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 of consequence relation standard in abstract treatments of logic to accommodate intuitions of relevance. The guiding idea follows the use criterion, according to which in order for some premises to have some conclusion(s) as consequence(s), the premises must each be used in 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 between multisets. 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)

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)

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.

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)

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.

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.

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.

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 (understood as pairs of sets of formulas) 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)

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.

In lieu of an abstract, here is a brief excerpt of the content:On Franco-Ferraz, Theism and the Theatre of the Mind MiguelA. Badia-Cabrera In "Theatre andReligiousHypothesis,"1 MariaFranco-Ferraz offersan eloquent and reasoned argument in favour ofa fresh and different sort of hermeneutic approach to the Dialogues Concerning Natural Religion as a suitable means to disentangle the web of proverbially difficult philosophical questions posed by Hume in that work. In order to arrive at a coherent understanding ofthe Dialogues as a whole (...) and perhaps also to calm the sense of bafflement which the confrontation with this enigmatical text always provokes, many Hume scholars have taken as a point of departure and clue for unravelling it either one or two of Hume's principal philosophical doctrines or a presumedly unitary view of Hume's philosophy in general. Equipped with one or both of these instruments, they proceed to dissect the arguments of the Dialogues with the intention of determining what Hume's position really is. This task also almost invariably ends by assigning the victory in the dialectical struggle and the representation of the author's 'true' opinions to the character (sometimes even more than one) in whom they are more or less able to find the purest expression ofwhat theyhave previously decided is the kernel oressence ofHume's philosophy. Franco-Ferraz has taken a very different route. For her the key point of the Dialogues lies in the fictional setting in which the theological discussion takes place. She goes on to show that the literary form of the Dialogues is intrinsically connected with its philosophical content so as to serve to elucidate the precise nature ofthe conclusion which is established at the end, in Part XII. In other words, the theatrical structure facilitates a more profound and detailed investigation ofthe problems dealing with the existence and nature of God. In fact, Hume offers no final solutions to these problems and thereby none ofthe characters ofthe Dialogues can properly be said to present the victorious point of view or the one counting with the complete authority ofthe author ofthe play: The most important aspect of this text is the interaction between the different characters' points of view, which, by being in confrontation with each other, undergo reciprocal transformations. (Franco-Ferraz, 228) Volume XVI Number 2 131 MIGUEL A. BADÍA-CABRERA With this one cannot but agree. In addition she clinches her case by 'ai-trull·/ and convincingly showing that there is a profound relation between this theatrical form and some substantive philosophical theses which Hume developed in the Treatise, the first Enquiry, and The Natural History ofReligion. In reality, the theatre metaphor clarifies the sense ofthose philosophical doctrines, which in turn throw light on and corroborate her main contention about the conclusion of the Dialogues. For according to her, any definitive solution to the problems ofnatural theology is not presented at the end by any ofthe characters on behalfofa God-like author (Dieu auteur), and furthermore this deus ex machina solution would conflict especially with Hume's conception of the mind as being une pièce sans théâtre, a drama without a stage (Franco-Ferraz, 234). Such is the substance ofher thesis, if I have succeeded in giving a fairrepresentationofit. I thinkthatherinterpretation providesa novel outlook and one fruitful means ofaccess to an adequate understanding of the Dialogues. But I am quite hesitant with some particular and important aspects ofher reading of that text for the 'inconveniences', in Philo's sense ofthe word, which seem to follow from it, and alsohave a few difficulties in assessing the exact import of one or two of her central claims due to the somewhat vague manner in which, I think, these are formulated. First, at the verybeginningofher paper, Franco-Ferraz quotes the followingpassagefromPartIIIoftheDialoguesin which Demeaobjects to Cleanthes' claim, defended by two very ingenious analogies, that an orderly world like a coherent, articulate speech will always be taken as a sign ofintelligent design: When I read a volume, I enter into the mind and intention of the author: I become him, in a manner, for the instant; and have an immediate feeling and conception of those ideas, which revolved in his imagination, while employed in that composition. (D 155-56) I take issue with... (shrink)

