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Logical Reasoning with Diagrams
... On the other hand, this research can also be included within a line of research that aims to illustrate how use visualization can be a powerful tool for better understanding some logical concepts. The line of research that explores visual arguments is known as proofs without words [10][11][12]. The images created in this line should help understand mathematical ideas, demonstrations, and arguments. ...
... The ability to confirm this fact at a glance can be exceptionally valuable. Consequently, this work opens up new avenues for research, much like the research direction that Nelsen [10] and others [11,12] are currently pursuing in the realm of theorem proving through images. ...
This article presents different artistic raster images as a resource for correcting misconceptions about different laws and assumptions that underlie the propositional systems of binary logic, Łukasiewicz's trivalent logic, Peirce's trivalent logic, Post's n-valent logic, and Black and Zadeh's infinite-valent logic. Recognizing similarities and differences in how images are constructed allows us to deepen, through comparison, the laws of bivalence, non-contradiction, and excluded middle, as well as understanding other multivalent logic assumptions from another perspective, such as their number of truth values. Consequently, the first goal of this article is to illustrate how the use of visualization can be a powerful tool for better understanding some logic systems. To demonstrate the utility of this objective, we illustrate how a deeper understanding of logic systems helps us appreciate the necessity of employing Likert scales based on the logic of Post or Zadeh, which is the second goal of the article.
... It can be applied more simply and systematically than other pre-existing methods. Up until now one of the most commonly used approaches to determine the validity of categorical syllogisms has been that of using diverse types of diagrams [1], [2], [3], [4], [5], and [6]. A previous contribution by the authors of this article, with that objective, can be classified within that approach [7]. ...
... According to (2), ∅ can be replaced in (5) with U. Thus the following equation is obtained: ...
A presentation is provided of a method — the Membership Table Method (MTM) — to determine the validity of categorical syllogisms. This method makes it possible for each syllogism to be assigned a specific set. If this set is equal to the universal set U, then the categorical syllogism considered is valid, and if that set is not equal to U, then that categorical syllogism is not valid. In other words, any categorical syllogism is valid if and only if its respective set, according to the MTM, is equal to the universal set U. The conclusion of a valid categorical syllogism whose premises are true is true.
... It can be applied more simply and systematically than other pre-existing methods. Up until now one of the most commonly used approaches to determine the validity of categorical syllogisms has been that of using diverse types of diagrams [1], [2], [3], [4], [5], and [6]. A previous contribution by the authors of this article, with that objective, can be classified within that approach [7]. ...
... According to (2), ∅ can be replaced in (5) with U. Thus the following equation is obtained: ...
Apresentation is provided of a method – the Membership Table Method (MTM) – to determine the validity of categorical syllogisms. This method makes it possible for each syllogism to be assigned a specific set. If this set is equal to the universal set U, then the categorical syllogism considered is valid, and if that set is not equal to U, then that categorical syllogism is not valid. In other words, any categorical syllogism is valid if and only if its respective set, according to the MTM, is equal to the universal set U. The conclusion of a valid categorical syllogism whose premises are true is true.
... However, recently, after the pioneering works of Barwise, Etchemendy, and Shin (e.g., [1,5,21]), diagrams have been investigated as counterparts of logical formulas, which constitute formal proofs. Diagrams are rigorously defined as syntactic objects, allowing set-theoretical semantics to be defined. ...
... Thus, diagrammatic reasoning consists of (1) construction of a maximal diagram by unifying pieces of information contained in given assumptions, and (2) extraction of a conclusion from the unified diagram. This observation is also supported by various cognitive studies on diagrammatic reasoning, for example, [1,22]. ...
We extend natural deduction for first-order logic (FOL) by introducing diagrams as components of formal proofs. From the viewpoint of FOL, we regard a diagram as a deductively closed conjunction of certain FOL formulas. On the basis of this observation, we first investigate basic heterogeneous logic (HL) wherein heterogeneous inference rules are defined in the styles of conjunction introduction and elimination rules of FOL. By examining what is a detour in our heterogeneous proofs, we discuss that an elimination-introduction pair of rules constitutes a redex in our HL, which is opposite the usual redex in FOL. In terms of the notion of a redex, we prove the normalization theorem for HL, and we give a characterization of the structure of heterogeneous proofs. Every normal proof in our HL consists of applications of introduction rules followed by applications of elimination rules, which is also opposite the usual form of normal proofs in FOL. Thereafter, we extend the basic HL by extending the heterogeneous rule in the style of general elimination rules to include a wider range of heterogeneous systems.
... According to the literature, the notion of heterogeneous inferences is drawn at the representational level and does not imply a distinction at the level of content. Along this line, Allwein and Barwise (1996) and Barceló (2012) claim that heterogeneous inferences involve transitions between propositional contents. In contrast to this view, we have argued that maps have non-propositional (cartographic) content; thus, the kind of inference we are focused on involves transitions between propositional and non-propositional content. ...
... able studies on heterogeneous inferences in the literature (see Barwise & Ethemendy's Hyperproof in 1996). However, most of these studies have focused on the interaction between linguistic representations and diagrams (see Shin 1994 andAllwein & and images (see Barceló 2012). Based on these studies, we want to contribute to the case of heterogeneous inferences involving cartographic representations. ...
