No mathematical background is necessary to appreciate this classic of probability theory, which remains unsurpassed in its clarity, readability, and sheer charm. Its author, British logician John Venn (1834-1923), popularized the famous Venn Diagrams that are commonly used in teaching elementary mathematics.

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)

In this work, formulas are inclusions \ and non-inclusions \ between Boolean terms \ and \. We present a set of rules through which one can transform a term t in a diagram \ and, consequently, each inclusion \ ) in an inclusion \ ) between diagrams. Also, by applying the rules just to the diagrams we are able to solve the problem of verifying if a formula \ is consequence of a, possibly empty, set \ of formulas (...) taken as hypotheses. Our system has a diagrammatic language based on Venn diagrams that are read as sets, and not as statements about sets, as usual. We present syntax and semantics of the diagrammatic language, define a set of rules for proving consequence, and prove that our set of rules is strongly sound and complete in the following sense: given a set \ of formulas, \ is a consequence of \ iff there is a proof of this fact that is based only on the rules of the system and involves only diagrams associated to \ and to the members of \. (shrink)

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.

In this paper we argue that there were several currents, ideas and problems in 19th-century logic that motivated John Venn to develop his famous logic diagrams. To this end, we first examine the problem of uncertainty or over-specification in syllogistic that became obvious in Euler diagrams. In the 19th century, numerous logicians tried to solve this problem. The most famous was the attempt to introduce dashed circles into Euler diagrams. The solution that John Venn developed (...) for this problem, however, came from a completely different area of logic: instead of orienting to syllogistic like Euler diagrams, Venn applied Boolean algebra to improve visual reasoning. Venn’s contribution to solving the problem of elimination also played an important role. The result of this development is still known today as the ‘Venn Diagram’. (shrink)

The following technique has proven effective in helping beginning logic students locate the sections of a three-circled Venn Diagram in which they are to represent a categorical sentence. Very often theses students are unable to identify the parts of the diagram they are to shade or bar.

When interpreting the meanings of visual features in information visualizations, observers have expectations about how visual features map onto concepts (inferred mappings.) In this study, we examined whether aspects of inferred mappings that have been previously identified for colormap data visualizations generalize to a different type of visualization, Venn diagrams. Venn diagrams offer an interesting test case because empirical evidence about the nature of inferred mappings for colormaps suggests that established conventions for Venn diagrams (...) are counterintuitive. Venn diagrams represent classes using overlapping circles and express logical relationships between those classes by shading out regions to encode the concept of non-existence, or none. We propose that people do not simply expect shading to signify non-existence, but rather they expect regions that appear as holes to signify non-existence (the hole hypothesis.) The appearance of a hole depends on perceptual properties in the diagram in relation to its background. Across three experiments, results supported the hole hypothesis, underscoring the importance of configural processing for interpreting the meanings of visual features in information visualizations. (shrink)

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. (shrink)

This paper offers two reasons why students frequently struggle to read Venn diagrams and discusses the semantic knowledge requisite for the appropriate perceptual experience of them. The first reason students struggle is that Venn diagrams require one to intentionally effect a certain figure-background organization that runs counter to the tendency of perceptual organization which attends to similarity in line over similarity in form. The second reason is that, while Venn diagrams rely on Boolean truth (...) conditions to determine shading patterns, Venn diagrams are not capable of literally representing Boolean truth conditions. It is therefore not something within the diagram that tells us what to attend to perceptually, but rather knowledge of the semantic relation between Boolean truth conditions and Venn diagrams. After explaining the semantics of each of these and the discrepancy between them, the author argues that Venn diagrams are nevertheless capable of metaphorically representing Boolean truth conditions. This metaphorical representation is addressed in detail and the author concludes that this form of representation does not tell against teaching Venn diagrams as a method for testing validity, especially since their legibility is greatly increased with the semantic knowledge expounded here. (shrink)

Concern about high-profile ethical lapses by business managers has led to an increasing emphasis on ethics instruction in business schools. Various pedagogical methods are used to expose business students to real-world ethical dilemmas, yet students may not readily grasp the linkages between ethical theories and dilemmas to identify possible ethical solutions. Venn diagrams are a valuable instructional tool in business ethics classes when used with other teaching methodologies such as case studies. We describe how the use of (...) class='Hi'>Venn diagrams assists students in visualizing the key points of and the connections between ethical theories and dilemmas to shed light on possible ethical solutions. Examples of teaching exercises are provided along with ideas for future research in the use of Venn diagrams in activating moral imagination and improving ethical reasoning. Overall, positive student reactions to the introduction of Venn diagrams in business ethics classrooms support the use of this methodology. (shrink)

