In this article we show that bi-intuitionistic predicate logic lacks the Craig Interpolation Property. We proceed by adapting the counterexample given by Mints, Olkhovikov and Urquhart for intuitionistic predicate logic with constant domains [13]. More precisely, we show that there is a valid implication $\phi \rightarrow \psi $ with no interpolant. Importantly, this result does not contradict the unfortunately named ‘Craig interpolation’ theorem established by Rauszer in [24] since that article is about the property more correctly named ‘deductive interpolation’ (see (...) Galatos, Jipsen, Kowalski and Ono’s use of this term in [5]) for global consequence. Given that the deduction theorem fails for bi-intuitionistic logic with global consequence, the two formulations of the property are not equivalent. (shrink)

ABSTRACTDo truth tables—the ordinary sort that we use in teaching and explaining basic propositional logic—require an assumption of consistency for their construction? In this essay we show that truth tables can be built in a consistency-independent paraconsistent setting, without any appeal to classical logic. This is evidence for a more general claim—that when we write down the orthodox semantic clauses for a logic, whatever logic we presuppose in the background will be the logic that appears in the foreground. Rather than (...) any one logic being privileged, then, on this count partisans across the logical spectrum are in relatively similar dialectical positions. (shrink)

Bi-intuitionistic logic is the result of adding the dual of intuitionistic implication to intuitionistic logic. In this note, we characterize the expressive power of this logic by showing that the first order formulas equivalent to translations of bi-intuitionistic propositional formulas are exactly those preserved under bi-intuitionistic directed bisimulations. The proof technique is originally due to Lindstrom and, in contrast to the most common proofs of this kind of result, it does not use the machinery of neither saturated models nor elementary (...) chains. (shrink)

We generalize the notion ofconsequence relationstandard in abstract treatments of logic to accommodate intuitions ofrelevance. The guiding idea follows theuse criterion, according to which in order for some premises to have some conclusion(s) as consequence(s), the premises must each beusedin some way to obtain the conclusion(s). This relevance intuition turns out to require not just a failure of monotonicity, but also a move to considering consequence relations as obtaining betweenmultisets. We motivate and state basic definitions of relevant consequence relations, both (...) in single conclusion (asymmetric) and multiple conclusion (symmetric) settings, as well as derivations and theories, guided by the use intuitions, and prove a number of results indicating that the definitions capture the desired results (at least in many cases). (shrink)

In this note we study a counterpart in predicate logic of the notion of logical friendliness, introduced into propositional logic in [15]. The result is a new consequence relation for predicate languages with equality using first-order models. While compactness, interpolation and axiomatizability fail dramatically, several other properties are preserved from the propositional case. Divergence is diminished when the language does not contain equality with its standard interpretation.

Lindström’s theorem obviously fails as a characterization of first-order logic without identity ( $\mathcal {L}_{\omega \omega }^{-} $ ). In this note, we provide a fix: we show that $\mathcal {L}_{\omega \omega }^{-} $ is a maximal abstract logic satisfying a weak form of the isomorphism property (suitable for identity-free languages and studied in [11]), the Löwenheim–Skolem property, and compactness. Furthermore, we show that compactness can be replaced by being recursively enumerable for validity under certain conditions. In the proofs, we (...) use a form of strong upwards Löwenheim–Skolem theorem not available in the framework with identity. (shrink)

This paper is a step toward showing what is achievable using non-classical metatheory—particularly, a substructural paraconsistent framework. What standard results, or analogues thereof, from the classical metatheory of first order logic can be obtained? We reconstruct some of the originals proofs for Completeness, Löwenheim-Skolem and Compactness theorems in the context of a substructural logic with the naive comprehension schema. The main result is that paraconsistent metatheory can ‘re-capture’ versions of standard theorems, given suitable restrictions and background assumptions; but the shift (...) to non-classical logic may recast the meanings of these apparently ‘absolute’ theorems. (shrink)

This paper defines a Sahlqvist fragment for relevant logic and establishes that each class of frames in the Routley-Meyer semantics which is definable by a Sahlqvist formula is also elementary, that is, it coincides with the class of structures satisfying a given first order property calculable by a Sahlqvist-van Benthem algorithm. Furthermore, we show that some classes of Routley-Meyer frames definable by a relevant formula are not elementary.

