This paper examines truth diagrams for some non-classical, modal and dynamic logics. Truth diagrams are diagrammatic and visual ways to represent logical truth akin to truth tables, developed by Peter C.-H. Cheng. Currently, it is only given for classical propositional logic. In this paper, we establish truth diagrams for Priest's Logic of Paradox, Belnap–Dunn's Four-Valued Logic, MacColl's Connexive Logic, Bochvar–Halldén's Logic of Non-Sense, Carnielli–Coniglio's logic of formal inconsistency as well as classical modal (...) logic and its dynamic extension to shed light on the semantic behaviour of some non-classical and modal logics. (shrink)

This paper examines truth diagrams for some non-classical, modal and dynamic logics. Truth diagrams are diagrammatic and visual ways to represent logical truth akin to truth tables, developed by Peter C.-H. Cheng. Currently, it is only given for classical propositional logic. In this paper, we establish truth diagrams for Priest's Logic of Paradox, Belnap–Dunn's Four-Valued Logic, MacColl's Connexive Logic, Bochvar–Halldén's Logic of Non-Sense, Carnielli–Coniglio's logic of formal inconsistency as well as classical modal (...) logic and its dynamic extension to shed light on the semantic behaviour of some non-classical and modal logics. (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)

This paper examines truth diagrams for some non-classical, modal and dynamic logics. Truth diagrams are diagrammatic and visual ways to represent logical truth akin to truth tables, developed by Peter C.-H. Cheng. Currently, it is only given for classical propositional logic. In this paper, we establish truth diagrams for Priest's Logic of Paradox, Belnap–Dunn's Four-Valued Logic, MacColl's Connexive Logic, Bochvar–Halldén's Logic of Non-Sense, Carnielli–Coniglio's logic of formal inconsistency as well as classical modal (...) logic and its dynamic extension to shed light on the semantic behaviour of some non-classical and modal logics. (shrink)

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. (shrink)

This article offers an analysis of the cognitive role of diagrammatic movements in the theater. Based on the recognition of a theatrical work’s inherent ability to provide new insights concerning reality, the article concentrates on the way by which actors’ movements on stage create spatial diagrams that can provide new insights into the spectators’ world. The suggested model of theater’s epistemology results from a combination of Charles S. Peirce’s doctrine of diagrammatic reasoning and David Lewis’s theoretical account of the (...) truth value of counterfactual conditionals. I argue that in several theatrical works – in particular those whose central image is dominated by movements – the relation of what Lewis names “comparative overall similarity” between the fictional and the actual world is based on diagrammatic homology. The cognitive process involved in deciphering them is, hence, based on diagrammatic reasoning. The main emphasis of the analysis is on the previously unnoticed but important cognitive role of observation in the theater: the idea that observation takes an active role in the reasoning process that enables the spectators to form new knowledge about their actual world. Samuel Beckett’s plays Quad and Come and Go serve here as case studies. (shrink)

This book investigates a number of central problems in the philosophy of Charles Peirce grouped around the realism of his semiotics: the issue of how sign systems are developed and used in the investigation of reality. Thus, it deals with the precise character of Peirce's realism; with Peirce's special notion of propositions as signs which, at the same time, denote and describe the same object. It deals with diagrams as signs which depict more or less abstract states-of-affairs, facilitating reasoning (...) about them; with assertions as public claims about the truth of propositions. It deals with iconicity in logic, the issue of self-control in reasoning, dependences between phenomena in their realist descriptions. A number of chapters deal with applied semiotics: with biosemiotic sign use among pre-human organisms: the multimedia combination of pictorial and linguistic information in human semiotic genres like cartoons, posters, poetry, monuments. All in all, the book makes a strong case for the actual relevance of Peirce's realist semiotics. (shrink)

What is logical relevance? Anderson and Belnap say that the “modern classical tradition [,] stemming from Frege and Whitehead-Russell, gave no consideration whatsoever to the classical notion of relevance.” But just what is this classical notion? I argue that the relevance tradition is implicitly most deeply concerned with the containment of truth-grounds, less deeply with the containment of classes, and least of all with variable sharing in the Anderson–Belnap manner. Thus modern classical logicians such as Peirce, Frege, Russell, Wittgenstein, (...) and Quine are implicit relevantists on the deepest level. In showing this, I reunite two fields of logic which, strangely from the traditional point of view, have become basically separated from each other: relevance logic and diagram logic. I argue that there are two main concepts of relevance, intensional and extensional. The first is that of the relevantists, who overlook the presence of the second in modern classical logic. The second is the concept of truth-ground containment as following from in Wittgenstein’s Tractatus. I show that this second concept belongs to the diagram tradition of showing that the premisses contain the conclusion by the fact that the conclusion is diagrammed in the very act of diagramming the premisses. I argue that the extensional concept is primary, with at least five usable modern classical filters or constraints and indefinitely many secondary intensional filters or constraints. For the extensional concept is the genus of deductive relevance, and the filters define species. Also following the Tractatus, deductive relevance, or full truth-ground containment, is the limit of inductive relevance, or partial truth-ground containment. Purely extensional inductive or partial relevance has its filters or species too. Thus extensional relevance is more properly a universal concept of relevance or summum genus with modern classical deductive logic, relevantist deductive logic, and inductive logic as its three main domains. (shrink)

In this paper, I take up the following puzzle: If Feynman diagrams represent states of affairs, but do not do so truthfully what can their epistemic value be? I argue that Feynman diagrams have been epistemically powerful (at least in part) because, as pictorial representations, they facilitate an understanding of quantum electrodynamics, and quantum field theories more generally. Drawing on Richard Feynman’s own remarks and Catherine Z. Elgin’s account of the role of understanding in science, I tease out (...) what it might mean to have an understanding of something that is not factive. Although my approach allows for a thin sense of substantively non-factive epistemic success, it is continuous with a factive sense of understanding that is more familiar in the sciences. (shrink)

