Although Fuzzy logic and Fuzzy Mathematics is a widespread subject and there is a vast literature about it, yet the use of Fuzzy issues like Fuzzy sets and Fuzzy numbers was relatively rare in time concept. This could be seen in the Fuzzy time series. In addition, some attempts are done in fuzzing Turing Machines but seemingly there is no need to fuzzy time. Throughout this article, we try to change this picture and show why it is helpful to consider (...) the instants of time as Fuzzy numbers. In physics, though there are revolutionary ideas on the time concept like B theories in contrast to A theory also about central concepts like space, momentum… it is a long time that these concepts are changed, but time is considered classically in all well-known and established physics theories. Seemingly, we stick to the classical time concept in all fields of science and we have a vast inertia to change it. Our goal in this article is to provide some bases why it is rational and reasonable to change and modify this picture. Here, the central point is the modified version of “Unexpected Hanging” paradox as it is described in "Is classical Mathematics appropriate for theory of Computation".This modified version leads us to a contradiction and based on that it is presented there why some problems in Theory of Computation are not solved yet. To resolve the difficulties arising there, we have two choices. Either “choosing” a new type of Logic like “Para-consistent Logic” to tolerate contradiction or changing and improving the time concept and consequently to modify the “Turing Computational Model”. Throughout this paper, we select the second way for benefiting from saving some aspects of Classical Logic. In chapter 2, by applying quantum Mechanics and Schrodinger equation we compute the associated fuzzy number to time. (shrink)

The main objective o f this descriptive paper is to present the general notion of translation between logical systems as studied by the GTAL research group, as well as its main results, questions, problems and indagations. Logical systems here are defined in the most general sense, as sets endowed with consequence relations; translations between logical systems are characterized as maps which preserve consequence relations (that is, as continuous functions between those sets). In this sense, logics together with translations form (...) a bicomplete category of which topological spaces with topological continuous functions constitute a full subcategory. We also describe other uses of translations in providing new semantics for non-classical logics and in investigating duality between them. An important subclass of translations, the conservative translations, which strongly preserve consequence relations, is introduced and studied. Some specific new examples of translations involving modal logics, many-valued logics, para- consistent logics, intuitionistic and classical logics are also described. (shrink)

In order to reason in a non-trivializing way with contradictions, para- consistent logics reject some classically valid inferences. As a way of re- covering some of these inferences, Graham Priest ([Priest, 1991]) proposed to nonmonotonically strengthen the Logic of Paradox by allowing the se- lection of “less inconsistent” models via a comparison of their respective inconsistent parts. This move recaptures a good portion of classical logic in that it does not block, e.g., disjunctive syllogism, unless it is (...) applied to contradictory assumptions. In Priest’s approach the inconsistent parts of models are compared in an extensional way by considering their inconsis- tent objects. This distinguishes his system from the standard format of (inconsistency-)adaptive logics pioneered by Diderik Batens, according to which (atomic) contradictions validated in models form the basis of their comparison. A well-known problem for Priest’s extensional approach is its lack of the Strong Reassurance property, i.e., for specific settings there may be infinitely descending chains of less and less inconsistent models, thus never reaching a minimally inconsistent model. In this paper, we demonstrate that Strong Reassurance holds for the extensional approach under a cardinality-based comparison of the incon- sistent parts of models. Furthermore, we introduce and investigate the metatheory of the class of first-order nonmonotonic inconsistency-tolerant construct over the extensional or quantitative comparisons of their respec- tive models. Core model-theoretic properties for these logics, such as the Löwenheim-Skolem theorems, along with other nonmonotonic properties, are further studied. (shrink)

I discuss this collection of essays on Hegel and the sciences while stressing the interest of Hegel's philosophy of nature in the light of later non-mainstream developments in the life-sciences and medicine. I compare then the chapters dedicated to Hegel's logic with recent literature on para-consistent logic and re-interpretations of Hegel's own logic.

Although Fuzzy logic and Fuzzy Mathematics is a widespread subject and there is a vast literature about it, yet the use of Fuzzy issues like Fuzzy sets and Fuzzy numbers was relatively rare in time concept. This could be seen in the Fuzzy time series. In addition, some attempts are done in fuzzing Turing Machines but seemingly there is no need to fuzzy time. Throughout this article, we try to change this picture and show why it is helpful to consider (...) the instants of time as Fuzzy numbers. In physics, though there are revolutionary ideas on the time concept like B theories in contrast to A theory also about central concepts like space, momentum… it is a long time that these concepts are changed, but time is considered classically in all well-known and established physics theories. Seemingly, we stick to the classical time concept in all fields of science and we have a vast inertia to change it. Our goal in this article is to provide some bases why it is rational and reasonable to change and modify this picture. Here, the central point is the modified version of “Unexpected Hanging” paradox as it is described in "Is classical Mathematics appropriate for theory of Computation".This modified version leads us to a contradiction and based on that it is presented there why some problems in Theory of Computation are not solved yet. To resolve the difficulties arising there, we have two choices. Either “choosing” a new type of Logic like “Para-consistent Logic” to tolerate contradiction or changing and improving the time concept and consequently to modify the “Turing Computational Model”. Throughout this paper, we select the second way for benefiting from saving some aspects of Classical Logic. In chapter 2, by applying quantum Mechanics and Schrodinger equation we compute the associated fuzzy number to time. These, provides a new interpretation of Quantum Mechanics.More exactly what we see here is "Particle-Fuzzy time" interpretation of quantum Mechanics, in contrast to some other interpretations of Quantum Mechanics like " Wave-Particle" interpretation. At the end, we propound a question about the possible solution of a paradox in Physics, the contradiction between General Relativity and Quantum Mechanics. (shrink)

En este texto retomo una pregunta importante que había quedado abierta al final del segundo capítulo de mi (2019): ¿cuáles son los límites del análisis? Sabemos que por obvias limitaciones materiales, todo análisis siempre será incompleto y por lo tanto, limitado. A estos límites les hemos llamado “metodológicos”, para distinguirlos de los metafísicos. Si el análisis tiene límites en este segundo sentido, aun si termináramos de analizar por completo un concepto, no podremos estar seguros de conocer todas sus propiedades (...) esenciales (ya que hay propiedades de este tipo que no las capta ningún análisis) o de ser capaces de distinguirlo de todos los conceptos (pues lo que lo distingue de otros conceptos puede ser inanalizable). Comúnmente se dice que el análisis no nos da todo el contenido de los conceptos, sino sólo su forma lógica o estructura semántica. Por lo tanto, otra manera de reformular la pregunta por los límites del análisis es ¿cuál es la relación entre forma y contenido? (shrink)

