constructive number classes and segments of the cumulative hierarchy of sets. -/- This dissertation gives a mathematical characterization of a species of inductively defined structures, called accessible domains, which include all of the above examples except the (...) set of theorems. The concept of an accessible domain comes from Wilfried Sieg's analysis of proof-theoretic practices, starting with his dissertation. In particular, he noticed the special epistemological character of elements of an accessible domain: they can always be uniquely identified with their build-up. Generally, the unique build-up of elements justifies the principles of induction and recursion. -/- I use category theory to give an abstract characterization of accessible domains. I claim that accessible domains are all instances of initial algebras for endofunctors. Grounded in the historical roots of Sieg's discussions, this dissertation shows how the properties of initial algebras for endofunctors and accessible domains coincide in a satisfying and natural way. -/- Filling out this characterization, I show how important examples of accessible domains fit into this broad characterization. I first characterize some accessible domains by relatively simple functors (e.g. finite, polynomial, those that preserve certain colimits) where we can see how iterating the functor can produce the accessible domain. Then I describe accessible domains that result from more involved specifications (e.g. ordinals and segments of the cumulative hierarchies associated with CZF, IZF, and ZF) by relying heavily on algebraic set theory. I end with a discussion of some of the methodological features of category theory in particular that helped characterize accessible domains. (shrink)

We evaluated the reliability, validity, and differential item functioning (DIF) of a shorter version of the Defining Issues Test-1 (DIT-1), the behavioral DIT (bDIT), measuring the development of moral reasoning. 353 college students (81 males, 271 females, 1 not reported; age M = 18.64 years, SD = 1.20 years) who were taking introductory psychology classes at a public University in a suburb area in the Southern United States participated in the present study. First, we examined the reliability of the bDIT (...) using Cronbach’s α and its concurrent validity with the original DIT-1 using disattenuated correlation. Second, we compared the test duration between the two measures. Third, we tested the DIF of each question between males and females. Findings reported that first, the bDIT showed acceptable reliability and good concurrent validity. Second, the test duration could be significantly shortened by employing the bDIT. Third, DIF results indicated that the bDIT items did not favour any gender. Practical implications of the present study based on the reported findings are discussed. (shrink)

The functional approach to scientific progress has been mainly developed by Kuhn, Lakatos, Popper, Laudan, and more recently by Shan. The basic idea is that science progresses if key functions of science are fulfilled in a better way. This chapter defends the function approach. It begins with an overview of the two old versions of the functional approach by examining the work of Kuhn, Laudan, Popper, and Lakatos. It then argues for Shan’s new functional approach, in which scientific progress (...) is defined as an increase of usefulness of exemplary practices. (shrink)

A common kind of explanation in cognitive neuroscience might be called functiontheoretic: with some target cognitive capacity in view, the theorist hypothesizes that the system computes a well-defined function (in the mathematical sense) and explains how computing this function constitutes (in the system’s normal environment) the exercise of the cognitive capacity. Recently, proponents of the so-called ‘new mechanist’ approach in philosophy of science have argued that a model of a cognitive capacity is explanatory only to the extent (...) that it reveals the causal structure of the mechanism underlying the capacity. If they are right, then a cognitive model that resists a transparent mapping to known neural mechanisms fails to be explanatory. I argue that a functiontheoretic characterization of a cognitive capacity can be genuinely explanatory even absent an account of how the capacity is realized in neural hardware. (shrink)

The notion of function is indispensable to our understanding of distinctions such as that between being broken and being in working order (for artifacts) and between being diseased and being healthy (for organisms). A clear account of the ontology of functions and functioning is thus an important desideratum for any top-level ontology intended for application to domains such as engineering or medicine. The benefit of using top-level ontologies in applied ontology can only be realized when each of the categories (...) identified and defined by a top-level ontology is integrated with the others in a coherent fashion. Basic Formal Ontology (BFO) has from the beginning included function as one of its categories, exploiting a version of the etiological account of function that is framed at a level of generality sufficient to accommodate both biological and artifactual functions. This account has been subjected to a series of criticisms and refinements. We here articulate BFO’s account of function, provide some reasons for favoring it over competing views, and defend it against objections. (shrink)

Functionalism would be mistaken if there existed a system of deviant relations (an “anti-mind”) that had the same functional roles as the standard mental relations. In this paper such a system is constructed, using “Quinean transformations” of the sort associated with Quine’s thesis of the indeterminacy of translation. For example, a mapping m from particularistic propositions (e.g., that there exists a rabbit) to universalistic propositions (that rabbithood is manifested). Using m, a deviant relation thinking* is defined: x thinks* p (...) iff x thinks m(p). Such deviant relations satisfy the commonly discussed functionalist psychological principles. Finally, a more complicated system of deviant relations is constructed, one satisfying sophisticated principles dealing with the self-conscious rational mind. (shrink)