Concern about high-profile ethical lapses by business managers has led to an increasing emphasis on ethics instruction in business schools. Various pedagogical methods are used to expose business students to real-world ethical dilemmas, yet students may not readily grasp the linkages between ethical theories and dilemmas to identify possible ethical solutions. Venn diagrams are a valuable instructional tool in business ethics classes when used with other teaching methodologies such as case studies. We describe how the use of (...) class='Hi'>Venn diagrams assists students in visualizing the key points of and the connections between ethical theories and dilemmas to shed light on possible ethical solutions. Examples of teaching exercises are provided along with ideas for future research in the use of Venn diagrams in activating moral imagination and improving ethical reasoning. Overall, positive student reactions to the introduction of Venn diagrams in business ethics classrooms support the use of this methodology. (shrink)

It is of common use in modern Venn diagrams to mark a compartment with a cross to express its non-emptiness. Modern scholars seem to derive this convention from Charles S. Peirce, with the assumption that it was unknown to John Venn. This paper demonstrates that Venn actually introduced several methods to represent existentials but felt uneasy with them. The resistance to formalize existentials was not limited to diagrammatic systems, as George Boole and his followers also failed (...) to provide a satisfactory symbolic representation for them. This difficulty points out issues that are inherent to the very nature of existentials. This paper assesses the various methods designed for the representation of existential statements with Venn diagrams. First, Venn’s own attempts are discussed and compared with other solutions proposed by his contemporaries and successors, notably Lewis Carroll and Peirce. Since disjunctives hold an important role in an effective representation of existentials, their representation is also discussed. Finally, recent methods for the diagrammatic representation of existing individuals, rather than mere existence, are surveyed. (shrink)

This paperback is a programed text designed for teaching introductory logic, either in conjunction with a standard text based upon traditional logic or as a do-it-yourself supplement for students taking courses stressing symbolic logic. The student learns logical theory by answering a variety of short answer, objective type exercises. The correct answer is given directly below each question or exercise, and the student is required to cover the answer while working the exercise; the purpose of this immediate access to the (...) answer is to enable the student to determine quickly whether or not he comprehends the material. The content of the book concentrates primarily upon categorical statements, presenting both the existential and hypothetical interpretations of the square of opposition, but unfortunately continues to promulgate the historically inaccurate terminology of "Aristotelian" and "Boolean." [Categorical statements were first expressed as existential statements by Franz Brentano in 1874.] Categorical statements not in standard form are standardized, e.g., "No roaches feel despair" is standardized by being rewritten as "No roaches are entities who feel despair." In addition to considering the logical relations of opposition with respect to A, E, I, O statements, these same relations are extended to non-categorical statements, e.g., "George Washington did not die in Europe" and "George Washington did not die in Asia" are subcontraries, while "There is philosophical activity on Venus" and "There is philosophical activity in the universe" are alterns. Determining validity of syllogisms is based upon the distribution of terms, and also by the use of Venn diagrams. The traditional figures and moods of the syllogism are ignored, as is the distinction between major and minor terms, with these latter two being lumped together as end terms. Immediate inferences and non-syllogistic arguments are treated by Venn diagrams. One-, two-, three- and four-term Venn diagrams are utilized throughout a substantial part of the book.--T. G. N. (shrink)

This paper describes and argues for the inclusion of a modified form of Venn diagrams in critical thinking courses and textbooks. The modified Venn Diagrams, it is argued, are easier to learn as they more clearly show the meanings of inclusion and exclusion, easier to use when solving problems (including those found on LSAT exams), are often included in LSAT preparatory material, and students tend to have a more thorough understanding of the concept of logical possibility (...) after having used modified Venn diagrams. (shrink)

In an article in the Journal of Philosophical Logic in 1996, "Towards a Model Theory of Venn Diagrams," (Vol. 25, No. 5, pp. 463-482), Hammer and Danner proved the full completeness of Shin's formal system for reasoning with Venn Diagrams. Their proof is eight pages long. This note gives a brief five line proof of this same result, using connections between diagrammatic and sentential representations.

In an article in the Journal of Philosophical Logic in 1996, "Towards a Model Theory of Venn Diagrams,", Hammer and Danner proved the full completeness of Shin's formal system for reasoning with Venn Diagrams. Their proof is eight pages long. This note gives a brief five line proof of this same result, using connections between diagrammatic and sentential representations.

