First published Tue Aug 28, 2001; substantive revision Thu Dec 13, 2018
All of us engage in and make use of valid reasoning, but the reasoning we actually perform differs in various ways from the inferences studied by most (formal) logicians. Reasoning as performed by human beings typically involves information obtained through more than one medium. Formal logic, by contrast, has thus far been primarily concerned with valid reasoning which is based on information in one form only, i.e., in the form of sentences. Recently, many philosophers, psychologists, logicians, mathematicians, and computer scientists have become increasingly aware of the importance of multi-modal reasoning and, moreover, much research has been undertaken in the area of non-symbolic, especially diagrammatic, representation systems.[1] This entry outlines the overall directions of this new research area and focuses on the logical status of diagrams in proofs, their representational function and adequacy, different kinds of diagrammatic systems, and the role of diagrams in human cognition.
- 1. Introduction
- 2. Diagrams as Representational Systems
- 3. Consequences of Spatial Properties of Diagrams
- 4. Diagrammatic Systems in Geometry
- 5. Diagrams and Cognition, Applications
- Summary
- Bibliography
- Academic Tools
- Other Internet Resources
- Related Entries
1. Introduction
Diagrams or pictures probably rank among the oldest forms of human communication. They are not only used for representation but can also be used to carry out certain types of reasoning, and hence play a particular role in logic and mathematics. However, sentential representation systems (e.g., first-order logic) have been dominant in the modern history of logic, while diagrams have largely been seen as only of marginal interest. Diagrams are usually adopted as a heuristic tool in exploring a proof, but not as part of a proof.[2] It is a quite recent movement among philosophers, logicians, cognitive scientists and computer scientists to focus on different types of representation systems, and much research has been focussed on diagrammatic representation systems in particular.
Challenging a long-standing prejudice against diagrammatic representation, those working on multi-modal reasoning have taken different kinds of approaches which we may categorize into three distinct groups. One branch of research can be found in philosophy of mind and cognitive science. Since the limits of linguistic forms are clear to those who have been working on mental representation and reasoning, some philosophers and cognitive scientists have embraced this new direction of multi-modal reasoning with enthusiasm and have explored human reasoning and mental representation involving non-linguistic forms (Cummins 1996; Chandrasekaran et al. 1995). Another strand of work on diagrammatic reasoning shows that there is no intrinsic difference between symbolic and diagrammatic systems as far as their logical status goes. Some logicians have presented case studies to prove that diagrammatic systems can be sound and complete in the same sense as symbolic systems. This type of result directly refuted a widely-held assumption that diagrams are inherently misleading, and abolished theoretical objections to diagrams being used in proofs (Shin 1994; Hammer 1995a). A third direction in multi-modal reasoning has been taken by computer scientists, whose interest is much more practical than those of the other groups. Not so surprisingly, those working in many areas in computer science—for example, knowledge representation, systems design, visual programming, GUI design, and so on—found new and exciting opportunities in this new concept of ‘heterogeneous system’ and have implemented diagrammatic representations in their research areas.
We have the following goals for this entry. First of all, we would like to acquaint the reader with the details of some specific diagrammatic systems. At the same time, the entry will address theoretical issues, by exploring the nature of diagrammatic representation and reasoning in terms of expressive power and correctness. The case study of the second section will not only satisfy our first goal but also provide us with solid material for the more theoretical and general discussion in the third section. The fourth section presents another case study and considers it in light of the third section’s general discussion. As mentioned above, the topic of diagrams has attracted much attention with important results from many different research areas. Hence, our fifth section aims to introduce various approaches to diagrammatic reasoning taken in different areas.
For further discussion, we need to clarify two related but distinct uses of the word ‘diagram’: diagram as internal mental representation and diagram as external representation. The following quotation from Chandrasekaran et al. (1995: p. xvii) succinctly sums up the distinction between internal versus external diagrammatic representations:
- External diagrammatic representations: These are constructed by the agent in a medium in the external world (paper, etc), but are meant as representations by the agent.
- Internal diagrams or images: These comprise the (controversial) internal representations that are posited to have some pictorial properties.
As we will see below, logicians focus on external diagrammatic systems, the imagery debate among philosophers of mind and cognitive scientists is mainly about internal diagrams, and research on the cognitive role of diagrams touches on both forms.
2. Diagrams as Representational Systems
The dominance of sentential representation systems in the history of modern logic has obscured several important facts about diagrammatic systems. One of them is that several well-known diagrammatic systems were available as a heuristic tool before the era of modern logic. Euler circles, Venn diagrams, and Lewis Carroll’s squares have been widely used for certain types of syllogistic reasoning (Euler 1768; Venn 1881; Carroll 1896). Another interesting, but neglected, story is that a founder of modern symbolic logic, Charles Peirce, not only revised Venn diagrams but also invented a graphical system, Existential Graphs, which has been proven to be equivalent to a predicate language (Peirce 1933; Roberts 1973; Zeman 1964).
These existing diagrams have inspired those researchers who have recently drawn our attention to multi-modal representation. Logicians who participate in the project have explored the subject in two distinct ways. First, their interest has focused exclusively on externally-drawn representation systems, as opposed to internal mental representations. Second, their aim has been to establish the logical status of a system, rather than to explain its heuristic power, by testing the correctness and the expressive power of selective representation systems. If a system fails to justify its soundness or if its expressive power is too limited, a logician’s interest in that language will fade (Sowa 1984; Shin 1994).