The function of a trait token is usually defined in terms of some properties of other (past, present, future) tokens of the same trait type. I argue that this strategy is problematic, as trait types are (at least partly) individuated by their functional properties, which would lead to circularity. In order to avoid this problem, I suggest a way to define the function of a trait token in terms of the properties of the very same trait token. (...) To able to allow for the possibility of malfunctioning, some of these properties need to be modal ones: a function of a trait is to do F just in case its doing F would contribute to the inclusive fitness of the organism whose trait it is. Function attributions have modal force. Finally, I explore whether and how this theory of biological function could be modified to cover artifact function. (shrink)

Accounts of the concepts of function and dysfunction have not adequately explained what factors determine the line between low‐normal function and dysfunction. I call the challenge of doing so the line‐drawing problem. Previous approaches emphasize facts involving the action of natural selection (Wakefield 1992a, 1999a, 1999b) or the statistical distribution of levels of functioning in the current population (Boorse 1977, 1997). I point out limitations of these two approaches and present a solution to the line‐drawing problem that builds (...) on the second one. (shrink)

The Defining Issues Test (DIT) has been widely used in psychological experiments to assess one’s developmental level of moral reasoning in terms of postconventional reasoning. However, there have been concerns regarding whether the tool is biased across people with different genders and political and religious views. To address the limitations, in the present study, I tested the validity of the brief version of the test, that is, the behavioral DIT, in terms of the measurement invariance and differential item functioning (DIF). (...) I could not find any significant non-invariance at the test level or any item demonstrating practically significant DIF at the item level. The findings indicate that neither the test nor any of its items showed a significant bias toward any particular group. As a result, the collected validity evidence supports the use of test scores across different groups, enabling researchers who intend to examine participants’ moral reasoning development across heterogeneous groups to draw conclusions based on the scores. (shrink)

This paper reconstructs the function argument of Aristotle’s Eudemian Ethics 2.1. The argument seeks to define happiness through the method of division; shows that the highest good is better than all four of the goods of the soul, not only two, as commentators have thought; and unlike the Nicomachean argument, makes the highest good definitionally independent of the human function.

Language defines human existence. Yet defining language is a fraught project. I use the term "language" to refer to a specific mode of information transfer. First, it is a communicative mode. By communication I mean the information transfer serves a function, that is, an activity that occurs because it has increased the evolutionary fitness of ancestors. Secondly, while all communication is governed by norms, human communication, as opposed to biological communication, is governed by norms that have evolved within the (...) learned traditions of individual cultures. The meaning of an assertion in a culture’s language is the set of commitments to which that culture holds its speakers when they utter that assertion. Syntax has appeared in recent evolution to facilitate and enrich the communicative function, but it is a secondary aspect of language. The defining characteristic of human communication, of "language," -- is its capacity to constitute meaning. (shrink)

Functional reductionism concerning mental properties has recently been advocated by Jaegwon Kim in order to solve the problem of the 'causal exclusion' of the mental. Adopting a reductionist strategy first proposed by David Lewis, he regards psychological properties as being 'higher-order' properties functionally defined over 'lower-order' properties, which are causally efficacious. Though functional reductionism is compatible with the multiple realizability of psychological properties, it is blocked if psychological properties are subdivided or crosscut by neurophysiological properties. I argue that there (...) is recent evidence from cognitive neuroscience that shows that this is the case for the psychological property of fear. Though this may suggest that some psychological properties should be revised in order to conform to those of neurophysiology, the history of science demonstrates that this is not always the outcome, particularly with properties that play an important role in our folk theories and are central to human concerns. (shrink)

The Spohnian paradigm of ranking functions is in many respects like an order-of-magnitude reverse of subjective probability theory. Unlike probabilities, however, ranking functions are only indirectly—via a pointwise ranking function on the underlying set of possibilities W —defined on a field of propositions A over W. This research note shows under which conditions ranking functions on a field of propositions A over W and rankings on a language L are induced by pointwise ranking functions on W and the (...) set of models for L, ModL, respectively. (shrink)

This article develops and defends a new functional approach to scientific progress. I begin with a review of the problems of the traditional functional approach. Then I propose a new functional account of scientific progress, in which scientific progress is defined in terms of usefulness of problem defining and problem solving. I illustrate and defend my account by applying it to the history of genetics. Finally, I highlight the advantages of my new functional approach over the epistemic and semantic (...) approaches and dismiss some potential objections to my approach. (shrink)

Discussions of the functions of biological traits generally take the notion of a trait for granted. Defining this notion is a non-trivial problem. Different approaches to function place different constraints on adequate accounts of the notion of a trait. Accounts of function based on engineering-style analyses allow trait boundaries to be a matter of human interest. Accounts of function based on natural selection have typically been taken to require trait boundaries that are objectively real. After canvassing problems (...) raised by each approach, I conclude with some facts that satisfactory notions of trait must respect. (shrink)

