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Analogy and axiomatics
Int. J. Man-Machine Studies (1983) 18, 161-173
Analogy and axiomatics JOHN STELZER
501 Callahan Road, Roseburg, Oregon 97470, U.S.A. (Received 19 January 1981, and in revised form 12 April 1982) The concept of an analogy can be viewed from two perspectives. One active, being the process of drawing an analogy. The other passive, being the relationship of one thing being an analogy for another. The focus in this paper is on the latter meaning, or the static view of analogy. To be investigated are the types of analogies there are, not where they come from. Examples of analogies abound: LISP is analogous to SLIP; logic is analogous to Boolean algebra; electricity is analogous to hydraulics; home brewing is analogous to bread baking. The scope of this paper is further restricted to a consideration of analogies between subject matters. The subject matter may be logic, Boolean algebra, electricity, brewing, baking or some other. It need not be the complete subject matter but only some sub-part.
To be discussed are the types of analogies that might exist between subject matters. To this end, some m e t h o d of representing the subject matters of the knowledge spaces is required. The axiomatic method of representing subject matter has been investigated by Stelzer & Kingsley (1975). The analysis of axiomatic structures given below is an extension of this earlier work. The concept of an analogy will be viewed herein as a relationship that exists between subject matters themselves. The analogous relationships will be analysed in terms of functional relationships. Requiring analogies to be functional relationships is m o r e restrictive than, for example, the approach take by Gaines & Shaw (1982), who base their analysis on the notion of partial correspondences. The concept of an axiomatic theory is based on eight primitive concepts: L the underlying language in which the axiomatization proceeds, PE the set of primitive expressions representing primitive concepts, SE the set of secondary expressions representing defined concepts, PS the set of possible statements in the language L 1, an extension of L, TS the set of theoretical statements actually formulated in L 1 about the subject matter, Fxs the relationship "expression x is used to formulate statement s", A ~- s the relationship " s t a t e m e n t s is derived f r o m the set of statements A " and A ~ s the relationship "statement s can be derived from the set of statements ~' h where F __%_(PE t3 SE) x (PS), ~- G ~ (TS) x (TS), ___~ (PS) • (PS). 161 0020-7373/83/020161+ 13503.00/0
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DEFINITION 1. 91 = (L, PE, SE, PS, TS, F, ~-, ~) is an axiomatically structured theory for a subfect matter if and only if 91 satisfies the following 12 axioms. Axiom 1. PS is the set of well formed statements within some language L 1, which is a modification of the underlying language L, effected as follows. (i) Expressions appearing in PE or SE that do not appear in the lexicon of L are added to the lexicon of L; and transformational rules appropriate for these terms are added to the grammar for L. (ii) Transformational rules in the grammar for L for expressions appearing in either PE or SE, and in the lexicon of L, are modified as required to permit new transformational possibilities as appropriate. (iii) Step (ii) is carried out in such a way that nontheoretical uses of these expressions remain unchanged so that every statement expressible in L remains expressible in L 1. (iv) (Vx E P E w SE)(x E PS). That is, primitive and secondary expressions as members of the lexicon of L are not in and of themselves possible statements. Axiom 2. T S _ PS, TS is finite and the elements of TS are distinctly labelled so that different elements have different labels. Axiom 3. P E # 0 and PE is finite. Axiom 4. PE ~ SE = 0. That is, the same expression is not used to represent both a primitive and a defined concept. Axiom 5. T h e r e is a one-to-one function, h, from SE into TS. DEFINITION 2. For all x ESE, h(x) is referred to as the definition of x and is denoted bye. That is, h(x) =~. Axiom 6. (Vx E SE)Fxs That is, secondary expressions are used to formulate their own definitions. Axiom 7. (Vx ~ S E ) 7 (3A E ~ ( T S ) ) A ~- s That is, definitions are not proved. Axiom 8. There is no sequence of elements of SE, x a , . . . , x,(n >-1) such that the following hold: Fx 1J72, Fx2x3~
Fxn.~l. That is, there are no circular definition chains. Axiom 9. There is no sequence of subsets of TS, A I . . . . . An (n - 1) and elements of TS, Sl . . . . . s, such that the following hold:
s~EA2
A1 ~ s2, A2~-s3,
s, E A ,
A,~sl.
