Diagrams as sketches

This article puts forward the notion of ??evolving diagram?? as an important case of mathematical diagram. An evolving diagram combines, through a dynamic graphical enrichment, the representation of an object and the representation of a piece of reasoning based on the representation of that object. Evolving diagrams can be illustrated in particular with category-theoretic diagrams (hereafter ??diagrams*??) in the context of ??sketch theory,?? a branch of modern category theory. It is argued that sketch theory provides a diagrammatic* theory of diagrams*, that it helps to overcome the rivalry between set theory and category theory as a general semantical framework, and that it suggests a more flexible understanding of the opposition between formal proofs and diagrammatic reasoning. Thus, the aim of the paper is twofold. First, it claims that diagrams* provide a clear example of evolving diagrams, and shed light on them as a general phenomenon. Second, in return, it uses sketches, understood as evolving diagrams, to show how diagrams* in general should be re-evaluated positively.     

How Children and Adults Represent God's Mind

For centuries, humans have contemplated the minds of gods. Research on religious cognition is spread across sub‐disciplines, making it difficult to gain a complete understanding of how people reason about gods' minds. We integrate approaches from cognitive, developmental, and social psychology and neuroscience to illuminate the origins of religious cognition. First, we show that although adults explicitly discriminate supernatural minds from human minds, their implicit responses reveal far less discrimination. Next, we demonstrate that children's religious cognition often matches adults' implicit responses, revealing anthropomorphic notions of God's mind. Together, data from children and adults suggest the intuitive nature of perceiving God's mind as human‐like. We then propose three complementary explanations for why anthropomorphism persists in adulthood, suggesting that anthropomorphism may be (a) an instance of the anchoring and adjustment heuristic; (b) a reflection of early testimony; and/or (c) an evolutionary byproduct.     

Atomic discourse inThe Feynman Lectures on Physics

We examine the compromises that are actually made in modern scientific discourse concerning atoms. We conclude that even the ideals of clarity and consistency can be legitimately compromised in order to obtain other advantages.     

Scientific Discovery With Law-Encoding Diagrams

This article introduces the concept of law-encoding diagrams (LEDs) and presents the argument that they have had a role in scientific discovery that has not been previously recognized. An LED is a representation that correctly encodes the underlying relations of a law, or a system of simultaneous laws, in the structure of a diagram by the means of geometric, topological, and spatial constraints, such that the instantiation of a particular diagram represents a single instance of the phenomena or a particular case of the law(s). Examples of LEDs in the history of science are discussed, and the benefits of using LEDs in discovery are considered. LEDs are distinguished from other forms of diagrammatic representation. Previous work on the computational modeling of diagrammatic law induction is reinterpreted in terms of the search for diagrammatic constraints of LEDs. A general characterization of the role of LEDs in discovery is considered, and a framework for classifying processes of discovery based on LEDs is proposed.     

Diagrams: Socrates and Meno's slave

Abstract

When do two mental items belong to the same life? We could be content with the answer ‐just when they have certain volitional qualities in common. An affinity is noted between that theory and Berkeley's early doctrine of the self. Some rivals of the volitional theory invoke a spiritual or physical owner of mental items. They run a risk either of empty formality or of causal superstition. Other rivals postulate a non‐transitive and symmetrical relation in the set of mental items. They must allow in consequence either for joint ownership of one and the same mental item, or for incompatible simultaneous decisions by one and the same person, or for new forms of death. This makes them disquieting. Another rival invokes a transitive and symmetrical relation defined in terms of co‐consciousness. Even that allows for incongruous simultaneities. The volitional theory is free from such disadvantages.     

Word Diagrams in Teaching Classical Conditioning

The Psychological Record - The present experiment examined the value of word diagrams on the concept learning of university students. All students learned the concept of classical conditioning by...     

Teaching Legal Theory with Venn Diagrams

Venn diagrams, which are widely used in introductory logic courses, provide a convenient and illuminating way of presenting the various theories concerning the nature of law. When combined with the Aristotelian square of opposition, these diagrams show not only how the theories are related to one another, logically, which is essential to understanding them, but also which theories are compossible. One surprising result of this approach is that it shows the substantive compatibility of the theories of law set forth by H. L. A. Hart and Ronald Dworkin, who are usually pitted against one another. I show, through quotation and visual representation, that there is no essential disagreement between these jurisprudential stalwarts concerning the relation of law and morality.     