Heck (2007) draws on an apparent dichotomy between linguistic and iconic representations. According to Heck, whereas linguistic representations have conceptual content, the content of iconic representations is non-conceptual. Based on the case of cartographic systems, we criticize Heck's distinction and argue that maps are composed of semantically arbitrary elements that play different syntactic roles. Upon this, we claim that maps convey conceptual content. Finally, we argue that, despite their differences, maps and sentences can logically interact with each other. These considerations challenge the view that conceptual content and inferential processes necessarily involve linguistic representations; furthermore, they provide a new perspective for thinking about maps, their semantics and syntax, and their interaction with linguistic systems.
... There are remarkable studies on heterogeneous inferences in the literature (see Barwise and Ethemendy's Hyperproof in 1996). However, most of these studies focused on the logical interaction between linguistic representations and diagrams (see Shin 1994;Allwein and Barwise 1996) and images (see Barceló 2012). Based on these studies, I want to advance the case for heterogeneous inferences involving cartographic representations. ...
... e. logical form) required to stand in logical relations to one another is exclusive to linguistic representations (Bermúdez 1998;Brandom 2000;Davidson 1983;Fodor 2008;McDowell 1994;Peacocke 1992). This view of reasoning has been subject to much criticism (Aguilera 2016;Allwein and Barwise 1996;Hurley and Nudds 2006;Vigo and Allen 2009). In this section, I want to challenge the traditional view of reasoning by appealing to heterogeneous inferences, that is, logical inferences that combine representations that belong to different representational systems. ...
Since Tolman’s paper in 1948, psychologists and neuroscientists have argued that cartographic representations play an important role in cognition. These empirical findings align with some theoretical works developed by philosophers who promote a pluralist view of representational vehicles, stating that cognitive processes involve representations with different formats. However, the inferential relations between maps and representations with different formats have not been sufficiently explored. Thus, this paper is focused on the inferential relations between cartographic and linguistic representations. To that effect, I appeal to heterogeneous inference with ordinary maps and sentences. In doing so, I aim to build a model to bridge the gap between cartographic and linguistic thought.
... This bias is based upon the assumption that diagrams are naturally prone to fallacies, mistakes, and are not susceptible of generalization. Nevertheless, specially since the work of Shin (1994), and Allwein and Barwise (1996), these objections have been handled with care and the relation between logic and diagrams has been recast with more caution and detail. ...
... Consequently, today we have a research program about diagrammatic inference that promotes different studies and model theoretic schemes that help us represent and better understand diagrams in logical terms, thus allowing logicians to look back into the history of logic for instances of diagrammatic inference (cf. Gardner, 1958;Swoyer, 1991;Shin, 1994;Glasgow, Narayanan and Chandrasekaran, 1995;Stenning and Oberlander, 1995;Allwein and Barwise, 1996;Nakatsu, 2010;Moktefi and Shin, 2013). This process has proven to be very successful and has produced a revised history of logic diagrams that includes, for example, explanatory diagrams for the square of opposition (Londey and Johanson, 1987, p.109), syllogistic (Hamilton, Mansel, and Veitch, 1865, p.420), the pons asinorum (Hamblin, 1976), and the medieval summulae logicales (Murner, 1509). ...
In this contribution we pursue a simple goal: to bring more attention on the representative attributes of the Kant-Jäsche diagrams in order to assess their logical properties. After reviewing and reconstructing the diagrams that seem to be available in the Jäsche Logik, we provide an assessment of their logical qualities by comparing them with VENN.
... Siguiendo algunos planteamientos del programa de investigación sobre razonamiento diagramático de Shin (1994) y Allwein y Barwise (1996), en otros trabajos hemos ofrecido análisis, reconstrucciones y comparaciones de varios sistemas diagramáticos (Castro-Manzano, 2016;2017a;2017b;2018;2021). Asumiendo dicho programa, en este trabajo ofrecemos una revisión de los diagramas lógicos de Murphree para la silogística asertórica y numérica (Murphree, 1991;1998), los cuales, en nuestra opinión, tienen propiedades interesantes pero no son tan populares como deberían serlo. ...
Aunque la literatura sobre diagramas lógicos es amplia y profunda, los diagramas de Murphree para la silogística asertórica y numérica no son tan conocidos; sin embargo, creemos que merecen más atención en virtud de que tienen propiedades lógicas (corrección y completitud) y representativas (comprensión y claridad) interesantes. Por ello, siguiendo algunos de los planteamientos de un programa de investigación sobre razonamiento diagramático, en este trabajo ofrecemos una revisión de estos diagramas, los cuales, en nuestra opinión, no son tan populares como deberían serlo.
... This brief discussion illustrates one of the main points emphasized in this chapter, that is, that working with diagrams, or more specifically making diagrammatic abductions, is intrinsically related with the act of working with relations (preserving, manipulating, unfolding, and packaging together relations). In other words, diagrammatic abduction can be recast in a sort of calculus of relations, of which category theory represents the precise mathematical formalization: Further research in this area of investigation may be found in Allwein and Barwise (1996), Anderson et al. (2002), Boutilier and Becher (1995), Brown (1988), Gangle (2016), Fann (1970), Gabbay and Woods (2005), Gerner and Pombo (2010), Harman (1965), Hoffmann (2018), Larkin and Simon (1987), Magnani (2004), Moktefi and Shin (2013), Park (2015), Park (2017), Pietarinen (2020), Stjernfelt (2007), Stjernfelt (2014), Walton (2005). ...