Sl E A1
That is, there are no circular proof chains. Axiom 10. (Vs E TS)(VAE ~(TS))(A ~ s ~ A ~ s). That is, if a theoretical statement is proved, it can be proved. Axiom 11. ( V x E S E ) ( V s E P S ) [ F x s ~ ( 3 s l ~ P S ) ( V w ~ P E u S E ) (fwsl~--~w~x & (Fws v FwY)) & {~} ~ s<--~Sl]. That is, the definition of any secondary expression,
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x, is formulated in such a way that any possible statement, s, formulated in terms of x is provably equivalent to another possible statement, sl, that is formulated in terms of expressions other than x, and that appear in s or ~; the equivalence proof using only s (Eliminability). Axiom 12. ( V x E S E ) ( V s E P S ) [ ~ F x s ~ ( V A E ~ ( T S ) ) ( A w { s & (V~slEA)-n FXSl ~ A ~ s)]. That is, the definition of any secondary expression, x, is formulated in such a way that any possible statement, s, not formulated in terms of x and derivable from ~ and a set of theoretical statements, A, no members of which are formulated in terms of x, remains derivable from A only (Non-creativity). DEFINITION 3. The set of definitions, theorems and axioms for an A S T , (i)
Def(91) ={s ~ TS: (3x ~ S E ) s =s
(ii)
Th(91) --{s c TS: ( 3 A ~ ~ ( T S ) ) A ~- s},
(iii)
Ax(')l) = TS ~ [ D E F ( g ) u TH(91)].
We can show in an AST, that SE is finite and that Def (91), Th (91), Ax(91) partition TS. Stelzer & Kingsley (1975) investigate the notion of dependency in an AST. A matrix method that is deterministic is developed that can be used to identify all dependency relationships. It can also be proved that all secondary expressions representing defined concepts can be eliminated from any AST for a subject matter. In essence, axioms 1-12 satisfy our expectations concerning the results that should be forthcoming concerning ASTs. They provide an adequate characterization of the concept on an AST. Below we assume throughout that 91 = (L, PE, SE, PS, TS, F, ~-, ~), 91" = (L*, PE*, SE*, PS*, TS*, F, ~*, ~) and 91"* = (L**, PE**, SE**, PS**, TS**, F, ~**, %) are all ASTs. It is not necessary to differentiate between F, F*, F** and ~, ~*, ~** since the context of use suffices to determine which AST is being referenced. DEFINITION 4. For all expressions t, t* in P E w PE* u S E & SE*, t, t* have the same logical structure if and only if (i)
t, t* are both individual names, or
(ii)
t, t* are both names of sets, or
(iii)
t, t* are both n-place predicates.
Besides comparing the logical structure of primitive and secondary expressions, it is necessary to be able to discuss how primary and secondary expressions function syntactically. Two expressions t, t* are said to function syntactically the same way (or have the same syntactic functions) if and only if for all well formed statements s such that Fts, if every occurrence of t in s is replaced by t*, then the resulting statement is also well formed. Alternatively, it might be said that t and t* are in the same transformational categories (noun phrase, verb phrase, etc.), or that transformational rules applicable to t apply as well to t* and, conversely.
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DEFINITION 5. The set of expressions used to formulate the axioms for an AST. Ax exp (9/) = {t ~ PE u SE: (3s r Ax (9/))Fts}. As will be seen, an analogy results when certain axioms for one subject matter are re-interpreted in another subject matter. To do so means that the primitive and secondary expressions appearing in the axioms selected from one AST are replaced by primitive and secondary expressions from the analogous AST. The replacement of expressions must be done in such a way that every replacement expression has the same logical structure as the expression it replaces. The statement that results from the replacement of expressions must be comparable with selected axioms and theorems in the analogous subject matter in specified ways based on the notion of implication. More precisely, suppose 9/* is intended to be an analogy for 9/. On the one hand we must be able to describe how much of 9/'s structure carries over to 9/*. Three possible categories describing three general cases of relative structural characteristics will be distinguished. On the other hand, we must be able to compare the relative strengths of statements analogous to the axioms of 9/. That is, we must be able to describe how strong (derivationally) 9/'s axioms are in their analogous form. T h r e e possible categories describing relative derivational characteristics will be defined. A nine-fold description of the concept of analogy will result from these definitions. The defintion of the concept of analogy that provides the most general category describing relative structural characteristics will be given first. DEFINITION 6. A n analogy for an A S T 9/ is an A S T 9/* where L = L*, and a pair
of functions f, g satisfying the following four conditions. (i) f i s a one-to-one function from a non-empty subset. A , of A x (9/) into Ax(9/*) u Th(9/*). (ii) g is a one-to-one function from B ={t~PEuSE:
(3s~A)Fts}
intoPE*uSE*.
