It is common to distinguish two great families of representation. Symbolic representations include logical and mathematical symbols, words, and complex linguistic expressions. Iconic representations include dials, diagrams, maps, pictures, 3-dimensional models, and depictive gestures. This essay describes and motivates a new way of distinguishing iconic from symbolic representation. It locates the difference not in the signs themselves, nor in the contents they express, but in the semantic rules by which signs are associated with contents. The two kinds of rule have (...) divergent forms, occupying opposite poles on a spectrum of naturalness. Symbolic rules are composed entirely of primitive juxtapositions of sign types with contents, while iconic rules determine contents entirely by uniform natural relations with sign types. This distinction is marked explicitly in the formal semantics of familiar sign systems, both for atomic first-order representations, like words, pixel colors, and dials, and for complex second-order representations, like sentences, diagrams, and pictures. (shrink)

What is the relationship between the world and logic, between intuition and language, between objects and their quantitative determinations? Rationalists, on the one hand, hold that the world is structured in a rational way. Representationalists, on the other hand, assume that language, logic, and mathematics are only the means to order and describe the intuitively given world. In World and Logic, Jens Lemanski takes up three surprising arguments from Arthur Schopenhauer’s hitherto undiscovered Berlin Lectures, which concern the philosophy of language, (...) logic, and mathematics. Based on these arguments, Lemanski develops a new position entitled ‘rational representationalism’: the world is always structured by human beings according to linguistic, logical, and mathematical principles, but the basic vocabulary of these structural descriptions already contains metaphors taken from the world around us. (shrink)

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

I argue that maps do not feature predication, as analyzed by Frege and Tarski. I take as my foil (Casati and Varzi, Parts and places, 1999), which attributes predication to maps. I argue that the details of Casati and Varzi’s own semantics militate against this attribution. Casati and Varzi emphasize what I call the Absence Intuition: if a marker representing some property (such as mountainous terrain) appears on a map, then absence of that marker from a map coordinate signifies absence (...) of the corresponding property from the corresponding location. Predication elicits nothing like the Absence Intuition. “F(a)” does not, in general, signify that objects other than a lack property F. On the basis of this asymmetry, I argue that attaching a marker to map coordinates is a different mode of semantic composition than attaching a predicate to a singular term. (shrink)

In this paper, we introduce Speedith which is an interactive diagrammatic theorem prover for the well-known language of spider diagrams. Speedith provides a way to input spider diagrams, transform them via the diagrammatic inference rules, and prove diagrammatic theorems. Speedith’s inference rules are sound and complete, extending previous research by including all the classical logic connectives. In addition to being a stand-alone proof system, Speedith is also designed as a program that plugs into existing general purpose theorem provers. This allows (...) for other systems to access diagrammatic reasoning via Speedith, as well as a formal verification of diagrammatic proof steps within standard sentential proof assistants. We describe the general structure of Speedith, the diagrammatic language, the automatic mechanism that draws the diagrams when inference rules are applied on them, and how formal diagrammatic proofs are constructed. (shrink)

We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic (Peirce’s Existential Graphs α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}). The straightforward construction of a presheaf category EGα∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}{{\mathcal {G}}}_{\alpha ^{*}}$$\end{document} of cuts-only Existential Graphs (equivalent to the well-studied category of finite forests) provides a basis for the further construction of the category EGα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} (...) \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}{{\mathcal {G}}}_\alpha $$\end{document} which introduces variables in a reconstructedly generic, or label-free, mode. Morphisms in these categories represent syntactical embeddings or, equivalently but dually, extensions. Through the example of Peirce’s system, it is shown how the generic figures approach facilitates the formal investigation of relations between syntax and semantics in such diagrammatic systems. (shrink)

In this paper I offer a theory about the nature of diagrammatic representation in geometry. On my view, diagrammatic representaiton differs from pictorial representation in that neither the resemblance between the diagram and its object nor the experience of such a resemblance plays an essential role. Instead, the diagrammatic representation is arises from the role the components of the diagram play in a diagramatic practice that allows us to draws inferences based on them about the ojbects they represent.

We present a reading of the traditional syllogistics in a fragment of the propositional intuitionistic multiplicative linear logic and prove that with respect to a diagrammatic logical calculus that we introduced in a previous paper, a syllogism is provable in such a fragment if and only if it is diagrammatically provable. We extend this result to syllogistics with complemented terms à la De Morgan, with respect to a suitable extension of the diagrammatic reasoning system for the traditional case and a (...) corresponding reading of such De Morgan style syllogistics in the previously referred to fragment of linear logic. (shrink)

Proof-theory has traditionally been developed based on linguistic (symbolic) representations of logical proofs. Recently, however, logical reasoning based on diagrammatic or graphical representations has been investigated by logicians. Euler diagrams were introduced in the eighteenth century. But it is quite recent (more precisely, in the 1990s) that logicians started to study them from a formal logical viewpoint. We propose a novel approach to the formalization of Euler diagrammatic reasoning, in which diagrams are defined not in terms of regions as in (...) the standard approach, but in terms of topological relations between diagrammatic objects. We formalize the unification rule, which plays a central role in Euler diagrammatic reasoning, in a style of natural deduction. We prove the soundness and completeness theorems with respect to a formal set-theoretical semantics. We also investigate structure of diagrammatic proofs and prove a normal form theorem. (shrink)

