In recent years, semiotics has become an innovative theoretical framework in mathematics education. The purpose of this article is to show that semiotics can be used to explain learning as a process of experimenting with and communicating about one's own representations of mathematical problems. As a paradigmatic example, we apply a Peircean semiotic framework to answer the question of how students learned the concept of "distribution" in a statistics course by "diagrammatic reasoning" and by developing "hypostatic abstractions," that (...) is by forming new mathematical objects which can be used as means for communication and further reasoning. Peirce's semiotic terminology is used as an alternative for notions such as modeling, symbolizing, and reification. We will show that it is a precise instrument of analysis with regard to the complexity of learning and of communication in mathematics classroom. (shrink)

Just before the Scientific Revolution, there was a "Mathematical Revolution", heavily based on geometrical and machine diagrams. The "faculty of imagination" (now called scientific visualization) was developed to allow 3D understanding of planetary motion, human anatomy and the workings of machines. 1543 saw the publication of the heavily geometrical work of Copernicus and Vesalius, as well as the first Italian translation of Euclid.

A Mathematical Review by John Corcoran, SUNY/Buffalo -/- Macbeth, Danielle Diagrammatic reasoning in Frege's Begriffsschrift. Synthese 186 (2012), no. 1, 289–314. ABSTRACT This review begins with two quotations from the paper: its abstract and the first paragraph of the conclusion. The point of the quotations is to make clear by the “give-them-enough-rope” strategy how murky, incompetent, and badly written the paper is. I know I am asking a lot, but I have to ask you to read the quoted (...) passages—aloud if possible. Don’t miss the silly attempt to recycle Kant’s quip “Concepts without intuitions are empty; intuitions without concepts are blind”. What the paper was aiming at includes the absurdity: “Proofs without definitions are empty; definitions without proofs are, if not blind, then dumb.” But the author even bollixed this. The editor didn’t even notice. The copy-editor missed it. And the author’s proof-reading did not catch it. In order not to torment you I will quote the sentence as it appears: “In a slogan: proofs without definitions are empty, merely the aimless manipulation of signs according to rules; and definitions without proofs are, if no blind, then dumb.”[sic] The rest of my review discusses the paper’s astounding misattribution to contemporary logicians of the information-theoretic approach. This approach was cruelly trashed by Quine in his 1970 Philosophy of Logic, and thereafter ignored by every text I know of. The paper under review attributes generally to modern philosophers and logicians views that were never espoused by any of the prominent logicians—such as Hilbert, Gödel, Tarski, Church, and Quine—apparently in an attempt to distance them from Frege: the focus of the article. On page 310 we find the following paragraph. “In our logics it is assumed that inference potential is given by truth-conditions. Hence, we think, deduction can be nothing more than a matter of making explicit information that is already contained in one’s premises. If the deduction is valid then the information contained in the conclusion must be contained already in the premises; if that information is not contained already in the premises […], then the argument cannot be valid.” Although the paper is meticulous in citing supporting literature for less questionable points, no references are given for this. In fact, the view that deduction is the making explicit of information that is only implicit in premises has not been espoused by any standard symbolic logic books. It has only recently been articulated by a small number of philosophical logicians from a younger generation, for example, in the prize-winning essay by J. Sagüillo, Methodological practice and complementary concepts of logical consequence: Tarski’s model-theoretic consequence and Corcoran’s information-theoretic consequence, History and Philosophy of Logic, 30 (2009), pp. 21–48. The paper omits definitions of key terms including ‘ampliative’, ‘explicatory’, ‘inference potential’, ‘truth-condition’, and ‘information’. The definition of prime number on page 292 is as follows: “To say that a number is prime is to say that it is not divisible without remainder by another number”. This would make one be the only prime number. The paper being reviewed had the benefit of two anonymous referees who contributed “very helpful comments on an earlier draft”. Could these anonymous referees have read the paper? -/- J. Corcoran, U of Buffalo, SUNY -/- PS By the way, if anyone has a paper that has been turned down by other journals, any journal that would publish something like this might be worth trying. (shrink)

The approach to the question of philosophical practice has been dominated by a subordination of practice to theory corresponding in general to a representational conception of philosophy. Methods of diagrammatic reasoning developed within philosophical semiotics provide a more effective approach. Inparticular, Peirce’s system of existential graphs exemplifies how diagrammatic reasoning is able formally to express the processes through which philosophical dialogue and cooperation actually take place and to link such processes to the methods and practices arising (...) in other disciplines and traditions, including those of science, politics, art, religion and technological production. (shrink)

In the graphical representation of ontologies, it is customary to use graph theory as the representational background. We claim here that the standard graph-based approach has a number of limitations. We focus here on a problem in the graph-based representation of ontologies in complex domains such as biomedical, engineering and manufacturing: lack of mereotopological representation. Based on such limitation, we proposed a diagrammatic way to represent an entity’s structure and various forms of mereotopological relationships between the entities.