In his article “Insight: Genesis and Ongoing Context,” Fred Crowe calls out Lonergan’s line “the diagram is more important than…is ordinarily believed” as the “philosophical understatement of the century.” Sixteen pages later he identifies elaborating an invariant cognitional theory to underlie generalized emergent probability and thus “the immanent order of the universe of proportionate being,” as “our challenge,” “but given the difficulty” he does not “see any prospect for an immediate answer.” Could this have something to do with the lack (...) of a comprehensive diagram of cognitional theory? Appendix A of The Boston College Lectures on Mathematical Logic and Existentialism offers diagrams of the dynamics of knowing and doing perhaps copied from Lonergan’s own blackboard work, but they do not distinguish explanatory and descriptive insights, let alone statistical insights, and do not illustrate the pull upwards or the fusing of routinized insights. Before we can effectively relate cognitional theory to generalized emergent probability, we must have an adequately rigorous and precise cognitional theory. I firmly believe in the truth of Lonergan’s fundamental insights, but in order to rigorously undergird generalized emergent probability there are many pertinent questions about his cognitional theory which must be asked and answered. In this paper I (1) review some of Lonergan’s attempts to diagram cognitional theory and discuss what insights they do and do not express, (2) elaborate and defend principles for making our formulations of insight into insight rigorous and clear, and (3) attempt to build up a diagram which makes full use of those principles in a maximally expressive way. (shrink)

"Davidson begins by harking back to an early interest in the classics, and an even earlier engagement with the workings of grammar. In the pleasures of diagramming sentences in grade school, he locates his first glimpse into the mechanics of how we conduct the most important activities in our life - such as declaring love, asking directions, issuing orders, and telling stories. Davidson connects these essential questions with the most basic and yet hard to understand mysteries of language use - (...) how we connect noun to verb. This is a problem that Plato and Aristotle wrestled with, and Davidson draws on their thinking to show how an understanding of linguistic behavior is critical to the formulating of a workable concept of truth. Against a whole army of contemporary philosophers, Davidson argues that the concept of truth is not ambiguous.". (shrink)

Inheritance diagrams are directed acyclic graphs with two types of connections between nodes: x → y and x ↛ y . Given a diagram D, one can ask the formal question of “is there a valid path between node x and node y?” Depending on the existence of a valid path we can answer the question “x is a y” or “x is not a y”. The answer to the above question is determined through a complex inductive algorithm on (...) paths between arbitrary pairs of points in the graph. This paper aims to simplify and interpret such diagrams and their algorithms. We approach the area on two fronts. Suggest reactive arrows to simplify the algorithms for the winning paths. We give a conceptual analysis of inheritance diagrams, and compare our analysis to the “small” and “big sets” of preferential and related reasoning. In our analysis, we consider nodes as information sources and truth values, direct links as information, and valid paths as information channels and comparisons of truth values. This results in an upward chaining, split validity, off-path preclusion inheritance formalism of a particularly simple type. We show that the small and big sets of preferential reasoning have to be relativized if we want them to conform to inheritance theory, resulting in a more cautious approach, perhaps closer to actual human reasoning. We will also interpret inheritance diagrams as theories of prototypical reasoning, based on two distances: set difference, and information difference. We will see that some of the major distinctions between inheritance formalisms are consequences of deeper and more general problems of treating conflicting information. It is easily seen that inheritance diagrams can also be analysed in terms of reactive diagrams - as can all argumentation systems. AMS Classification: 68T27, 68T30. (shrink)

One way visual representations might function in scientific reasoning is to convey content that is true or false, analogous to making a claim. An alternative way that visual representations might function is as an object that may make statements true or false, but is not itself true or false, analogous to a scientific model. In this paper I evaluate the most recent and extended defense of this latter position and show that the case study involved does not in fact support (...) the view that the diagrams discussed function as truth-makers rather than truth bearers. (shrink)

Introduction: contemporary conditions and diagrammatic trajectory -- From joy to the gap: the accessing of the infinite by the finite (Spinoza, Nietzsche, Bergson) -- The care of the self versus the ethics of desire: two diagrams of the production of subjectivity (and of the subject's relation to truth) (Foucault versus Lacan) -- The aesthetic paradigm: from the folding of the finite-infinite relation to schizoanalytic metamodelisation (to biopolitics) (Guattari) -- The strange temporality of the subject: life in-between the infinite (...) and the finite (Deleuze contra Badiou) -- Desiring-machines, chaoids, probeheads: towards a speculative production of subjectivity (Deleuze and Guattari) -- Conclusion: composite diagram and relations of adjacency. (shrink)

Abductive conclusions are drawn in a special, co-hortative mood. Abductive conclusions are representative interpretants that represent abduction as a form of reasoning that can convey a general conception of the truth. The truth is not asserted; abduction merely delivers the idea of a matter of course, rendering that idea comparatively simple and natural, hence assuring us of its justified assertibility. Hence abductive reasoning is at home in addressing ‘How Possible’-questions in science. Abductive reasoning concerns the question of how (...) things might, could or would conceivably be such that they can be plausibly asserted. Peirce took all reasoning to be diagrammatic and representable using the graphical method of logic. Yet no examples have previously been found in his large manuscript corpus of what such non-deductive graphs might look like. This paper proposes a new interpretation of a sole exception, a sketch of two graphs from a rejected page from 1903, which might be the only surviving example of Peirce’s abductive graphs. The proposed interpretation takes them to be representative interpretants of this special inverse type of inference. (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)

We address in this paper the problem of counting the models of a propositional theory under incremental changes to its literals. Specifcally, we show that if a propositional theory Δ is in a special form that we call smooth, deterministic, decomposable negation normal form, then for any consistent set of literals S, we can simultaneously count the models of Δ ∪ S and the models of every theory Δ ∪ T where T results from adding, removing or flipping a literal (...) in S. We present two results relating to the time and space complexity of compiling propositional theories into sd-DNNF. First, we show that if a conjunctive normal form has a bounded treewidth, then it can be compiled into an sd-DNNF in time and space which are linear in its size. Second, we show that sd-DNNF is a strictly more space efficient representation than Free Binary Decision Diagrams. Finally, we discuss some applications of the counting results to truth maintenance systems, belief revision, and model-based diagnosis. (shrink)