This paper is devoted to the problem of existence of saturated models for first-order many-valued logics. We consider a general notion of type as pairs of sets of formulas in one free variable that express properties that an element of a model should, respectively, satisfy and falsify. By means of an elementary chains construction, we prove that each model can be elementarily extended to a $\kappa $-saturated model, i.e. a model where as many types as possible are realized. In order (...) to prove this theorem we obtain, as by-products, some results on tableaux and their consistency and satisfiability and a generalization of the Tarski–Vaught theorem on unions of elementary chains. Finally, we provide a structural characterization of $\kappa $-saturation in terms of the completion of a diagram representing a certain configuration of models and mappings. (shrink)

Under a proper translation, the languages of propositional (and quantified relevant logic) with an absurdity constant are characterized as the fragments of first order logic preserved under (world-object) relevant directed bisimulations. Furthermore, the properties of pointed models axiomatizable by sets of propositional relevant formulas have a purely algebraic characterization. Finally, a form of the interpolation property holds for the relevant fragment of first order logic.

Stemming from the works of Petr Hájek on mathematical fuzzy logic, graded model theory has been developed by several authors in the last two decades as an extension of classical model theory that studies the semantics of many-valued predicate logics. In this paper we take the first steps towards an abstract formulation of this model theory. We give a general notion of abstract logic based on many-valued models and prove six Lindström-style characterizations of maximality of first-order logics in terms of (...) metalogical properties such as compactness, abstract completeness, the Löwenheim–Skolem property, the Tarski union property, and the Robinson property, among others. As necessary technical restrictions, we assume that the models are valued on finite MTL-chains and the language has a constant for each truth-value. (shrink)

We study classes of graded structures satisfying the properties of amalgamation, joint embedding and hereditariness. Given appropriate conditions, we can build a graded analogue of the Fraïssé limit. Some examples such as the class of all finite weighted graphs or the class of all finite fuzzy orders (evaluated on a particular countable algebra) will be examined.

We show that propositional intuitionistic logic is the maximal abstract logic satisfying a certain form of compactness, the Tarski union property, and preservation under asimulations.

We characterize the non-trivial substructural logics having the variable sharing property as well as its strong version. To this end, we find the algebraic counterparts over varieties of these logical properties. -/- .

This is an introduction to the special issue of the AJL on Val Plumwood's manuscript "False Laws of Logic" and her other work in logic.

Analogues of Scott's isomorphism theorem, Karp's theorem as well as results on lack of compactness and strong completeness are established for infinitary propositional relevant logics. An "interpolation theorem" for the infinitary quantificational boolean logic L-infinity omega. holds. This yields a preservation result characterizing the expressive power of infinitary relevant languages with absurdity using the model-theoretic relation of relevant directed bisimulation as well as a Beth definability property.

We provide a sucient frame-theoretic condition for a super bi-intuitionistic logic to have Maksimova's variable separation property. We conclude that bi-intuitionistic logic enjoys the property. Furthermore, we offer an algebraic characterization of the super-bi-intuitionistic logics with Maksimova's property.

Beall and Cotnoir (2017) argue that theists may accept the claim that God's omnipotence is fully unrestricted if they also adopt a suitable nonclassical logic. Their primary focus is on the infamous Stone problem (i.e., whether God can create a stone too heavy for God to lift). We show how unrestricted omnipotence generates Curry‐like paradoxes. The upshot is that Beall and Cotnoir only provide a solution to one version of the Stone problem, but that unrestricted omnipotence generates other problems which (...) they do not adequately address. (shrink)

This paper is a contribution to graded model theory, in the context of mathematical fuzzy logic. We study characterizations of classes of graded structures in terms of the syntactic form of their first-order axiomatization. We focus on classes given by universal and universal-existential sentences. In particular, we prove two amalgamation results using the technique of diagrams in the setting of structures valued on a finite MTL-algebra, from which analogues of the Łoś–Tarski and the Chang–Łoś–Suszko preservation theorems follow.