Schemes of diagrammatic representation have been so familiarly introduced into logical treatises during the last century or so, that many readers, even of those who have made no professional study of logic, may be supposed to be acquainted with the general nature and object of such devices. Of these schemes one only, viz. that commonly called "Eulerian circles," has met with any general acceptance. A variety of others indeed have been proposed by ingenious and celebrated logicians, several of which (...) would claim notice in a historical treatment of the subject; but they mostly do not seem to me to differ in any essential respect from that of Euler. They rest upon the same leading principle, and are subject all alike to the same restrictions and defects. We must therefore cast about for some new scheme of diagrammatic representation which shall be competent to indicate imperfect knowledge on our part; for this will at once enable us to appeal to it step by step in the process of working out our conclusions. I have never seen any hint at such a scheme, though the want seems so evident that one would suppose that something of the kind must have been proposed before. (shrink)

The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that (...) one of the reasons why CDs form a good notation is that they are highly mathematically tractable: experts can obtain valid results by ‘calculating’ with CDs. These calculations, take the form of ‘diagram chases’. In order to draw inferences, experts move algebraic elements around the diagrams. It will be argued that these diagrams are dynamic. It is thanks to their dynamicity that CDs can externalize the relevant reasoning and allow experts to draw conclusions directly by manipulating them. Lastly, it will be shown that CDs play essential roles in the context of proof as well as in other phases of the mathematical enterprise, such as discovery and conjecture formation. (shrink)

Commonsense reasoning is one of the main open problems in the field of Artificial Intelligence (AI) while, on the other hand, seems to be a very intuitive and default reasoning mode in humans and other animals. In this talk, we discuss the different paradigms that have been developed in AI and Computational Cognitive Science to deal with this problem (ranging from logic-based methods, to diagrammatic-based ones). In particular, we discuss - via two different case studies concerning commonsense (...) categorization and knowledge invention tasks - how cognitively inspired heuristics can help (both in terms of efficiency and efficacy) in the realization of intelligent artificial systems able to reason in a human-like fashion, with results comparable to human-level performances. (shrink)

The recent wave of data on exoplanets lends support to METI ventures (Messaging to Extra-Terrestrial Intelligence), insofar as the more exoplanets we find, the more likely it is that “exominds” await our messages. Yet, despite these astronomical advances, there are presently no well-confirmed tests against which to check the design of interstellar messages. In the meantime, the best we can do is distance ourselves from terracentric assumptions. There is no reason, for example, to assume that all inferential abilities are language-like. (...) With that in mind, I argue that logical reasoning does not have to be couched in symbolic notation. In diagrammatic reasoning, inferences are underwritten, not by rules, but by transformations of self-same qualitative signs. I use the Existential Graphs of C. S. Peirce to show this. Since diagrams are less dependent on convention and might even be generalized to cover non-visual senses, I argue that METI researchers should add some form of diagrammatic representations to their repertoire. Doing so can shed light, not just on alien minds, but on the deepest structures of reasoning itself. (shrink)

We offer a model of moral reform and regeneration that involves a wrong-doer making two movements: on the one hand, he identifies with himself as the one who did the act, while he also intentionally moves away from that self (or set of desires and intentions) and moves toward a transformed identity. We see this model at work in the formal practice of contrition and reform in Christian and Buddhist rites. This paper is part of a broader project we are (...) undertaking on the philosophy of forgiveness. (shrink)

In many diagrams one seems to perceive necessity – one sees not only that something is so, but that it must be so. That conflicts with a certain empiricism largely taken for granted in contemporary philosophy, which believes perception is not capable of such feats. The reason for this belief is often thought well-summarized in Hume's maxim: ‘there are no necessary connections between distinct existences’. It is also thought that even if there were such necessities, perception is too passive or (...) localized a faculty to register them. We defend the perception of necessity against such Humeanism, drawing on examples from mathematics. (shrink)