A presente dissertação tem como objetivo uma apresentação da proposta de Charles Kielkopf, de tradução da lógica deôntica standard em uma lógica normal alética e de seusresultados quanto à construção de um sistema de lógica deôntica que capture conceitos eprincípios kantianos como necessidade causal e as formulações do Imperativo Categórico acerca do Reino da Natureza e do Reino dos Fins. Uma vez que este processo resulta em uma interpretação de aspectos da filosofia kantiana, optou-se inicialmente por uma apresentação em linhas (...) gerais destas concepções e, tendo em vista as dificuldades referentes a aplicabilidade de um processo de tradução entre princípios deônticos e ônticos,fez-se necessária também uma exposição acerca do problema das barreiras inferenciais, bem como de sua conseqüência mais imediata, a saber, o Dilema de Jörgensen. Num segundo momento, também foi feita uma caracterização dos sistemas modais normais, tanto deônticos quanto aléticos, bem como da noção de tradução entre lógicas e dos modelos de Dawson. O capítulo final consiste justamente num exame crítico da proposta de Kielkopf, o qual faz uso dos modelos de Dawson para desenvolver uma lógica deôntica a partir do sistema lógico K1. Tais modelos de Dawson permitem a definição de uma lógica deôntica em termos de modalidades aléticas iteradas, com o que este modelo constitui-se em uma maneira de evitar os problemas referentes às barreiras inferenciais. O desenvolvimento de uma alternativa para a atribuição de um status lógico a concepções deônticas não constitui,contudo, o aspecto inovador desta proposta, tal aspecto encontra-se justamente na utilização por Kielkopf, de seu modelo formal como uma ferramenta para a investigação de concepções filosóficas, no caso, as concepções kantianas já citadas. -/- The current dissertation has as its main objective an exposition and critical analysis of Charles Kielkopf's translation of the Standard Deontic Logic (SDL) into a normal alethic logic, and the resulting construction of a system of deontic logic that captures Kantian fundamental concepts and principles such as the concept of causal necessity and the formulations of the Categorical Imperative concerning Kingdom of Nature and Kingdom of Ends. Since this process results in an interpretation of aspects of the Kantian philosophy, it has been chosen,initially, a presentation in general lines of these concepts and, considering the difficulties regarding the applicability of a translation process between deontic and ontic principles, it was done necessary also an exhibition concerning the problem of the inferential barriers, as well as of his more immediate consequence, i. e., Jörgensen's Dilemma. In a secon dmoment, it was also made a characterization of normal modal systems, deontic ones andalethic ones, as well as of the notions of translation between logics and of Dawson modelling. The final chapter consists of a critical examination of Kielkopf's proposal, which uses Dawson modelling to develop a deontic logic based on the alethic system K1. Such Dawson modelling allows the definition of a deontic modalities in terms of iterated alethic modalities,therefore this model constitutes a way of avoiding the problems regarding inferential barriers. The development of an alternative for attribution of logical status to deontic concepts doesn't constitute, however, in an innovative aspect of this proposal, such aspect can be founded inthe use made by Kielkopf of his formal model as a tool for the investigation of philosophical concepts, such as the Kantian concepts already mentioned. (shrink)

Although logical consistency is desirable in scientific research, standard statistical hypothesis tests are typically logically inconsistent. To address this issue, previous work introduced agnostic hypothesis tests and proved that they can be logically consistent while retaining statistical optimality properties. This article characterizes the credal modalities in agnostic hypothesis tests and uses the hexagon of oppositions to explain the logical relations between these modalities. Geometric solids that are composed of hexagons of oppositions illustrate the conditions for these modalities to be (...) logically consistent. Prisms composed of hexagons of oppositions show how the credal modalities obtained from two agnostic tests vary according to their threshold values. Nested hexagons of oppositions summarize logical relations between the credal modalities in these tests and prove new relations. (shrink)

Critico la teoría hermenéutica de Landgrebe sobre el cartesianismo de Husserl mostrando la estructura argumentativa en la que se inserta, lo que pretende y lo que está forzada a pretender. Su núcleo duro radica en los conceptos de crítica inmanente y lógica interna y tiene como meta final promover una idea de trascendental no-entitativa voluntarista y correlacional opuesta a la de un yo absoluto. Para probar esto, analizo tres niveles del cartesianismo de Landgrebe: primero, la síntesis contradictoria entre apodicticidad (...) y experiencia; segundo, su interpretación de Filosofía Primera, y, tercero, el objetivismo de Husserl. Concluyo de la función que juega el cartesianismo en este esquema y del fracaso de documentar con ejemplos la idea que el concepto no describe una noción en la filosofía de Husserl, sino que es una herramienta hermenéutica para controlar el sentido de trascendental en esta filosofía.I criticize Landgrebe’s hermeneutical theory about Husserl’s Cartesianism describing the structure of the argument where this concept is rooted, what it claims and what it is forced to claim. The theory’s hard core is based on the concepts of immanent critique and internal logic and it’s final goal consists in advancing a non-entitative voluntaristic and correlational definition of transcendental opposite to that of an absolute I. To prove this, I analyze three levels inside Landgrebe’s concept of Cartesianism: first, the contradictory synthesis of experience and apodicticity; second, his interpretation of First Philosophy, and, third, Husserl’s objectivism. I conclude from the goal that Cartesianism plays in this argument and the failure in documenting this idea with examples that Landgrebe’s Cartesianism is rather an hermeneutical tool made to control the definition of transcendental in Husserl’s philosophy than a concept describing Husserl’s thought. (shrink)