1. Setting off: An introduction. 1.1. A measure of motivation. 1.2. Computable mathematics. 1.3. Reverse mathematics. 1.4. An overview. 1.5. Further reading -- 2. Gathering our tools: Basic concepts and notation. 2.1. Computability theory. 2.2. Computability theoretic reductions. 2.3. Forcing -- 3. Finding our path: Konig's lemma and computability. 3.1. II[symbol] classes, basis theorems, and PA degrees. 3.2. Versions of Konig's lemma -- 4. Gauging our strength: Reverse mathematics. 4.1. RCA[symbol]. 4.2. Working in RCA[symbol]. 4.3. ACA[symbol]. 4.4. WKL[symbol]. 4.5. [symbol]-models. (...) 4.6. First order axioms. 4.7. Further remarks -- 5. In defense of disarray -- 6. Achieving consensus: Ramsey's theorem. 6.1. Three proofs of Ramsey's theorem. 6.2. Ramsey's theorem and the arithmetic hierarchy. 6.3. RT, ACA[symbol], and the Paris-Harrington theorem. 6.4. Stability and cohesiveness. 6.5. Mathias forcing and cohesive sets. 6.6. Mathias forcing and stable colorings. 6.7. Seetapun's theorem and its extensions. 6.8. Ramsey's theorem and first order axioms. 6.9. Uniformity -- 7. Preserving our power: Conservativity. 7.1. Conservativity over first order systems. 7.2. WKL[symbol] and II[symbol]-conservativity. 7.3. COH and r-II[symbol]-conservativity -- 8. Drawing a map: Five diagrams -- 9. Exploring our surroundings: The world below RT[symbol]. 9.1. Ascending and descending sequences. 9.2. Other combinatorial principles provable from RT[symbol]. 9.3. Atomic models and omitting types -- 10. Charging ahead: Further topics. 10.1. The Dushnik-Miller theorem. 10.2. Linearizing well-founded partial orders. 10.3. The world above ACA[symbol]. 10.4. Still further topics, and a final exercise. (shrink)

Recent literary methods have opened new possibilities in reading and understanding the logic of narratives. The Greimassian approach offers such a possibility. Though Greimas’ approach is by now accepted as part of the canon of narratology, some of its components have not yet received due attention. This is the case with his ‘veridictory square’, a diagram that applies especially to texts where oppositions such as truth-falsehood, hero-villain and subject-anti-subject are prominent themes. This article aims to demonstrate that these kinds (...) of narratives, such as Susanna, do not concern themselves with objective truth but persuasion about truth, that is, veridiction. Truth telling in those stories corresponds to manipulation, exercising a particular cognitive doing or causing to appear as true. In other words, such a manipulation of truth aims at causing people to believe. Using Greimas’ veridictory square built on opposing modalities of being and seeming, the contrast between reality and appearance, this article provides insight into the games of truth in Susanna and thus offers a new inspiring way of reading these kinds of stories. (shrink)

The concept of approximate truth plays a prominent role in most versions of scientific realism. However, adequately conceptualizing ?approximate truth? has proved challenging. This article argues that the goal of articulating the concept of approximate truth can be advanced by first investigating the processes by which science accords theories the status of accepted or rejected. Accordingly, this article uses a path diagram model as a visual heuristic for the purpose of showing the processes in science that are (...) involved in determining a theory's status. This ?inductive realist? model of theory status then serves as a starting point for explicating an inductive realist view of approximate truth that, it is argued, can explain instances of the success of science, but does not (1) require science's theories to be strictly true in any world or (2) require a metric for measuring how close an approximately true theory is to some strictly true theory. To show the advantages of the inductive realist approach to approximate truth, an example of a major success story of science, the successful eradication of smallpox, is reviewed and then explained. (shrink)

Psychologists take two propositions for granted. Specifically, empirical verification of predictions derived from a theory support that the theory is more likely to be true and support that additional predictions derived from the theory have an increased probability of being sustained if subjected to empirical testing. In contrast, I argue that both propositions depend strongly on whether auxiliary assumptions are taken into account. When auxiliary assumptions are not taken into account, the first proposition is valid but the second is not. (...) When auxiliary assumptions are taken into account, the first proposition is not valid, and the second proposition encounters additional problems. I use Venn diagrams and Bayesian principles to demonstrate these conclusions. (shrink)

This article focuses on two different styles of profanation in Spinoza and Badiou. Notwithstanding the significant differences between them, their shared desire for profanation testifies to an interesting convergence. I deal with this convergence in divergence as a case of disjunctive synthesis through a comparison of the different understandings of religion in Spinoza and Badiou’s truth procedures. It is commonly held that Spinoza operates with three understandings of religion. But I argue that Spinoza’s thought opens up the space for (...) a fourth understanding of ‘religion’. Then I discuss the formal similarity between Spinoza’s four religions and Badiou’s four truth procedures. I illustrate this discussion through two diagrams. I claim that Badiou’s truth procedures could be perceived as the Spinozist diagram’s re-entry into itself. (shrink)

This chapter contains sections titled: Where we are When truth at a world depends on more than how that world is A diagram to give the key idea A language where truth at a world is, by stipulation, a function of which world is actual On looking for examples of two–dimensional sentences in the English of the folk How should we approach questions like, How do we use the word “water”? and, How do we use the word “Gödel”? (...) More on the evidential role of intuitions about possible cases What information do we impart with the word “water”–first pass With enemies like these, who needs friends?16 What information do we impart with the word “water”–second pass The issue about narrow content18 Proper names and information (I) Proper names and information (II) Coda. (shrink)