It is argued, on the basis of ideas derived from Wittgenstein's Tractatus and Husserl's Logical Investigations, that the formal comprehends more than the logical. More specifically: that there exist certain formal-ontological constants (part, whole, overlapping, etc.) which do not fall within the province of logic. A two-dimensional directly depicting language is developed for the representation of the constants of formal ontology, and means are provided for the extension of this language to enable the representation of certain materially necessary relations. The (...) paper concludes with a discussion of the relationship between formal logic, formal ontology, and mathematics. (shrink)

Logicians commonly speak in a relatively undifferentiated way about pre-euler diagrams. The thesis of this paper, however, is that there were three periods in the early modern era in which euler-type diagrams (line diagrams as well as circle diagrams) were expansively used. Expansive periods are characterized by continuity, and regressive periods by discontinuity: While on the one hand an ongoing awareness of the use of euler-type diagrams occurred within an expansive period, after a subsequent phase of regression the entire knowledge (...) about the systematic application and the history of euler-type diagrams was lost. I will argue that the first expansive period lasted from Vives (1531) to Alsted (1614). The second period began around 1660 with Weigel and ended in 1712 with lange. The third period of expansion started around 1760 with the works of Ploucquet, euler and lambert. Finally, it is shown that euler-type diagrams became popular in the debate about intuition which took place in the 1790s between leibnizians and Kantians. The article is thus limited to the historical periodization between 1530 and 1800. (shrink)

Robert Brandom holds that what we mean is best understood in terms of what inferences we are prepared to defend, and that such a defence is best understood in terms of rule-governed social interactions. This manages to explain quite a lot. However, for those who think that there is more to making correct/incorrect inferences than obeying/breaking accepted rules, Brandom’s account fails to adequately capture what it means to reason properly. Thus, in an effort to sketch an alternative that does not (...) rely primarily on peer pressure, I draw on the work of C. S. Peirce. Peirce argued that, when we reason, we manipulate abstract diagrams in order to observe what results. Since some manipulations are barred by the self-same nature of the diagrams, I try to show that this qualitative incompatibility, which I dub “contrapiction,” is a good reason to regard some reasoning as bad. (shrink)

Why study notations, diagrams, or more broadly the variety of nonverbal “representations” or “signs” that are used in mathematical practice? This chapter maps out recent work on the topic by distinguishing three main philosophical motivations for doing so. First, some work (like that on diagrammatic reasoning) studies signs to recover norms of informal or historical mathematical practices that would get lost if the particular signs that these practices rely on were translated away; work in this vein has the (...) potential to broaden our view of the specific normativity of mathematics beyond logical proof. Second, some work focuses on signs to highlight the open-ended and “nonrigorous” aspects of mathematical practice, and the role of these aspects in the historical development of mathematics. The third motivation is based on the following observation: the reason differences in signs matter in mathematics is that humans are finite agents, with cognitive and computational limitations. Accordingly, studying signs can be a way to study how human computational constraints shape the mathematics that we do, and to tackle the topic of mathematical understanding. (shrink)

Some have suggested that images can be arguments. Images can certainly bolster the acceptability of individual premises. We worry, though, that the static nature of images prevents them from ever playing a genuinely argumentative role. To show this, we call attention to a dilemma. The conclusion of a visual argument will either be explicit or implicit. If a visual argument includes its conclusion, then that conclusion must be demarcated from the premise or otherwise the argument will beg the question. If (...) a visual argument does not include its conclusion, then the premises on display must license that specific conclusion and not its opposite, in accordance with some demonstrable rationale. We show how major examples from the literature fail to escape this dilemma. Drawing inspiration from the graphical logic of C. S. Peirce, we suggest instead that images can be manipulated in a way that overcomes the dilemma. Diagrammatic reasoning can take one stepwise from an initial visual layout to a conclusion—thereby providing a principled rationale that bars opposite conclusions—and the visual inscription of this correct conclusion can come afterward in time—thereby distinguishing the conclusion from the premises. Even though this practical application of Peirce’s logical ideas to informal contexts requires that one make adjustments, we believe it points to a dynamic conception of visual argumentation that will prove more fertile in the long run. (shrink)

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. (shrink)