There’s a general assumption that teleology and function do not exist in inanimate nature. Throughout biology, it is generally taken as granted that teleology (or teleonomy) and functions are not only unique to life, but perhaps even a defining quality of life. For many, it’s obvious that rocks, water, and the like, are not teleological, nor could they possibly have stand-alone functions. This idea - that teleology and function are unique to life - is the target of this (...) paper. I begin with an overview of McShea’s field theoretic account of teleology. I start with the field theoretic account because it presents a promising analysis of teleological systems. It is promising because, in not making any assumptions about life’s special status in teleological systems, it avoids counterexamples that have problematized other accounts. I then consider some of the prominent efforts that have been made to attempt to avoid ascribing functions or teleology to some form of inanimate nature. In my assessment, none of the efforts are successful. I conclude by offering mineral evolution as a case study to show how inanimate nature can be both teleological and functional. The evolution of mineral species reveals that teleology and function extend to inanimate nature, and that teleology and function come in degrees. (shrink)

It is well known that the following features hold of AR + T under the strong Kleene scheme, regardless of the way the language is Gödel numbered: 1. There exist sentences that are neither paradoxical nor grounded. 2. There are 2ℵ0 fixed points. 3. In the minimal fixed point the weakly definable sets (i.e., sets definable as {n∣ A(n) is true in the minimal fixed point where A(x) is a formula of AR + T) are precisely the Π1 1 sets. (...) 4. In the minimal fixed point the totally defined sets (sets weakly defined by formulae all of whose instances are true or false) are precisely the ▵1 1 sets. 5. The closure ordinal for Kripke's construction of the minimal fixed point is ωCK 1. In contrast, we show that under the weak Kleene scheme, depending on the way the Gödel numbering is chosen: 1. There may or may not exist nonparadoxical, ungrounded sentences. 2. The number of fixed points may be any positive finite number, ℵ0, or 2ℵ0 . 3. In the minimal fixed point, the sets that are weakly definable may range from a subclass of the sets 1-1 reducible to the truth set of AR to the Π1 1 sets, including intermediate cases. 4. Similarly, the totally definable sets in the minimal fixed point range from precisely the arithmetical sets up to precisely the ▵1 1 sets. 5. The closure ordinal for the construction of the minimal fixed point may be ω, ωCK 1, or any successor limit ordinal in between. In addition we suggest how one may supplement AR + T with a function symbol interpreted by a certain primitive recursive function so that, irrespective of the choice of the Godel numbering, the resulting language based on the weak Kleene scheme has the five features noted above for the strong Kleene language. (shrink)

This is a non-technical version of "The Decision Problem for Effective Procedures." The “somewhat vague, intuitive” notion from computability theory of an effective procedure (method) or algorithm can be fairly precisely defined, even if it does not have a purely mathematical definition—and even if (as many have asserted) for that reason, the Church–Turing thesis (that the effectively calculable functions on natural numbers are exactly the general recursive functions), cannot be proved. However, it is logically provable from the notion of (...) an effective procedure, without reliance on any (partially) mathematical thesis or conjecture concerning effective procedures, such as the Church–Turing thesis, that the class of effective procedures is undecidable, i.e., that there is no effective procedure for ascertaining whether a given procedure is effective. The proof does not even appeal to a precise definition of ‘effective procedure’. Instead, it relies solely and entirely on a basic grasp of the intuitive notion of such a procedure. Though the result itself is not surprising, it is also not without significance. It has the consequence, for example, that the solution to a decision problem, if it is to be complete, must be accompanied by a separate argument that the proposed ascertainment procedure is, in fact, a decision procedure, i.e., effective—for example, that it invariably terminates with the correct verdict. (shrink)

This paper defines and defends a notion of teleological function which is fit to figure in explanations concerning how organic systems, and the items which compose them, are able to perform certain activities, such as surviving and reproducing or pumping blood. According to this notion, a teleological function of an item (such as the heart) is a typical way in which items of that type contribute to some containing system's ability to do some activity. An account of what (...) it is for an item to contribute to a containing system's ability to perform an activity is provided. I argue that the view acquires its normative status in virtue of the fact that it obeys a function-accident distinction and obeys a function-dysfunction distinction; that the view is ahistorical; and, that its ahistoricity provides it with an advantage over one of its main competitors. (shrink)