逻辑图表的发展

长期以来,逻辑图表被逻辑学家们仅仅视为教学辅助工具;但逻辑图表从Euler图到Venn图再到存在图表和概念图表的发展表明,它已经超越出教学工具这一狭隘界限,从而获得了更为深刻,广泛的意义,实际上逻辑图表是与我们标准数理逻辑记法相区别的另一种有效的逻辑记法,而且有着更强的直观性和可应用性。     

Bull's Theorem by the Method of Diagrams

We show how to use diagrams in order to obtain straightforward completeness theorems for extensions of K4.3 and a very simple and constructive proof of Bull's theorem: every normal extension of S4.3 has the finite model property.     

Formalization of Reliability Block Diagrams in Higher-order Logic

Reliability Block Diagrams (RBDs) allow us to model the failure relationships of complex systems and their sub-components and are extensively used for system reliability, availability and maintainability analyses. Traditionally, these RBD-based analyses are done using paper-and-pencil proofs or computer simulations, which cannot ascertain absolute correctness due to their inaccuracy limitations. As a complementary approach, we propose to use the higher-order logic theorem prover HOL to conduct RBD-based analysis. For this purpose, we present a higher-order logic formalization of commonly used RBD configurations, such as series, parallel, parallel-series and series-parallel, and the formal verification of their equivalent mathematical expressions. A distinguishing feature of the proposed RBD formalization is the ability to model nested RBD configurations, which are RBDs having blocks that also represent RBD configurations. This generality allows us to formally analyze the reliability of many real-world systems. For illustration purposes, we formally analyze the reliability of a generic Virtual Data Center (VDC) in a cloud computing infrastructure exhibiting the nested series-parallel RBD configuration.     

Euler-type Diagrams and the Quantification of the Predicate

Journal of Philosophical Logic - Logicians have often suggested that the use of Euler-type diagrams has influenced the idea of the quantification of the predicate. This is mainly due to the fact...     

Logic Diagrams as Argument Maps in Eristic Dialectics

This paper analyses a hitherto unknown technique of using logic diagrams to create argument maps in eristic dialectics. The method was invented in the 1810s and -20s by Arthur Schopenhauer, who is considered the originator of modern eristic. This technique of Schopenhauer could be interesting for several branches of research in the field of argumentation: Firstly, for the field of argument mapping, since here a hitherto unknown diagrammatic technique is shown in order to visualise possible situations of arguments in a dialogical controversy. Secondly, the art of controversy or eristic, since the diagrams do not analyse the truth of judgements and the validity of inferences, but the persuasiveness of arguments in a dialogue.

     

Why horizontalism

Philosophical Studies - Horizontalism is the thesis that what a speaker asserts in literally and sincerely uttering an indicative sentence is some horizontal proposition of her utterance;...     

Translating Toulmin Diagrams: Theory Neutrality in Argument Representation

The Toulmin diagram layout is very familiar and widely used, particularly in the teaching of critical thinking skills. The conventional box-and-arrow diagram is equally familiar and widespread. Translation between the two throws up a number of interesting challenges. Some of these challenges (such as the relationship between Toulmin warrants and their counterparts in traditional diagrams) represent slightly different ways of looking at old and deep theoretical questions. Others (such as how to allow Toulmin diagrams to be recursive) are diagrammatic versions of questions that have already been addressed in artificial intelligence models of argument. But there are further questions (such as the relationships between refutations, rebuttals and undercutters, and the roles of multiple warrants) that are posed as a specific result of examining the diagram inter-translation problem. These three classes of problems are discussed. To the first class are addressed solutions based on engineering pragmatism; to the second class, are addressed solutions drawn from the appropriate literature; and to the third class, fuller exploration is offered justifying the approaches taken in developing solutions that offer both pragmatic utility and theoretical interest. Finally, these solutions are explored briefly in the context of the Araucaria system, showing the ways in which analysts can tackle arguments either using one diagrammatic style or another, or even a combination of the two.