The attempt to find usable diagrams fornterms of the sort devised by John Venn seems to have originated with Venn himself, who published diagrams for up to five classes (the fifth class, however, was shaped like a doughnut, and contained an area outside itself — like the hole in the doughnut). Venn then suggested that “if we wanted to use a diagram forsixterms (x, y, z, w, v, u) the best plan would probably be to (...) taketwofive-term figures, one for theupart and one for the non-upart of all the other combinations …” Such a method would, as Venn admits, be somewhat confusing to the eye, and thus fail to fulfil one of the major purposes of the diagram; nevertheless, it is a method which could be extended to covernclasses.Venn also suggests (but without illustration) another method for drawing diagrams fornterms: “… the rule of formation would be very simple. We should merely have to begin by drawing any closed figure, and then proceed to draw others subject to the one condition that each intersect once, and once only, all the existing subdivisions produced by those which had gone before.” Then, in a rather surprising footnote, he adds: “It will be found that when we adhere to continuous figures, instead of the discontinuous five-term figure given above, there is a tendency for the resultant outlines thus successively drawn to assume a comb-like shape after the first four or five. If we begin by circles or other rounded figures the teeth are curved, if by parallelograms then they are straight. (shrink)

A new diagram system \ where properties are fundamental and an object exists only w.r.t a property is presented. This work modifies both in syntax and semantics the system \ proposed by Choudhury and Chakraborty to picturise and address issues connected with open universe. Semantics for the current system is given. Soundness and completeness w.r.t the semantics are established.

It is not rare to meet in scientific literature with a figure made of three circles, intersecting in such a way as to delineate all the combinations of the components that they stand for. This figure is commonly known as a ‘Venn diagram’ or ‘Venn’s three circles’. In this paper, we argue that many so-called Venn diagrams found in modern scientific literature do not truly depict intersections, and hence, are not true Venn diagrams.

In elementary logic textbooks, Venn diagrams are used to analyze and evaluate the validity of syllogistic arguments. Although the method of Venn diagrams is shown to be a powerful analytical tool in these textbooks, it still has limitations. On the one hand, such method fails to represent singular statements of the form, “a is F.” On other hand, it also fails to represent identity statements of the form, “a is b.” Because of this, it also fails (...) to give an account of the validity of some obviously valid arguments that contain these types of statements as constituents. In this paper, owing to the developments in the literature on Venn diagrams, we offer a way of supplementing the rules of the Venn diagram found in textbooks, and show how this retooled Venn diagram technique could account for the problem cases. (shrink)