Until the late 19th century scientists almost always assumed that the world could be described as a rule-based and hence deterministic system or as a set of such systems. The assumption is maintained in many 20th century theories although it has also been doubted because of the breakthrough of statistical theories in thermodynamics (Boltzmann and Gibbs) and other fields, unsolved questions in quantum mechanics as well as several theories forwarded within the social sciences. Until recently it has furthermore been assumed (...) that a rule-based and deterministic system was also predictable if only the rules were known, but this assumption has now been undermined by modern chaos-theory describing rule-based and deterministic, but unpredictable systems, while catastrophe-theory delivers a set of types describing various kinds of instability and conditions for the stability of a given system. Hence the main trait in the theoretical development in the 20th-century science can be described as a basic modification and limitation of some of the fundamental and strong assumptions forwarded in the previous epochs of modern science. Ironically, the very same process has been a process in which the human capacity to intervene in nature has expanded dramatically and mainly with the help of the very same theories, and not least because they allow nature to be described and made manipulable on a lower level and a more fine-grained scale. While the overall theoretical consistency between the various theories has gone, the reach of human intervention in nature has increased based on quite new dimensions whether in the area of physics (e.g.: energy technologies, chemical technologies, nanotechnologies etc.) or biology (ge¬netic manipulation) or in the area of psychology, sociology and culture (artificial simulations of mental processes, new means of communication, implying changes in the social infrastructure and cultural behaviour etc.). While some of these changes and new conditions can be reflected from within the conceptual framework of rule-based systems, albeit more complex than formerly recognized, others seem to give rise to the question of whether there are »systems« and relations between different systems in the world which are not rule-based? For instance, it seems to be obvious that the notion of instability represents a major conceptual break with former theories of rule-based systems, as the stability of the latter is an axiomatically given property implied in the very notion of rule-based systems, while instability can only be the result of external influence which should be explained as the result of another rule-based sy¬stem. While there are no difficulties implied concerning the stability of rule-based systems, the notion of unstable states of a system raises the question of how there can be a system at all if there are no invariant stabilising principles? This is the first question which I will address. And I shall do so by taking two examples of such systems as my point of departure. The first example will be the computer and the second will be ordinary language. In both cases I will argue that the stability of these systems (which are both defined by the existence/presence of human intentions) are provided by the help of - differently organised - redundancy functions which both allow the maintenance of systems in unstable macro-states, suspension of previous rules, underdetermination and overdetermination and generation/emergence/creation of new rules more or less independent of previous rules by the help of optional recursions to the permanently accessible underlying levels as for instance the level of binary representation in computers. Since the notion of redundancy is both controversial as such and often avoided, the concept is discussed (as defined in Claude Shannon's mathematical theory of information and in the semiotic framework of J. J. Greimas) and leading to a more general definition in which the redundancy functions serve to overcome noisy conditions, but at the cost of rule-based stability, determination, and predictability. A second question will be how the notion of rule-generating systems relates to the notion of anticipatory systems and it will be argued that rule-generating systems share some features with an¬ticipatory systems and that the former from a certain viewpoint can be seen as a subclass of the latter, although anticipative features are not necessarily a part of the definition of rule-generating systems. On the other hand, it will be discussed whether anticipatory systems which are not rule-generating systems can exist and it will be argued that the capacity to anticipate is strongly limited if it is not part of a rule-generating system. Therefore, it is concluded that the most powerful anticipatory systems need to be rule-generating systems. (shrink)

This essay examines the concept of leadership as it is commonly understood within the field of leadership studies today. The inquiry is framed by an analysis of three generally accepted definitions of leadership. I look at the selected definitions from four angles, which I call the four dimensions of leadership: the behavioral (what the leader does, or ought to do, that makes it leadership), the asymmetrical (in what sense a leader is different from the others in the group), the social (...) (what it is that the leader is leading), and the teleological (what the direction is). By doing this, I find that these definitions can be problematic, if instructively so. I identify two underlying concerns. The first problem is the tendency to fuse informal and formal leadership into a single concept. I argue that informal and formal leadership can be more helpfully understood, rather, as two separate, if sometimes overlapping, concepts, each needing its own definition and analysis. This is because formal leadership is a functional concept, while informal leadership is not; this entails not only that each type of leader is leading a different type of entity, but also that each is judged by different standards. The second problem is, paradoxically, a bias against leadership. To put this less polemically: scholars who recognize that organizational success can be hindered by excessive emphasis upon hierarchy, and who thus study the dynamics of egalitarian collaboration, can inadvertently generate confusion by allowing the honorific of “leadership” to be applied also to that work. So, I propose definitions of the two key concepts, formal and informal leadership, with the suggestion that this provides a helpful method of sorting leadership scholarship into three categories: formal leadership; informal leadership; and egalitarianism. (shrink)

Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS : the fact that all numbers can be generated by (...) a successor function and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Munduruc (an Amazonian language), and young Western children (3-4 years old) understand these fundamental properties of numbers. (shrink)

Replicability is widely regarded as one of the defining features of science and its pursuit is one of the main postulates of meta-research, a discipline emerging in response to the replicability crisis. At the same time, replicability is typically treated with caution by philosophers of science. In this paper, we reassess the value of replicability from an epistemic perspective. We defend the orthodox view, according to which replications are always epistemically useful, against the more prudent view that claims that it (...) is useful in very limited circumstances. Additionally, we argue that we can learn more about the original experiment and the limits of the discovered effect from replications at different levels. We hold that replicability is a crucial feature of experimental results and scientists should continue to strive to secure it. (shrink)