Simultaneous hypothesis tests can fail to provide results that meet logical requirements. For example, if A and B are two statements such that A implies B, there exist tests that, based on the same data, reject B but not A. Such outcomes are generally inconvenient to statisticians (who want to communicate the results to practitioners in a simple fashion) and non-statisticians (confused by conflicting pieces of information). Based on this inconvenience, one might want to use tests that satisfy logical requirements. (...) However, Izbicki and Esteves shows that the only tests that are in accordance with three logical requirements (monotonicity, invertibility and consonance) are trivial tests based on point estimation, which generally lack statistical optimality. As a possible solution to this dilemma, this paper adapts the above logical requirements to agnostic tests, in which one can accept, reject or remain agnostic with respect to a given hypothesis. Each of the logical requirements is characterized in terms of a Bayesian decision theoretic perspective. Contrary to the results obtained for regular hypothesis tests, there exist agnostic tests that satisfy all logical requirements and also perform well statistically. In particular, agnostic tests that fulfill all logical requirements are characterized as region estimator-based tests. Examples of such tests are provided. (shrink)

A fundamental question asked in modal logic is whether a given theory is consistent. But consistent with what? A typical way to address this question identifies a choice of background knowledge axioms (say, S4, D, etc.) and then shows the assumptions codified by the theory in question to be consistent with those background axioms. But determining the specific choice and division of background axioms is, at least sometimes, little more than tradition. This paper introduces generic theories for (...) propositional modal logic to address consistency results in a more robust way. As building blocks for background knowledge, generic theories provide a standard for categorical determinations of consistency. We argue that the results and methods of this paper help to elucidate problems in epistemology and enjoy sufficient scope and power to have purchase on problems bearing on modalities in judgement, inference, and decision making. (shrink)

Como tantas veces en filosofía, el título no sólo establece la línea de batalla, sino que expone los prejuicios y errores del autor, ya que si podemos o no dar sentido al juego de idiomas 'Ver las cosas como son' y si es posible tener una 'teoría filosófica' de percepción' (que sólo puede ser sobre cómo funciona el lenguaje de la percepción), a diferencia de uno científico, que es una teoría sobre cómo funciona el cerebro, son exactamente los problemas. Este (...) es el Clásico Searle, excelente y probablemente al menos tan bueno como cualquier otro puede producir, pero carente de una comprensión completa de las ideas fundamentales del Wittgenstein posterior y sin comprender los dos sistemas de marco de pensamiento, lo que podría haberlo hecho brillante. Al igual que en su trabajo anterior, Searle evita en gran medida la ciencia, pero hay lapsos frecuentes y no entiende que los problemas son siempre sobre los juegos de idiomas, un fracaso que comparte con casi todo el mundo. Después de proporcionar un marco que consiste en una Tabla de Intencionalidad basada en los dos sistemas de pensamiento y pensamiento y investigación de decisiones, doy un análisis detallado del libro. Aquellos que deseen un marco completo hasta la fecha para el comportamiento humano de la moderna dos sistemas punto de vista puede consultar mi libros Talking Monkeys 3ª ed (2019), Estructura Logica de Filosofia, Psicología, Mente y Lenguaje en Ludwig Wittgenstein y John Searle 2a ed (2019), Suicidio pela Democracia 4ª ed (2019), La Estructura Logica del Comportamiento Humano (2019), The Logical Structure de la Conciencia (2019), Comprender las Conexiones entre Ciencia, Filosofía, Psicología, Religión, Política , Economía, Hitoria y Literatura (2020) y Delirios Utópicos Suicidas en el siglo 21 5ª ed (2019), Observaciones sobre Imposibilidad, Incompletitud, Paraconsistencia, Indecidibilidad, Aleatoriedad, Computabilidad, Paradoja e Incertidumbre en Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, da Costa, Godel, Searle, Rodych Berto, Floyd, Moyal-Sharrock y Yanofsky (2019) y otras. Al igual que con Wittgenstein (en adelante W), todo lo que Searle (en adelante S) escribe es un tesoro y es maravilloso que permanezca afilado a medida que se acerca a los 80. A diferencia de la mayoría, incluso sus primeros trabajos siguen siendo relevantes y está trabajando en varios otros libros. También sugiero sus más de 100 conferencias y entrevistas en youtube, vimeo, etc., que, aunque inevitablemente un poco repetitivas, contienen muchas declaraciones que no están en sus escritos. He leído casi toda su obra, y he escuchado todas las conferencias, la mayoría de ellas 2 o 3 veces. Estos son de especial interés como (como Wittgenstein) que no lee de las notas, por lo que cada uno es único y no una réplica de un papel, y es un excelente orador extemporáneo que en su mayoría utiliza un lenguaje sin pretensiones (ambos tan diferentes de la mayoría de los demás). Las conferencias recientes dadas en las Universidades Europeas son magníficas, pero no te pierdas las antiguas como la conferencia de la BBC "Una realidad cambiante, la ciencia del comportamiento humano", que da un excelente relato de por qué la causalidad legítima y repetitiva del rápido cerebro sistema automático, no linguístico 1 (S1) es fundamentalmente diferente y no destacable de la misma manera que la complejidad ilimitada de razones que caracterizan el lento sistema deliberativo, consciente de la lengua 2 (S2), que genera una explosión combinatoria no generalmente representable de una manera útil por las leyes científicas. El sistema dual (S1, S2) método de describir el pensamiento utilizado en esta revisión, común a la investigación de razonamiento desde hace unos 20 años, es mío y no de Searle. Dado que recientemente he escrito un artículo de 75p analizando el trabajo de Searle en comparación con el de Wittgenstein (La estructura lógica de la filosofía, la psicología, la mente y el lenguaje como revelado por Ludwig Wittgenstein y John Searle) no lo repetiré y lo repetiré concentrarse en este libro solamente. (shrink)

In this paper I present a new way of understanding Dutch Book Arguments: the idea is that an agent is shown to be incoherent iff he would accept as fair a set of bets that would result in a loss under any interpretation of the claims involved. This draws on a standard definition of logical inconsistency. On this new understanding, the Dutch Book Arguments for the probability axioms go through, but the Dutch Book Argument for Reflection fails. The question of (...) whether we have a Dutch Book Argument for Conditionalization is left open. (shrink)

Moral conflicts are real, and while a deontic logic containing a modified "agglomeration rule" may be able to accommodate this fact, even the most sophisticated logic will still overlook much of what everyday normative reasoning involves.

