Name der Zeitschrift: Zeitschrift für Religionswissenschaft Jahrgang: 23 Heft: 1 Seiten: 188-206.

Using a variation of the rainbow construction and various pebble and colouring games, we prove that RRA, the class of all representable relation algebras, cannot be axiomatised by any first-order relation algebra theory of bounded quantifier depth. We also prove that the class At(RRA) of atom structures of representable, atomic relation algebras cannot be defined by any set of sentences in the language of RA atom structures that uses only a finite number of variables.

When engaging in community-based research, it is important to consider ethical research practices throughout the project. While current research practices require many investigators to obtain approval from an ethics review board before starting a project, more is required to ensure that ethical principles are applied once the investigations begin and after the investigations are complete. In response to this concern, as expressed by workers at a feminist non-profit during a community placement, we developed a tool to foster both greater ethical (...) and feminist research practice in community-based research. Using feminist theories, methodologies, and concepts such as epistemic justice, epistemic trust, and coauthorship, a tool was developed to support researchers and other collaborators in building relationships of reciprocity. This tool, called the Research Responsibility Agreement (RRA) invites all members of a research project to explicitly reflect on their role in the research, their relationships with other collaborators, their responsibility to contributing meaningfully in the project, and their plans to remain accountable to one another. In doing so, the RRA adds to existing tools that support ethical research by sharing explicit reflections from all collaborators on how to prevent harm and by asking them to reflect on ethical practices beyond the initial stages of the project. The RRA also encourages greater engagement from researchers and collaborators toward building meaningful relationships with each other, and with participants, to work together in advancing social change. As a practical tool that promotes reflection, that builds relationships, and that holds all parties accountable to ethical and feminist research practices, the RRA has the potential to generate impactful change in community-based research projects and beyond. While the RRA is tailored to community-based research, it can be applied widely to any research project and has the potential to revolutionize how research relationships are built across disciplines. (shrink)

This paper is a survey of recent results concerning connections between relation algebras , cylindric algebras and polyadic equality algebras . We describe exactly which subsets of the standard axioms for RA are needed for axiomatizing RA over the RA-reducts of CA3's, and we do the same for the class SA of semi-associative relation algebras. We also characterize the class of RA-reducts of PEA3's. We investigate the interconnections between the RA-axioms within CA3 in more detail, and show that only four (...) implications hold between them . In the other direction, we introduce a natural CA-theoretic equation MGR+, generalization of the well-known Merrry-Go-Round equation MGR of CA-theory. We show that MGR+ is equivalent to the RA-reduct being an SA, and that MGR+ implies that the RA-reduct determines the algebra itself, while MGR is not sufficient for either of these to hold. Then we investigate how different CA's a single RRA can 'generate' in the general case. We solve the first part of Problem 11 from the 'Problem Session Paper' of [2].While proving some of the statements, for others we give only outline of proof. The paper contains several open problems. A full version of this paper is under preparation.Keywords: relation algebras, cylindric algebras, polyadic algebras, algebraic logic, arrow logic, proof theory, finite variable fragments, provability with 3 variables, non-finitizability, twisting, non-standard models, neat reducts, representability. (shrink)

We study algebras whose elements are relations, and the operations are natural “manipulations” of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schröder . Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of n-ary relations, RPEAn of polyadic equality algebras of n-ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCAn has to be very complex in the following sense: (...) for every natural number k there is an equation in E containing more than k distinct variables and all the operation symbols, if 2 < n < ω. Completely analogous statement holds for the case n ω. This improves Monk's famous non-finitizability theorem for which we give here a simple proof. We prove analogous nonfinitizability properties of the larger varieties SNrnCAn + k. We prove that the complementation-free subreducts of RCAn do not form a variety. We also investigate the reason for the above “non-finite axiomatizability” behaviour of RCAn. We look at all the possible reducts of RCAn and investigate which are finitely axiomatizable. We obtain several positive results in this direction. Finally, we summarize the results and remaining questions in a figure. We carry through the same programme for RPEAn and for RRA. By looking into the reducts we also investigate what other kinds of natural algebras of relations are possible with more positive behaviour than that of the well known ones. Our investigations have direct consequences for the logical properties of the n-variable fragment Ln of first order logic. The reason for this is that RCAn and RPEAn are the natural algebraic counterparts of Ln while the varieties SNrnCAn + k are in connection with the proof theory of Ln. This paper appears in two parts. The first part contains the non-finite axiomatizability results. The present second part contains finite axiomatizations of some fragments together with a figure summarizing the finite and non-finite axiomatizability results in this area and the problems left open. (shrink)