The discussions which follow rest on a distinction, first expounded by Husserl, between formal logic and formal ontology. The former concerns itself with (formal) meaning-structures; the latter with formal structures amongst objects and their parts. The paper attempts to show how, when formal ontological considerations are brought into play, contemporary extensionalist theories of part and whole, and above all the mereology of Leniewski, can be generalised to embrace not only relations between concrete objects and object-pieces, but also relations between what (...) we shall call dependent parts or moments. A two-dimensional formal language is canvassed for the resultant ontological theory, a language which owes more to the tradition of Euler, Boole and Venn than to the quantifier-centred languages which have predominated amongst analytic philosophers since the time of Frege and Russell. Analytic philosophical arguments against moments, and against the entire project of a formal ontology, are considered and rejected. The paper concludes with a brief account of some applications of the theory presented. (shrink)

The icon is the type of sign connected to efficient representational features, and its manipulation reveals more information about its object. The London Underground Diagram (LUD) is an iconic artifact and a well-known example of representational efficiency, having been copied by urban transportation systems worldwide. This paper investigates the efficiency of the LUD in the light of different conceptions of iconicity. We stress that a specialized representation is an icon of the formal structure of the problem for which it has (...) been specialized. By embedding such rules of action and behavior, the icon acts as a semiotic artifact distributing cognitive effort and participating in niche construction. (shrink)

This article presents some results of a research on computational strategies for the visualization of sign classification structures and sign processes. The focus of this research is the various classifications of signs described by Peirce. Two models are presented. One of them concerns specifically the 10-fold classification as described in the 1903 Syllabus (MS 540, EP 2: 289–299), while the other deals with the deep structure of Peirce’s various trichotomic classifications. The first is 10cubes, an interactive 3-D model of Peirce’s (...) 10-fold classification, as described in the Syllabus. The second is 3N3, a computer program that builds equivalent diagrams for any n-trichotomic classification of signs. We are specially interested in how a graphic design methodology, associated with computer graphic resources and techniques, can contribute to the construction of interactive models that serve as tools for the investigation of C. S. Peirce’s theory of signs. (shrink)

The following text is divided in four parts. The first presents the inner relation between the phenomenological concept of intentionality and space in a general mathematical sense. Following this train of though the second part brie_ly characterizes the use of the geometrical concept of manifold (Mannigfaltigkeit) in Husserl’s work. In the third part we present some examples of the use of the concept in Husserl’s analyses of space, time and intersubjectivity, pointing out some dif_iculties in his endeavor. In the fourth (...) and _inal part we offer some points of coincidence between phenomenology and category theory suggesting that the latter can work as a formal frame for ontology in the former. Our thesis is that intentionality operates in different levels as a morphism, functor and natural transformation. (shrink)

This paper argues that the theory of structured propositions is not undermined by the Russell-Myhill paradox. I develop a theory of structured propositions in which the Russell-Myhill paradox doesn't arise: the theory does not involve ramification or compromises to the underlying logic, but rather rejects common assumptions, encoded in the notation of the $\lambda$-calculus, about what properties and relations can be built. I argue that the structuralist had independent reasons to reject these underlying assumptions. The theory is given both a (...) diagrammatic representation, and a logical representation in a novel language. In the latter half of the paper I turn to some technical questions concerning the treatment of quantification, and demonstrate various equivalences between the diagrammatic and logical representations, and a fragment of the $\lambda$-calculus. (shrink)

Charles Peirce's diagrammatic logic — the Existential Graphs — is presented as a tool for illuminating how we know necessity, in answer to Benacerraf's famous challenge that most ‘semantics for mathematics’ do not ‘fit an acceptable epistemology’. It is suggested that necessary reasoning is in essence a recognition that a certain structure has the particular structure that it has. This means that, contra Hume and his contemporary heirs, necessity is observable. One just needs to pay attention, not merely (...) to individual things but to how those things are related in larger structures, certain aspects of which relations force certain other aspects to be a certain way. (shrink)

The aim of this article is to explain why knot diagrams are an effective notation in topology. Their cognitive features and epistemic roles will be assessed. First, it will be argued that different interpretations of a figure give rise to different diagrams and as a consequence various levels of representation for knots will be identified. Second, it will be shown that knot diagrams are dynamic by pointing at the moves which are commonly applied to them. For this reason, experts must (...) develop a specific form of enhanced manipulative imagination, in order to draw inferences from knot diagrams by performing epistemic actions. Moreover, it will be argued that knot diagrams not only can promote discovery, but also provide evidence. This case study is an experimentation ground to evaluate the role of space and action in making inferences by reasoning diagrammatically. (shrink)