... Another object of study that will not be in the focus of this survey consists of diagrams, which have been analyzed from the perspectives of syntax and semantics, and have been discussed as early evidence for non-linguistic representations (see e.g.Shin, 1994;Shimojima, 2015, and the papers inAllwein and Barwise, 1996). ...
We argue that formal linguistic theory, properly extended, can provide a unifying framework for diverse phenomena beyond traditional linguistic objects. We display applications to pictorial meanings, visual narratives, music, dance, animal communication, and, more abstractly, to logical and non-logical concepts in the ‘language of thought’ and reasoning. In many of these cases, a careful analysis reveals that classic linguistic notions are pervasive across these domains, such as for instance the constituency (or grouping) core principle of syntax, the use of logical variables (for object tracking), or the variety of inference types investigated in semantics/pragmatics. The aim of this overview is to show how the application of formal linguistic concepts and methodology to non-linguistic objects yields non-trivial insights, thus opening the possibility of a general, precise theory of signs. (An appendix, found in the online supplements to this article, surveys applications of Super Linguistics to animal communication.)
... This brief discussion illustrates one of the main points emphasized in this chapter, that is, that working with diagrams, or more specifically making diagrammatic abductions, is intrinsically related with the act of working with relations (preserving, manipulating, unfolding, and packaging together relations). In other words, diagrammatic abduction can be recast in a sort of calculus of relations, of which category theory represents the precise mathematical formalization: Further research in this area of investigation may be found in Allwein and Barwise (1996), Anderson et al. (2002), Boutilier and Becher (1995), Brown (1988), Gangle (2016), Fann (1970), Gabbay and Woods (2005), Gerner and Pombo (2010), Harman (1965), Hoffmann (2018), Larkin and Simon (1987), Magnani (2004), Moktefi and Shin (2013), Park (2015), Park (2017), Pietarinen (2020), Stjernfelt (2007), Stjernfelt (2014), Walton (2005). ...
This chapter engages the question of abductive inferences in diagrammatic reasoning by coordinating a semiotic approach through C.S. Peirce’s triadic conception of the sign with the relational and compositional framework of the mathematics of category theory. A straightforward triadic diagram of composed functors models abductive reasoning with diagrams in general. More abstract considerations give rise to abductive models using categorical constructions of limits and adjunctions. The use of mathematics in natural science is shown to be representable as itself diagrammatic from this relational semiotic and categorical perspective.
... Hence, for example, one of the methods which may be used to solve that problem is presented in chapter 16, "Syllogisms", of W. V. Quines' Method of Logic [1]. Other methods used with that same objective have been presented, for example, in [2], [3], [4], [5], [6], and [7]. The authors of this article have introduced two more of those methods: the Inclusion Diagrams Method (IDM) [8] and the Membership Table Method (MTM) [9]. ...
A description is provided of a method -- the Set Equality Method (SEM) -- to determine the validity, or lack of validity, of each categorical syllogism. A justification is given for the presentation of a new method to solve a problem which has already been solved using different approaches. First, the SEM assigns an equality of certain sets (or two of those equalities in specific cases as will be indicated) to each of the categorical propositions composing each syllogism considered -- that is, to each of the two premises and to the conclusion. Each syllogism considered is valid if and only if a) it is possible to select one of those equalities corresponding to one of the premises such that one of its members is a certain set and the other of those equalities corresponding to the other premise such that one of its members is a subset of the set mentioned, and b) it is possible to deduce an equality corresponding to the conclusion of the two equalities corresponding to the premises. In some cases, as will be specified, it is possible to provide a second test for the validity of a syllogism whose validity was already proven, thus providing information about the logical form of categorical syllogisms.
... Hence, for example, one of the methods which may be used to solve that problem is presented in chapter 16, "Syllogisms", of W. V. Quines' Method of Logic [1]. Other methods used with that same objective have been presented, for example, in [2], [3], [4], [5], [6], and [7]. The authors of this article have introduced two more of those methods: the Inclusion Diagrams Method (IDM) [8] and the Membership Table Method (MTM) [9]. ...
A description is provided of a method -- the Set Equality Method (SEM) -- to determine the validity, or lack of validity, of each categorical syllogism. A justification is given for the presentation of a new method to solve a problem which has already been solved using different approaches. First, the SEM assigns an equality of certain sets (or two of those equalities in specific cases as will be indicated) to each of the categorical propositions composing each syllogism considered -- that is, to each of the two premises and to the conclusion. Each syllogism considered is valid if and only if a) it is possible to select one of those equalities corresponding to one of the premises such that one of its members is a certain set and the other of those equalities corresponding to the other premise such that one of its members is a subset of the set mentioned, and b) it is possible to deduce an equality corresponding to the conclusion of the two equalities corresponding to the premises. In some cases, as will be specified, it is possible to provide a second test for the validity of a syllogism whose validity was already proven, thus providing information about the logical form of categorical syllogisms.
... Hence, for example, one of the methods which may be used to solve that problem is presented in chapter 16, "Syllogisms", of W. V. Quines' Method of Logic [1]. Other methods used with that same objective have been presented, for example, in [2], [3], [4], [5], [6], and [7]. The authors of this article have introduced two more of those methods: the Inclusion Diagrams Method (IDM) [8] and the Membership Table Method (MTM) [9]. ...
