In this paper we show that the class of fork squares has a complete orthodox axiomatization in fork arrow logic (FAL). This result may be seen as an orthodox counterpart of Venema's non-orthodox axiomatization for the class of squares in arrow logic. FAL is the modal logic of fork algebras (FAs) just as arrow logic is the modal logic of relation algebras (RAs). FAs extend RAs by a binary fork operator and are axiomatized by adding three equations to RAs equational (...) axiomatization. A proper FA is an algebra of relations where the fork is induced by an injective operation coding pair formation. In contrast to RAs, FAs are representable by proper ones and their equational theory has the expressive power of full first-order logic. A square semantics (the set of arrows is ᵎ x ᵎ for some set ᵎ) for arrow logic was defined by Y. Venema. Due to the negative results about the finite axiomatizability of representable RAs, Venema provided a non-orthodox finite axiomatization for arrow logic by adding a new rule governing the applications of a difference operator. We address here the question of extending the type of relational structures to define orthodox axiomatizations for the class of squares. Given the connections between this problem and the finitization problem addressed by I. Németi, we suspect that this cannot be done by using only logical operations. The modal version of the FA equations provides an orthodox axiomatization for FAL which is complete in view of the representability of FAs. Here we review this result and carry it further to prove that this orthodox axiomatization for FAL also axiomatizes the class of fork squares. (shrink)

In this paper we show that the class of fork squares has a complete orthodox axiomatization in fork arrow logic (FAL). This result may be seen as an orthodox counterpart of Venema's non-orthodox axiomatization for the class of squares in arrow logic. FAL is the modal logic of fork algebras (FAs) just as arrow logic is the modal logic of relation algebras (RAs). FAs extend RAs by a binary fork operator and are axiomatized by adding three equations to RAs equational (...) axiomatization. A proper FA is an algebra of relations where the fork is induced by an injective operation coding pair formation. In contrast to RAs, FAs are representable by proper ones and their equational theory has the expressive power of full first-order logic. A square semantics (the set of arrows is U×U for some set U) for arrow logic was defined by Y. Venema. Due to the negative results about the finite axiomatizability of representable RAs, Venema provided a non-orthodox finite axiomatization for arrow logic by adding a new rule governing the applications of a difference operator. We address here the question of extending the type of relational structures to define orthodox axiomatizations for the class of squares. Given the connections between this problem and the finitization problem addressed by I. Németi, we suspect that this cannot be done by using only logical operations. The modal version of the FA equations provides an orthodox axiomatization for FAL which is complete in view of the representability of FAs. Here we review this result and carry it further to prove that this orthodox axiomatization for FAL also axiomatizes the class of fork squares. (shrink)

We compare fork arrow logic, an extension of arrow logic, and its natural first-order counterpart (the correspondence language) and show that both have the same expressive power. Arrow logic is a modal logic for reasoning about arrow structures, its expressive power is limited to a bounded fragment of first-order logic. Fork arrow logic is obtained by adding to arrow logic the fork modality (related to parallelism and synchronization). As a result, fork arrow logic attains the expressive power of its first-order (...) correspondence language, so both can express the same input–output behavior of processes. (shrink)

We compare fork arrow logic, an extension of arrow logic, and its natural first-order counterpart (the correspondence language) and show that both have the same expressive power. Arrow logic is a modal logic for reasoning about arrow structures, its expressive power is limited to a bounded fragment of first-order logic. Fork arrow logic is obtained by adding to arrow logic the fork modality (related to parallelism and synchronization). As a result, fork arrow logic attains the expressive power of its first-order (...) correspondence language, so both can express the same input-output behavior of processes. (shrink)