The recent encyclical of Pope Francisco has been classified by many as the encyclical on the climate and the environment. However, father Francisco not only mentions several of the environmental problems of today’s world, including the more dramatic, but analyses the causes of such problems and seeks to shed plenty of light to find solutions. In the present investigation, it delves into the importance of the message of hope from Pope Francisco facing the serious crisis which describes in his Encyclical (...) insists on hope as a core element of an ethic that will help to get out of these problems, because without hope there is no possible ethics. The Encyclical is situated in the field of bioethics demonstrating the concern of the Pope at the general situation of the planet Earth, placing the human person at the Centre of this reflection, to then describe the positive elements of the diagnosis of the reality and bets that allow you to build a vision of hope. -/- La reciente encíclica del Papa Francisco ha sido catalogada por muchos como la encíclica sobre el clima y el medio ambiente. Sin embargo, el Papa Francisco no solo menciona varios de los problemas ambientales del mundo actual, incluyen-do los más dramáticos, sino que analiza las causas de dicha problemática y busca arrojar abundante luz para encontrarle soluciones. En la presente investigación, se ahonda en la importancia del mensaje de esperanza del Papa Francisco frente a la grave crisis que describe en su encíclica y se insiste en la esperanza como elemento medular de una ética que ayude a salir de dichos problemas, pues sin esperanza no hay ética posible. Se sitúa la encíclica en el ámbito de la bioética evidenciando la preocupación del Papa por la situación general del planeta Tie-rra, poniendo a la persona humana en el centro de dicha reflexión, para luego describir los elementos positivos del diagnóstico de la realidad y las apuestas que permiten construir una visión de esperanza. (shrink)

Extending a construction of Andreka, Givant, and Nemeti (2019), we construct some finite vector spaces and use them to build finite non-representable relation algebras. They are simple, measurable, and persistently finite, and they validate arbitrary finite sets of equations that are valid in the variety RRA of representable relation algebras. It follows that there is no finitely axiomatisable class of relation algebras that contains RRA and validates every equation that is both valid in RRA and preserved by completions of relation (...) algebras. Consequently, the variety generated by the completions of representable relation algebras is not finitely axiomatisable. This answers a question of Maddux (2018). (shrink)

We study algebras whose elements are relations, and the operations are natural “manipulations” of relations. This area goes back to 140 years ago to works of De Morgan, Peirce, Schröder . Well known examples of algebras of relations are the varieties RCAn of cylindric algebras of n-ary relations, RPEAn of polyadic equality algebras of n-ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E, of RCAn has to be very complex in the following sense: (...) for every natural number k there is an equation in E containing more than k distinct variables and all the operation symbols, if 2 < n < ω. Completely analogous statement holds for the case n ω. This improves Monk's famous non-finitizability theorem for which we give here a simple proof. We prove analogous nonfinitizability properties of the larger varieties SNrnCAn + k. We prove that the complementation-free subreducts of RCAn do not form a variety. We also investigate the reason for the above “non-finite axiomatizability” behaviour of RCAn. We look at all the possible reducts of RCAn and investigate which are finitely axiomatizable. We obtain several positive results in this direction. Finally, we summarize the results and remaining questions in a figure. We carry through the same programme for RPEAn and for RRA. By looking into the reducts we also investigate what other kinds of natural algebras of relations are possible with more positive behaviour than that of the well known ones. Our investigations have direct consequences for the logical properties of the n-variable fragment Ln of first order logic. The reason for this is that RCAn and RPEAn are the natural algebraic counterparts of Ln while the varieties SNrnCAn + k are in connection with the proof theory of Ln. This paper appears in two parts. The first part contains the non-finite axiomatizability results. The present second part contains finite axiomatizations of some fragments together with a figure summarizing the finite and non-finite axiomatizability results in this area and the problems left open. (shrink)

Logics of binary relations corresponding, among others, to the class RRA of representable relation algebras and the class FRA of full relation algebras are presented together with the proof systems in the style of dual tableaux. Next, the logics are extended with relational constants interpreted as point relations. Applications of these logics to reasoning in non-classical logics are recalled. An example is given of a dual tableau proof of an equation which is RRA-valid, while not RA-valid.

Logics of binary relations corresponding, among others, to the class RRA of representable relation algebras and the class FRA of full relation algebras are presented together with the proof systems in the style of dual tableaux. Next, the logics are extended with relational constants interpreted as point relations. Applications of these logics to reasoning in non-classical logics are recalled. An example is given of a dual tableau proof of an equation which is RRA-valid, while not RA-valid.