While mechanistic explanation and, to a lesser extent, nomological explanation are well-explored topics in the philosophy of biology, topological explanation is not. Nor is the role of diagrams in topological explanations. These explanations do not appeal to the operation of mechanisms or laws, and extant accounts of the role of diagrams in biological science explain neither why scientists might prefer diagrammatic representations of topological information to sentential equivalents nor how such representations might facilitate important processes of explanatory reasoning (...) unavailable to scientists who restrict themselves to sentential representations. Accordingly, relying upon a case study about immune system vulnerability to attacks on CD4+ T-cells, I argue that diagrams group together information in a way that avoids repetition in representing topological structure, facilitate identification of specific topological properties of those structures, and make available to controlled processing explanatorily salient counterfactual information about topological structures, all in ways that sentential counterparts of diagrams do not. (shrink)

During the last decades, many cognitive architectures (CAs) have been realized adopting different assumptions about the organization and the representation of their knowledge level. Some of them (e.g. SOAR [35]) adopt a classical symbolic approach, some (e.g. LEABRA[ 48]) are based on a purely connectionist model, while others (e.g. CLARION [59]) adopt a hybrid approach combining connectionist and symbolic representational levels. Additionally, some attempts (e.g. biSOAR) trying to extend the representational capacities of CAs by integrating diagrammatical representations and reasoning (...) are also available [34]. In this paper we propose a reflection on the role that Conceptual Spaces, a framework developed by Peter G¨ardenfors [24] more than fifteen years ago, can play in the current development of the Knowledge Level in Cognitive Systems and Architectures. In particular, we claim that Conceptual Spaces offer a lingua franca that allows to unify and generalize many aspects of the symbolic, sub-symbolic and diagrammatic approaches (by overcoming some of their typical problems) and to integrate them on a common ground. In doing so we extend and detail some of the arguments explored by G¨ardenfors [23] for defending the need of a conceptual, intermediate, representation level between the symbolic and the sub-symbolic one. In particular we focus on the advantages offered by Conceptual Spaces (w.r.t. symbolic and sub-symbolic approaches) in dealing with the problem of compositionality of representations based on typicality traits. Additionally, we argue that Conceptual Spaces could offer a unifying framework for interpreting many kinds of diagrammatic and analogical representations. As a consequence, their adoption could also favor the integration of diagrammatical representation and reasoning in CAs. (shrink)

Logic diagrams have seen a resurgence in their application in a range of fields, including logic, biology, media science, computer science and philosophy. Consequently, understanding the history and philosophy of these diagrams has become crucial. As many current diagrammatic systems in logic are based on ideas that originated in the 18th and 19th centuries, it is important to consider what motivated the use of logic diagrams in the past and whether these reasons are still valid today. This paper proposes (...) that transcendental philosophy was a key inspiration for the development of logic diagrams and that such diagrams can be employed in transcendental arguments, even after the linguistic turn. (shrink)

Argument-mapping software abounds, and one of the reasons is that using the software has been shown to teach/promote/improve critical thinking skills. These positive results are very encouraging, but they also raise the question of whether the computer tutorial environment is producing these results, or whether learning argument mapping, even with just paper and pencil, is sufficient. Based on the results of two empirical studies, I argue that the basic skill of being able to represent an argument diagrammatically plays an important (...) role in the improvement of critical-thinking skills. While these studies do not offer a direct comparison between the two methods, it is important for anyone wishing to employ argument mapping in the classroom to know that significant results can be obtained even with the most rudimentary of tools. (shrink)

It is tough to relate visual perception or other achievements to physiological processing in the central nervous system. The diagrammatic, algebraic, and verbal pictures of how sights seem to Lehar do not advance understanding of how we manage to see what is in the world. There are well-known conceptual reasons why no such purely introspective approach can be productive.