This paper studies which truth-values are most likely to be taken on finite models by arbitrary sentences of a many-valued predicate logic. The classical zero-one law (independently proved by Fagin and Glebskiĭ et al.) states that every sentence in a purely relational language is almost surely false or almost surely true, meaning that the probability that the formula is true in a randomly chosen finite structures of cardinal n is asymptotically $0$ or $1$ as n grows to infinity. We obtain (...) generalizations of this result for any logic with values in a finite lattice-ordered algebra, and for some infinitely valued logics, including Łukasiewicz logic. The finitely valued case is reduced to the classical one through a uniform translation and Oberschelp’s generalization of the zero-one law. Moreover, it is shown that the complexity of determining the almost sure value of a given sentence is PSPACE-complete (generalizing Grandjean’s result for the classical case), and for some logics we describe completely the set of truth-values that can be taken by sentences almost surely. (shrink)

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 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.

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.

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)

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.

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.

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.

O presente artigo emergiu de uma reflexão sobre os percalços que afligem e chocam os professores, sobretudo no início da carreira, entre os quais se destaca tanto nos estudos quanto nos relatos propalados pelos próprios professores, a violência manifestada dentro e fora das salas de aula. Dada essa constatação e a evidente necessidade de sua superação para além de medidas efêmeras, esta pesquisa traça dois objetivos igualmente relevantes: a) incitar a discussão da violência no momento da formação docente inicial, bem (...) como o surgimento de novos estudos acerca dessa temática no âmbito da pesquisa acadêmica; e b) transgredir a tradição de pesquisas que consistem em recomendar prescrições a serem aplicadas sobre como banir a violência escolar, desconsiderando as peculiaridades de cada contexto. Tais objetivos são provenientes da seguinte questão norteadora: quais estratégias são desenvolvidas nos cursos de licenciatura que visam à preparação do futuro professor para lidar com o fenômeno da violência na escola básica? Intentando responder a essa questão e, portanto, contemplar os objetivos delineados optou-se pela análise de material bibliográfico especializado, sistematicamente selecionado, como procedimento metodológico. Como principal resultado, verificou-se que a discussão da temática da violência escolar na formação inicial é algo ainda escasso na prática e incipiente na esfera da pesquisa. Assim sendo, enfatiza-se a importância do fortalecimento de tal discussão a fim de que, ao ingressarem na carreira docente, futuros professores sintam-se devidamente subsidiados para enfrentarem os desafios cotidianos que venham a se manifestar ao longo do exercício da profissão. (shrink)

O presente texto trata das relações que podem existir entre filosofia e vida. Ao lado de uma filosofia do imaginário que procura o reencantamento da cultura escolar pela arte, é possível criar um novo imaginário da filosofia com a instauração de um outro “logos” que se afaste do paradigma clássico e do racionalismo. Michel Onfray, com seu “projeto existencial” e a criação da Universidade Popular, aparece como um exemplo dessa possibilidade que é aqui apresentada.

El estudio de la obra de Aristóteles es imprescindible para acercarnos al modo en que Heidegger responde a la pregunta de qué es el tiempo1. El objetivo de la presente comunicación es, en primer lugar, analizar la peculiar radicalización del concepto de tiempo aristotélico efectuada por el joven Heidegger; en segundo lugar, mostrar la estrecha relación entre la comprensión vulgar del tiempo ofrecida por Aristóteles y el tiempo originario, centrándonos en el papel que tiene el momento temporal del futuro en (...) la constitución de nuestro propio ser. (shrink)

Hume did not criticize Hutcheson’s moral-empirical argument in his published philosophical works, even though he forcefully denied, especially in Parts X and XI of the Dialogues Concerning Natural Religion, that we could empirically prove the moral attributes of the Deity. Yet he seemingly rejected this particular reasoning in a famous letter to Hutcheson, dated March 16, 1740. Hutcheson’s claim that our moral sense is a likely to be expected effect of divine benevolence and Hume’s critique of this claim are analyzed (...) in this essay. It is argued that the particular criticism presented by Hume in that letter is not sufficient to refute Hutcheson’s argument. It is also suggested that Hume may be right after all in rejecting Hutcheson’s inference about God’s benevolent design on other grounds, that is, our deep-rooted knowledge of the annoying conspicuousness of evil in the world. (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.