A description is provided of a method – the Set Equality Method (SEM) – to determine the validity, or lack of validity, of each categorical syllogism. A justification is given for the presentation of a new method to solve a problem which has already been solved using different approaches. First, the SEM assigns an equality of certain sets (or two of those equalities in specific cases as will be indicated) to each of the categorical propositions composing each syllogism considered – that is, to each of the two premises and to the conclusion. Each syllogism considered is valid if and only if a) it is possible to select one of those equalities corresponding to one of the premises such that one of its members is a certain set and the other of those equalities corresponding to the other premise such that one of its members is a subset of the set mentioned, and b) it is possible to deduce an equality corresponding to the conclusion of the two equalities corresponding to the premises. In some cases, as will be specified, it is possible to provide a second test for the validity of a syllogism whose validity was already proven, thus providing information about the logical form of categorical syllogisms.
... References [15,16] were not Euler diagrams, but have similarities with this study in that they constructed data for use in the Semantic Web by utilizing each structured model. Likewise, there are many studies that try to plot reasoning as a Euler diagram [17,18]. However, in the case of the above studies, this is close to replacing the existing expression language such as the RDF with a Euler diagram form by focusing on viewing the Euler diagram as a single expression style. ...
On the Semantic Web, resources are connected to each other by the IRI. As the basic unit is comprised of linked data, machines can use semantic data and reason their relations without additional intervention on the Semantic Web. However, it is necessary for users who first encounter the Semantic Web to understand its underlying structure and some grammatical rules. This study suggests linking data sets of the Semantic Web through the Euler diagram, which does not require any prior knowledge. We performed a user study with our relationship-building system and verified that users could better understand linked data through the usage of the system. Users can indirectly be guided by using our Euler diagram-based data relationship-building system to understand the Semantic Web and its data linkage system. We also expect that the data sets defined through our system can be used in various applications.
... Lagrange 1997, vi;Leibniz, Remnant and Bennett 1996, 309;Dieudonné 1960, v;Tennant 1986, 303ff;Hammack 2013, 20ff). However, today we have several models and theories that help us represent and better understand the concepts of logic diagram and diagrammatic inference (Gardner 1958;Swoyer 1991;Shin 1994;Chandrasekaran, Glasgow and Narayanan 1995;Stenning and Oberlander, 1995;Allwein and Barwise 1996;Nakatsu 2009;Moktefi and Shin 2013). Following some of the tenets of this research program, in other places we have performed comparative studies of different systems of diagrams with the purpose of answering the question of what is that makes a diagram a bona fide logic diagram. ...
... Lagrange 1997, vi;Leibniz, Remnant and Bennett 1996, 309;Dieudonné 1960, v;Tennant 1986, 303ff;Hammack 2013, 20ff). However, today we have several models and theories that help us represent and better understand the concepts of logic diagram and diagrammatic inference (Gardner 1958;Swoyer 1991;Shin 1994;Chandrasekaran, Glasgow and Narayanan 1995;Stenning and Oberlander, 1995;Allwein and Barwise 1996;Nakatsu 2009;Moktefi and Shin 2013). Following some of the tenets of this research program, in other places we have performed comparative studies of different systems of diagrams with the purpose of answering the question of what is that makes a diagram a bona fide logic diagram. ...
This is. a Trilingual collection of contributions made by members of the Peirce Latin-American Society to their inaugural event in 2019. it is a historical book since there is no such comprehensive approach to the variety of Peirce studies in Latin-America that shows the history of the reception of Peirce and his Pragmatism, a dialogue with the philosophy and thought of the region and, furthermore, contributions to the study of Peirce's thought made from the Americas.
... Some authors have argued that maps can participate in inferences (see Aguilera 2016, Camp 2007, Casati and Varzi 1999 16 According to the literature, the notion of heterogeneous inferences is drawn at the representational level and does not imply a distinction at the level of content. Along this line, Allwein and Barwise (1996) and Barceló (2012) claim that heterogeneous inferences involve transitions between propositional contents. ...
Abstract:
To make a case for non-conceptualism, Heck (2007) turns to an apparent dichotomy between linguistic and iconic representations. According to Heck, whereas linguistic representations have conceptual content, the content of iconic representations is non-conceptual. Although we deeply sympathize with Heck’s representationalist shift, we argue that it does not tip the balance in favor of non-conceptualism. Based on the case of cartographic systems, we criticize Heck’s dichotomous distinction between linguistic and iconic representations. Firstly, we argue that, despite being non-linguistic, maps are not iconic as they are composed of semantically arbitrary elements that play different syntactic roles. Upon this, we claim that maps have a predicative structure. Secondly, we argue that maps satisfy the generality constraint and have conceptual content by having a predicative structure. Finally, based on the previous conclusions, we argue that, despite their differences, maps and sentences can logically interact with each other through heterogeneous inferences. These considerations challenge the view that conceptual content and inferential processes necessarily involve linguistic representations. They also provide a new perspective for thinking about maps, their semantics and syntax, and their interaction with linguistic systems.