Ein Beispiel für die Rezeption und Fortführung der schopenhauerschen Logik findet man in den Mathematiklehrbüchern Friedrich Adolph Wilhelm Diesterwegs (1790–1866), In diesem Aufsatz werden die historische und systematische Dimension dieser Anwendung von Logikdiagramme auf die Mathematik skizziert. In Kapitel 2 wird zunächst die frühe Rezeption der schopenhauerschen Logik und Philosophie der Mathematik vorgestellt. Dabei werden einige oftmals tradierte Vorurteile, die das Werk Schopenhauers betreffen, in Frage gestellt oder sogar ausgeräumt. In Kapitel 3 wird dann die Philosophie der Mathematik und der (...) Logik Schopenhauers vorgestellt. Es wird gezeigt, dass Schopenhauers Logikdiagramme auf Ideen der Mathematiker Leonhard Euler (1707–1783), Bernhard Friedrich Thibaut (1775–1832) und Franz Ferdinand Schweins (1780–1856) zurückgehen und dass Schopenhauer diese Logikdiagramme auch bereits zur Analyse mathematischer Begriffe verwendet. Kapitel 4 beschäftigt sich mit der Schopenhauer-Rezeption Diesterwegs. Es wird die Frage behandelt, inwieweit Diesterweg in der Mathematik von Schopenhauer beeinflusst war und welche Rolle dieser Einfluss auf sein Verständnis von Logik und Mathematik hatte. In Kapitel 5 wird schließlich gezeigt, dass Diesterweg die schopenhauersche Begriffsanalyse der Mathematik mit Hilfe von Logikdiagrammen aufgreift und weiterführt. Da Diesterweg einer der einflussreichsten Pädagogen und Mathematikdidaktiker des 20. Jahrhunderts war, ist anzunehmen, dass sich Spuren dieser Methode noch heute finden lassen. Inwiefern dies wahrscheinlich ist, wird in der Zusammenfassung (Kapitel 6) thematisiert. (shrink)

This paper presents a diagrammatic notation for visualizing epistemic entities and relations. The notation was created during the Visualizing Worldviews project funded by the University of Toronto’s Jackman Humanities Institute and has been further developed by the scholars participating in the university’s Research Opportunity Program. Since any systematic diagrammatic notation should be based on a solid ontology of the respective domain, we first outline the current state of the scientonomic ontology. We then proceed to providing diagrammatic tools (...) for visualizing the epistemic entities and relations of this ontology. These basic diagramming techniques allow us to construct diagrams of various types for both synchronic and diachronic visualizations. The paper concludes by highlighting some future research directions. As the notation presented here is de facto accepted and used in scientonomy, the paper suggests no modifications. (shrink)

It has been stated that the notion of cause and effect is one object of study that sciences and engineering revolve around. Lately, in software engineering, diagrammatic causal inference methods (e.g., Pearl’s model) have gained popularity (e.g., analyzing causes and effects of change in software requirement development). This paper concerns diagrammatical (graphic) models of causal relationships. Specifically, we experiment with using the conceptual language of thinging machines (TMs) as a tool in this context. This would benefit works on causal (...) relationships in requirements engineering, enhance our understanding of the TM modeling, and contribute to the study of the philosophical notion of causality. To specify the causality in a system’s description is to constrain the system’s behavior and thus exclude some possible chronologies of events. The notion of causality has been studied based on tools to express causal questions in diagrammatic and algebraic forms. Causal models deploy diagrammatic models, structural equations, and counterfactual and interventional logic. Diagrammatic models serve as a language for representing what we know about the world. The research methodology in the paper focuses on converting causal graphs into TM models and contrasts the two types of representation. The results show that the TM depiction of causality is more complete and therefore can provide a foundation for causal graphs. (shrink)

Charles S. Peirce’s semiotics uniquely divides signs into: i) symbols, which pick out their objects by arbitrary convention or habit, ii) indices, which pick out their objects by unmediated ‘pointing’, and iii) icons, which pick out their objects by resembling them (as Peirce put it: an icon’s parts are related in the same way that the objects represented by those parts are themselves related). Thus representing structure is one of the icon’s greatest strengths. It is argued that the implications of (...) scaffolding education iconically are profound: for providing learners with a navigable road-map of a subject matter, for enabling them to see further connections of their own in what is taught, and for supporting meaningful active learning. Potential objections that iconic teaching is excessively entertaining and overly susceptible to misleading rhetorical manipulation are addressed. (shrink)

This is a discussion of the state of discussion on pragmatic reasons for belief.

Some sensory experiences are pleasant, some unpleasant. This is a truism. But understanding what makes these experiences pleasant and unpleasant is not an easy job. Various difficulties and puzzles arise as soon as we start theorizing. There are various philosophical theories on offer that seem to give different accounts for the positive or negative affective valences of sensory experiences. In this paper, we will look at the current state of art in the philosophy of mind, present the main contenders, critically (...) compare and contrast them. In particular, we want to examine how they handle the reason-giving power of affective states. We will look into two representationalist proposals (Evaluativism and Imperativism) and a functionalist proposal, and argue that, contrary to their own advertisements, the representationalist proposals don’t have good accounts of why and how sensory affect can motivate, rationalize, and justify subsequent behavior and intentional mental activity. We will show that our own functionalist proposal does a much better job in this regard, and that when the representationalist proposals are modified to do a better job, they fare better not because of their representationalist credentials but due to their functionalist ones. (shrink)