This paper describes Peirce's systems of logic diagrams, focusing on the so-called ''existential'' graphs, which are equivalent to the first-order predicate calculus. It analyses their implications for the nature of mental representations, particularly mental models with which they have many characteristics in common. The graphs are intended to be iconic, i.e., to have a structure analogous to the structure of what they represent. They have emergent logical consequences and a single graph can capture all the different ways in which a (...) possibility can occur. Mental models share these properties. But, as the graphs show, certain aspects of propositions cannot be represented in an iconic or visualisable way. They include negation, and the representation of possibilities qua possibilities, which both require representations that do not depend on a perceptual modality. Peirce took his graphs to reveal the fundamental operations of reasoning, and the paper concludes with an analysis of different hypotheses about these operations. (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)

We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic ). The straightforward construction of a presheaf category \ of cuts-only Existential Graphs provides a basis for the further construction of the category \ which introduces variables in a reconstructedly generic, or label-free, mode. Morphisms in these categories represent syntactical embeddings or, equivalently but dually, extensions. Through the example of Peirce’s system, it is shown how the generic figures approach facilitates the (...) formal investigation of relations between syntax and semantics in such diagrammatic systems. (shrink)

We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic ). The straightforward construction of a presheaf category \ of cuts-only Existential Graphs provides a basis for the further construction of the category \ which introduces variables in a reconstructedly generic, or label-free, mode. Morphisms in these categories represent syntactical embeddings or, equivalently but dually, extensions. Through the example of Peirce’s system, it is shown how the generic figures approach facilitates the (...) formal investigation of relations between syntax and semantics in such diagrammatic systems. (shrink)

Although scientific images have begun to receive significant attention from philosophers, one type of image has thus far been ignored: moving images. As techniques such as live cell imaging and videomicroscopy are becoming increasingly important in many areas of biology, however, this oversight needs to be corrected. Biologists often claim that there are relevant differences between video and static images. Most interesting is the idea that video images allow us to see causal relationships. By identifying the conditions that would be (...) required for this to be true and showing that they are not satisfied at the micro level, I will show that videomicroscopy does not provide us with special access to causal information. †To contact the author, please write to: Department of Philosophy, University of Calgary, 2500 University Dr. NW, Calgary, Alberta T2N 1N4; e-mail: [email protected]. (shrink)

We investigate the major mathematical theories of space from a modal standpoint: topology, affine geometry, metric geometry, and vector algebra. This allows us to see new fine-structure in spatial patterns which suggests analogies across these mathematical theories in terms of modal, temporal, and conditional logics. Throughout the modal walk through space, expressive power is analyzed in terms of language design, bisimulations, and correspondence phenomena. The result is both unification across the areas visited, and the uncovering of interesting new questions.

In the vein of a renewed interest in diagrammatic reasoning, this paper challenges an opposition between logic diagrams and formal languages that has traditionally been the common view in philosophy of logic and linguistics. We examine, from a philosophical point of view, what we call five dogmas of logic diagrams. These are as follows: (1) diagrams are non-linguistic; (2) diagrams are visual representations; (3) diagrams are iconic, and not symbolic; (4) diagrams are non-linear; (5) diagrams are heterogenous, and not homogenous. (...) Using historical examples, we argue that none of these dogmas is an adequate criterion to distinguish logic diagrams from formal languages. Instead, we advocate that there is a common core between linguistic and diagrammatic representation and reasoning. (shrink)

A partir de la conformación de un modelo semántico fundado por la relación de presuposición se construye un entramado algebraico de carácter diagramático y articulación reticular, que considera y se apoya en la noción de orden como dimensión semiótica. Esta articulación relacional se realiza al abarcar diversos niveles de análisis, que van desde un nivel grafémico a un nivel discursivo. Se destacan de modo significativo los procesos complejos de composicionalidad semántica (asociados a modalidades de síntesis) en contraposición con procesos elementales (...) de aditividad (asociados a sumas analíticas). Asimismo, a lo largo del análisis, quedarán establecidos los elementos configuracionales para la articulación reticular de los sucesos pertenecientes a un relato, formando unidades discursivas complejas a través de esquemas narrativos. (shrink)

Historians occasionally use timelines, but many seem to regard such signs merely as ways of visually summarizing results that are presumably better expressed in prose. Challenging this language-centered view, I suggest that timelines might assist the generation of novel historical insights. To show this, I begin by looking at studies confirming the cognitive benefits of diagrams like timelines. I then try to survey the remarkable diversity of timelines by analyzing actual examples. Finally, having conveyed this (mostly untapped) potential, I argue (...) that neglecting timelines might mean neglecting significant aspects of reality that are revealed only by those signs. My overall message is that once we accept that relations are as important for the mind as what they relate, we have to pay closer attention to any semiotic device that enables or facilitates the discernment of new relations. (shrink)