Forthcoming in Grazer Philosophische Studien
... Esto se conoció rápidamente como "El puente de los asnos" (Pons asinorum) 1 . Bochenski (1984) señala que el término "Puente de los Asnos" solo aparece en Tartaretus que afirma en su Introducción: 1 El propósito de este articulo no detallar el desarrollo de este diagrama como lo realiza Allwein (1996), sino más bien obtener una versión informática para propósitos de enseñanza como sirva de estrategia para fomentar la alfabetización digital y los análisis de las reglas lógica. Para que el arte de encontrar el término medio de manera fácil, clara y directa, se propone como explicación la siguiente figura que, por su aparente dificultad, se le domina corrientemente como el "Puente de los Asnos" (Pons asinorum) a pesar de que este arte puede ser entendido de manera concisa y clara para todos en el caso de que las palabras de dicha sección (passu) sean lo suficientemente claras. ...
Se presenta una nueva versión simplificada del “Puente de los Asnos” (Pons asinorum) desarrollado a partir de la interpretación de fuentes clásicas y del reajuste del mismo propuesto por Hamblin (1976). Se describe el sentido y alcance del Pons asinorum así como otros intentos históricos de la automatización del silogismo. Posteriormente se compara a Juan de Santo Tomás y Aristóteles para explicar los criterios para la elaboración de la aplicación informática señalando las mejoras técnicas y conceptuales dando lugar al software del Puente de los Asnos. Se mues- tra cómo el software y la enseñanza por medio del mismo puede ser un detonador de nuevos modos de razonar a los estudiantes potenciales, en consecuencia, se demuestran nuevas formas de enseñanza a través de la lógica tradicional.
... Nevertheless, in the last few decades there has been a revival of interest in diagrams in mathematics. But this revival-at least at its origin, which can be traced back to Allwein and Barwise (1996), Hammer (1995), and Shin (1994)-has been motivated by adherence to the view that the method of mathematics is the axiomatic method, and specifically by the attempt to fit diagrams into the axiomatic method, translating particular diagrams into statements and inference rules of a formal system. Thus, Barwise and Etchemendy say that "diagrams and other forms of visual representation can be essential and legitimate components in valid deductive reasoning" (Barwise and Etchemeny 1996a, 12). ...
In the last few decades there has been a revival of interest in diagrams in mathematics. But the revival, at least at its origin, has been motivated by adherence to the view that the method of mathematics is the axiomatic method, and specifically by the attempt to fit diagrams into the axiomatic method, translating particular diagrams into statements and inference rules of a formal system. This approach does not deal with diagrams qua diagrams, and is incapable of accounting for the role diagrams play as means of discovery and understanding. Alternatively, this paper purports to show that the view that the method of mathematics is the analytic method is capable of dealing with diagrams qua diagrams, and of accounting for such role.
... Por outro lado, desde o século 18, os silogismos veem sendo representados, de maneira mais ou menos eficiente, no domínio das chamadas lógicas heterogêneas, que são aquelas que combinam notação simbólica e diagramática (ALLWEIN & BARWISE, 1996). Os círculos de Euler e os diagramas de Venn são os primeiros sistemas representacionais diagramáticos conhecidos e ainda usados na demonstração de argumentos compostos por proposições categoriais. ...
A lógica aristotélica tem sido reconstruída em diferentes sistemas formais desde o século passado. Hoje, a interpretação da silogística como um sistema de dedução natural, proposta por Corcoran e Smiley nos anos 1970, é considerada a mais coerente com os Primeiros Analíticos. Por outro lado, a prova de validade dos silogismos categóricos em diagramas remonta aos séculos 18 e 19, com os círculos de Euler e Venn. Pouca atenção foi dada, contudo, aos Grafos Existenciais (GE) de Charles S. Peirce, reconhecidos como um dos mais prolíficos e completos sistemas diagramáticos já inventados. A proposta deste artigo é traduzir o modelo de Corcoran-Smiley para a sintaxe do sistema Beta dos GE, que corresponde à lógica de primeira ordem. Objetiva-se, com isso, elucidar vantagens do sistema diagramático peirciano em relação aos tradicionais de Euler e Venn. Argumentamos que os GE seriam mais adequados para a silogística pelo fato de terem sido concebidos como instrumento de análise do raciocínio dedutivo. Sugerimos, por fim, seu uso heurístico e pedagógico.
... As such they obey their own set of constraints." ( Allwein et al, 1996: 23). A physical situation is a situation in the world, in this case, it is a portion of paper (i.e. a space) on which we draw the diagrams (Shimojima, 1996: 38). ...
p>Recibido: 18/06/2016 • Aceptado: 02/02/2017
Luego de una breve revisión de la noción de inferencia diagramática, mostramos en qué sentido VENN provee un marco lógico capaz de modelar cierta forma de inferencia diagramática no-monotónica. Este resultado indica que, así como existen sistemas lógicos sentenciales clásicos y no-clásicos, pueden existir sistemas lógicos diagramáticos clásicos y no-clásicos.</p
... In the last two decades, these images gradually-yet in a rather sporadic manner-became subject of analyses in philosophy of science. Scientific images have been understood as models (Giere 1996), as stimuli for visual thinking (Bredekamp 2005), and as tools for scientific reasoning (Kulvicki 2010; see also articles in Allwein and Barwise 1996). Furthermore, the capacity of visual representations to 'bear truth' and thus to justify or confirm scientific hypothesis (Perini 2005b(Perini , c, 2012 as well as their ability to facilitate (or impede) communication (Doyle 2007;Mahony and Hulme 2012) have been emphasized. ...