I defend the view that a reason for someone to do something is just a reason why she ought to do it. This simple view has been thought incompatible with the existence of reasons to do things that we may refrain from doing or even ought not to do. For it is widely assumed that there are reasons why we ought to do something only if we ought to do it. I present several counterexamples to this principle and reject some (...) ways of understanding "ought" so that the principle is compatible with my examples. I conclude with a hypothesis for when and why the principle should be expected to fail. (shrink)

This paper explores various subtleties in our ordinary thought and talk about normative reasons—subtleties which, if taken seriously, have various upshots, both substantive and methodological. I focus on two subtleties in particular. The first concerns the use of reason (in its normative sense) as both a count noun and as a mass noun, and the second concerns the context-sensitivity of normative reasons-claims. The more carefully we look at the language of reasons, I argue, the clearer its limitations and liabilities become. (...) The cumulative upshot is that although talk of reasons is intelligible and useful for the purposes of communication, we should be wary of placing much weight on it when engaging in substantive normative inquiry. By way of illustration, I consider some potential pitfalls of taking our talk of reasons too seriously, explaining how careful attention to the language of reasons undermines the main argument for moral particularism, Mark Schroeder’s recent defense of Humeanism about practical reasons, and the “reasons-first” program in metanormativity. (shrink)

Many philosophers have been attracted to the view that reasons are premises of good reasoning – that reasons to φ are premises of good reasoning towards φ-ing. However, while this reasoning view is indeed attractive, it faces a problem accommodating outweighed reasons. In this article, I argue that the standard solution to this problem is unsuccessful and propose an alternative, which draws on the idea that good patterns of reasoning can be defeasible. I conclude by drawing (...) out implications for the debate over pragmatic reasons for belief and other attitudes and for one influential form of reductionism about the normative. (shrink)

Here are considered the conditions under which the method of diagrams is liable to include non-classical logics, among which the spatial representation of non-bivalent negation. This will be done with two intended purposes, namely: a review of the main concepts involved in the definition of logical negation; an explanation of the epistemological obstacles against the introduction of non-classical negations within diagrammatic logic.

This article gives an overview of some recent debates about the relationship between reasons and rational requirements of coherence - e.g. the requirements to be consistent in our beliefs and intentions, and to intend what we take to be the necessary means to our ends.

What reasons do we have to use certain concepts and conceptions rather than others? Approaching that question in a methodologically humanistic rather than Platonic spirit, one might seek “reasons for concept use” in how well concepts serve the contingent human concerns of those who live by them. But appealing to the instrumentality of concepts in meeting our concerns invites the worry that this yields the wrong kind of reasons, especially if the relevant concerns are nonmoral ones. Drawing on Susan Wolf’s (...) work on the moral/nonmoral distinction and the neglected role of reasons of love, I argue that this worry is misplaced, and in fact overlooks some of our most important reasons to prefer certain concepts over others. Yet a lingering worry remains, namely that the value of concepts does not just lie in what they are good for. Drawing on another strand in Wolf’s work, I explore the question whether concepts can be valuable good-for-nothings, and show how this ultimately also underscores the importance of reasons of love as reasons for concept use. (shrink)

This chapter develops and defends an account of reason: to reason is to scrutinize one’s attitudes by consulting the perspectives of other persons. The principal attraction of this account is its ability to vindicate the unique of authority of reason. The chapter argues that this conception entails that reasoning is a robustly social endeavor—that it is, in the first instance, something we do with other people. It is further argued that such social endeavors presuppose mutual respect on the part (...) of those participating in them. The account therefore yields a form of Kantian constructivism: we have an unconditional duty of respect for persons because such a duty is implicit in the very nature of reasoning. -/- . (shrink)

This paper is a survey of the generalism-particularism debate and related issues concerning the relationship between normative reasons and moral principles.