_How to Think Critically_ begins with the premise that we are all, every day, engaged in critical thinking. But just as we may develop bad habits in daily life if we don’t scrutinize our practices, so are we apt to develop bad habits in critical thinking if we are careless in our reasoning. Readers are presented with a traditional step-by-step method for analysis that can be applied to all argument forms. Hundreds of exercises (with solutions) are included, as are several (...) random statement generators that can be used to create thousands of additional examples. Truth tables, Venn diagrams, and other essential concepts are introduced not merely as objects for academic study but also as tools for better thinking and living. At a time when the value of critical thinking is recognized to be greater than ever, this book is an important resource both inside and outside the classroom. (shrink)

This book marks the inauguration of the Historia Logicae book series, which seeks to publish high-quality monographs, dissertations, textbooks, proceedings, and anthologies on the history of logic in either German or English. Serving as the inaugural volume in this series, the book explores the contemporary interpretation of logic across many centuries and cultures. The first section of the volume comprises a compilation of papers dedicated to ancient and medieval logic, examining prominent thinkers such as Plato, Aristotle, Seneca, Porphyry, Proclus, Boethius, (...) Buridan, and Kilwardby. The second section shifts focus towards early-modern and contemporary logics, including the works of Caramuel, Kant, Drobisch, Husserl, and Jaskowski. Topics of discussion encompass the relationship between mathematics and logic, natural deduction, intuitionism, graph theory, categorical judgments, theories of truth, modal logic, opposition relations, logic diagrams, definitions of formal logic, many-sorted logic, model theory, and paraconsistent logic. (shrink)

I focus on three preoccupations of recent writings on proof. -/- I. The role and possible effects of empirical reasoning in mathematics. Do recent developments (specifically, the computer-assisted proof of the 4CT) point to something essentially new as regards the need for and/or effects of using broadly empirical and inductive reasoning in mathematics? In particular, should we see such things as the computer-assisted proof of the 4CT as pointing to the existence of mathematical truths of which we cannot have a (...) priori knowledge? -/- 2. The role of formalization in proof. What are the patterns ofinference according to which mathematical reasoning naturally proceeds? Are they of 'local' character (i.e. sensitive to the subject-matter of the reasoning concerned) or 'global' character (i.e. invariant across all subject-matters)? Finally, what if any relationship is there (a) between the patterns of inference manifest in a proof and its explanatory capacity and (b) between explanatory capacity and rigor? -/- 3. Diagrams and their role in mathematical reasoning. What essentially is diagrammatic reasoning, and what is the nature and basis of its usefulness? Can it play a justificative role in the development of mathematical knowledge and, more particularly, in genuine proof? Finally, does the use of diagrammatic reasoning force an adjustment either in our conception of rigor or in our view of its importance? (shrink)

The duality of truth and falsity in a Boolean algebra of propositions is used to generate a duality of belief and disbelief. To each additive probability measure that represents belief there corresponds a dual additive measure that represents disbelief. The dual measure has its own peculiar calculus, in which, for example, measures are added when propositions are combined under conjunction. A Venn diagram of the measure has the contradiction as its total space. While additive measures are not self-dual, the (...) epistemic state of complete ignorance is represented by the unique, monotonic, non-additive measure that is self-dual in its contingent propositions. Convex sets of additive measures fail to represent complete ignorance since they are not self-dual. (shrink)

This book is written for those who wish to learn some basic principles of formal logic but more importantly learn some easy methods to unpick arguments and assess their value for truth and validity. -/- The first section explains the ideas behind traditional logic which was formed well over two thousand years ago by the ancient Greeks. Terms such as ‘categorical syllogism’, ‘premise’, ‘deduction’ and ‘validity’ may appear at first sight to be inscrutable but will easily be understood with (...) examples bringing the subjects to life. Traditionally, Venn diagrams have been employed to test arguments. These are very useful but their application is limited and they are not open to quantification. The mid-section of this book introduces a methodology that makes the analysis of arguments accessible with the use of a new form of diagram, modified from those of the mathematician Leonhard Euler. These new diagrammatic methods will be employed to demonstrate an addition to the basic form of syllogism. This includes a refined definition of the terms ‘most’ and ‘some’ within propositions. This may seem a little obscure at the moment but one will readily apprehend these new methods and principles of a more modern logic. (shrink)

The article affixes a resolutely structuralist view to Alain Badiou’s proposals on the infinite, around the theory of sets. Structuralism is not what is often criticized, to administer mathematical theories, imitating rather more or less philosophical problems. It is rather an attitude in mathematical thinking proper, consisting in solving mathematical problems by structuring data, despite the questions as to foundation. It is the mathematical theory of categories that supports this attitude, thus focusing on the functioning of mathematical work. From this (...) perspective, the thought of infinity will be grasped as that of mathematical work itself, which consists in the deployment of dualities, where it begins the question of the discrete and the continuous, Zeno’s paradoxes. It is, in our opinion, in the interval of each duality ― “between two ends”, as our title states ― that infinity is at work. This is confronted with the idea that mathematics produces theories of infinity, infinitesimal calculus or set theory, which is also true. But these theories only give us a grasp of the question of infinity if we put ourselves into them, if we practice them; then it is indeed mathematical activity itself that represents infinity, which presents it to thought. We show that tools such as algebraic universes, sketches, and diagrams, allow, on the one hand, to dispense with the “calculations” together with cardinals and ordinals, and on the other hand, to describe at leisure the structures and their manipulations thereof, the indefinite work of pasting or glueing data, work that constitutes an object the actual infinity of which the theory of structures is a calculation. Through these technical details it is therefore proposed that Badiou envisages ontology by returning to the phenomenology of his “logic of worlds”, by shifting the question of Being towards the worlds where truths are produced, and hence where the subsequent question of infinity arises. (shrink)