Recent philosophical analyses of the epistemic dimension of images in the sciences show a certain trend in acknowledging potential roles of these images beyond their merely decorative or pedagogical functions. We argue, however, that this new debate has yet paid little attention to a special type of pictures, we call ‘visual metaphor’, and its versatile heuristic potential in organizing data, supporting communication, and guiding research, modeling, and theory formation. Based on a case study of Conrad Hal Waddington’s epigenetic landscape images in biology, we develop a descriptive framework applicable to heuristic roles of various visual metaphors in the sciences.
En la enseñanza de las matemáticas influyen diferentes variables y se espera que después de completar los ciclos de formación en matemáticas los alumnos tengan diferentes habilidades útiles para la vida diaria. Si com- prendemos tanto lo que influye en la enseñanza como lo que se consigue verdaderamente con ella podremos mejorar las clases de matemáticas. Basado en otros instrumentos, se diseñó uno que intenta medir el razonamiento lógico con base en la habilidad deductiva y la capacidad de abstracción. El estudio se realizó con una muestra correspondiente a licenciaturas con enfoque en la enseñanza, en una escuela privada del norte de México, con una edad promedio de 21 años. Se encontró que el razonamiento lógico no tiene correlación con el rendimiento académico general (ρ = .262, p = .061) pero sí con el rendimiento académico matemático (ρ = .303, p = .041). Esta última correlación lineal resultó positiva y nos indica que al aumentar el razonamiento lógico aumenta el rendimiento académico en matemáticas y, al disminuir una de ellas, la otra también.
In his well-known paper on Euclid’s geometry, Ken Manders sketches an argument against conceiving the diagrams of the Elements in ‘semantic’ terms, that is, against treating them as representations—resting his case on Euclid’s striking use of ‘impossible’ diagrams in some proofs by contradiction. This paper spells out, clarifies and assesses Manders’s argument, showing that it only succeeds against a particular semantic view of diagrams and can be evaded by adopting others, but arguing that Manders nevertheless makes a compelling case that semantic analyses ought to be relegated to a secondary role for the study of mathematical practices.
Philosophers have long debated the relative priority of thought and language, both at the deepest level, in asking what makes us distinctively human, and more superficially, in explaining why we find it so natural to communicate with words. The “linguistic turn” in analytic philosophy accorded pride of place to language in the order of investigation, but only because it treated language as a window onto thought, which it took to be fundamental in the order of explanation. The Chomskian linguistic program tips the balance further toward language, by construing the language faculty as an independent, distinctively human biological mechanism. In Ignorance of Language, Devitt attempts to swing the pendulum back toward the other extreme, by proposing that thought itself is fundamentally sentential, and that there is little or nothing for language to do beyond reflecting the structure and content of thought. I argue that both thought and language involve a greater diversity of function and form than either the Chomskian model or Devitt’s antithesis acknowledge. Both thought and language are better seen as complex, mutually supporting suites of interacting abilities.
Educational linguistics is a dyadic science. The noun, linguistics , is a broad term which includes neuro-, psycho-, socio-, pragma-, ethno-linguistics and communication studies: areas where national ‘schools’ non longer exist. Educational , on the contrary, is a culture-bound term: language teaching is carried out according to laws which concern syllabi, exams and certifications, the language(s) of instruction, the teaching of the host language to migrant students, teacher training programmes etc. These juridical and administrative acts are meant for the local educational systems. We propose that it is possible to find a number of principles and models (we call them “hypotheses”) which can be accepted by culture-bound educational decision-makers, thus increasing consistency within language teaching and research throughout the world.
A key 2005 collection of papers (Royer 2008) showed how complex the study of mathematical cognition (MC) had become already in the early 2000s, incorporating a broad range of scientific, educational, and humanistic perspectives into its modus operandi. Studies published in the journal Mathematical Cognition have also revealed how truly expansive the field is, bringing together researchers and scholars from diverse disciplines, from neuroscience to semiotics. This volume has aimed to provide a contemporary snapshot of how the study of MC is developing. In this final chapter, the objective is to provide a selective overview of different approaches from the past as a concluding historical assessment.
Proofs contribute to mathematical knowledge in a richer way than through exclusively of their results. Then, a philosophically relevant task is to inquire how diverse demonstrations of the same result concretize that contribution. This essay compares (following a recent work by John Dawson) various demonstrations of an elementary result of number theory, regarding a specific relation: “. . . is more perspicuous than . . .”. The main conclusion of this work aims to highlight (in the cases considered) the relevance of the analysis of the strategic and expressive contrasts and its peculiar dynamics, in the understanding of the relationship of perspicuity between proofs. © 2017 Federal University of Santa Catarina. All rights reserved.
Venn diagram system has been extended by introducing names of individuals and their absence. Absence gives a kind of negation of singular propositions. We have offered here a non-classical interpretation of this negation. Soundness and completeness of the present diagram system have been established with respect to this interpretation.
Spider diagrams are based on Euler and Venn/Peirce diagrams, forming a system which is as expressive as monadic first order logic with equality. Rather than being primarily intended for logicians, spider diagrams were developed at the end of the 1990s in the context of visual modelling and software specification. We examine the original goals of the designers, the ways in which the notation has evolved and its connection with the philosophical origins of the logical diagrams of Euler, Venn and Peirce on which spider diagrams are based. Using Peirce’s concepts and classification of signs, we analyse the ways in which different sign types are exploited in the notation. Our hope is that this analysis may be of interest beyond those readers particularly interested in spider diagrams, and act as a case study in deconstructing a simple visual logic. Along the way, we discuss the need for a deeper semiotic engagement in visual modelling.