Assessing maturity is one of the main ethical challenges for bioethics. It is even more complex when it comes to adolescents, as they are still in their maturing process. Three difficulties emerge: a conceptual difficulty as maturity is related to autonomy, competence and capacity; a methodological difficulty regarding which indicators should be used to assess it; and a practical difficulty in how to properly deal with it, as it is not easy to respect a maturity that only for a little (...) while now is being recognized and promoted. This paper begins by answering the question of why it is important to assess and promote maturity. Then, a chronology will be offered of the evolution of maturity attribution to minors as well as of the main variables that have been defined to asses it. Finally, some concepts will be clarified in order to develop an agglutinating proposal on indicators, requirements and conditioning factors of mature decision-making processes. (shrink)

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.

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.

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)

Thepresentwork is arevisedand enlarged English versionofa book originally writtenin Spanish and published in late 1996, La rejlexion de DavidHume en lorno a /a religion. SinceDavidHume is arguablynot only the most important philosopherwhohaseverwrittenintheEnglishlanguage,butthemoststudiedand influential,itisonlynaturalthatsoonerthanlaterIwouldfeelthe urgencyto bring totheattentionofamuchwiderpublicaworkwhoseoutlook is, I think, signifi cantlydifferentfrom that ofother books which deal with the Scottish thinker's worksonreligionandnaturaltheology.Thisdesirewassostrongastoallowmeto overcome the all-too-natural fear that my wavering and uncertain command of English wouldmakethe few valuableinsights theworkmight containappearun clear,andmyphilosophicalerrors,evenmoreastonishing. This book is addressednot only to scholarswhomay beinterested in modem philosophy in (...) generalorHume'sphilosophyofreligion in particular,but also to themoreextensivecompassofreaderseitherintriguedortroubledby religion and themyriadofissuesandproblemsitposes,whichare, as it were, the primematter forphilosophicalanalysisandtheorizing.Inspiteofitsphilosophicaland linguistic limitations,forwhichIamentirelyresponsible,Iferventlyhopethatthisworkmay befoundtocontainsomethingofthatelusivetruthafterwhichHumestrove, andto whichheremainedconstantto theend, particularlyatatime whenthe priceto be paidforthepublicationofcontroversialphilosophicalandreligious views wascon siderablyhigherthanthatofbecominganeasytargetforuniversalridicule. Thecompletionofthisworkhasbeenmadepossible principally by asabbatical leavethattheUniversityofPuertoRico grantedme fortheyear 1999. This leave gavemethenecessarytimeto readandstudy thepertinentliterature,aswell asto writemostoftheEnglish manuscript.Italsoallowedmeto visit Edinburgh and spendmostofJuly andAugust 1999 attheNationalLibraryofScotlandand the EdinburghUniversityLibrary.Although Ihaveincorporatedinto thepresentwork onlyasmallpartofmyresearchatthose libraries,ifthis bookshows agreaterap preciation of the immediate social and religious context of Hume's thought than its Spanish predecessor, it is in no small measure due to the books, articles, and xi xii Preface manuscritsIwasabletoconsultatthe time. Inthis respect, Iwouldliketo thank Professor Peter Jones, Director of the Institute for Advanced Studies in the HumanitiesattheUniversityofEdinburgh, who was very instrumental in making myvisittoEdinburghareality,andgenerouslyputthefacilitiesoftheInstitute at mydisposal. Manythoughts andlinesofargumentcontained in this bookdatebackto my doctoraldissertation,andsomeearlierversionsofanumberofchaptersorsections ofchaptershavebeenpublished in differentphilosophyjournalsor anthologies. I oftheminthebibliographyundermy name. But sincemy previousworks listall onthe subjecthavebeenrevisedandmodifiedsomanytimes in the lightofnew fmdingsortomeetactualorpossiblecriticism,Ican saythatthey havebeeneffec tivelysupersededbythepresentwriting. (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)

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)