Rapid individual cognitive phenotyping holds the potential to revolutionize domains as wide‐ranging as personalized learning, employment practices, and precision psychiatry. Going beyond limitations imposed by traditional lab‐based experiments, new efforts have been underway toward greater ecological validity and participant diversity to capture the full range of individual differences in cognitive abilities and behaviors across the general population. Building on this, we developed Skill Lab, a novel game‐based tool that simultaneously assesses a broad suite of cognitive abilities while providing an engaging (...) narrative. Skill Lab consists of six mini‐games as well as 14 established cognitive ability tasks. Using a popular citizen science platform (N = 10,725), we conducted a comprehensive validation in the wild of a game‐based cognitive assessment suite. Based on the game and validation task data, we constructed reliable models to simultaneously predict eight cognitive abilities based on the users’ in‐game behavior. Follow‐up validation tests revealed that the models can discriminate nuances contained within each separate cognitive ability as well as capture a shared main factor of generalized cognitive ability. Our game‐based measures are five times faster to complete than the equivalent task‐based measures and replicate previous findings on the decline of certain cognitive abilities with age in our large cross‐sectional population sample (N = 6369). Taken together, our results demonstrate the feasibility of rapid in‐the‐wild systematic assessment of cognitive abilities as a promising first step toward population‐scale benchmarking and individualized mental health diagnostics. (shrink)

We investigate the possibility to characterize (multi) functions that are Σ b i -definable with small i (i = 1, 2, 3) in fragments of bounded arithmetic T 2 in terms of natural search problems defined over polynomial-time structures. We obtain the following results: (1) A reformulation of known characterizations of (multi)functions that are Σ b 1 - and Σ b 2 -definable in the theories S 1 2 and T 1 2 . (2) New characterizations of (multi)functions that are (...) Σ b 2 - and Σ b 3 -definable in the theory T 2 2 . (3) A new non-conservation result: the theory T 2 2 (α) is not ∀Σ b 1 (α)-conservative over the theory S 2 2 (α). To prove that the theory T 2 2 (α) is not ∀Σ b 1 (α)-conservative over the theory S 2 2 (α), we present two examples of a Σ b 1 (α)-principle separating the two theories: (a) the weak pigeonhole principle WPHP (a 2 , f, g) formalizing that no function f is a bijection between a 2 and a with the inverse g, (b) the iteration principle Iter (a, R, f) formalizing that no function f defined on a strict partial order ( $\{0,\dots,a\}$ , R) can have increasing iterates. (shrink)

pt. 1. A critique of current educational aims.--pt. 2. Reason.--pt. 3. Education and reason.

This collection of essays, written on four continents by scientists, philosophers and humanists, was initially presented to Karl R. Popper on his sixtieth birthday as a token of critical admiration and in recognition of his work. But the volume also stands on its own as a remarkable series of statements utilizing Popper's critical vision in the study of philosophy proper, logic, mathematics, science as method and theory, and finally to the study of society and history. What is remarkable is that (...) Popper worked in all of these areas, not in a cursory or discursive way, but with the utmost clarity and rigor.. The core position of this volume and its contributors is that the progress of knowledge is not a linear accumulation of definitive acquisitions but a zigzagging process in which counterexamples and unfavorable evidence ruin generalizations and prompt the invention of more comprehensive and sometimes deeper generalizations, to be criticized in their turn. A critical approach to problems, procedures, and results in every field of inquiry is therefore a necessary condition for the continuance of progress. The title of this volume then is, in a sense, an homage to Popper's critical rationalism and critical empiricism. The essays are a tribute to his unceasing and uncompromising quest, not for final certainty, but for closer truth and increased clarity. Among the contributors are outstanding figures in philosophy and the exact sciences in their own right, including Herbert Feigl, R. M. Hare, J.O. Wisdom, Nicholas Rescher, David Bohm, Paul K. Feyerabend, F. A. Hayek, and Adolf Grunbaum. Social science contributions include Hans Albert on social science and moral philosophy, W. B. Gallie, on the critical philosophy of history, Pieter Geyl on The Open Society and its Enemies, and George H. Nadel on the philosophy of History. (shrink)

A critical and constructive discussion of philosophical questions which have particular bearing on the formulation of educational aims.