(iii) For all t in B, t and g(t) have the same logical structure and have the same syntactic function. (iv) For all s in A and for all t in B, Fts +-~Fg(t)f(s). Condition (i) of Definition 6 asserts that a subset of the set of axioms for 9/2[is mapped by f into a subset of the union of the axioms and theorems for 9/*. The subset of Ax(9/) on which f is defined is assumed to be non-empty to ensure that the analogy to 9/* is non-trivial. Thus, f associates some axioms for 9/with unique statements in 9/* that are known to be true of 9/*'s subject matter. T o say that f is one-to-one means that unique axioms in 9 / a r e associated with unique statements in 9/*. It can be shown that the set B in condition (ii) of Definition 6 is a subset of A x exp (9/) Thus, g is a mapping from a subset of Ax exp (9/) into PE* u SE*. The function g associates each element in this subset of Ax exp (9/) with a unique element in PE* u SE* that has the same logical structure [condition (iii), Definition 6]. Finally, condition (iv) asserts that an expression is used to formulate an axiom in A just in case the image under g of the expression is used to formulate the image of the axiom in ~* under f. The notion of analogy can be represented diagramatically as shown in Fig. 1. In Figure 1, 9/ and 9/* are represented by closed spaces. It is important to realize that
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FIG. 1. ~*, f and g provide an analogy for 9.1.
an analogy is a triple consisting of an AST, 9.t*, and the two functions f and g. That is, one AST, 9~*, can serve as an analogy for another, 9~, in various ways depending upon the definitions of f and g. Below, we distinguish three types of analogy. The AST 9~* with two specific functions f, g cannot be more than one type of analogy. But by redefining f and g, while keeping 9~* constant, more than one type of analogy can be generated. Thus, the structure of 9~* alone does not classify uniquely the kind of analogy. When classifying analogies the functions f and g must be taken into consideration also. Below ~ ( f ) , ~ ( g ) and Y~(f), ~ ( g ) denote the domain and range of f and g, respectively. DEFINITION 7. Let ~*, f, g be an analogy for 9~. (i) 9~*, f, g is partial analogy if and only if ~ ( f ) = Ax (9~). (ii) 9~*, f, g is a complete analogy if and only if ~ ( f ) =Ax(9~). (iii) 9~*, f, g, is an exact analogy if and only if 9.1", f, g is a complete analogy and (f) = A x (9~*). That is, in a partial analogy some of the axioms do not find analogous interpretations. In a complete analogy all axioms carry over to the analogous subject matter. In an exact analogy all axioms are reinterpreted as axioms in the analogous domain. Figure 1 illustrates a partial analogy while Figs 2 and 3 illustrate complete and exact analogies, respectively. ~1"
Ax(,tl)
f
y
FIG. 2. ~*, f, g provide a complete analogyfor 9~.
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J. STELZER
f
Axexp('~)
Axexp(~ * }
FIG. 3. 9~*,f, g provide an exact analogy for 9A.
Proposition I (i) If 92*, f, g are a partial analogy for 92, then
~ ( g ) ~ Ax exp (92). (ii) If 92", f, g are a complete analogy for 92, then ~ ( g ) = A x exp (92). (iii) If 92*, f, g are an exact analogy for 92, then ~ ( g ) = A x exp(92)
and
~ ( g ) = A x exp(92*).