In this paper, I set out some basic elements of a deontic logic with an implementation appropriate for handling conflicting legal obligations for purposes of programming autonomous machine agents. Kantian justice demands that the prescriptive system of enforceable public laws be consistent, yet statutes or case holdings may often describe legal obligations that contradict; moreover, even fundamental constitutional rights may come into conflict. I argue that a deontic logic of the law should not try to work around such conflicts (...) but, instead, identify and expose them so that the rights and duties that generate inconsistencies in public law can be explicitly qualified and the conflicts resolved. I then argue that a credulous, non-monotonic deontic logic can describe inconsistent legal obligations while meeting Kant’s demand for consistency in the prescriptive system of public law. Finally, I propose an implementation of this logic via a modified form of “answer set programming,” which I demonstrate with some simple examples. (shrink)

Imperatives cannot be true or false, so they are shunned by logicians. And yet imperatives can be combined by logical connectives: "kiss me and hug me" is the conjunction of "kiss me" with "hug me". This example may suggest that declarative and imperative logic are isomorphic: just as the conjunction of two declaratives is true exactly if both conjuncts are true, the conjunction of two imperatives is satisfied exactly if both conjuncts are satisfied—what more is there to say? Much more, (...) I argue. "If you love me, kiss me", a conditional imperative, mixes a declarative antecedent ("you love me") with an imperative consequent ("kiss me"); it is satisfied if you love and kiss me, violated if you love but don't kiss me, and avoided if you don't love me. So we need a logic of three -valued imperatives which mixes declaratives with imperatives. I develop such a logic. (shrink)

El presente trabajo pretende explicitar que los operadores modales, como construcciones lógicas, insertos en las interpretaciones modales (MI) de la mecánica cuántica son usados de manera informal sin una semántica modal adecuada. Primero se estudiarán en detalle los motivos por los que ninguna lógica cuántica puede ofrecer una base apropiada para formalizar estos operadores en contextos mecánico-cuánticos. A continuación, se presentará el enfoque de las historias cuánticas como una nueva lógica cuántica (NQL) intrínsecamente booleana como posible herramienta para (...) formalizar operadores modales en el contexto de las MI. Tras comentar brevemente las principales MI, se propondrá abordar un estudio de las modalidades presentes en ellas desde esta NQL prestando siempre especial atención al problema de la medida como hilo conductor. Finalmente, se pretende señalar cuáles han de ser los primeros pasos necesarios para ofrecer una exposición lógicamente consistente de las modalidades en contextos cuánticos relevantes con NQL como herramienta, así como señalar las ventajas que esta perspectiva podría llegar a ofrecer. (shrink)

Recent work in formal semantics suggests that the language system includes not only a structure building device, as standardly assumed, but also a natural deductive system which can determine when expressions have trivial truth-conditions (e.g., are logically true/false) and mark them as unacceptable. This hypothesis, called the `logicality of language', accounts for many acceptability patterns, including systematic restrictions on the distribution of quantifiers. To deal with apparent counter-examples consisting of acceptable tautologies and contradictions, the logicality of language is often paired (...) with an additional assumption according to which logical forms are radically underspecified: i.e., the language system can see functional terms but is `blind' to open class terms to the extent that different tokens of the same term are treated as if independent. This conception of logical form has profound implications: it suggests an extreme version of the modularity of language, and can only be paired with non-classical---indeed quite exotic---kinds of deductive systems. The aim of this paper is to show that we can pair the logicality of language with a different and ultimately more traditional account of logical form. This framework accounts for the basic acceptability patterns which motivated the logicality of language, can explain why some tautologies and contradictions are acceptable, and makes better predictions in key cases. As a result, we can pursue versions of the logicality of language in frameworks compatible with the view that the language system is not radically modular vis-a-vis its open class terms and employs a deductive system that is basically classical. (shrink)

We explore the view that Frege's puzzle is a source of straightforward counterexamples to Leibniz's law. Taking this seriously requires us to revise the classical logic of quantifiers and identity; we work out the options, in the context of higher-order logic. The logics we arrive at provide the resources for a straightforward semantics of attitude reports that is consistent with the Millian thesis that the meaning of a name is just the thing it stands for. We provide models (...) to show that some of these logics are non-degenerate. (shrink)

The Swiss psychologist Jean Piaget contends that children below the age of 12 see no necessity for the logical law of non-contradiction. I argue this view is problematic. First of all, Piaget's dialogues with children which are considered supportive of this position are not clearly so. Secondly, Piaget underestimates the necessary nature of following the logical law of non-contradiction in everyday discourse. The mere possibility of saying something significant and informative at all presupposes that the law of non-contradiction is enforced.

Epistemologists have begun paying attention to the phenomenon of _core cognition_ from developmental psychology. Core cognition posits innate automatic cognitive modules that enable children to quickly grasp and learn certain concepts. A key element of core cognition is sometimes named _core knowledge_ because it encodes the constraints, parameters, and concepts that are required for core cognition modules to function. Until now, no successful epistemological account of it has been presented, and it is difficult to integrate into standard accounts of epistemology (...) given that it is only implicitly believed, not accessible to explicit cognitive processing, and innate. In this paper I propose an account of the epistemology of core cognition, focussing on the epistemic status of this core knowledge. I argue that, rather than being knowledge, or some ordinary justified belief, it consists of Wittgensteinian _hinge certainties_. These are the implicit presupposition that we need for our epistemology to function. I illustrate the argument with the core cognition of causality. Finally, I propose that even though core knowledge consists of unjustified hinges, we are epistemically entitled to trust them to be accurate. (shrink)

We present a philosophical motivation for the logics of formal inconsistency, a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency within the object language. We shall defend the view according to which logics of formal inconsistency are theories of logical consequence of normative and epistemic character. This approach not only allows us to make inferences in the presence of contradictions, but offers a philosophically acceptable account of paraconsistency.