This volume critically and constructively discusses philosophical questions which have particular bearing on the formulation of educational aims. The book is divided into three major parts: the first deals with the nature of education, and discusses the various general aims, such as 'mental health', 'socialization' and 'creativity' which have been thought to characterize it; the second section is concerned with the nature of reason and its relationship to feeling, will and action; finally the development of different aspects of reason in (...) an educational context is considered. (shrink)

R F Dearden; Theory and Practice in Education, Journal of Philosophy of Education, Volume 14, Issue 1, 30 May 2006, Pages 17–29, https://doi.org/10.1111/j.1467-.

ALTHOUGH THE IDEA OF A VIOLATION OF NATURAL LAW IS NOT NECESSARILY INVOLVED IN THE IDEA OF THE MIRACULOUS, THERE IS "ONE KIND" OF MIRACLE WHICH SEEMS TO INVOLVE IT. HUME’S DISCUSSION OF THE EVIDENCE FOR MIRACLES RELATES TO THIS KIND AND IS INTERPRETABLE AS AN ARGUMENT AGAINST ITS POSSIBILITY. ALSO THERE IS AN ARGUMENT THAT THE EXPRESSION "VIOLATION OF NATURAL LAW" SIGNIFIES A CONFUSION IN WHICH THE IDEAS OF NATURAL LAW AND LEGAL LAW COLLAPSE INTO EACH OTHER. NEITHER OF (...) THESE ARGUMENTS IS EFFICACIOUS. FURTHERMORE, THE CONTENTION THAT THERE CAN BE NO SUCH THING AS ESTABLISHING THE "ABSENCE" OF A NATURAL CAUSE IS OPEN TO OBJECTION. HOWEVER, TO BE CONCEIVED AS A VIOLATION OF NATURAL LAW, A MIRACLE MUST BE THOUGHT OF AS AN OCCURRENCE WHICH IS BOTH EMPIRICALLY CERTAIN AND CONCEPTUALLY IMPOSSIBLE--WHICH OUGHT TO MAKE THE NOTION RIDICULOUS. AND YET IT NEED NOT. (shrink)

Reading the Republic without reference to the less familiar Laws can lead to a distorted view of Plato's political theory. In the Republic the philosopher describes his ideal city; in his last and longest work he deals with the more detailed considerations involved in setting up a second-best 'practical utopia.' The relative neglect of the Laws has stemmed largely from the obscurity of its style and the apparent chaos of its organization so that, although good translations now exist, students of (...) philosophy and political science still find the text inaccessible. This first full-length philosophical introduction to the Laws will therefore prove invaluable. The opening chapters describe the general character of the dialogue and set it in the context of Plato's political philosophy as a whole. Each of the remaining chapters deals with a single topic, ranging over material scattered through the text and so drawing together the threads of the argument in a stimulating and readily comprehensible way. Those topics include education, punishment, responsibility, religion, virtue and pleasure as well as political matters and law itself. Throughout, the author encourages the reader to think critically about Plato's ideas and to see their relevance to present-day philosophical debate. No knowledge of Greek is required and only a limited background in philosophy. Although aimed primarily at students, the book will also be of interest to more advanced readers since it provides for the first time a philosophical, as opposed to linguistic or historical, commentary on the Laws in English. (shrink)

Arvind Thomas has written a remarkable book. That said, however, it must be quickly added that it is not a book for everyone, not even for all students of medieval literature. It is a very thoughtf...

We introduce a linear analogue of Läuchli's semantics for intuitionistic logic. In fact, our result is a strengthening of Läuchli's work to the level of proofs, rather than provability. This is obtained by considering continuous actions of the additive group of integers on a category of topological vector spaces. The semantics, based on functorial polymorphism, consists of dinatural transformations which are equivariant with respect to all such actions. Such dinatural transformations are called uniform. To any sequent in Multiplicative Linear Logic (...) , we associate a vector space of“diadditive” uniform transformations. We then show that this space is generated by denotations of cut-free proofs of the sequent in the theory MLL + MIX. Thus we obtain a full completeness theorem in the sense of Abramsky and Jagadeesan, although our result differs from theirs in the use of dinatural transformations.As corollaries, we show that these dinatural transformations compose, and obtain a conservativity result: diadditive dinatural transformations which are uniform with respect to actions of the additive group of integers are also uniform with respect to the actions of arbitrary cocommutative Hopf algebras. Finally, we discuss several possible extensions of this work to noncommutative logic.It is well known that the intuitionistic version of Läuchli's semantics is a special case of the theory of logical relations, due to Plotkin and Statman. Thus, our work can also be viewed as a first step towards developing a theory of logical relations for linear logic and concurrency. (shrink)