There have been attempts to get some logic out of belief dynamics, i.e. attempts to define the constants of propositional logic in terms of functions from sets of beliefs to sets of beliefs. It is interesting to see whether something similar can be done for ontological categories, i.e. ontological constants. The theory presented here will be a (modest) expansion of belief dynamics: it will not only incorporate beliefs, but also parts of beliefs, so called belief fragments. On the basis of (...) this we will give a belief-dynamical account of the ontological categories of states of affairs, individuals, properties of arbitrary adicities and properties of arbitrary orders. (shrink)

Functional reductionism characterises inter-theoretic reduction as the recovery of the upper-level behaviour described by the reduced theory in terms of the lower-level reducing theory. For instance, finding a statistical mechanical realiser that plays the functional role of thermodynamic entropy allows for establishing a reductive link between thermodynamics and statistical mechanics. This view constitutes a unique approach to reduction that enjoys a number of positive features, but has received limited attention in the philosophy of science. -/- This paper aims to clarify (...) the meaning of functional reductionism in science, with a focus on physics, to define both its place with respect to other approaches to reduction and its connection to ontology. To do so, we develop and explore two alternative frameworks for functional reduction, called Syntactic Functional Reductionism and Semantic Functional Reductionism. They represent two different possible stances on the view that expand and improve the basic functional reductionist approach along different lines, and make clear how the approach works in practice. The former elaborates on David Lewis’ account, is connected with the syntactic view of theories, is committed to a logical characterisation of functional roles, and is embedded within Nagelian reductionism. The latter adopts a semantic view of theories, spells out functional roles mainly in terms of mathematical roles within the models of theories, and is expressed in terms of the related structuralist approach to reduction. The development of these frameworks has the final goal of advancing functional reductionism, to make it a fully-fledged alternative account for reduction in science. (shrink)

The mathematical structure of realist quantum theories has given rise to a debate about how our ordinary 3-dimensional space is related to the 3N-dimensional configuration space on which the wave function is defined. Which of the two spaces is our (more) fundamental physical space? I review the debate between 3N-Fundamentalists and 3D-Fundamentalists and evaluate it based on three criteria. I argue that when we consider which view leads to a deeper understanding of the physical world, especially given the (...) deeper topological explanation from the unordered configurations to the Symmetrization Postulate, we have strong reasons in favor of 3D-Fundamentalism. I conclude that our evidence favors the view that our fundamental physical space in a quantum world is 3-dimensional rather than 3N-dimensional. I outline lines of future research where the evidential balance can be restored or reversed. Finally, I draw lessons from this case study to the debate about theoretical equivalence. (shrink)

In Zettel, Wittgenstein considered a modified version of Cantor’s diagonal argument. According to Wittgenstein, Cantor’s number, different with other numbers, is defined based on a countable set. If Cantor’s number belongs to the countable set, the definition of Cantor’s number become incomplete. Therefore, Cantor’s number is not a number at all in this context. We can see some examples in the form of recursive functions. The definition "f(a)=f(a)" can not decide anything about the value of f(a). The definiton is (...) incomplete. The definition of "f(a)=1+f(a)" can not decide anything about the value of f(a) too. The definiton is incomplete.<br><br>According to Wittgenstein, the contradiction, in Cantor's proof, originates from the hidden presumption that the definition of Cantor’s number is complete. The contradiction shows that the definition of Cantor’s number is incomplete. <br><br>According to Wittgenstein’s analysis, Cantor’s diagonal argument is invalid. But different with Intuitionistic analysis, Wittgenstein did not reject other parts of classical mathematics. Wittgenstein did not reject definitions using self-reference, but showed that this kind of definitions is incomplete.<br><br>Based on Thomson’s diagonal lemma, there is a close relation between a majority of paradoxes and Cantor’s diagonal argument. Therefore, Wittgenstein’s analysis on Cantor’s diagonal argument can be applied to provide a unified solution to paradoxes. (shrink)

In this paper, first the term 'prioritarianism' is defined, with some mathematical precision, on the basis of intuitive conceptions of prioritarianism, especially the idea that "benefiting people matters more the worse off these people are". (The prioritarian weighting function is monotonously ascending and concave, while its first derivation is smoothly descending and convex but positive throughout.) Furthermore, (moderate welfare) egalitarianism is characterized. In particular a new symmetry condition is defended, i.e. that egalitarianism evaluates upper and lower deviations from (...) the social middle symmetrically and equally negatively (as do e.g. variance and Gini). Finally, it is shown that this feature distinguishes egalitarianism also extensionally from prioritarianism. (shrink)