(iv) 92", f, g provide an exact analogy for 92 if and only if 92, f - l , g-1 provide an exact analogy for 92*. (v) If 92", f, g provide an exact analogy for 92 and if 92"*, fx, gl provide an exact analogy for 92*, then 92"*, f o f 1 and g o gl provide an exact analogy for 92. A result similar to (iv), above, is not forthcoming for analogy, complete analogy, or partial analogy. While f-~ and g-~ have the appropriate structure (one-to-one) in these cases, there is no guarantee that ~ ( f - i ) c h x (~[:~). Indeed, it is entirely possible for ~ ( f - x ) = ~(f)__-Th (92*). It is assumed here that 92", f and g provide either an analogy, a complete analogy, or a partial analogy for 92. Now consider (v), above, where 92", f, g provide an exact analogy for 92; and 92"*, fx, g~ provide an exact analogy for 92*. Then 92** along with the composite functions f o f~ and g o gl provide an exact analogy for 92. That is, the composite functions f o f~ and g o g l provide the requisite mappings from 92 to 92**. Composites of one-to-one functions are known to be one-to-one and so f o f~ and g o gl satisfy conditions (i), (ii) and (iii) of Definition 6. For condition (iv) we have both Fts ~->Fg(t)f(s)
and Fg(t)f(s) ~->Fgl(g(t))fl(s)).
Thus, Fts +->Fg o gl(t)f ~f l(s).
Finally ~([ oA) = ~(f) = Ax(~)
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and
~(f
oA) = ~ ( A ) = A x (92"*)
and, thus, Definition 7 (iii) is satisfied as required. As with (iv), a result similar to (v) is not forthcoming for analogy, complete analogy, or partial analogy. As before, the reason is that we do not know in these cases that (f) _ Ax (92*). Thus we are not assured that the composite function f o jq is defined. That is, for f o f~ to be defined, it must be the case that
(f) -~ ~ (A). But, we only know that (fl) c_ Ax (92*) and we cannot guarantee that
(f) =_Ax (92*) holds. We can establish the following proposition. Proposition 2 (i) If 92", f, g provide an exact analogy for 92, and 92"*, fl, gl provide a complete analogy for 92", then 92**, f ~ g ~ gl provide a complete analogy for 92. (ii) If 92*, f, g provide an exact analogy for 92, and 92"*, fl, gl provide an analogy (or a partial analogy) for 92", then 92"*, f ~ g ~ provide an analogy (or partial analogy) for 92. The difference between (i) and (ii) in Proposition 2 is quite simple. In (i), every element in Ax(92) is mapped into Ax(92**)uTh (92**) by f ~ In (ii), only a subset of Ax(92) is mapped into this set. That is, in (ii), at best, (fi) ___Ax (92*).
Thus, only those elements, s, in Ax (92) such that
f(s) ~ ~ ( A ) have images in Ax (92**)w Th (92**). Similar remarks apply to Ax exp (92) as well. This completes the discussion of the purely structural characteristics of analogies. Now turn to the problem of comparing the derivational strengths of analogies. Let 92", f, g provide an analogy for 92. Suppose s is both in Ax (92) and in ~ ( f ) . This means that s has an image under f,f(s), in Ax(92*)v0Th (92*). We will refer to f(s) as the analogous interpretation (in 92*) of s. It is of interest to compare all s in ~ ( f ) to their analogous interpretations in 92*. Three types of analogies can be distinguished in this way: proper, approximate, and enveloping. One preliminary definition is required. DEFINITION 8. Let 92", f, g be an analogy for 92. For all s E ~ ( f ) , Subs(s/g(t), t) denotes the result of substituting in statement s, g(t) for every occurrence of every term t in ~(g).
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From Definition 8 it follows that if ~*, f, g is an analogy for 9~ and if s a ~ ( f ) , then: (i) (ii)
Subs (s/g(t), t) ~ PS* and Subs (Subs (s/g(t), t)/g-l(g(t)), g(t)) -- s.