In modern philosophy acquires relevance a problem of confrontation of linguistic and visual practices of philosophical knowledge. Since Antiquity, ways of contemplation are formed as ways of recognition characterized as “ocularcentrism”. In the historical and philosophical prospective “sight” and “vision” are metaphors of thinking. They are considered as internal contemplation, and as a means of establishing human cognitive abilities. It is shown that technology and models of representation in Western culture of thinking are based on practices that appeal to “visualization”. (...) The create a “videological matrix” and make certain common stratagems of visual discourse like metaphors, diagrams, representation, etc. The article emphasizes that “the ocular dictionary” of philosophy represents a wide range of options for visual metaphors that occupy an important place in a number of key philosophical concepts. On the one hand, there are metaphors that appeal to visual perception, visual experience. But on the other hand there is a traditional interpretation of philosophical activity through the prism of visual metaphors. These includes a metaphor of vision, metaphor of light, metaphor of mirror and others. The author highlights a metaphor of vision as a key metaphor of philosophical activity, in which the philosopher is a spectator, contemplating the truth, just as the spectator of a play, sport event or religious mysteries. In antiquity this kind of contemplation was a specific activity, called “theory”. Contemporary studies show that a term “theory” can establish an association between theory as specific activity in social life and a theory as a specific activity of a philosopher. Philosophical theory is an act of “contemplation”, as evidenced by the writings of Plato and Aristotle. “Theory” was also associated with a certain contemplative work, public instruction, pilgrimage. The “theory” was the action, which took place in three steps: leaving home, contemplation, returning home. A contemplation was in the center of theory as an activity. Consequently, the philosophical theory, the article says, is born with cultural practices, the origins of which are located primarily in the social life of ancient Greek. Thus, contemplation as a “theory” is understood as the process of getting information about images, ideas which is transformed into something new in the processes if intellectual activity. Consequently, the visibility immanently contained in the philosophy and describes the philosophical activity, as a way of philosophical research and is associated with philosophical thinking. Activity of a philosopher in the contest of contemplation is essentially a “metaphor of vision”, which acts as a certain type of philosophical activity. (shrink)

Many do not doubt that Peirce's Existential Graphs are diagrammatic, as opposed to symbolic. However, when we are pressured to draw a distinction between the two different forms of representation, we find ourselves at a loss and our intuition quite vague. In this paper, I locate fundamental differences between two logically equivalent systems, Peirce's Alpha system and propositional languages. Suppose we have only two sentential connectives, ¬ and ^. In spite of its truth-functional completeness, we don't want to use (...) this language for the translation of English sentences or as a deductive calculus. We would adopt this language only when we intend to develop logical theories. That is, it is convenient to have fewer connectives for a meta-theory, but not for practical use. So, there seems to be a trade-off between a language with fewer connectives and a language with more connectives. This view has been accepted without question. In this paper, I will argue that this trade-off is limited to linear symbolic systems and that we could have a diagrammatic system with fewer operations but no need to suffer from problems like those of a sentential language with fewer connectives. How is that possible? A comparison between Peirce's Alpha system and a propositional language is presented to answer this question. The case study identifies the following unique property of a non-linear diagrammatic system as a main source of the discrepancy between two different types of representation: One and the same diagram can be read off in more than one way by carving it up in many ways, but without ambiguity. (shrink)

The species problem is one of the oldest controversies in natural history. Its persistence suggests that it is something more than a problem of fact or definition. Considerable light is shed on the species problem when it is viewed as a problem in the representation of the natural system (sensu Griffiths, 1974, Acta Biotheor. 23: 85–131; de Queiroz, 1998, Philos. Sci. 55: 238–259). Just as maps are representations of the earth, and are subject to what is called cartographic generalization, so (...) diagrams of the natural system (evolutionary trees) are representations of the evolutionary chronicle, and are subject to a temporal version of cartographic generalization which may be termed systematic generalization. Cartographic generalization is based on judgements of geographical importance, and systematic generalization is based on judgements of historical importance, judgements expressed in narrative sentences (sensu Danto, 1985, Narration and knowledge, Columbia Univ. Press, New York). At higher systematic levels these narrative sentences are conventional and retrospective, but near the "species" level they become prospective, that is, dependent upon expectations of the future. The truth of prospective narrative sentences is logically indeterminable in the present, and since all the common species concepts depend upon prospective narration, it is impossible for any of them to be applied with precision. (shrink)

The first systematic exposition of all the central topics in the philosophy of logic, Susan Haack's book has established an international reputation for its accessibility, clarity, conciseness, orderliness, and range as well as for its thorough scholarship and careful analyses. Haack discusses the scope and purpose of logic, validity, truth-functions, quantification and ontology, names, descriptions, truth, truth-bearers, the set-theoretical and semantic paradoxes, and modality. She also explores the motivations for a whole range of non-classical systems of logic, (...) including many-valued logics, fuzzy logic, moddal and tense logics, and relevance logics. Persupposing only an elementary knowledge of formal logic, this book includes many useful summary tables and diagrams, as well as a helpful glossary of technical terms. Wide-ranging, informative, and eminently readable, this book has proven a valuable resource for generations of students and scholars in a variety of disciplines outside philosophy needing guidance on the philosophy of logic. (shrink)

Theorems in automated theorem proving are usually proved by formal logical proofs. However, there is a subset of problems which humans can prove by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is often more clearly perceived in these proofs than in the corresponding algebraic proofs; they capture an intuitive notion of truthfulness that humans find easy to see and understand. We are investigating and automating such diagrammatic reasoning about mathematical theorems. Concrete, rather than general (...) diagrams are used to prove particular concrete instances of the universally quantified theorem. The diagrammatic proof is captured by the use of geometric operations on the diagram. These operations are the inference steps of the proof. An abstracted schematic proof of the universally quantified theorem is induced from these proof instances. The constructive -rule provides the mathematical basis for this step from schematic proofs to theoremhood. In this way we avoid the difficulty of treating a general case in a diagram. One method of confirming that the abstraction of the schematic proof from the proof instances is sound is proving the correctness of schematic proofs in the meta-theory of diagrams. These ideas have been implemented in the system, called Diamond, which is presented here. (shrink)