It is often argued that the indispensability of continuum models comes from their empirical adequacy despite their decoupling from the microscopic details of the modelled physical system. There is thus a commonly held misconception that temperature varying across a region of space or time can always be accurately represented as a continuous function. We discuss three inter-related cases of temperature modelling — in phase transitions, thermal boundary resistance and slip flows — and show that the continuum view is fallacious (...) on the ground that the microscopic details of a physical system are not necessarily decoupled from continuum models. We show how temperature discontinuities are present in both data (experiments and simulations) and phenomena (theory and models) and how discontinuum models of temperature variation may have greater empirical adequacy and explanatory power. The conclusions of our paper are: a) continuum idealisations are not indispensable to modelling physical phenomena and both continuous and discontinuous representations of phenomena work depending on the context; b) temperature is not necessarily a continuously defined function in our best scientific representations of the world; and c) that its continuity, where applicable, is a contingent matter. We also raise a question as to whether discontinuous representations should be considered truly de-idealised descriptions of physical phenomena. (shrink)

In quantum mechanics, the wave function of a N-body system is a mathematical function defined in a 3N-dimensional configuration space. We argue that wave function realism implies particle ontology when assuming: (1) the wave function of a N-body system describes N physical entities; (2) each triple of the 3N coordinates of a point in configuration space that relates to one physical entity represents a point in ordinary three-dimensional space. Moreover, the motion of particles is random (...) and discontinuous. (shrink)

Abu Nasr Muhammad Al-Farabi (870–950 AD), the second outstanding representative of the Muslim peripatetic after al Kindi (801–873 AD), was born in Turkestan about 870 AD. Al-Farabi’s studies commenced in Farab, then he travelled to Baghdad, where he studied logic with a Christian scholar named Yuhanna b. Hailan. Al-Farabi wrote numerous works dealing with almost every branch of science in the medieval world. In addition to a large number of books on logic and other sciences, he came to be known (...) as the “Second Teacher” (al-Mou’allim al-Thani), Aristotle being the first. One of Al-Farabi’s most important contributions was clarifying the func- tions of logic as follows: 1. He defined logic and compared it with grammar, and discussed the clas- sification and fundamental principles of science in a unique and useful manner. 2. He made the study of logic easier by dividing it into two categories: Takhayyul (idea) and Thubut (proof). 3. He believed that the objective of logic is to correct faults we may find in ourselves and in others, and faults that others find in us. 4. He said that if we do not comprehend logic, we must either have faith in all people, or mistrust all people, or differentiate between them. Such actions would be undertaken without a basis of evidence or experimen- tation. In this paper, I will analyse the functions of logic in Al-Farabi’s works, Enumeration of the Sciences, Book on the Syllogism, Book on Dialectic, Book on Demonstration and Ring Stones of Wisdom, in order to present his contributions in the field of logic. (shrink)

The goal of this paper is to develop a systematic taxonomy of cognitive artifacts, i.e., human-made, physical objects that functionally contribute to performing a cognitive task. First, I identify the target domain by conceptualizing the category of cognitive artifacts as a functional kind: a kind of artifact that is defined purely by its function. Next, on the basis of their informational properties, I develop a set of related subcategories in which cognitive artifacts with similar properties can be grouped. (...) In this taxonomy, I distinguish between three taxa, those of family, genus, and species. The family includes all cognitive artifacts without further specifying their informational properties. Two genera are then distinguished: representational and non-representational (or ecological) cognitive artifacts. These genera are further divided into species. In case of representational artifacts, these species are iconic, indexical, or symbolic. In case of ecological artifacts, these species are spatial or structural. Within species, token artifacts are identified. The proposed taxonomy is an important first step towards a better understanding of the range and variety of cognitive artifacts and is a helpful point of departure, both for conceptualizing how different artifacts augment or impair cognitive performance and how they transform and are integrated into our cognitive system and practices. (shrink)

The article reveals a conceptual basis of the cryptocurrency functioning. The main types of cryptocurrencies are featured and analyzed as well as their general strengths and weaknesses. Based on the price dynamics correlation analysis of some cryptocurrency types, a general low level of dependence between digital assets is established. The main functions of the cryptocurrency are formulated in the form of transformed money functions. Also, additional functions of cryptocurrencies are defined on the basis of their innovative nature, as well (...) as the role in the modern financial system and world economic relations. (shrink)

Classical interpretations of Goedels formal reasoning, and of his conclusions, implicitly imply that mathematical languages are essentially incomplete, in the sense that the truth of some arithmetical propositions of any formal mathematical language, under any interpretation, is, both, non-algorithmic, and essentially unverifiable. However, a language of general, scientific, discourse, which intends to mathematically express, and unambiguously communicate, intuitive concepts that correspond to scientific investigations, cannot allow its mathematical propositions to be interpreted ambiguously. Such a language must, therefore, define mathematical truth (...) verifiably. We consider a constructive interpretation of classical, Tarskian, truth, and of Goedel's reasoning, under which any formal system of Peano Arithmetic---classically accepted as the foundation of all our mathematical Languages---is verifiably complete in the above sense. We show how some paradoxical concepts of Quantum mechanics can, then, be expressed, and interpreted, naturally under a constructive definition of mathematical truth. (shrink)