In this paper we explore the following question: how weak can a logic be for Rosser's essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson's Q is essentially undecidable in intuitionistic logic, and P. Hajek proved it in the fuzzy logic BL for Grzegorczyk's variant of Q which interprets the arithmetic operations as non-total non-functional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a much (...) weaker arithmetic theory, a version of Robinson's R (with arithmetic operations also interpreted as mere relations). Our result is based on a structural version of the undecidability argument introduced by Kleene and we show that it goes well beyond the scope of the Boolean, intuitionistic, or fuzzy logic. (shrink)

Following a result by De Rijke for modal logic, it is shown that the basic weak entailment model-theoretic language with absurdity is the maximal model-theoretic language having the finite occurrence property, preservation under relevant directed bisimulations and the finite depth property. This can be seen as a generalized preservation theorem characterizing propositional weak entailment formulas among formulas of other model-theoretic languages.

It is a great pleasure to present this special issue celebrating the 85th birthday in 2022 of British–Australian logician John Newsome Crossley (JNC). John’s mu.

La definición de un “mundo posible” de Robert Adams es paradójica, de acuerdo con Selmer Bringsjord, Patrick Grim y Cristopher Menzel. Las pruebas de Bringsjord y Grim utilizaban el axioma del Conjunto Potencia; Cristopher Menzel objetó que, mientras este fuese el caso, todavía existía esperanza para la definición de Adams, pero Menzel desempolvó una vieja paradoja de Russell para demostrar que podíamos obtener las mismas conclusiones sin apelar a otra teoría de conjuntos que el Axioma de Separación. Sin embargo, el (...) resultado de Menzel mostraba solo que no existía el mundo actual. En este trabajo intentamos generalizar la paradoja de Russell a mundos posibles arbitrarios sin necesidad de introducir conceptos modales en la discusión. (shrink)

Elimination of quantifiers is shown to fail dramatically for a group of well‐known mathematical theories (classically enjoying the property) against a wide range of relevant logical backgrounds. Furthermore, it is suggested that only by moving to more extensional underlying logics can we get the property back.

We consider a modal language over crisp frames and formulas evaluated on a finite MTL-chain (a linearly ordered commutative integral residuated lattice). We first show that the basic modal abstract logic with constants for the values of the MTL-chain is the maximal abstract logic satisfying Compactness, the Tarski Union Property and strong invariance for bisimulations. Finally, we improve this result by replacing the Tarski Union Property by a relativization property.

In this paper we explore the following question: how weak can a logic be for Rosser’s essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson’s Q is essentially undecidable in intuitionistic logic, and P. Hájek proved it in the fuzzy logic BL for Grzegorczyk’s variant of Q which interprets the arithmetic operations as nontotal nonfunctional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a much (...) weaker arithmetic theory, a version of Robinson’s R (with arithmetic operations also interpreted as mere relations). Our result is based on a structural version of the undecidability argument introduced by Kleene and we show that it goes well beyond the scope of the Boolean, intuitionistic, or fuzzy logic. (shrink)

In this note we provide a simple proof of the incompactness over Routley-Meyer B-frames of the A1 fragment of the second order propositional relevant language.

It is shown that the classes of Routley-Meyer models which are axiomatizable by a theory in a propositional relevant language with fusion and the Ackermann constant can be characterized by their closure under certain model-theoretic operations involving prime filter extensions, relevant directed bisimulations and disjoint unions.

A traditional aspect of model theory has been the interplay between formal languages and mathematical structures. This dissertation is concerned, in particular, with the relationship between the languages of relevant logic and Routley-Meyer models. One fundamental question is treated: what is the expressive power of relevant languages in the Routley-Meyer framework? In the case of finitary relevant propositional languages, two answers are provided. The first is that finitary propositional relevant languages are the fragments of first order logic preserved under relevant (...) directed bisimulations. The second is that, when we restrict our attention to what can be labelled as De Morgan models, we can obtain an analogue of Lindström's theorem for finitary propositional relevant languages. Furthermore, it is shown that a preservation theorem characterizing the expressive power of infinitary relevant languages in classical infinitary languages follows as a consequence of an interpolation theorem for classical infinitary logic. In addition, algebraic characterizations of the classes of Routley-Meyer models axiomatizable in relevant propositional languages, incompactness of infinitary relevant propositional languages and the expressive power of quantificational relevant languages are discussed. A final chapter is devoted to the study of relevant languages as second order frame languages. In particular we devote our attention to the problem of which properties expressible by relevant languages are elementary and which are not. An algebraic characterization of such elementary properties together with some examples of non first order properties axiomatizable in relevant logic are given. Finally, a Sahlqvist-van Benthem algorithm showing that relevant formulas with a certain syntactic form express calculable first order properties at the level of frames is established. (shrink)