DEFINITION 9. Let 9~*, f, g be an analogy for 91. (i) 9~*, f, g is a proper analogy if and only if for all s ~ ~ ( f )
PS* ~ Subs (s/g(t), t),~--~f(s). (ii) 9X*, f, g is an approximate analogy if and only if for all s ~ ~ ( f )
PS* ~ Subs (s/g(t), t)-->f(s). (iii) 9.I*, f, g is an enveloping analogy if and only if for all s ~ @ (f)
PS* ~ f(s) ~ Subs (s/g(t), t). In an approximate analogy the images are weaker than the substitution instances and in an enveloping analogy the image statements are stronger. It should be clear that Definitions 7 and 9 are combinable yielding the concepts of: proper partial analogies, proper complete analogies, proper exact analogies, approximate partial analogies, approximate complete analogies, approximate exact analogies, enveloping partial analogies, enveloping complete analogies and enveloping exact analogies. In Proposition 1 (iv) an elementary result was given concerning exact analogies based on properties of inverses of one-to-one functions. Proposition l(v) and Proposition 2(i) and (ii) give results based on composites of functions. Due to the definition given for the concept Subs (s/g(t), t), and due to the fact that the definitions of proper, approximate and enveloping analogies are based on the notion of implication; further results along these lines are forthcoming. That is, based on elementary properties of inverses, composites and implication, reflexive and transitive properties of analogies can be established. The following three tables summarize these results. The first table simply summarizes elementary results that follow immediately from the basic TABLE 1 ASSUME 96*, f, g is the following type analogy for 96
THEN 96", f, g is also the following type analogy for 96.I
exact proper exact approximate exact enveloping exact proper proper partial proper complete proper exact proper proper partial proper complete proper exact
complete proper complete approximate complete enveloping complete approximate approximate partial approximate complete approximate exact enveloping enveloping partial enveloping complete enveloping exact
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ANALOGY AND AXIOMATICS TABLE 2 ASSUME 92", f, g is the following type analogy for 92
THEN 92, f-l, g-1 is the following type analogy for 92*
exact proper exact enveloping exact approximate exact
exact proper exact approximate exact enveloping exact
TABLE 3 ASSUME 92*, f, g is the following type analogy for 92
ASSUME 92**, fl, g~ is the following type analogy for 92*
THEN 92"*, f ~ g ~ gl is the following type analogy for 92
exact exact exact proper exact proper exact proper exact proper exact proper exact proper exact approximate exact approximate exact approximate exact approximate exact approximate exact approximate exact proper exact proper exact proper exact enveloping exact enveloping exact enveloping exact enveloping exact enveloping exact enveloping exact exact proper exact proper exact proper exact approximate exact approximate exact enveloping exact enveloping exact
exact complete partial proper exact proper complete proper partial approximate exact approximate complete approximate partial proper exact proper complete proper partial approximate exact approximate complete approximate partial enveloping exact enveloping complete enveloping partial proper exact proper complete proper partial enveloping exact enveloping complete enveloping partial analogy proper analogy approximate analogy enveloping analogy proper analogy approximate analogy proper analogy enveloping analogy
exact complete partial proper exact proper complete proper partial approximate exact approximate complete approximate partial approximate exact approximate complete approximate partial approximate exact approximate complete approximate partial enveloping exact enveloping complete enveloping partial enveloping exact enveloping complete enveloping partial enveloping exact enveloping complete enveloping partial analogy proper analogy approximate analogy enveloping analogy approximate analogy approximate analogy enveloping analogy enveloping analogy
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definitions. Notice also that the results given in Proposition l(iv) and Proposition 2(i) and (ii) are included for completeness. Table 3 can be interpreted as follows. If an analogy of the type in column 1 is followed by an analogy of the type in column 2, then the resultant (composite) type is the type listed in column 3. For example, the last line means an enveloping exact analogy (from 92 to 92*) followed by a proper partial analogy (92* to 92**) results in an enveloping partial analogy (92 to 92**). Notice that the only exact analogies that appear, appear on the left in this schema. Notice also that in combining analogies in this way, approximate analogies never follow enveloping analogies; nor do enveloping analogies follow approximate analogies. A close look at the definitions of these concepts show that this must be the case due to the implication that may hold between elements. That is, from sl implies s2 (s2 approximates sl) and from s3 implies s2 (s3 envelops s2) we cannot infer any relationship between sl and s3. Similarly, knowing that s2 implies sl (s2 envelops s~) and s2 implies s3 (s3 approximates s2) we can infer nothing about the relationship between sx and sa. It is important to realize that analogies have been characterized solely in terms of axioms and subject-matter-related expressions appearing in axioms. Analogies are independent of definitions and theorems concerning subject matters. This is appropriate as is now suggested. Since all definitions are eliminable from an AST, it can be asserted that given the primitive expressions, all required definitions can be generated. That is, given some type of an analogy, we can proceed to define, within the analogy itself, concepts that are analogous to interesting concepts within the original subject matter. More specifically, g maps a set of expressions from 92 into 92*. Any definition in 92 definable only with expressions in the domain of g will have an analogous definition formulated solely with expressions in the range of g. Thus, having the analogies for (some) axioms and their expressions can result in analogies for other definitions in the subject matter. By way of an example of some of these concepts consider the analogy between elementary electronics and hydraulics. A simple circuit with a battery and resistor, both suitably grounded, is analogous to having a pond with a pump circulating water through a pipe containing a restriction. The restriction may be a reduction from a larger to a smaller pipe in the same direction as the water flow. Voltage is analogous to the potential for water flow as established by the pump: water pressure. Furthermore, current is analogous to water flow in the pipe: water flow. Resistance is analogous to a restriction impeding water flow. Finally, grounding of the ends of the circuit is analogous to having the pump pump water out of the pond through the pipe back into the pond. This situation is depicted graphically in Fig. 4. 92 will be the electronics AST and 92* will be the hydraulics AST. We being to specify 92 and 92* as follows. L = English, PE = {E, I, R, series circuit, parallel circuit}, SE = {total resistance (RT)}, L* = English,
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ANALOGY AND AXIOMATICS Flow --->
Restriction
f--.-~ 'VVV~A~
FIG. 4. Simple circuit/pond pumping analogy.