Epistemic reasons are mental states. They are not propositions or non-mental facts. The discussion proceeds as follows. Section 1 introduces the topic. Section 2 gives two concrete examples of how our topic directly affects the internalism/externalism debate in normative epistemology. Section 3 responds to an argument against the view that reasons are mental states. Section 4 presents two problems for the view that reasons are propositions. Section 5 presents two problems for the view that reasons are non-mental facts. Section 6 (...) argues that reasons are mental states. Section 7 responds to objections. (shrink)

Often, when one faces a choice between alternative actions, there are reasons both for and against each alternative. On one way of understanding these words, what one “ought to do all things considered (ATC)” is determined by the totality of these reasons. So, these reasons can somehow be “combined” or “aggregated” to yield an ATC verdict on these alternatives. First, various assumptions about this sort of aggregation of reasons are articulated. Then it is shown that these assumptions allow for the (...) proof of a “Reasons Aggregation Theorem” – parallel to John Harsanyi’s 1955 “Social Aggregation Theorem”. All reasons for action are grounded in reason-providing values; and, for every such reason-providing value, there is in principle a way of measuring how strongly this value counts against each alternative. The theorem tells us that the appropriate measure of how much reason ATC there is against each alternative is simply a weighted sum of the strengths with which these values count against that alternative; the agent ought not ATC to perform an action iff there is ATC more reason against it than against some available alternative. (shrink)

[Significantly updated in Chapter 6 of Awareness and the Substructure of Knowledge] In order for a reason to justify an action or attitude it must be one that is possessed by an agent. Knowledge-centric views of possession ground our possession of reasons, at least partially, either in our knowledge of them or in our being in a position to know them. On virtually all accounts, knowing P is some kind of non-accidental true belief that P. This entails that knowing P (...) is a kind of non-accidental true representation that P. I outline a novel theory of the epistemic requirement on possession in terms of this more general state of non-accidental true representation. It is just as well placed to explain the motivations behind knowledge-centric views of possession, and it is also better placed to explain the extent of the reasons we possess in certain cases of deductive belief-updates and cases involving environmental luck. I conclude with three reflections. First, I indicate how my arguments generate a dilemma for Errol Lord’s view that possessing reasons is just a matter of being in a position to manifest one’s knowledge how to use them. Second, I explain how my view can simultaneously manage cases of environmental luck without falling prey to lottery cases. Finally, I sketch the direction for a further range of counterexamples to knowledge-centric theories of possession. (shrink)

The notion of Simulative Reasoning in the study of propositional attitudes within Artificial Intelligence (AI) is strongly related to the Simulation Theory of mental ascription in Philosophy. Roughly speaking, when an AI system engages in Simulative Reasoning about a target agent, it reasons with that agent’s beliefs as temporary hypotheses of its own, thereby coming to conclusions about what the agent might conclude or might have concluded. The contrast is with non-simulative meta-reasoning, where the AI system reasons (...) within a detailed theory about the agent’s (conjectured) reasoning acts. The motive within AI for preferring Simulative Reasoning is that it is more convenient and efficient, because of a simplification of the representations and reasoning processes. The chapter discusses this advantage in detail. It also sketches the use of Simulative Reasoning in an AI natural language processing system, ATT-Meta, that is currently being implemented. This system is directed at the understanding of propositional attitude reports. In ATT-Meta, Simulative Reasoning is yoked to a somewhat independent set of ideas about how attitude reports should be treated. Central here are the claims that (a) speakers often employ commonsense (and largely metaphorical) models of mind in describing agents’ attitudes, (b) the listener accordingly needs often to reason within the terms of such models, rather than on the basis of any objectively justifiable characterization of the mind, and (c) the commonsense models filter the suggestions that Simulative Reasoning comes up with concerning target agents’ reasoning conclusions. There is a yet tighter connection between the commonsense models and the Simulative Reasoning. It turns out that Simulative Reasoning can be rationally reconstructed in terms of a more general type of reasoning about the possibly-counterfactual “world” that the target agent believes in, together with an assumption that that agent has a faithful representation of the world. In the ATT-Meta approach, the reasoner adopts that assumption when it views the target agent through a particular commonsense model (called IDEAS-AS-MODELS). (shrink)

This paper presents a series of studies (total N=579) which demonstrate that folk judgments concerning the reasonableness of decisions and actions depend strongly on whether they engender positive or negative consequences. A particular decision is deemed more reasonable in retrospect when it produces beneficial consequences than when it produces harmful consequences, even if the situation in which the decision was taken and the epistemic circumstances of the agent are held fixed across conditions. This finding is worrisome for the law, where (...) the reasonable person standard plays a prominent role. The legal concept of reasonableness is outcome-insensitive: whether the defendant acted in a reasonable fashion or not depends exclusively on her context of action, no matter how things play out. Folk judgments of reasonableness are thus inconsistent with the legal concept of reasonableness. Problematically, in common law jurisdictions, the decision whether a defendant’s behavior was reasonable or not is frequently (though not necessarily) delegated to a lay jury. (shrink)