“Trends in Logic XVI: Consistency, Contradiction, Paraconsistency, and Reasoning - 40 years of CLE” is being organized by the Centre for Logic, Epistemology and the History of Science at the State University of Campinas (CLEUnicamp) from September 12th to 15th, 2016, with the auspices of the Brazilian Logic Society, Studia Logica and the Polish Academy of Sciences. The conference is intended to celebrate the 40th anniversary of CLE, and is centered around the areas of logic, epistemology, philosophy and history of (...) science, while bringing together scholars in the fields of philosophy, logic, mathematics, computer science and other disciplines who have contributed significantly to what Studia Logica is today and to what CLE has achieved in its four decades of existence. It intends to celebrate CLE’s strong influence in Brazil and Latin America and the tradition of investigating formal methods inspired by, and devoted to, philosophical views, as well as philosophical problems approached by means of formal methods. The title of the event commemorates one of the three main areas of CLE, what has been called the “Brazilian school of paraconsistency”, combining such a pluralist view about logic and reasoning. (shrink)

In this paper we introduce a Gentzen calculus for (a functionally complete variant of) Belnap's logic in which establishing the provability of a sequent in general requires \emph{two} proof trees, one establishing that whenever all premises are true some conclusion is true and one that guarantees the falsity of at least one premise if all conclusions are false. The calculus can also be put to use in proving that one statement \emph{necessarily approximates} another, where necessary approximation is a natural dual (...) of entailment. The calculus, and its tableau variant, not only capture the classical connectives, but also the `information' connectives of four-valued Belnap logics. This answers a question by Avron. (shrink)

ESP: El concepto de alma en las filosofías de Platón y Aristóteles, por la posición que ocupa en sus argumentos relativos a la esclavitud, fue central para la estructura ideológica sobre la que se sostuvo la institución esclavista durante la Colonia. Será importante remitirnos, en un primer momento, al estudio de Erwin Rohde sobre este concepto, para examinar la materialidad del hálito y plantear una relación entre la mirada y la configuración de imaginarios. Posteriormente, examinaremos el lugar que (...) tiene el alma en los argumentos de Platón y Aristóteles en favor de esclavitud, con énfasis en el papel de la mirada, el dominio sobre el cuerpo y la naturaleza de los cuerpos esclavizables. Finalmente, revisaremos en la obra de Frantz Fanon una crítica a la mirada europea y occidental que diseca el cuerpo negro al otorgarle un alma. -/- ENG: The concept of the soul in the philosophies of Plato and Aristotle, due to the posi-tion it occupies in their arguments concerning slavery, will be central to the ideo-logical structure on which the institution of slavery was sustained during colonial times. It will be important to refer, at first, to Erwin Rohde’s study of this concept, to examine the materiality of the breath and to propose a relationship between the gaze and the configuration of imaginaries. We will then examine the position of the soul in Plato’s and Aristotle’s arguments in favor of slavery, emphasizing the role of the gaze, the dominion over the body, and the nature of enslaved bodies. Finally, we will review in the work of Frantz Fanon a critique of the European and Western gaze that dissects the black body by granting it a soul. (shrink)

(...) However, whether we chose a weak or strong approximation, the set would not make any sense at all, if (once more) this choice would not be justified in either temporal or spatial sense or given the context of possible applicability of the set in different circumstances. This would obviously represent a dualism in itself as we would (for instance) posit and apply a full identity-equality-equivalence of x and y when applying Newtonian physics to certain observations we make (it would (...) be the case of neural correlates), and we would posit and apply a non - identity-equality-equivalence of x and y when applying Quantum mechanics to other observations. Following this dualism in and of theories, the same sate would need to be slightly modified: U2:(x~y)∪(x≈y); U3:(x~y)∩(x≈y); U4a:(x~y)⊆(x≈y) vs. U4b:(x~y)⊇(x≈y). (shrink)

In this paper the propositional logic LTop is introduced, as an extension of classical propositional logic by adding a paraconsistent negation. This logic has a very natural interpretation in terms of topological models. The logic LTop is nothing more than an alternative presentation of modal logic S4, but in the language of a paraconsistent logic. Moreover, LTop is a logic of formal inconsistency in which the consistency and inconsistency operators have a nice topological interpretation. This constitutes a new proof of (...) S4 as being "the logic of topological spaces", but now under the perspective of paraconsistency. (shrink)

This paper attempts to address the question what logical strength theories of truth have by considering such questions as: If you take a theory T and add a theory of truth to it, how strong is the resulting theory, as compared to T? Once the question has been properly formulated, the answer turns out to be about as elegant as one could want: Adding a theory of truth to a finitely axiomatized theory T is more or less equivalent to a (...) kind of abstract consistency statement. A large part of the interest of the paper lies in the way syntactic theories are 'disentangled' from object theories. (shrink)

NOTE TO THE READER - December, 2021 ●●●●● -/- After a long period of time devoted to research in other areas, the author returned to the subject of this paper in a book-length study, CRITIQUE OF IMPURE REASON: Horizons of Possibility and Meaning. In this book (Chapter 11, “The Metalogic of Meaning”), the position developed in the 1982 paper, "Referential Consistency as a Criterion of Meaning", has been substantively revised and several important corrections made. It is recommended that readers read (...) the revised and corrected version that appears in the book, CRITIQUE OF IMPURE REASON. ●●●●● -/- The book is available in a printed edition, ISBN 978-0-578-88646-6, from online booksellers such as Barnes & Noble, Amazon, and through independent bookstores. The book is also available as a freely downloadable _corrected second edition_ eBook, DOI: 10.5281/zenodo.5458352, through PhilSci, PhilPapers, HAL, and other online archives. ●●●●● -/- In addition to the above-mentioned book, readers may find "A Primer on Bartlett's CRITIQUE OF IMPURE REASON" to be useful as a guide to the nearly 900-page CRITIQUE. The primer is available as a free download through PhilPapers. ●●●●● -/- "Referential Consistency as a Criterion of Meaning" describes a logically compelling criterion of meaning — that is, a necessary condition of meaning, one which is non-arbitrary and compelling. One cannot _not_ accept the proposed criterion without self-referential inconsistency. This “metalogical” variety of self-referential inconsistency is new, opening a third category beyond semantical and pragmatical forms of self-referential inconsistency. ●●●●● -/- It is argued that such a criterion of meaning can serve as an instrument of internal criticism for any theoretical framework that permits reference to a class of objects. The paper combines the concern of the logical empiricists to formulate a rigorous meaning criterion, with the analytical interest in identifying and eliminating self-defeating statements through an analysis of the referential structure of theories. ●●●●● -/- The paper is followed by a list of other publications by the author that further develop and extend the ideas presented here. (shrink)