John Venn is remembered today as the inventor of the famous "Venn diagram." The postmortem fame of the namesake diagram has until now eclipsed Venn's own status as one of the most accomplished logicians in his day. Praised by John Stuart Mill as a "highly successful thinker" with much "power of original thought," Venn profoundly influenced nineteenth-century philosophers, ranging from Mill and Henry Sidgwick to Charles Sanders Peirce. Venn was heir to a clerical, Evangelical dynasty (...) but religious doubts led him to resign Holy Orders and instead turn to an academic life, writing influential textbooks on probability theory and logic and advocating for education reform, including for women's education. Venn also collaborated with Francis Galton in the unofficial Anthropometrical Laboratory, and, through his writing and teaching, a direct line can be traced from Venn and his circle to the development of analytic philosophy in the work of W. E. Johnson, G. E. Moore, Bertrand Russell, and C. D. Broad. Through family descent, Venn was connected to eminent Victorians like Leslie Stephen and Virginia Woolf. This essential book explores Venn's life and work in context, taking readers on his journey from Evangelical son to Cambridge don. The picture that emerges of Venn, the person, is of a man with many sympathies-sometimes mutually reinforcing and at other times outwardly contradictory. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:John Venn. A Life in Logic by Lukas M. VerburgtClaudia CristalliLukas M. VerburgtJohn Venn. A Life in Logic Chicago and London: The University of Chicago Press, 2022. 411 pp., incl. indexThis is the first intellectual biography of John Venn (1834–1923), British logician, “philosopher and antiquarian” (DNB). Until now, Venn had not been studied as a philosophical figure in its own right. He is mostly (...) remembered today for the “Venn Diagrams,” a graphical device illustrating logical relations such as inclusion, exclusion, conjunction, disjunction, and for having systematically articulated a frequentist theory of probability. Together with an in-depth account of Venn’s logical developments, of his studies in probability theory, and of its applications to history and to anthropometry, Lukas M. Verburgt brings to life Venn’s spiritual struggles within—and eventually out of— his family’s religion, evangelicalism. Verburgt’s reconstruction is substantiated by many extracts from Venn’s correspondence and private notes, which add substantially to the value of this biography. (Of note is Verburgt’s 2022 editon of Venn’s unpublished writings and slected correspondence, which can be considered a companion to this biography).While showing the connections of epistemology, religion and academic politics in Victorian Cambridge, Verburgt contributes an important chapter in the history of logic and of empiricist philosophy.The book opens with a chapter on Venn’s childhood and early youth (1834–53), largely drawn from manuscript material of the Venn family—the “Parentalia”—and from Venn’s Annals of a Clerical Family, where he included “autobiographical recollections” (p. 2). The chapter introduces the reader to the values of a prominent evangelical family and to their peculiar way of understanding Christianity as a faith to be embedded in habit and to be taught by example. The following two [End Page 385] chapters follow Venn’s development, first as a student at Caius college, Cambridge (1853–57) and then as a country curate; the chapters elaborate on the institutional history of the college and on parish life from the perspective of evangelical orthodoxy. Chapter 4 is still chronologically situated within Venn’s early clerical service but is arranged thematically around his first publication (1862), “Science of History,” which contains some elements of the fundamental ideas elaborated in the Logic of Chance (1866): namely, that probability theory could only make predictions about long-run frequencies of events, and never about individual cases (p. 77). The core of Venn’s argument in 1862 revolved around possible atheistic implications of a science of history. If at some point it would be possible to predict future events, would there still be free will? Or a God? In his reply, Venn maintained that any prediction concerning individual agents could be defeated by assuming that the agent gets to know the prediction and changes her actions accordingly (pp. 74–76).After the publication of the “Science of History” essay, Venn left the parish for Cambridge, where he worked first as a Catechist (1862– 1867) and then as a moral sciences lecturer. Verburgt devotes chapter 5 to that and to the Cambridge milieu, where Venn sides for university reforms and engages with the Grote Club, all while drifting away from evangelical orthodoxy. After laying out Venn’s theory of probability (Chapter 6), Verburgt returns to an integrated analysis of Venn’s scientific and religious thought in chapter 7. This chapter may be particularly relevant for pragmatist readers, since here Verburgt highlights the commonalities between Venn’s “religious logic” and the notion of belief that will be expressed in more detail by pragmatism. Verburgt focuses on Venn’s logic in three subsequent chapters (Chapters 8, 9 and 11). Considered together, they provide a comprehensive history of Venn’s logical development, which Verburgt proudly announces as the first of this kind (p. xx). The last chapter on Venn’s logic (Chapter 11) also includes Verburgt’s argument for “includ[ing] Venn in the local history of early analytic philosophy at Cambridge” (p. 262). Chapters 10 and 12 recount Venn’s last years with particular emphasis on his interests in natural and historical sciences; they provide relevant material for scholars interested in the history and philosophy of... (shrink)

This paper explores the question of what makes diagrammatic representations effective for human logical reasoning, focusing on how Euler diagrams support syllogistic reasoning. It is widely held that diagrammatic representations aid intuitive understanding of logical reasoning. In the psychological literature, however, it is still controversial whether and how Euler diagrams can aid untrained people to successfully conduct logical reasoning such as set-theoretic and syllogistic reasoning. To challenge the negative view, we build on the findings of modern diagrammatic logic (...) and introduce an Euler-style diagrammatic representation system that is designed to avoid problems inherent to a traditional version of Euler diagrams. It is hypothesized that Euler diagrams are effective not only in interpreting sentential premises but also in reasoning about semantic structures implicit in given sentences. To test the hypothesis, we compared Euler diagrams with other types of diagrams having different syntactic or semantic properties. Experiment compared the difference in performance between syllogistic reasoning with Euler diagrams and Venn diagrams. Additional analysis examined the case of a linear variant of Euler diagrams, in which set-relationships are represented by one-dimensional lines. The experimental results provide evidence supporting our hypothesis. It is argued that the efficacy of diagrams in supporting syllogistic reasoning crucially depends on the way they represent the relational information contained in categorical sentences. (shrink)

Logic diagrams have been increasingly studied and applied for a few decades, not only in logic, but also in many other fields of science. The history of logic diagrams is an important subject, as many current systems and applications of logic diagrams are based on historical predecessors. While traditional histories of logic diagrams cite pioneers such as Leibniz, Euler, Venn, and Peirce, it is not widely known that Kant and the early Kantians in Germany and (...) England played a crucial role in popularising Euler(-type) diagrams. In this paper, the role of the Kantians in the late eighteenth and early nineteenth centuries will be analysed in more detail. It shows that diagrams (or intuition in general) were a highly contentious topic that depend on the philosophical attitude and went beyond logic to touch on issues of physics, metaphysics, linguistics and, above all, mathematics. (shrink)