In this section, we examine the historical development of Euler and Venn diagrams as a case study to illustrate the following aspects: First, this process will show us how one mathematician’s simple intuition about diagramming syllogistic reasoning has gradually been developed into a formal representation system. Second, we will observe different emphases given to different stages of extension and modification of a diagrammatic system. Thirdly and relatedly, this historical development illustrates an interesting tension and trade-off between the expressive power and visual clarity of diagrammatic systems. Most importantly, the reader will witness logicians tackle the issue of whether there is any intrinsic reason that sentential systems, but not diagrammatic systems, could provide us with rigorous proofs, and their success in answering this question in the negative.
Hence, the reader will not be surprised by the following conclusion drawn by Barwise and Etchemendy, the first logicians to launch an inquiry into diagrammatic proofs in logic,
there is no principled distinction between inference formalisms that use text and those that use diagrams. One can have rigorous, logically sound (and complete) formal systems based on diagrams. (Barwise & Etchemendy 1995: 214)
This conviction was necessary for the birth of their innovative computer program Hyperproof, which adopts both first-order languages and diagrams (in a multi-modal system) to teach elementary logic courses (Barwise & Etchemendy 1993 and Barwise & Etchemendy 1994).
2.1 Euler Diagrams
Leonhard Euler, an 18th century mathematician, adopted closed curves to illustrate syllogistic reasoning (Euler 1768). The four kinds of categorical sentences are represented by him as shown in Figure 1.
Figure 1: Euler Diagrams
For the two universal statements, the system adopts spatial relations among circles in an intuitive way: If the circle labelled ‘A’ is included in the circle labelled ‘B,’ then the diagram represents the information that all A is B. If there is no overlapping part between two circles, then the diagram conveys the information that no A is B.
This representation is governed by the following convention:[3]
Every object x in the domain is assigned a unique location, say l(x), in the plane such that l(x) is in region R if and only if x is a member of the set that the region R represents.
The power of this representation lies in the fact that an object being a member of a set is easily conceptualized as the object falling inside the set, just as locations on the page are thought of as falling inside or outside drawn circles. The system’s power also lies in the fact that no additional conventions are needed to establish the meanings of diagrams involving more than one circle: relationships holding among sets are asserted by means of the same relationships holding among the circles representing them. The representations of the two universal statements, ‘All Aare B’ and ‘No A is B,’ illustrate this strength of the system.
Moving on to two existential statements, this clarity is not preserved. Euler justifies the diagram of “Some A is B” saying that we can infer visually that something in A is also contained in B since part of area A is contained in area B (Euler 1768: 233). Obviously, Euler himself believed that the same kind of visual containment relation among areas can be used in this case as well as in the case of universal statements. However, Euler’s belief is not correct and this representation raises a damaging ambiguity. In this diagram, not only is part of circle A contained in area B (as Euler describes), but the following are true: (i) part of circle B is contained in area A (ii) part of circle A is not contained in circle B (iii) part of circle B is not contained in circle A. That is, the third diagram could be read off as “Some B is A,” “Some A is not B,” and “Some B is not A” as well as “Some A is B.” In order to avoid this ambiguity, we need to set up several more conventions.[4]
Euler’s own examples nicely illustrate the strengths and weaknesses of his diagrammatic system.
Example 1. All A are B. All C are A. Therefore, all C are B.
Example 2. No A is B. All C are B. Therefore, no C is A.
In both examples, the reader can easily infer the conclusion, and this illustrates visually powerful features of Euler diagrams. However, when existential statements are represented, things become more complicated, as explained above. For instance:
Example 3. No A is B. Some C is A. Therefore, Some C is not B.No single diagram can represent the two premises, since the relationship between sets B and Ccannot be fully specified in one single diagram. Instead, Euler suggests the following three possible cases:
Euler claims that the proposition ‘Some C is not B’ can be read off from all these diagrams. However, it is far from being visually clear how the first two cases lead a user to reading off this proposition, since a user might read off “No C is B” from case 1 and “All B is C” from case 2.
Hence, the representation of existential statements not only obscures the visual clarity of Euler Circles but also raises serious interpretational problems for the system. Euler himself seemed to recognize this potential problem and introduced a new syntactic device, ‘*’ (representing non-emptiness) as an attempt to repair this flaw (1768: Letter 105).
However, a more serious drawback is found when this system fails to represent certain compatible (that is, consistent) pieces of information in a single diagram. For example, Euler’s system prevents us from drawing a single diagram representing the following pairs of statements: (i) “All A are B” and “No A is B” (which are consistent if A is an empty set). (ii) “All A are B” and “All B are A” (which are consistent when A = B). (iii) “Some A is B” and “All A are B”. (Suppose we drew an Euler diagram for the former proposition and try to add a new compatible piece of information, i.e., the latter, to this existing diagram.) This shortcoming is closely related to Venn’s motivation for his own diagrammatic system (see Section 3.1 for other shortcomings of Euler’s system). READ MORE
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