The main problem of this study is David Hume’s (1711-76) view on Metaphysical Realism (there are mind-independent, external, and continuous entities). This specific problem is part of two more general questions in Hume scholarship: his attitude to scepticism and the relation between naturalism and skepticism in his thinking. A novel interpretation of these problems is defended in this work. The chief thesis is that Hume is both a sceptic and a Metaphysical Realist. His philosophical attitude is to suspend his judgment (...) on Metaphysical Realism, whereas as a common man he firmly believes in the existence of mind-independent, external, and continuous entities. Therefore Hume does not have any one position; accordingly, a form of “no one Hume” interpretation (Richard Popkin, Robert J. Fogelin, Donald L.M. Baxter) is argued for in the book. The key point in this distinction is the temporal difference between Hume’s philosophical and everyday views. It is introduced in order to avoid attributing a conscious contradiction to him (a problem which has not attracted enough attention in the literature). The method of the work is modelled on Peter Millican’s work on Hume and induction. The approach to the main problem is to study the two “profound” arguments against the senses that Hume presents in the Section 12 of An Enquiry concerning Human Understanding (1748). These arguments are first reconstructed in detail resulting in Millican-type diagrams of them and then Hume’s endorsement of them is established on the basis of the diagrams. The first profound argument concludes that Metaphysical Realism and thus any Realistic theory of perception is unjustified as well as the existence of God and the soul. The second argument goes further having first conceptual conclusion: the very notions of Real entitity, material substance, and bodies are completely out of the reach of the faculty of understanding. Therefore they ought to be rejected according to Hume. This is a consequence of the consistent use of the Humean faculty of reason: idea-analysis and inductive inference. The second profound argument thus concludes that believing in Metaphysical Realism is inconsistent with the rational attitude that is to refrain from this belief. Hence, if we attributed both of them to Hume, we would end up with a great philosopher who embraces a manifest contradiction. The study is finished by arguing that this sceptical and Metaphysically Realistic interpretation concurs well with (1) Hume’s professed Academical philosophy and (2) project of the science of human nature. (1) According to Hume, Academical philosophy is in the first place diffidence, modesty, and uncertainty including suspension on certain issues. Secondly, it is restriction of the range of topics for which experience can provide a standard of truth. This kind of empiricist epistemological realism is coherent with the sceptical attitude on Metaphysical Realism because the latter does not rule out inter-subjective consensus on what we experience. (2) Suspension of judgment on Metaphysical Realism coheres with the mind-dependency of the objects of Hume’s science of human nature: the understanding, passions, morals, aesthetics, politics, and the human culture in all of its manifestations. Although the study takes the first Enquiry to be Hume’s authorised word on the understanding, his juvenile work A Treatise of Human Nature (1739-40) is argued to support this “no one Hume” interpretation. Hume’s other works are also discussed when needed. (shrink)

BERTRAND RUSSELL ON MODALITY AND LOGICAL RELEVANCE - SECOND EDITION of 2015. Praise for the first edition of 1999: "In the twenty-nine years since Russell's death, much of the major scholarship has drawn heavily on his manuscripts and unpublished correspondence. The author shows that the published Russell is capable of new interpretations; in particular, that modal notions such as possibility have a greater place in various aspects of his logical and philosophical thought than has been previously imagined." -Ivor Grattan-Guinness, Foreword (...) to the first edition. EXCERPT FROM PUBLISHED REVIEW: "Dejnožka's book is the first full-length study of modality in Russell. It is useful for its very full survey of passages in which Russell makes use of or alludes to modal notions. Dejnožka's command of Russell's huge output is indeed impressive and his utilization of it thorough...." - Nicholas Griffin, Studia Logica. EXCERPT FROM PUBLISHED REVIEW: "Dejnožka's book raises a very important point in the history of formal logic. Until now the major studies on this topic have drawn heavily on the development of classical logic as standardized by Gottlob Frege and Bertrand Russell. Dejnožka challenges the reader to open his mind for a new interpretation of Russell's work, in particular that modal and relevance notions have a greater place in his philosophy of logic than has been stressed before.... "Dejnožka rightly observes that many of Russell's insights on modality are a result of his discussions with Hugh MacColl, who was indeed the first to seriously attempt to develop formal modal logic. This particularly applies to Russell's conception of a modal logic without modal operators....That is, classical logic can be used to simulate modal expressions. Thus, the notions of (logical) necessity and possibility are not 'fundamental notions'....On this basis, Dejnožka develops a higher level of modality, where the quantification scope extends to the predicates yielding what Russell calls 'fully general propositions'.... "The best studied translation method is known as the standard translation, and it is quite compatible with Dejnožka's suggestions.... "Dejnožka's book is full of material which stimulates [one] to rethink Russell's philosophy of logic and...it is greatly to the author's credit that he brings to light such a wealth of crucial issues in the history and philosophy of logic. - Shahid Rahman, History and Philosophy of Logic. BOOK DESCRIPTION: This book is the only exhaustive study of Russell on modality and logical relevance ever written. This is the second edition, revised over a period of over sixteen years, and over twice as long. Russell initially rejects the possibility of a modal logic in an unpublished paper written ca. 1903-05 and read to an audience in 1905. But the very next year, he adopts the theory of modality he had rejected, and he repeats that theory in published works for many years to come. That theory, together with other texts scattered among Russell's writings, implies eight modal logics which Russell himself never expressly accepted. And Russell's express acceptance of the early Wittgenstein's theory of entailment ("following from") as truth-ground containment implies a deductive relevance logic in Russell's writings which Russell himself never expressly accepted. The book ends with a look at John Maynard Keynes and Anglo-American evidence law as the origins of Russell's theory of probability as degree of logical relevance. Thus the book is written toward the reunion of modern classical logicians with modal logicians and relevantists. On the reunion of modern classical logicians with relevantists, please see also The Concept of Relevance and the Logic Diagram Tradition (2012). Thus I hope to have created two new fields of Russell studies. I may have created a new legal field as well: the history of logical relevance in evidence law: 132 pages (368-500) in chapter 10. (shrink)

