What is a logical diagram?

Robert Brandom’s expressivism argues that not all semantic content may be made fully explicit. This view connects in interesting ways with recent movements in philosophy of mathematics and logic (e.g. Brown, Shin, Giaquinto) to take diagrams seriously - as more than a mere “heuristic aid” to proof, but either proofs themselves, or irreducible components of such. However what exactly is a diagram in logic? Does this constitute a semiotic natural kind? The paper will argue that such a natural kind does exist in Charles Peirce’s conception of iconic signs, but that fully understood, logical diagrams involve a structured array of normative reasoning practices, as well as just a “picture on a page”

PyZX: Large Scale Automated Diagrammatic Reasoning

The ZX-calculus is a graphical language for reasoning about ZX-diagrams, a type of tensor networks that can represent arbitrary linear maps between qubits. Using the ZX-calculus, we can intuitively reason about quantum theory, and optimise and validate quantum circuits. In this paper we introduce PyZX, an open source library for automated reasoning with large ZX-diagrams. We give a brief introduction to the ZX-calculus, then show how PyZX implements methods for circuit optimisation, equality validation, and visualisation and how it can be used in tandem with other software. We end with a set of challenges that when solved would enhance the utility of automated diagrammatic reasoning.Comment: In Proceedings QPL 2019, arXiv:2004.1475

Reflective Argumentation

Theories of argumentation usually focus on arguments as means of persuasion, finding consensus, or justifying knowledge claims. However, the construction and visualization of arguments can also be used to clarify one's own thinking and to stimulate change of this thinking if gaps, unjustified assumptions, contradictions, or open questions can be identified. This is what I call "reflective argumentation." The objective of this paper is, first, to clarify the conditions of reflective argumentation and, second, to discuss the possibilities of argument visualization methods in supporting reflection and cognitive change. After a discussion of the cognitive problems we are facing in conflicts--obviously the area where cognitive change is hardest--the second part will, based on this, determine a set of requirements argument visualization tools should fulfill if their main purpose is stimulating reflection and cognitive change. In the third part, I will evaluate available argument visualization methods with regard to these requirements and talk about their limitations. The fourth part, then, introduces a new method of argument visualization which I call Logical Argument Mapping (LAM). LAM has specifically been designed to support reflective argumentation. Since it uses primarily deductively valid argument schemes, this design decision has to be justified with regard to goals of reflective argumentation. The fifth part, finally, provides an example of how Logical Argument Mapping could be used as a method of reflective argumentation in a political controversy

Accessible reasoning with diagrams: From cognition to automation

High-tech systems are ubiquitous and often safety and se- curity critical: reasoning about their correctness is paramount. Thus, precise modelling and formal reasoning are necessary in order to convey knowledge unambiguously and accurately. Whilst mathematical mod- elling adds great rigour, it is opaque to many stakeholders which leads to errors in data handling, delays in product release, for example. This is a major motivation for the development of diagrammatic approaches to formalisation and reasoning about models of knowledge. In this paper, we present an interactive theorem prover, called iCon, for a highly expressive diagrammatic logic that is capable of modelling OWL 2 ontologies and, thus, has practical relevance. Significantly, this work is the first to design diagrammatic inference rules using insights into what humans find accessible. Specifically, we conducted an experiment about relative cognitive benefits of primitive (small step) and derived (big step) inferences, and use the results to guide the implementation of inference rules in iCon

Embodiment and embodied design

Picture this. A preverbal infant straddles the center of a seesaw. She gently tilts her weight back and forth from one side to the other, sensing as each side tips downward and then back up again. This child cannot articulate her observations in simple words, let alone in scientific jargon. Can she learn anything from this experience? If so, what is she learning, and what role might such learning play in her future interactions in the world? Of course, this is a nonverbal bodily experience, and any learning that occurs must be bodily, physical learning. But does this nonverbal bodily experience have anything to do with the sort of learning that takes place in schools - learning verbal and abstract concepts? In this chapter, we argue that the body has everything to do with learning, even learning of abstract concepts. Take mathematics, for example. Mathematical practice is thought to be about producing and manipulating arbitrary symbolic inscriptions that bear abstract, universal truisms untainted by human corporeality. Mathematics is thought to epitomize our species’ collective historical achievement of transcending and, perhaps, escaping the mundane, material condition of having a body governed by haphazard terrestrial circumstance. Surely mathematics is disembodied