The doctrine of the Trinity is central to mainstream Christianity. But insofar as it posits “three persons” (Father, Son and Holy Spirit), who are “one God,” it appears as inconsistent as the claim that 1+1+1=1. -/- Much of the literature on “The Logical Problem of the Trinity,” as this has been called, attacks or defends Trinitarianism with little regard to the fourth century theological controversies and the late Hellenistic and early Medieval philosophical background in which it took shape. I argue (...) that this methodol- ogy, which I call “the Puzzle Approach,” produces obviously invalid arguments, and it is unclear how to repair it without collapsing into my preferred method- ology, “the Historical Approach,” which sees history as essential to the debate. I also discuss “mysterianism,” arguing that, successful or not, it has a different goal from the other approaches. I further argue that any solution from the His- torical Approach satisfies the concerns of the Puzzle Approach and mysterianism anyway. -/- I then examine the solution to the Logical Problem of the Trinity found in St. Gregory of Nyssa’s writings, both due to his place in the history of the doctrine, and his clarity in explicating what I call “the metaphysics of synergy.” I recast his solution in standard predicate logic and provide a formal proof of its consistency. I end by considering the possibilities for attacking the broader philosophical context of his defense and conclude that the prospects for doing so are dim. In any case, if there should turn out to be any problem with the doctrine of the Trinity at all, it will not be one of mere logical inconsistency in saying that “These Three are One.”. (shrink)

This paper investigates a set of issues connected with the so-called conservativeness argument against deflationism. Although I do not defend that argument, I think the discussion of it has raised some interesting questions about whether what I call “compositional principles,” such as “a conjunction is true iff its conjuncts are true,” have substantial content or are in some sense logically trivial. The paper presents a series of results that purport to show that the compositional principles for a first-order language, taken (...) together, have substantial logical strength, amounting to a kind of abstract consistency statement. (shrink)

Resumo -/- A partir da experiência de produção de uma videoaula de Lógica em Libras (Testa, Moraes, Bizio e Caló, 2021) para o IFSP FILOLIBRAS, inserida no contexto do projeto ‘O Ensino de Filosofia para Surdos: elaboração de material didático em uma perspectiva de inclusão escolar’ (Moraes e Bizio, 2021), levantamos algumas questões relativas ao arcabouço teórico do projeto. Após introduzirmos as motivações do projeto, explicamos como sua metodologia foi tratada no contexto da aula de Lógica, expondo as (...) principais dificuldades e soluções encontradas. Através da análise desta experiência, pretendemos levantar subsídios teóricos para pesquisas futuras no tema, incluindo a produção de um curso inteiramente voltado para o ensino de Lógica em Libras, não apenas enquanto área da Filosofia, mas também enquanto área de estudo multi e transdisciplinar, propedêutica ao fomento do Pensamento Crítico. -/- Palavras-chave: Ensino de Lógica, Educação Especial, Pedagogia Bilíngue, Libras. -/- Abstract -/- Based on the experience of producing a video lesson on Logic in Libras (Testa, Moraes, Bizio and Caló, 2021) to the IFSP FILOLIBRAS, inserted in the context of the project ‘O Ensino de Filosofia para Surdos: elaboração de material didático em uma perspectiva de inclusão escolar’ (‘Teaching Philosophy for Deaf students: elaboration of didactic material in a perspective of inclusive education’, in a free translation) (Moraes and Bizio, 2021), we raise some questions regarding the theoretical framework of the project. After introducing the motivations of the project, we explain how the chosen methodology was considered in the context of the Logic class, exposing the main difficulties and solutions found. Through the analysis of this experience, we intend to raise theoretical subsidies for future research on the subject, including the production of a course entirely focused on the teaching of Logic in Libras, not only as an area of Philosophy but also as a multi and transdisciplinary area of study, propaedeutic to fostering Critical Thinking. -/- Keywords: Teaching Logic, Special Education, Bilingual Pedagogy, Libras. (shrink)

The aim of this paper is threefold. Firstly, §1 and §2 introduce the novel concept logical akrasia by analogy to epistemic akrasia. If successful, the initial sections will draw attention to an interesting akratic phenomenon which has not received much attention in the literature on akrasia (although it has been discussed by logicians in different terms). Secondly, §3 and §4 present a dilemma related to logical akrasia. From a case involving the consistency of Peano Arithmetic and Gödel’s Second Incompleteness Theorem (...) it’s shown that either we must be agnostic about the consistency of Peano Arithmetic or akratic in our arithmetical theorizing. If successful, these sections will underscore the pertinence and persistence of akrasia in arithmetic (by appeal to Gödel’s seminal work). Thirdly, §5 concludes by suggesting a way of translating the dilemma of arithmetical akrasia into a case of regular epistemic akrasia; and further how one might try to escape the dilemma when it’s framed this way. (shrink)

Does rationality require logical omniscience? Our best formal theories of rationality imply that it does, but our ordinary evaluations of rationality seem to suggest otherwise. This paper aims to resolve the tension by arguing that our ordinary evaluations of rationality are not only consistent with the thesis that rationality requires logical omniscience, but also provide a compelling rationale for accepting this thesis in the first place. This paper also defends an account of apriori justification for logical beliefs that is (...) designed to explain the rational requirement of logical omniscience. On this account, apriori justification for beliefs about logic has its source in logical facts, rather than psychological facts about experience, reasoning, or understanding. This account has important consequences for the epistemic role of experience in the logical domain. In a slogan, the epistemic role of experience in the apriori domain is not a justifying role, but rather an enabling and disabling role. (shrink)

The consistent histories reformulation of quantum mechanics was developed by Robert Griffiths, given a formal logical systematization by Roland Omn\`{e}s, and under the label `decoherent histories', was independently developed by Murray Gell-Mann and James Hartle and extended to quantum cosmology. Criticisms of CH involve issues of meaning, truth, objectivity, and coherence, a mixture of philosophy and physics. We will briefly consider the original formulation of CH and some basic objections. The reply to these objections, like the objections themselves, involves (...) a mixture of physics and philosophy. These replies support an evaluation of the CH formulation as a replacement for the measurement, or orthodox, interpretation. (shrink)

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert’s consistency program. Similar efforts continue, but have been (...) expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages. (shrink)

Review of the book "Paraconsistent Logic: Consistency, Contradiction, and Negation by Water Carnielli and Marcelo Coniglio.