In this paper, two axiomatic theories T− and T′ are constructed, which are dual to Tarski’s theory T+ (1930) of deductive systems based on classical propositional calculus. While in Tarski’s theory T+ the primitive notion is the classical consequence function (entailment) Cn+, in the dual theory T− it is replaced by the notion of Słupecki’s rejection consequence Cn− and in the dual theory T′ it is replaced by the notion of the family Incons of inconsistent sets. The author has (...) proved that the theories T+, T−, and T′ are equivalent. (shrink)

By formalizing some classical facts about provably total functions of intuitionistic primitive recursive arithmetic (iPRA), we prove that the set of decidable formulas of iPRA and of iΣ1+ (intuitionistic Σ1-induction in the language of PRA) coincides with the set of its provably ∆1-formulas and coincides with the set of its provably atomic formulas. By the same methods, we shall give another proof of a theorem of Marković and De Jongh: the decidable formulas of HA are its provably ∆1-formulas.

The functional theory of boredom maintains that boredom ought to be defined in terms of its role in our mental and behavioral economy. Although the functional theory has recently received considerable attention, presentations of this theory have not specified with sufficient precision either its commitments or its consequences for the ontology of boredom. This essay offers an in-depth examination of the functional theory. It explains what boredom is according to the functional view; it shows how the functional theory can (...) account for the known characteristics of boredom; and it articulates the theory’s basic commitments, virtues, and limitations. Ultimately, by furthering our understanding of the functional theory of boredom, the essay contributes to a better theoretical grounding of boredom. (shrink)

The selected effects or ‘etiological’ theory of Proper function is a naturalistic and realist account of biological teleology. It is used to analyse normativity in philosophy of language, philosophy of mind, philosophy of medicine and elsewhere. The theory has been developed with a simple and intuitive view of natural selection. Traits are selected because of their positive effects on the fitness of the organisms that have them. These ‘selected effects’ are the Proper functions of the traits. Proponents argue that (...) this analysis of biological teleology has the unique advantage that the selected effect function of a trait is also a causal explanation of the trait: the trait exists because it performs this function. We show, however, that selected effect functions as currently defined explain the existence of traits only under highly restrictive assumptions about evolutionary dynamics. In many common scenarios in which traits evolve by natural selection, selected effect functions do not explain those traits. This is because definitions of selected effect function extract from any evolutionary scenario only the information that would be explanatorily relevant in the simple evolutionary scenario implicit in those definitions. When applied to more complex scenarios selected effect functions omit the key information that is explanatorily relevant in those scenarios. The assumptions required for selected effect functions to be explanatory are particularly unlikely to hold in the domain that its proponents care most about - the evolution of representation. A more adequate selected effects theory of Proper functions may be possible, but will require much greater attention to the structure of actual evolutionary explanations. (shrink)

José Ángel Gascón’s essay "Where are dissent and reasons in epistemic justification?" is an exposition of a version of a social functionalist epistemology. I agree with Gascón's emphasis on reasons and on taking into account dissent as important parts of epistemology. But I think that these concerns do not require a social functionalist epistemology, but that, on the contrary, Gascón's social functionalist epistemology throws the baby out with the bathwater. It does so by excluding also a traditional, at its core (...) individualistic epistemology, which defines central concepts like 'justified', 'knowledge' still in individualistic terms as the result of a mental cognizing process but is open to social extensions, e.g. concerning cooperation in the acquisition of knowledge or the transfer of knowledge via argumentation. Such a socially open epistemology with an individualistic core – or "open individualistic epistemology" for short – is also the basis of the epistemological argumentation theory. In the following I want to explain and defend this open individualistic epistemology together with the epistemological argumentation theory (sect. 2) and explain on this basis some problems of Gascón’s theory (sect. 3). (shrink)

There are two senses of ‘what scientists know’: An individual sense (the separate opinions of individual scientists) and a collective sense (the state of the discipline). The latter is what matters for policy and planning, but it is not something that can be directly observed or reported. A function can be defined to map individual judgments onto an aggregate judgment. I argue that such a function cannot effectively capture community opinion, especially in cases that matter to us.

Human consciousness, as dealt with in the Upanishads, modeled as a mechanical oscillator of infrasonic frequency (the Atman/Brahman), the result of breathing process, is further advanced to get an insight of functions of mind. An analytical approach is followed in parallel to and separette from quantum mechanical, quantum field and other theoretical propositions, approaches and presentations. Pure consciousness, unoccupied awareness and occupied awareness are identified, defined, classified and discussed together with fresh insight about time-space and time. A reversible transformation (...) (vivartanam) of virtual mental energy reflection (maya), creating various consequential / parallel / simultaneous conscious-states, phases, cognitive and communicative states, modes of language acquisition and communication, and kinds of function of human mind, which forms and facilitates human mental acquisitions, functions and communications, is proposed and discussed. Alternative analytical insight of human consciousness and mental functions to other theoretical approaches is given. All this is presented in brain wave modulation / demodulation terms. -/- . (shrink)