In “God of the Gaps: A Neglected Reply to God’s Stone Problem”, Jc Beall and A. J. Cotnoir offer a gappy solution to the paradox of (unrestricted) omnipotence that is typified by the classic stone problem. Andrew Tedder and Guillermo Badia, however, have recently argued that this solution could not be extended to a more serious Curry-like version of the paradox. In this paper, we show that such a gappy solution does extend to it.

In ‘Currying omnipotence: A reply to Beall and Cotnoir’, Andrew Tedder and Guillermo Badia argue that Jc Beall and A. J. Cotnoir’s gappy solution to the traditional paradox of unrestricted omnipotence does not extend to a Curry-like version of the paradox. In this paper, we show that it does extend to it.

The purpose of this paper is to introduce everyday aesthetics in education. First, it presents everyday aesthetics as a subdiscipline within philosophical aesthetics, that revisits sensory perception as the backdrop of all experience, claims ordinary life is a proper venue for aesthetic inquiry, and problematizes the impact aesthetic preferences have on habitual decisions. Second, the paper argues that among the diverse matters students learn in school, they learn—explicitly or implicitly—what and how to perceive, as well as the pedagogical purposes of (...) daily perception. School models the relationships that students will develop between what they perceive, what they think about it, and how they act accordingly. Next, the paper elaborates on the notions of “school aesthetic matrixes” and “space and place in school”. School aesthetic matrixes refer to perceptual patterns that allow characterizing relationships between daily perception and the pedagogical project of a school. They manifest through expressions such as prison school, home school, mall school, etc. Space and place in school allude to concomitant experiences, of corporeal basis, that allow describing how we inhabit the world. What distinguishes them is the approach each one gives to the senses and the body. The paper concludes by calling attention to the comprehension and role of perception in education, urging to mind the aesthetic-pedagogical character that students develop throughout 12 years of schooling. Ultimately, the key relationships we develop with ourselves, others, and the world, entail a basic perceptual dimension that can be educated to become more purposefully and integrally present in life. (shrink)

The field of Philosophy and Education seems to be experiencing a renewed interest in the work of Aristotle. As recently reviewed by Curren (Oxf Rev Educ 36(5):543–559, 2010), most of this attention aligns with the virtue ethics movement where themes like moral development in education, and the inquiry on human flourishing as the aim of education are prevalent. For sources, this scholarship relies heavily and extensively on the Nicomachean Ethics and Politics’ Book VIII where Aristotle develops his single, clearly defined (...) account of education. Among the short list of scholars who include Poetics in their research, their work seems to return to issues of morality and education (Carr in J Aesthet Educ 44(3):1–15, 2010; Gupta in J Aesthet Educ 44(4):60–80, 2010). This paper is an attempt to rediscover Aristotle’s insights on the peculiar type of learning that the arts seem to favor. In order to carry on this investigation, I will first guide my argument towards Aristotle’s remarks on poiesis. Then, I will focus on what constitutes the heart of poiesis, that is, mimesis. I will pay special attention to Aristotle’s consideration of both poiesis and mimesis as ultimately dealing with what he calls “possibility.” I will argue that learning in the arts entails the coming together of these three qualities. This would be a way of learning characterized by seeing human likenesses emerge as such in our interactions with artworks. (shrink)