This paper presents proof that Buss's S22 can prove the consistency of a fragment of Cook and Urquhart's PV from which induction has been removed but substitution has been retained. This result improves Beckmann's result, which proves the consistency of such a system without substitution in bounded arithmetic S12. Our proof relies on the notion of "computation" of the terms of PV. In our work, we first prove that, in the system under consideration, if an equation is proved and either (...) its left- or right-hand side is computed, then there is a corresponding computation for its right- or left-hand side, respectively. By carefully computing the bound of the size of the computation, the proof of this theorem inside a bounded arithmetic is obtained, from which the consistency of the system is readily proven. This result apparently implies the separation of bounded arithmetic because Buss and Ignjatović stated that it is not possible to prove the consistency of a fragment of PV without induction but with substitution in Buss's S12. However, their proof actually shows that it is not possible to prove the consistency of the system, which is obtained by the addition of propositional logic and other axioms to a system such as ours. On the other hand, the system that we have considered is strictly equational, which is a property on which our proof relies. (shrink)

The paper provides an argument for the thesis that an agent’s degrees of disbelief should obey the ranking calculus. This Consistency Argument is based on the Consistency Theorem. The latter says that an agent’s belief set is and will always be consistent and deductively closed iff her degrees of entrenchment satisfy the ranking axioms and are updated according to the ranktheoretic update rules.

A rational man’s beliefs are not logically consistent, and he does not believe all the logical consequences of his beliefs. This is because in any situational context, we only accept certain believed sentences. Within that context, we insist that sentences be logically consistent, and we accept the logical consequences of the other sentences we accept in that context. But such sentences do not have to be consistent with sentences we accept in other contexts, nor will we always (...) accept in that context the logical consequences of sentences that we accept in other contexts. We go through many different situational contexts in the course of a day. For example, in trying to start a car with a dead battery, explaining why the car won’t start may form one context, while explaining how to start it may form a different context; sentences we accept in the first context may be logically inconsistent with sentences we accept in the second. (shrink)

The Logic of Causation: Definition, Induction and Deduction of Deterministic Causality is a treatise of formal logic and of aetiology. It is an original and wide-ranging investigation of the definition of causation (deterministic causality) in all its forms, and of the deduction and induction of such forms. The work was carried out in three phases over a dozen years (1998-2010), each phase introducing more sophisticated methods than the previous to solve outstanding problems. This study was intended as part of a (...) larger work on causal logic, which additionally treats volition and allied cause-effect relations (2004). The Logic of Causation deals with the main technicalities relating to reasoning about causation. Once all the deductive characteristics of causation in all its forms have been treated, and we have gained an understanding as to how it is induced, we are able to discuss more intelligently its epistemological and ontological status. In this context, past theories of causation are reviewed and evaluated (although some of the issues involved here can only be fully dealt with in a larger perspective, taking volition and other aspects of causality into consideration, as done in Volition and Allied Causal Concepts). Phase I: Macroanalysis. Starting with the paradigm of causation, its most obvious and strongest form, we can by abstraction of its defining components distinguish four genera of causation, or generic determinations, namely: complete, partial, necessary and contingent causation. When these genera and their negations are combined together in every which way, and tested for consistency, it is found that only four species of causation, or specific determinations, remain conceivable. The concept of causation thus gives rise to a number of positive and negative propositional forms, which can be studied in detail with relative ease because they are compounds of conjunctive and conditional propositions whose properties are already well known to logicians. The logical relations (oppositions) between the various determinations (and their negations) are investigated, as well as their respective implications (eductions). Thereafter, their interactions (in syllogistic reasoning) are treated in the most rigorous manner. The main question we try to answer here is: is (or when is) the cause of a cause of something itself a cause of that thing, and if so to what degree? The figures and moods of positive causative syllogism are listed exhaustively; and the resulting arguments validated or invalidated, as the case may be. In this context, a general and sure method of evaluation called ‘matricial analysis’ (macroanalysis) is introduced. Because this (initial) method is cumbersome, it is used as little as possible – the remaining cases being evaluated by means of reduction. Phase II: Microanalysis. Seeing various difficulties encountered in the first phase, and the fact that some issues were left unresolved in it, a more precise method is developed in the second phase, capable of systematically answering most outstanding questions. This improved matricial analysis (microanalysis) is based on tabular prediction of all logically conceivable combinations and permutations of conjunctions between two or more items and their negations (grand matrices). Each such possible combination is called a ‘modus’ and is assigned a permanent number within the framework concerned (for 2, 3, or more items). This allows us to identify each distinct (causative or other, positive or negative) propositional form with a number of alternative moduses. This technique greatly facilitates all work with causative and related forms, allowing us to systematically consider their eductions, oppositions, and syllogistic combinations. In fact, it constitutes a most radical approach not only to causative propositions and their derivatives, but perhaps more importantly to their constituent conditional propositions. Moreover, it is not limited to logical conditioning and causation, but is equally applicable to other modes of modality, including extensional, natural, temporal and spatial conditioning and causation. From the results obtained, we are able to settle with formal certainty most of the historically controversial issues relating to causation. Phase III: Software Assisted Analysis. The approach in the second phase was very ‘manual’ and time consuming; the third phase is intended to ‘mechanize’ much of the work involved by means of spreadsheets (to begin with). This increases reliability of calculations (though no errors were found, in fact) – but also allows for a wider scope. Indeed, we are now able to produce a larger, 4-item grand matrix, and on its basis find the moduses of causative and other forms needed to investigate 4-item syllogism. As well, now each modus can be interpreted with greater precision and causation can be more precisely defined and treated. In this latest phase, the research is brought to a successful finish! Its main ambition, to obtain a complete and reliable listing of all 3-item and 4-item causative syllogisms, being truly fulfilled. This was made technically feasible, in spite of limitations in computer software and hardware, by cutting up problems into smaller pieces. For every mood of the syllogism, it was thus possible to scan for conclusions ‘mechanically’ (using spreadsheets), testing all forms of causative and preventive conclusions. Until now, this job could only be done ‘manually’, and therefore not exhaustively and with certainty. It took over 72’000 pages of spreadsheets to generate the sought for conclusions. This is a historic breakthrough for causal logic and logic in general. Of course, not all conceivable issues are resolved. There is still some work that needs doing, notably with regard to 5-item causative syllogism. But what has been achieved solves the core problem. The method for the resolution of all outstanding issues has definitely now been found and proven. The only obstacle to solving most of them is the amount of labor needed to produce the remaining (less important) tables. As for 5-item syllogism, bigger computer resources are also needed. (shrink)