Truth diagrams are introduced as a novel graphical representation for propositional logic. To demonstrate their epistemic efficacy a set of 28 concepts are proposed that any comprehensive representation for PL should encompass. TDs address all the criteria whereas seven other existing representations for PL only provide partial coverage. These existing representations are: the linear formula notation, truth tables, a PL specific interpretation of Venn Diagrams, Frege’s conceptual notation, diagrams from Wittgenstein’s Tractatus, Pierce’s alpha graphs and Gardner’s (...) shuttle diagrams. The comparison of the representations succeeds in distinguishing ideas that are fundamental to PL from features of common PL representations that are somewhat arbitrary. (shrink)

Logic, the discipline that explores valid reasoning, does not need to be limited to a specific form of representation but should include any form as long as it allows us to draw sound conclusions from given information. The use of diagrams has a long but unequal history in logic: The golden age of diagrammatic logic of the 19th century thanks to Euler and Venn diagrams was followed by the early 20th century's symbolization of modern logic by Frege (...) and Russell. Recently, we have been witnessing a revival of interest in diagrams from various disciplines - mathematics, logic, philosophy, cognitive science, and computer science. This book aims to provide a space for this newly debated topic - the logical status of diagrams - in order to advance the goal of universal logic by exploring common and/or unique features of visual reasoning. (shrink)

This research lies within the scientific and pedagogic scope of Philosophy for Children, more specifically its implementation in a community of philosophical inquiry, the pedagogic and epistemological model of the Philosophy for Children, with pre-school children. The purpose was to research the type of answer this community of inquiry would give to some logic problems by resorting to Venn Diagrams. Specifically, the observation of problem-solving through the intersection of sets was under scrutiny, and the conclusion was that this (...) logic resource is not only viable in the context of pre-schooling, as well as relevant for the formation of thinking. (shrink)

From the mid-1600s to the beginning of the eighteenth century, there were two main circles of German scholars which focused extensively on diagrammatic reasoning and representation in logic. The first circle was formed around Erhard Weigel in Jena and consists primarily of Johann Christoph Sturm and Gottfried Wilhelm Leibniz; the second circle developed around Christian Weise in Zittau, with the support of his students, particularly Samuel Grosser and Johann Christian Lange. Each of these scholars developed an original form of using (...) geometric diagrams in logic. In this paper, I will trace the historical notes of John Venn and other modern logicians back to the original works published in the Weigel and Weise circles and describe the development of using geometric figures for logical reasoning and representation in that period of time. (shrink)

Mario Savio is widely known as the first spokesman for the Free Speech Movement. Having spent the summer of 1964 as a civil rights worker in segregationist Mississippi, Savio returned to the University of California at a time when students throughout the country were beginning to mobilize in support of racial justice and against the deepening American involvement in Vietnam. His moral clairty, his eloquence, and his democratic style of leadership inspired thousands of fellow Berkeley students to protest university regulations (...) that had severely limited political speech and activity on campus. The nonviolent campaign culminated in the largest mass arrest in American history, drew widespread faculty support, and resulted in a revision of university rules to permit political speech and organizing. This significant advance for student freedom rapidly spread to countless other colleges and universities across the country. Mario Savio went on to become a college teacher of physics, logic, and philosophy, to speak and organize in favor of immigrant rights and affirmative action and against U.S. intervention in Central America. He died on November 6, 1996, in the middle of a struggle against California State University fee hikes that hurt working-class students. Savio had submitted this article to the Notre Dame Journal of Formal Logic before he died. Final revisions were made by Philip Clayton with the assistance of Mario's colleagues at Sonoma State University. As reader for the Journal, George Englebretsen not only provided an extensive commentary on the article--much of which has been incorporated here--but also assisted in the difficult task of making revisions without changing the substance of Mario's style or thought. It is fitting that this, Savio's final publication, would be pedagogical in orientation. For him, moral considerations were no less pertinent in logic than in philosophy's less abstract fields. The usual student confusion with Venn diagrams led him to develop the new pictorial device presented in the following pages, which he believed was more sensitive to user psychology. It is hard to miss the political overtones in Savio's closing worry that in Venn diagrams "information of real significance may occasionally appear hidden and distorted." The decision by the Notre Dame Journal of Formal Logic to publish this piece posthumously is a testimony that logic, no less than other fields of philosophy, can be a tool of free speech and political change--as powerful in its way as the rhetorical brilliance of that young man standing on top of a police car who launched a worldwide movement with the words, "There is a time when the operation of the machine becomes so odious, makes you so sick at heart, you can't take part.". (shrink)

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. (shrink)