Let us imagine that a man loses his keys one night and starts looking for them under the light of a street lamp. When people join him to help him search, they ask where it was that he thinks he might have let them fall; with a frustrated look on his face, he then points into the dark distance and says, by way of explanation, “I am looking under the lamppost because this is where the light is!” This story, introduced (...) by Janice Ross, provides us with a metaphorical description of an ongoing situation in arts education advocacy1: Certainly, I can only look for the arts in education where there is something about them that can be seen. Yet there may be aspects of them concealed beyond the reach of the vision.. (shrink)

The goal of this study was to introduce a model explaining how managers' attitudes, subjective norms, attributions, and the individualism-collectivism cultural dimension affect the way managers' deal with employee bribery in organizations. Twenty-six internal and external attributions related to bribery were identified through a series of structured interviews with 65 subject matter experts. These attributions, together with the other variables in the model, were evaluated by 354 Ecuadorian managers. Hierarchical regression analyses indicated that attitudes and external attributions significantly predicted managers' (...) intentions to discipline employees who accepted a bribe. In addition, external attributions mediated the relationship between individualism-collectivism and intent to discipline corrupt employees. Implications for the management of bribery in Latin American organizations are discussed. (shrink)

En el escrito que aquí se rescata, publicado originalmente por la Universidad Nacional del Litoral en 1936, Vicente Fatone (Buenos aires, 1903-1962) concibe la danza y la arquitectura como modos privilegiados de acceso a lo absoluto. El filósofo argentino identifica la arquitectura como un infinito espacial y con una ascética, y a la danza, como infinito temporal, con una mística: mientras la primera es entendida como instauración de un mundo, la segunda es considerada como un ejercicio de libertad, que expresa (...) lo divino que habita en el ser humano a través de un principio dinámico. (shrink)

We present a mathematical approach to defeasible reasoning based on arguments. This approach integrates the notion of specificity introduced by Poole and the theory of warrant presented by Pollock. The main contribution of this paper is a precise, well-defined system which exhibits correct behavior when applied to the benchmark examples in the literature. It aims for usability rather than novelty. We prove that an order relation can be introduced among equivalence classes of arguments under the equi-specificity relation. We also prove (...) a theorem that ensures the termination of the process of finding the justified facts. Two more lemmas define a reduced search space for checking specificity. In order to implement the theoretical ideas, the language is restricted to Horn clauses for the evidential context. The language used to represent defeasible rules has been restricted in a similar way. The authors intend this work to unify the various existing approaches to argument-based defeasible reasoning. (shrink)

O artigo – desde uma perspectiva da educação como direito humano universal que reconhece a diversidade – realiza um recorrido panorâmico por uma série de dimensões da educação intercultural no ensino superior do Chile, especialmente universitária, discute criticamente a situação atual da interculturalização na universidade – entendida genericamente – e conclui com algumas perguntas que podem orientar pesquisas ou decisões que levem a que o ensino superior considere os direitos indígenas, permita a expressão de sua cultura, promova a interculturalidade, forme (...) os técnicos e profissionais que requerem o desenvolvimento da sociedade e das comunidades e enriqueça a docência e a ciência com outros paradigmas de conhecimento. Desse modo, que reconheça e valide a interculturalidade e os aportes das culturas indígenas a sua identidade, formação e produção de conhecimento. Trata das comunidades e dos estudantes indígenas em geral, mas aprofunda a situação desde o povo mapuche, embasando-se em projetos de pesquisa e desenvolvimento e em experiências em e desde a Região da Araucanía. A proposta se centra na ideia de que o direito à educação, desde a lei e inconstitucionalidade atual, se centra no sistema escolar formal de ensino básico e médio e não considera a educação superior ou a um sistema de educação continuado como direito universal, no qual poderia se especificar uma condição especial aos estudantes indígenas. Por isso, hoje se luta pela igualdade de oportunidades de inclusão entre indígenas e não indígenas. Palavras-chave: Ensino Superior. Interculturalidade. Cultura mapuche. (shrink)

This paper offers an exposition of Husserl's mature philosophy of mathematics, expounded for the first time in Logische Untersuchungen and maintained without any essential change throughout the rest of his life. It is shown that Husserl's views on mathematics were strongly influenced by Riemann, and had clear affinities with the much later Bourbaki school.