ABSTRACT: An introduction to Stoic logic. Stoic logic can in many respects be regarded as a fore-runner of modern propositional logic. I discuss: 1. the Stoic notion of sayables or meanings (lekta); the Stoic assertibles (axiomata) and their similarities and differences to modern propositions; the time-dependency of their truth; 2.-3. assertibles with demonstratives and quantified assertibles and their truth-conditions; truth-functionality of negations and conjunctions; non-truth-functionality of disjunctions and conditionals; language regimentation and ‘bracketing’ devices; Stoic basic principles of propositional logic; 4. (...) Stoic modal logic; 5. Stoic theory of arguments: two premisses requirement; validity and soundness; 6. Stoic syllogistic or theory of formally valid arguments: a reconstruction of the Stoic deductive system, which consisted of accounts of five types of indemonstrable syllogisms, which function as nullary argumental rules that identify indemonstrables or axioms of the system, and four deductive rules (themata) by which certain complex arguments can be reduced to indemonstrables and thus shown to be formally valid themselves; 7. arguments that were considered as non-syllogistically valid (subsyllogistic and unmethodically concluding arguments). Their validity was explained by recourse to formally valid arguments. (shrink)

Why are natural theories pre-well-ordered by consistency strength? In previous work, an approach to this question was proposed. This approach was inspired by Martin's Conjecture, one of the most prominent conjectures in recursion theory. Fixing a reasonable subsystem $T$ of arithmetic, the goal was to classify the recursive functions that are monotone with respect to the Lindenbaum algebra of $T$. According to an optimistic conjecture, roughly, every such function must be equivalent to an iterate $\mathsf{Con}_T^\alpha$ of the consistency operator ``in (...) the limit'' within the ultrafilter of sentences that are true in the standard model. In previous work the author established the first case of this optimistic conjecture; roughly, every recursive monotone function is either as weak as the identity operator in the limit or as strong as $\mathsf{Con}_T$ in the limit. Yet in this note we prove that this optimistic conjecture fails already at the next step; there are recursive monotone functions that are neither as weak as $\mathsf{Con}_T$ in the limit nor as strong as $\mathsf{Con}_T^2$ in the limit. In fact, for every $\alpha$, we produce a function that is cofinally equivalent to $\mathsf{Con}^\alpha_T$ yet cofinally equivalent to $\mathsf{Con}_T$. (shrink)

This piece explores the meaning of the following quote from Charles Peirce (1902), ". . . the main reason logic is unsettled is that thirteen different opinions are current as to the true aim of the science. Now this is not a logical difficulty, but an ethical difficulty; for ethics is the science of aims. Secondly, it is true that ethics has been, and always must be, a theatre of discussion for the reason that its study consists in the gradual (...) development of a distinct recognition of a satisfactory aim. It is a science of subtleties, no doubt; but it is not logic, but the development of the ideal, which really creates and resolves the problems of ethics.". (shrink)

In this paper, I examine Quine's views on the epistemology of logic. According to Quine's influential holistic account, logic is central in the “web of belief” that comprises our overall theory of the world. Because of this, revisions to logic would have devastating systematic consequences, and this explains why we are loath to make such revisions. In section1, I clarify this idea and thereby show that Quine actually takes the web of belief to have asymmetrical internal structure. This raises two (...) puzzles. First, as I show in section 2, Quine's mature thoroughly naturalized view has it that logic is simply obvious, and this is explains why we do not typically consider revising it. While Quine presents this naturalized view as a way to make good on his earlier metaphor of centrality in a web of belief, I argue that the resources of Quine's naturalized epistemology cannot adequately explain why we are reluctant to revise logic. And, Quine seems to recognize this point himself. In light of this, I explain in section 3 how Quine can allow that our overall scientific theory has systematic structure in a way that is consistent with his naturalistic strictures. Second, the asymmetrical internal structure of the web of belief seems to be inconsistent with its being a holistic web at all. I defuse this problem in section 4 by showing how Quine distinguishes between structural and confirmational considerations. I close by using this distinction to show how Quine's view can evade Michael Friedman’s criticisms, and allow for important methodological distinctions between areas of the web of belief. (shrink)

It is argued that the goal of Hilbert's program was to prove the model-theoretical consistency of different axiom systems. This Hilbert proposed to do by proving the deductive consistency of the relevant systems. In the extended independence-friendly logic there is a complete proof method for the contradictory negations of independence-friendly sentences, so the existence of a single proposition that is not disprovable from arithmetic axioms can be shown formally in the extended independence-friendly logic. It can also be proved by means (...) of independence-friendly logic that proof-theoretical consistency of a sentence S implies the existence of a model in which S is not false. Hence the consistency of the axioms of arithmetic in the sense of being not-false in a model can be proved. (shrink)

The Symbolic Logic Study Guide is designed to accompany the widely used symbolic logic textbook Language, Proof and Logic (LPL), by Jon Barwise and John Etchemendy (CSLI Publications 2003). The guide has two parts. The first part contains condensed, essential lecture notes, which streamline and systematize the first fourteen chapters of the book into seven teaching sections, and thus provide a clear, well-designed roadmap for the understanding of the text. The second part consists of twelve sample quizzes and solutions. The (...) Symbolic Logic Study Guide is essential for all instructors and students who use LPL in their symbolic logic classes. (shrink)

An old and well-known objection to non-classical logics is that they are too weak; in particular, they cannot prove a number of important mathematical results. A promising strategy to deal with this objection consists in proving so-called recapture results. Roughly, these results show that classical logic can be used in mathematics and other unproblematic contexts. However, the strategy faces some potential problems. First, typical recapture results are formulated in a purely logical language, and do not generalize nicely to languages (...) containing the kind of vocabulary that usually motivates non-classical theories—for example, a language containing a naïve truth predicate. Second, proofs of recapture results typically employ classical principles that are not valid in the targeted non-classical system; hence, non-classical theorists do not seem entitled to those results. In this paper, we analyze these problems and provide solutions on behalf of non-classical theorists. To address the first problem, we provide a novel kind of recapture result, which generalizes nicely to a truth-theoretic language. As for the second problem, we argue that it relies on an ambiguity and that, once the ambiguity is removed, there are no reasons to think that non-classical logicians are not entitled to their recapture results. (shrink)