In this work, author expressed new R-Synthesis specifically. Good and/or correct perspective that must be behind the definitions and administration generally expressed. New perspective of the philosophy explained generally. Philosophy of GodForm is defined and expressed as connected/related with the following concepts: (a) basic principles, (b) 17 upper constructional philosophies, (c) 14 lower constructional philosophies, (d) eight basic philosophies. As special cases, Philosophy of Engineering and Technology, Philosophy of Wireless Administration and others defined as hybrid philosophies. 17 specific (...) components/units which can be considered to define the basic principles of the related formation, are proposed to be designed by GodForm and defined by the author. Philosophy of GodForm and its relation with Values and Positions specifically defined. New Era Theory and New Era Belief explained generally. Some specific religions/beliefs are defined as variants/forms of the Progressive Religion. Good and/or correct power authorities (R-Power Authority) are defined due to 29 categories. Functional position levels defined for the R-Power Authorities with a unique table. As result of the R-Synthesis, functional position levels of some philosophers, some scientists, and of Prophet Jesus, Prophet Mohamed, Prophets Mouses, Buddha, and of Confucius generally/specifically defined. Integration and past/present effects of some R-Power Authorities in 5 x 5 Ideal Political Construction specifically explained. New Era Belief, Concepts of GodX and Science relation explained with the following concepts: (1) Religion of GodForm, (2) Knowledge of GodX, (3) Nature of GodX. With this respect, some other characteristics of 7 GodX power authorities expressed with details as complementary information. Following concepts/systems defined to arrange and solve some/most/all religious problems, which are determined with R-Synthesis: (a) Religious Responsibilities and Positions, (b) Ideal Religious Administration, (c) Sustainable/Continuable Political Administration System, (d) Community Values Council/Authority for each world country, (e) Progressive Councils. 15 general religious position categories defined to express their good and/or correct meanings/values/responsibilities in the religious system. 21 general/specific cases/programs defined due to New Era Belief to solve possible past/present/future community values problems, to have continuous, judicious, and progressive administration about the 36 kinds of community values. Progressive Councils, which are related with the CUS-WW-HO, are defined. Progressive Religion Council and some of the related religious responsibilities expressed for the kind progressive religion studies due to the Philosophy of Progressive Religions. (shrink)

The doctrine of the imago dei is foundational for Christian anthropology. It informs the nature of humanity as individuals and as a corporate entity. It is the basis of the Christian valuation of human life. And it defines the purpose of humans both now and in the world to come. This article will attempt to define the imago dei in a purely functional manner and examine the implications of such a definition in the light of the doctrines of the fall (...) and the incarnation. (shrink)

The preconditions and features of the formation of post-industrial society are defined in the article. The distinctive role of active innovation infrastructure of the country in integrating into the European community and society based on knowledge are proved. The characteristic features of the economy of post-industrial society are the increasing role of intangible resources in ensuring social reproduction, “softization” and “servization” the subjects of innovation infrastructure. The essence of economic category “innovation infrastructure“ is defined in the article. It (...) is a dynamic self-regulating system of markets and subjects that entering these markets in certain economic relations and it provides the necessary conditions for implementation of the innovation processes. Mechanism of functioning of the constituent elements of innovation infrastructure in market economy is proposed, relationships between them are defined. (shrink)

Essential for the concept of the law of nature is not only spatio-temporal universality, but also functionality in the sense of the dependency on physical conditions of natural entities. In the following it is explained in detail that just the neglect of this functional property is to be understood as the real reason for the occurrence of the Goodman paradox – with the consequence, that the behavior of things seems to be completely at the mercy of change of unique unrepeatable (...) temporal points. It is exactly this (mis-)understanding that also generated the induction problem. From the intrinsic connection between universality and functionality, however, – that is my claim – the ontological consequence of a nature results, for which lawfulness is coupled to essentially functionally defined time sequences, thereby implying a potentiality dimension of nature, too. (shrink)

This paper seeks to establish the reflexive fundamentals about the importance of language in initiave-functional college education, with the general objective of finding and developing categories that allow synthesizing schemes of different approaches to language, in order to contribute to the teaching practice with discourse analysis -/- techniques, which favor processes that promote the pupils’operative intellectual habits. To present this, we will show some propositions that underlie didactic strategies we have used, to motivate, stimulate, and provoke participation from the students (...) in the teaching-learning process of the topics raised in the different subjects that integrate the Sciences of Communication course at the National Autonomous University of Mexico (UNAM). -/- Therefore, the aim of this exposition of causes is to define: 1) our experiential context so to make explicit the limitations and challenges we experiment with and face in this academic exercise to provoke the pupils’ amazement or at least keep their interest; and 2) the theoretical assumptions that underlie our investigation about the relationship between observed language (system) and lived language (act). (shrink)