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<Paper uid="T75-2033">
  <Title>I I I I I I I I I I, I I I I I I I I I I I I I I I</Title>
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AFTERTHOUGHTS ON ANALOGICAL REPRESENTATIONS
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    <Paragraph position="0"> In 19711 wrote a paper attempting to relate some old philosophical issues about representation and reasoning to problems in Artificial Intelligence. A major theme of the paper was the importance of distinguishing &amp;quot;analogical&amp;quot; from &amp;quot;Fregean&amp;quot; representations. I still think the distinction is important, though perhaps not as important for current problems in A.I.</Paragraph>
    <Paragraph position="1"> as I used to think. In this paper I'll try to explain why.</Paragraph>
    <Paragraph position="2"> Throughout I'll use the term &amp;quot;representation&amp;quot; to refer to a more or less complex structure which has addressable and significant parts, and which as a whole is used to denote or refer to something else.</Paragraph>
    <Paragraph position="3"> Thus maps, sentences, and phrases like &amp;quot;The paternal grandfather of the present mayor of Brighton&amp;quot; are representations. There is much that's puzzling and complex about the concept of using something to &amp;quot;denote&amp;quot; or &amp;quot;refer to &amp;quot; something else, but for the present I'll dodge that issue and rely on our intuitive understanding thereof.</Paragraph>
    <Paragraph position="4"> The analogical/Fregean distinction is not new: people have been discovering and re-discovering it for a long time, though they rarely manage to say clearly and pecisely what it is, despite agreement on (most) examples: e.g. maps, photographs and family trees are analogical representations whereas many sentences, referring phrases, and most logical and mathematical formulae are Fregean. I. use the word &amp;quot;Fregean&amp;quot; because it was Gottleb Frege who first clearly saw that a great deal of natural-language syntax and semantics could be analysed in terms of the application of functions to arguments, and that this analysis was far superior to previous attempts to understand the structure of sentences. For instance, it enabled him to invent the logic of quantifiers and develop a notation which provided some of the essential ideas of Church's lambda-calculus, and thereby some of the goodies in programming languages like LISP, ALGOL and POP-2. I use the word &amp;quot;Fregean&amp;quot; not only to honour Frege but also because there is no unambiguous alternative. The most popular rivals - &amp;quot;symbolic&amp;quot; and &amp;quot;verbal&amp;quot; - are used in too many different ill-defined ways, and in addition the first seems too general, the second too narrow. People seem to have a lot of trouble seeing clearly what the distinction is, so I'll list and comment on some of the more common misrepresentations of what I wrote in the 1971 paper.</Paragraph>
    <Paragraph position="5"> Misrepresentations (I) &amp;quot;Analogical representations are continuous, Fregean representations discrete&amp;quot;. Comment: I gave examples of discrete analogical representations, e.g. a list whose elements are ordered  according to the order of what they represent.</Paragraph>
    <Paragraph position="6">  (2) &amp;quot;Analogical representations are 2-dimensional, Fregean representations l-dimensional.&amp;quot; Comment: I gave examples of 1-d analogical representations (e.g.</Paragraph>
    <Paragraph position="7"> the list example). Much mathematical notation is 2-dimensional and Fregean (e.g. integral or summation symbols, the normal representation of fractions).</Paragraph>
    <Paragraph position="8"> (3) &amp;quot;Analogical representations are isomorphic with what they represent.&amp;quot; Comment: I discussed 2-d prictures which are not isomorphic with the 3-d scenes they represent analogiCally.</Paragraph>
    <Paragraph position="9"> (4) &amp;quot;Fregean representations are symbolic, analogical representations non-symbolic.&amp;quot; Comment: I find this notion unintelligible. The only sense of &amp;quot;symbolic&amp;quot; which I can understand clearly includes both maps and sentences. People who arrive at this misinterpretation seem to be guilty of using &amp;quot;symbolic&amp;quot; in a sloppy, ill-defined sense, to contrast with some equally ill-defined alternative. Their excuse may be that this is frequently done (e.g. by Minsky and Papert in their Progress Report, and by Minsky in his more recent paper on frames - 1974.) (5) &amp;quot;Sentences in a natural Inaguage are all Fregean.&amp;quot; Comment: I pointed out that  some English sentences function in a partly analogical way, as is illustrated by the difference in meaning of &amp;quot;She shot him and kissed him&amp;quot; and &amp;quot;She kissed him and shot him&amp;quot;. Compare &amp;quot;Tom, Dick and Harry stood in that order&amp;quot;.</Paragraph>
    <Paragraph position="10"> Contrast &amp;quot;She shot him after she kissed him&amp;quot;, where a relation is explicitly named, and the semantics is Fregean.</Paragraph>