The article studies the tradicional locus of madhouse present in 2666 by Roberto Bolaño, but taking into consideration also the desert and the church as spaces where the enunciation of madness is expands: the madman’s truth. To this end, three aspects that configure the heuristic notion of lettered madhouse are analyzed: the digressive diagrams, the phobic philology, and enlightened hospice. In short, it shows how Bolaño elaborates in his novel a critical and ironic vision of Latin American literary (...) intelligentsia or the remnants of The lettered City of Angel Rama. (shrink)

In modern philosophy acquires relevance a problem of confrontation of linguistic and visual practices of philosophical knowledge. Since Antiquity, ways of contemplation are formed as ways of recognition characterized as “ocularcentrism”. In the historical and philosophical prospective “sight” and “vision” are metaphors of thinking. They are considered as internal contemplation, and as a means of establishing human cognitive abilities. It is shown that technology and models of representation in Western culture of thinking are based on practices that appeal to “visualization”. (...) The create a “videological matrix” and make certain common stratagems of visual discourse like metaphors, diagrams, representation, etc. The article emphasizes that “the ocular dictionary” of philosophy represents a wide range of options for visual metaphors that occupy an important place in a number of key philosophical concepts. On the one hand, there are metaphors that appeal to visual perception, visual experience. But on the other hand there is a traditional interpretation of philosophical activity through the prism of visual metaphors. These includes a metaphor of vision, metaphor of light, metaphor of mirror and others. The author highlights a metaphor of vision as a key metaphor of philosophical activity, in which the philosopher is a spectator, contemplating the truth, just as the spectator of a play, sport event or religious mysteries. In antiquity this kind of contemplation was a specific activity, called “theory”. Contemporary studies show that a term “theory” can establish an association between theory as specific activity in social life and a theory as a specific activity of a philosopher. Philosophical theory is an act of “contemplation”, as evidenced by the writings of Plato and Aristotle. “Theory” was also associated with a certain contemplative work, public instruction, pilgrimage. The “theory” was the action, which took place in three steps: leaving home, contemplation, returning home. A contemplation was in the center of theory as an activity. Consequently, the philosophical theory, the article says, is born with cultural practices, the origins of which are located primarily in the social life of ancient Greek. Thus, contemplation as a “theory” is understood as the process of getting information about images, ideas which is transformed into something new in the processes if intellectual activity. Consequently, the visibility immanently contained in the philosophy and describes the philosophical activity, as a way of philosophical research and is associated with philosophical thinking. Activity of a philosopher in the contest of contemplation is essentially a “metaphor of vision”, which acts as a certain type of philosophical activity. (shrink)

This article introduces, studies, and applies a new system of logic which is called ‘HYPE’. In HYPE, formulas are evaluated at states that may exhibit truth value gaps and truth value gluts. Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional (...) and the predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE’s logical connectives. Furthermore, HYPE’s first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work. (shrink)

Recent work in formal philosophy has concentrated over-whelmingly on the logical problems pertaining to epistemic shortfall - which is to say on the various ways in which partial and sometimes incorrect information may be stored and processed. A directly depicting language, in contrast, would reflect a condition of epistemic perfection. It would enable us to construct representations not of our knowledge but of the structures of reality itself, in much the way that chemical diagrams allow the representation (at a (...) certain level of abstractness) of the structures of molecules of different sorts. A diagram of such a language would be true if that which it sets out to depict exists in reality, i.e. if the structural relations between the names (and other bits and pieces in the diagram) map structural relations among the corresponding objects in the world. Otherwise it would be false. All of this should, of course, be perfectly familiar. (See, for example, Aristotle, Metaphysics, 1027 b 22, 1051 b 32ff.) The present paper seeks to go further than its predecessors, however, in offering a detailed account of the syntax of a working universal characteristic and of the ways in which it might be used. (shrink)

This paper explores some of the constructive dimensions and specifics of human theoretic cognition, combining perspectives from (Husserlian) genetic phenomenology and distributed cognition approaches. I further consult recent psychological research concerning spatial and numerical cognition. The focus is on the nexus between the theoretic development of abstract, idealized geometrical and mathematical notions of space and the development and effective use of environmental cognitive support systems. In my discussion, I show that the evolution of the theoretic cognition of space apparently follows (...) two opposing, but in truth, intrinsically aligned trajectories. On the epistemic plane, which is the main focus of Husserl’s genetic phenomenological investigations, theoretic conceptions of space are progressively constituted by way of an idealizing emancipation of spatial cognition from the concrete, embodied intentionality underlying the human organism’s perception of space. As a result of this emancipation, it ultimately becomes possible for the human mind to theoretically conceive of and posit space as an ideal entity that is universally geometrical and mathematical. At the same time, by synthesizing a range of literature on spatial and mathematical cognition, I illustrate that for the theoretic mind to undertake precisely this emancipating process successfully, and further, for an ideal and objective notion of geometrical and mathematical space to first of all become fully scientifically operative, the cognitive support provided by a range of specific symbolic technologies is central. These include lettered diagrams, notation systems, and more generally, the technique of formalization and require for their functioning various cognitively efficacious types of embodiment. Ultimately, this paper endeavors to understand the specific symbolic-technological dimensions that have been instrumental to major shifts in the development of idealized, scientific conceptions of space. The epistemic characteristics of these shifts have been previously discussed in genetic phenomenology, but without devoting sufficient attention to the constructive role of symbolic technologies. At the same time, this paper identifies some of the irreducible phenomenological and epistemic dimensions that characterize the functioning of the historically situated, embodied and distributed theoretic mind. (shrink)