In this paper we recover Leibniz's diagrammatic logic for syllogistic and we discover its computational and logical features by providing a formal approach to it in metalogical terms, which is something that, as far as we know, is yet to be accomplished. Thus, in this contribution we pursue, respectively, two goals, a historical and a logical one: i) to bring more attention on the algorithmic aspects of Leibniz's diagrammatic system for syllogistic, which be believe have been neglected because of a general bias against diagram-based reasoning;
and ii) to prove metalogical properties of the system in order to argue that such system is a bona fide logical system.
Mathematics and computer science are strongly intertwined, especially in using the language of set theory as a suitable base for transitions between formal (human-computer) and informal (human-human) communications and in supporting modeling, e.g., object oriented concepts. Yet, it seems that difficulties occur when set notations and diagrams are applied, which has motivated a search for an alternative to traditional ways of visually depicting set theory. Some researchers propose distinguishing between “set” and “collection.” Others claim that the fundamental error in metaphysical interpretations of set theory, the reification of a collection as a separate object, is a result of grammatical confusion. Another issue raised in this context with regard to set theory is the static state that has been imposed on its representation; e.g., sets are spaces of containers, and “movements” of members are conceptualized as “arches” called mapping and functions. This paper offers a visual representation of set theory that may benefit modeling in computer science based on the familiar notion of machines. A machine can be seen as an extended form of the input-process-output system with five basic “operations”: releasing, transferring, receiving, processing, and creating of things that flow. The paper employs such a schematization apparatus to construct a high-level description of sets and functions. The resultant diagrammatic representation seems a viable tool for enhancing the relationship between mathematics and computer science.
In his classic 1962 study, Max Black
showed convincingly how scientific theories are constructed through unconscious
metaphorical reasoning, thus linking them to the experiences
of the scientist, the social and historical
contexts
in which they emerge
, and the image
schemata that are established within specific scientific domains. Some works have looked at this representational
phenomenon within education
, but only sporadically. This chapter focuses on metaphorical arguments and how they guide the construction
of educational theories that lead to models and diagrammatic strategies, which in turn guide the derivative educational practices. It will then examine the possibility
that metaphor itself can be incorporated into actual teaching practices, illustrating how this can be done in the teaching of mathematics and second languages
. The chapter, by documenting the connection between metaphors, models, diagrams, and learning theories, addresses edusemiotics
in its both theoretical and empirical aspects.
The article proposes a model of intercultural communicative competence. The need for further reflection on this topic derives from two facts: existing models are not built on Dell Hymes' model of communicative competence (cc), although they use Hymes' term 'cc' is used; our discussion in this essay starts from a definition of cc in Hymes' tradition and studies what changes are needed to make it fit to describe intercultural cc; most models are, in fact, not 'models'. According to the theory of models, models aim at being universal, based on formal logic rather than empirical evidence, and this is extremely important in ICC studies where empirical research is always partial. The result is a new instrument to analyse, describe and teach ICC.
The book deals with a scientific approach to the theory of LTR, trying to shift this area of research from soft sciences to (at leat partially) hard sciences methodology of research.
Als „logisch“ bezeichnet man ein Denken, das bei Abwägung der verfügbaren Informationen als folgerichtig und widerspruchsfrei gelten kann. Beim sicheren logischen Schließen ist eine Schlussfolgerung sicher wahr, wenn man von wahren Voraussetzungen ausgeht. Beim unsicheren logischen Schließen können Schlüsse auch möglich, plausibel oder wahrscheinlich sein. Bei der Bewertung solcher Schlüsse wird auf Rationalitätsnormen Bezug genommen. Empirische Befunde zeigen, dass sich Menschen oft an diese normativen Vorgaben halten, aber manchmal auch von diesen Sollwerten abweichen. Die Theorien, die diese Abweichungen erklären wollen, werden diskutiert und ihre neuronalen Grundlagen dargestellt. Es werden außerdem einige wichtige Fragen des Forschungsgebiets diskutiert. Dabei geht es auch um die Beziehung zwischen logischem Denken und Rationalität und die Frage, was als vernünftiges Denken, Argumentieren, Urteilen und Entscheiden gelten soll.
Schlüsselwörter: Logik; Rationalität; Denken; Argumentation; Deduktion; Konditionales Schließen; Relationales Schließen; Syllogismen; Nichtmonotones Schließen; Logisches Gehirn
We propose a methodology for building consistent metadata for the description of scientific working documents. Our methodology is supported by distributed cognitive theoretical principles and implemented through a combination of anthropological, semantic, and linguistic approaches. Our analysis is applied to a particular pharmacist activity?namely, the adaptation of posology. Issues focus on the representation of situations that describe individuals, tools, and artifacts. Regular relations between these situations characterize types that allow the description of conceptual contents associated with these empirical objects. Because these situations and types are expressed by a set of metadata and then associated with current metadata, our proposal extends the nature of entities described by metadata to useful internal activity artifacts.
Many diagrammatic representations such as Venn, Euler, and Peirce diagrams have been used and intensively studied in set theory and in logic formalisms. It has been claimed that diagrammatic representations in general have advantages over linguistic ones; however, current visualizations of logical problems do not provide clarity on a basic static structure with elementary dynamic features, creating a conceptual gap that can cause misinterpretation and difficulty in understanding. This paper proposes a flow-based model to represent the fundamental structure of paradoxical problems in logic. The method can provide illustrations and models to facilitate general understanding and also be used in teaching such problems.