- How can anthropology improve our understanding of the interrelationship between nature and culture? - What can anthropology contribute to practical debates which depend on particular definitions of nature, such as that concerning sustainable development? Humankind has evolved over several million years by living in and utilizing 'nature' and by assimilating it into 'culture'. Indeed, the technological and cultural advancement of the species has been widely acknowledged to rest upon human domination and control of nature. Yet, by the 1960s, the (...) idea of culture in confrontation with nature was being challenged by science, philosophy and the environmental movement. Anthropology is increasingly concerned with such issues as they become more urgent for humankind as a whole. This important book reviews the current state of the concepts of 'nature' we use, both as scientific devices and ideological constructs, and is organised around three themes: - nature as a cultural construction; - the cultural management of the environment; and - relations between plants, animals and humans. (shrink)

The idea of freedom plays a key role in Plato's moral and political thought. In the Republic justice is shown to be beneficial because the just man alone is truly free. There are parallels here with modern discussions of freedom. The Laws argues that to be free a city must avoid the extremes of liberty and of authoritarianism. The legislator should rely on persuasion, not force, so that people willingly obey his laws. The underlying idea is that we are free (...) if we willingly follow the demands of reason rather than being coerced by external forces or by unruly desires. (shrink)

The question of what counts as a successful outcome of the process of genetics counselling has recently become central because of the increasing calls for efficiency in health care, and for means of measuring efficiency. Angus Clarke has drawn attention to this trend, and has argued against both a measure in terms of the number of terminations of pregnancy performed as a result of counselling, and an assessment in terms of the contribution of genetics counselling to a national eugenics policy. (...) He suggests instead a measure of workload. There are good arguments for supporting Clarke's position up to a point. In looking for an appropriate measure, much turns on how genetics counselling is defined. It is here understood in the context of an autonomy model of health care. It is argued that there is a contradiction between such an interpretation and the termination measure of outcome. The political philosophy underlying this outcome is also defective. Workload is not sufficient as a measure of outcome however; it is essential to look at the proper goals of an activity. It is argued that these must be connected in some way with the genetic health of the population; that the promotion of reproductive autonomy itself is not sufficient as a goal. The concern for genetic health, however, is interpreted in a way that avoids Clarke's concerns about a national eugenics policy. (shrink)

„My sole purpose in this paper is to try and correct what I take to be a common misinterpretation of Hume’s opinions on mathematics. I shall not enquire whether he was right or wrong in holding these opinions. Nor shall I offer opinions of my own.“.

The problem of teaching and learning of Russian language and literature in schools with native language of teaching related to the implementation of the principle of dialogue between cultures. The article draws on the results of the survey of graduates of the two high schools of Kazan: School #2 with teaching in Tatar language and school #37 with teaching in Russian-language. The results of the survey are associated with the problems of bilingualism, multiculturalism and bimentality. Graduates from Tatar language gymnasium (...) are bilingual and fluent in Russian and Tatar languages. 97% of graduates of Tatar gymnasium speak, read and write free in their native language and 96% in Russian. Bilingualism acts in this case like one of the manifestations of inter-ethnic interaction, a particular manifestation of biculturalism. In the National School dialogue of cultures passes through a system of parallel study of both Russian and Tatar languages and literatures: the formation of a bicultural, bilingual, bimental personality able to perceive not only the native culture, but also other culture as the native is taking place. (shrink)