Mathematics often seems incomprehensible, a melee of strange symbols thrown down on a page. But while formulae, theorems, and proofs can involve highly complex concepts, the math becomes transparent when viewed as part of a bigger picture. What Is a Number? provides that picture. Robert Tubbs examines how mathematical concepts like number, geometric truth, infinity, and proof have been employed by artists, theologians, philosophers, writers, and cosmologists from ancient times to the modern era. Looking at a broad range of (...) topics -- from Pythagoras's exploration of the connection between harmonious sounds and mathematical ratios to the understanding of time in both Western and pre-Columbian thought -- Tubbs ties together seemingly disparate ideas to demonstrate the relationship between the sometimes elusive thought of artists and philosophers and the concrete logic of mathematicians. He complements his textual arguments with diagrams and illustrations. This historic and thematic study refutes the received wisdom that mathematical concepts are esoteric and divorced from other intellectual pursuits -- revealing them instead as dynamic and intrinsic to almost every human endeavor. (shrink)

Not long ago, I received an email from a man who had been trying to get his seven-year-old son interested in science, and teach him a little bit about the workings of the brain. He had been showing his son one of those diagrams of a brain with various regions labeled as "speech center," vision center," and the like (something similar to this, I suppose), when the little boy suddenly asked, "Daddy, which part of the brain does imagination come (...) from?". It was not on the diagram, and the father, although he studied human biology at college, realized he did not know the answer. As we do these days, he got on the internet to try to find out. It was not as easy as he might have expected. "Imagination" is certainly a word that is found on a lot of web pages, and there are plenty that seem to be devoted to celebrating or promoting it, but very few of them seem to have anything at all to say about what it is, how it works, or where in the brain it might be implemented. Eventually, he found his way to this site. Even here, however, he could find no straightforward answer to his son's apparently straightforward question. (The truth is, the question is not nearly as straightforward as it appears.) He thought I might be a good a person to ask, however, and sent me an email. What follows is a lightly revised version of the reply I sent. It is not written in terms that a seven-year-old could understand (I am not clever enough to do that), but neither is it pitched at the professional, academic level of most of the material on this site. I would like to think that it answers the question (inasmuch as it can be answered in the current state of scientific knowledge) in a way that a layperson should be able to understand, and that they might then be able to explain to a curious and intelligent child. If you want to know the detailed reasons, and see the citations to the scientific literature, that justify the claims made here, you can find them in the other articles on this site.. (shrink)

Students of geometry do not prove Euclid's first theorem by examining an accompanying diagram, or by visualizing the construction of a figure. The original proof of Euclid's first theorem is incomplete, and this gap in logic is undetected by visual imagination. While cognition involves truth values, vision does not: the notions of inference and proof are foreign to vision.

Designed for use by philosophy students, this 2006 book provides an accessible, yet technically sound treatment of modal logic and its philosophical applications. Every effort has been made to simplify the presentation by using diagrams in place of more complex mathematical apparatus. These and other innovations provide philosophers with easy access to a rich variety of topics in modal logic, including a full coverage of quantified modal logic, non-rigid designators, definite descriptions, and the de-re de-dictio distinction. Discussion of philosophical (...) issues concerning the development of modal logic is woven into the text. The book uses natural deduction systems and also includes a diagram technique that extends the method of truth trees to modal logic. This feature provides a foundation for a novel method for showing completeness, one that is easy to extend to systems that include quantifiers. (shrink)

In this vivid and incisive guide, philosophy comes to life. Using the central idea of causality as a guiding principle, Brendan Wilson shows how the history of philosophy becomes a very clear and natural sequence of events. The resulting perspective reveals the deep connections between the problems of science, mind and reality, freedom and responsibility, knowledge, language, truth and religion. Newcomers to philosophy will be able to engage with the great questions and ideas of the western tradition. The writing (...) is clear and jargon-free, while the 'short-chapter' format gives the reader time to pause and think at each step along the way. Style and layout put accessibility first and the many illustrations include explanatory diagrams and drawings as well as pictures of philosophers from Aristotle to Wittgenstein. Key Features Encourages critical and independent thinking Written in an engaging, non-technical style Makes clear sense of philosophy's rich and complex history Presents a distinctive argument Provides a wide range of case studies and examples from philosophy, science and law Text Features Chapter overviews Chapter sub-headings for ease of use Carefully-chosen quotations from philosophers ancient and modern End of chapter discussion points An open and attractive page layout Lavish and relevant illustrations, including photographs, diagrams, explanatory boxes and drawings. (shrink)

I argue that emojis are essentially little pictures, rather than words, gestures, expressives, or diagrams. ???? means that the world looks like that, from some viewpoint. I flesh out a pictorial semantics in terms of geometric projection with abstraction and stylization. Since such a semantics delivers only very minimal contents I add an account of pragmatic enrichment, driven by coherence and nonliteral interpretation. The apparent semantic distinction between emojis depicting entities (like ????) and those depicting facial expressions (like ????) (...) I analyze as a difference between truth-conditional and use-conditional pictorial content: ???? depicts what the world of evaluation looks like, while ???? depicts what the utterance context looks like. (shrink)

This entry traces the historical development of the Square of Opposition, a collection of logical relationships traditionally embodied in a square diagram. This body of doctrine provided a foundation for work in logic for over two millenia. For most of this history, logicians assumed that negative particular propositions ("Some S is not P") are vacuously true if their subjects are empty. This validates the logical laws embodied in the diagram, and preserves the doctrine against modern criticisms. Certain additional principles ("contraposition" (...) and "obversion") were sometimes adopted along with the Square, and they genuinely yielded inconsistency. By the nineteenth century an inconsistent set of doctrines was widely adopted. Strawson's 1952 attempt to rehabilitate the Square does not apply to the traditional doctrine; it does salvage the nineteenth century version but at the cost of yielding inferences that lead from truth to falsity when strung together. (shrink)