At this point, we are very close to applied aspects of research in multi-modal reasoning—design theory and AI research—by providing these disciplines with computational support for visual reasoning. Related to the issue of imagistic mental representation is the examination of the semantics of various diagrammatic systems and what they can teach us about the nature of languages in general e.
We believe that the considerations presented above give us some empirical handle on this type of claim at least—depending on the imagistic objects and relations used, patterns of incorrect inference should be predictable and detectable. An important, if little-known, article on this theme is Malinas We began by motivating the philosophical interest of diagrams, by way of their role in human reasoning and their relation to the study of language in general, and multi-modal information processing.
It was argued that diagrammatic systems can be afforded the same logical status as traditional linear proof calculi. We then explained some of the potential pitfalls of diagrammatic representation and reasoning, by examining spatial constraints on diagrammatic systems and how they can affect correctness and expressive power.
We closed by surveying other diagrams systems, the interest in diagrams generated in computer science and cognitive science, and gave an introduction to the imagery debate in the philosophy of mind.
Shin yale. Introduction 2. Diagrams as Representational Systems 2. Consequences of Spatial Properties of Diagrams 3. Diagrammatic Systems in Geometry 4. Diagrams and Cognition, Applications 5. Introduction Diagrams or pictures probably rank among the oldest forms of human communication. Internal diagrams or images : These comprise the controversial internal representations that are posited to have some pictorial properties.
Diagrams as Representational Systems The dominance of sentential representation systems in the history of modern logic has obscured several important facts about diagrammatic systems.
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. 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. Example 1. All A are B.
All C are A. Therefore, all C are B. For instance: Example 3. Peirce : 4. Consequences of Spatial Properties of Diagrams While it is often possible to afford diagrams the same logical status as formulae as argued above , there are still important differences which can have ramifications for correctness of the system between diagrams and traditional linear proof calculi.
To understand the ramifications of this, consider the following problem: Example 4. The variance in inferential potential of different modes of representation is largely attributable to different ways in which these structural constraints on representations match with the constraints on targets of representation Shimojima a , Diagrammatic Systems in Geometry Mathematicians have used, and continue to use, diagrams extensively.
As an illustration, consider the proof for proposition 16 of book I of the Elements. The proposition is: In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles.
The role of diagrams as representations is remarked upon by Aristotle in book A, chapter 10 of the Posterior Analytics : The geometer bases no conclusion on the particular line he has drawn being that which he has described, but [refers to] what is illustrated by the figures. Proclus asserts that passing from a particular instance to a universal conclusion is justified because geometers … use the objects set out in the diagram not as these particular figures, but as figures resembling others of the same sort.
Anticipating the predominate modern view, Leibniz asserts: …it is not the figures which furnish the proof with geometers, though the style of the exposition may make you think so. In reasoning in this way, the geometer considers the concept in concreto , although non-empirically, but rather solely as one it has exhibited a priori , i.
The methodological norm guiding the work is nicely expressed in the following often-quoted passage: In fact, if geometry is genuinely deductive, the process of deducing must be in all respects independent of the sense of the geometrical concepts, just as it must be independent of figures; only the relations set out between the geometrical concepts used in the propositions respectively definitions concerned ought to be taken into account.
The translation here is from Schlimm The norm has since entrenched itself both in mathematics and in philosophical discussions of mathematics. As Manders puts it The practice has resources to limit the risk of disagreement on explicit co-exact attributions from a diagram; but it lacks such resources for exact attributions, and therefore could not allow them without dissolving into a disarray of irresolvably conflicting judgements. Then the diagram serves to represent the co-exact properties of this type.
Then the result of carrying this step out on the diagram ceases to be representative. For example, with the triangle the result of applying the construction step is 4. Diagrams and Cognition, Applications Despite the formal limitations of some diagrammatic systems noted above, many different systems are currently used in a wide variety of contexts; logic teaching, automated reasoning, specifying computer programs, reasoning about situations in physics, graphical user interfaces to computer programs, and so on.
Summary We began by motivating the philosophical interest of diagrams, by way of their role in human reasoning and their relation to the study of language in general, and multi-modal information processing. Bibliography References Allwein, G. Barwise eds. Avigad, J. Dean and J.
Barker-Plummer, D. Special Issue on Diagrammatic Representation and Reasoning. Beaver, J. Barwise, J. Mineau, B. Moulin, and J. Washington: Mathematical Association of America. Glasgow, N. Hari Narayanan, and B. Chandrasekaran, eds , Diagrammatic Reasoning: Cognitive and Computational Perspectives , pages — Berkeley, G.
Block, N. Rumbaugh, and I. Coecke, B. Carroll, L. Chandrasekaran, B. Glasgow, and N. Hari Narayanan , eds. Cummins, R. De Toffoli, S. Dennett, D. Englebretsen, G. Translated with introduction and commentary by Sir Thomas L. Heath, from the text of Heiberg.
Euler, L. Fodor, J. Frege, G. Gardner, M. Goodman, N. Greaves, M. Grigni, M. Papadias, and C. Gurr, C. Lee, and K. Halimi, B. Hamami Y. Hammer, E. Harel, D.
Howell, R. Jamnik, M. Bundy, and I. Kant, I. Guyer and A. Wood, Cambridge: Cambridge University Press, Kauffman, L. Kosslyn, S. Lambert, J. Larkin, J. Leibniz, G. Lemon, O. Malinas, G. Manders, K. Mancosu ed. First circulated as a manuscript in Morrow, G. Mumma, J. Narayanan, N. Pasch, M. The conclusion is All x are y' which is equivalent to the combination No xy exist and Some xy' exist. Contact Front page Contents Up.
What is what? Trilateral Diagrams Lewis Carroll invented trilateral diagrams for the express purpose of resolving syllogisms. Follow 3 rules: If a square contains two gray counters, transfer one gray counter to the bilateral diagram. If a square contains at least one red counter, transfer one red counter to the bilateral diagram. Oxford University Press. Crane, A. Corporate social responsibility: readings and cases in a global context 2nd ed.
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Carroll was on the faculty of the Terry College for 40 years. Brown and A. Lipartito, J. Post, P. Werhane, and Executive Editor K. Reprints and Permissions. Int J Corporate Soc Responsibility 1, 3 Download citation. Received : 16 March Who invented the schlegel diagram? How old was Lewis Carroll when he wrote Alice in Wonderland? Who invented the word chortled? Diagram of various types of microscopes invented?
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