    <Paragraph position="11"> (5) &amp;quot;Analogical representations are complete: whatever is not represented in a picture or map is thereby represented as not existing. By contrast Fregean representations may abstract from as many or as few features of a situation as desired: if I say &amp;quot;Tom stood between Dick and Harry&amp;quot;, then nothing is implied about whether anyone else was there or not.&amp;quot; Comment: there may be an important distinction between descriptions or representations which are complete (relative to the resources of a language) and those which are incomplete, but this has nothing to do with the analogical/Fregean distinction.</Paragraph>
    <Paragraph position="12"> E.g. a map showing only some of the towns and roads of Britain is still an analogical representation. We are free to specify for some pictures or maps that they are to be interpreted as complete, and for others that they depict relations between some but not all parts of a situation or object.</Paragraph>
    <Paragraph position="13"> Similarly a LISP llst might contain items representing events in the order in which the events occurred, yet be incomplete in that new items are added as new knowledge about the tlme-order of events is acquired.</Paragraph>
    <Paragraph position="14"> (7) &amp;quot;Fregean representations have a grammar, analogical representations do not.&amp;quot; Comment: it is easy to define a grammar for lists and trees, frequently used as analogical representions in computing.</Paragraph>
    <Paragraph position="16"> One can also define a grammar for a  class of line-drawings which includes pictures of polyhedral scenes. (8) &amp;quot;Although digital computers can use Fregean representations, only analog computers can handle analogical representations.&amp;quot; Comment: see (I) and (2) above.</Paragraph>
    <Paragraph position="17">  Explanation of the Distinction What then is the distinction? Both Fregean and analogical representations are complex, i.e. they have parts and relations between parts, and therefore a syntax. They may both be used to represent, refer to, or denote, things which are complex, i.e. have parts and relations between parts. The difference is that in the case of analogical represenations both must be complex (i.e. representation and thing) and there must be some correspondence between their structure, whereas in the case of Fregean representations there need be no corresondence. Roughly, in a complex Fregean symbol the structure of the symbol corresponds not to the structure of the thing denoted, but to the structure of the procedure by which that thing is identified, or computed.</Paragraph>
    <Paragraph position="18"> We can be a bit more precise about analogical representations. If R is an analogical representation of T, then (a) there must be parts of R representing parts of T, as dots and squiggles on a map represent towns and rivers in a country, or lines and regions in a picture represent edges and faces in a scene, and (b) it must be possible to specify some sort of correspondence, possibly context-dependent, between properties or relations of parts of R and properties or relations of parts of T, e.g. size, shape, direction and distance of marks on a map may represent size, shape, direction and distance of towns, and different 2-d relationships of lines meeting at a Junction in a picture may represent (possibly ambiguously) 3-d configurations of edges and surfaces in a scene. The relationship between R and T need not be an isomorphism, for instance when a relation between parts of R (such as direction or distance) represents different relations between parts of T in different contexts.</Paragraph>
    <Paragraph position="19"> In a perspective drawing there is no simple, context independent, rule for translating angles between lines into angles between edges or surfaces in the scene depicted. In such cases the task of interpreting R, i.e. working out what T is, may involve solving quite complex problems in order to find a globally consistent interpretation. (See Clowes 1971.) From (a) and (b) it follows that in analogical representations, relationships within T do not need to be explicitly named in R, i.e. there need not be a Dart of R corresponding to relations like &amp;quot;above&amp;quot;, &amp;quot;behind&amp;quot;, &amp;quot;intersects&amp;quot; in T. The conditions (a) and (b) do not hold for Fregean representations.</Paragraph>
    <Paragraph position="20"> A Fregean formula may be very complex, with many parts and relationships, but none of the parts or relationships need corresond to parts or relations within the thing denoted. The phrase &amp;quot;the city 53 miles north of Brighton&amp;quot; contains the symbol &amp;quot;Brighton&amp;quot; as a part, but the thing denoted does not contain the town Brighton as a part. The thing denoted, London, has a complex structure of its own, which bears no relation whatsoever to the structure of the phrase. Similarly &amp;quot;the father of Fred&amp;quot;,</Paragraph>
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