Historians occasionally use timelines, but many seem to regard such signs merely as ways of visually summarizing results that are presumably better expressed in prose. Challenging this language-centered view, I suggest that timelines might assist the generation of novel historical insights. To show this, I begin by looking at studies confirming the cognitive benefits of diagrams like timelines. I then try to survey the remarkable diversity of timelines by analyzing actual examples. Finally, having conveyed this (mostly untapped) potential, I argue that neglecting timelines might mean neglecting significant aspects of reality that are revealed only by those signs. My overall message is that once we accept that relations are as important for the mind as what they relate, we have to pay closer attention to any semiotic device that enables or facilitates the discernment of new relations.
Diagrams probably rank among the oldest forms of human communication. Traditional logic diagrams (e.g., Venn diagrams, Euler diagrams, Peirce existential diagrams) have been utilized as conceptual representations, and it is claimed that these diagrammatic representations, in general, have advantages over linguistic ones. Nevertheless, current representations are not satisfactory. Diagrams of logic problems incompletely depict their underlying semantics and fail to provide a clear, basic, static structure with elementary dynamic features, creating a conceptual gap that sometimes causes misinterpretation. This paper proposes a conceptual apparatus to represent mathematical structure, and, without loss of generality, it focuses on sets. Set theory is described as one of the greatest achievements of modern mathematics. Nevertheless, its metaphysical interpretations raise paradoxes, and the notion of a collection, in terms of which sets are defined, is inconsistent. Accordingly, exploring a new view, albeit tentative, attuned to basic notions such as the definition of set is justifiable. This paper aims at providing an alternative graphical representation of a set as a machine with five basic “operations”: releasing, transferring, receiving, processing, and creating of things. Here, a depiction of sets is presented, as in the case of Venn-like diagrams, and is not intended to be a set theory contribution. The paper employs schematization as an apparatus of descriptive specification, and the resultant high-level description seems a viable tool for enhancing the relationship between set theory and computer science.
Some conceptions of logic claim that they are universal. By contrast, I assume that the applications of any logic are central to its conception, so that it has to comprise a hierarchy of its metalogics, metametalogics, …, indefinitely extended but never capped off with some universal logic. I also advocate for the distinction between parts and moments of a multitude as key to this conception, and I query the assumption that set theory provides the most general means of handling collections of objects.
We present and discuss a diagrammatic visualization and reasoning language coming about by augmenting Euler diagrams with higraphs. The diagrams serve (hierarchical as well as trans-hierarchical) classification and specification of various logical relationships between classes. The diagrams rely on a well-defined underlying class-relationship logic, called CRL, being a fragment of predicate logic. The inference rules at the level of diagrams take form of simple diagrammatic ipso facto rules. The diagrams are intended for computerization by offering navigation and zooming facilities as known from road maps. As such they may facilitate ontological engineering, which often involves larger amounts of data. The underlying inference process is expressible in function-free definite clauses, datalog. We also discuss the relationship to similar diagram and logic proposals.
This article is a contribution to the study of representation from the experimental point of view. It is based on the observation of design by means of diagrams in information systems and shows that representation is a process using an abductive reasoning mode. The article offers a general description, independent from the domain and the designers, of the properties and behavioral regularities inherent to a representing system. The description puts in evidence a major function of diagrams, the externalization of an inference step in design reasoning. Building a software architecture for design assistance and representation phenomena study thus relies upon a principle of interaction between a designer and a machine for which the diagram consists in the intermediary object.
I share with Schoenfeld and Törner the experience of being educated as a mathematician, then learning to work in mathematics education research. I write this chapter from this perspective, drawing on my experiences of work with both mathematicians and education researchers. In this chapter, I describe how being a mathematician might shape one’s perspective on mathematics education research.
The author had previously developed a framework called Conposit for implementing symbolic representation and reasoning in a connectionist framework. Early theoretical work towards the system placed strong emphasis on handling spatial-imagistic and diagrammatic forms of representation as well as abstract symbolic forms. In the present context the word “diagrammatic” conveys, roughly speaking, both the mixing together of (schematic) spatial-imagistic representational forms and abstract symbolic forms, and the metaphorical use of space and the physical entities to couch abstract information. However, the imagistic/diagrammatic motivation was not adhered to in the framework that was actually developed over time—and which therefore concentrated entirely on abstract symbolic representation—except that the representational primitives in the final framework owe much, implicitly, to the original spatial/diagrammatic orientation. The present talk, abstracted here, is an initial, speculative step towards resuscitation of the original spatial/diagrammatic aims. Such a step is timely because of a surge of interest in diagrammatic representation and reasoning in recent years. Suggestions are made about what it means for internal representations (including connectionist ones) to be diagrammatic, and about modifications to the Conposit framework that would be needed to take it (back) towards imagistic/diagrammatic representation and reasoning.
An Information-Theoretic of Valid Reasoning with Venn Diagrams, in Sit-nation theory and its applications, Barwise et. al
- Sun
- Shin
Sun-Joo Shin, An Information-Theoretic of Valid Reasoning with Venn Diagrams, in Sit-nation theory and its applications, Barwise et. al., CSLI Lecture Notes, University of Chicago Press, 199