P E * = {pr, fl, re, series pipe, parallel pipe}, SE* = {total resistance (ReT)}. T h e function g will be: g(E) = pr (pressure), g(I) = fl (flow), g (R) = re (resistance), g(RT) = reT (total resistance), g(series circuit) = series pipe, g (parallel circuit) = parallel pipe. TS in electronics contains four statements. sl: E = I x R , s2: In a series circuit Ra- = R1 + R2, s3: In a parallel circuit RT is p r o p o r t i o n a l to R1 x R2 RI• s4: In a parallel circuit RT = R1 + R----~" F u r t h e r m o r e , TS is partitioned by:
Ax(9~)={sl, s2, s3} and In hydraulics TS* has four statements. s*:pr=flxre, s* : In a series pipe reT = re1 + re2, s* : In a parallel pipe reT = re~ x re2 re1 9 re2'
Th(9.I)={s4}.
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J. STELZER s4* : In a parallel pipe reT is proportional to re1 x re2, Ax(92*)=
* s*}, {sl,* s~,
Th(92*) = {s* }. Successive analogy types are generated by varying f. (1) Partial analogy: no image under f for s3:
/(Sl)
= sL
f(s~) = s L
(2) Complete analogy. Add to f above f(s3) = s L
This is not an exact analogy since $4 ~ A x (92").
(3) Exact analogy: Change the image of s3: f(s3) = s L
(4) Proper analogy. Both (1) and (2), above, are proper analogies: (1) is a proper partial analogy and (2) is a proper complete analogy; (3) is not a proper analogy. Change Ax (92*) and define f as follows to get a proper exact analogy: Ax(92*) ={s*, s*2, s 4*lJ', f(Sl) = S*,
f(s2) = S~,
f(s3) = S*.
(5) Approximate analogy. Define Ax (92), Ax (92*) and f as follows: Ax (92) = {sl, s2, $4}, Ax(92*) = { s * , s*2, s 3.1I, f ( s l ) = S~,
f ( s 4 ) = 84~.
This yields an approximate partial analogy. Extending f to include f(s~) = s*
results in an approximate complete analogy. Change Ax (92*) so that Ax (92") = (s*, s*, s*} and leave f as defined to get an approximate exact analogy. (6) Enveloping analogy. Restore Ax (92) so that A x (92) = {sl, sz, s3}.
Define f as follows: f(Sl) = s*,
f(s~) = s*,
to get an enveloping partial analogy. To make this enveloping analogy complete, add: f(s~) = s*.
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Restore Ax (~*) so that Ax (gA*)={s*, s*2, s .13s and the analogy becomes an enveloping exact analogy. Research performed under contract (DAHC19-74-C-0066) to the U.S. Army Research Institute for the Behavioral and Social Sciences; Organization and Systems Laboratory, Dr Joseph Zeidner, Director; while the author was a Senior Scientist at the Human Resources Research Organization, 1974/75. The author thanks Dr Leon Nawrocki and Dr Joe Ward of the former Institute, and Dr Robert Seidel of the latter for their support.
References GAINES, B. R. & SHAW, M. L. G. (1982). Analysing analogy. In TRAPPL, R., RICCIARDI, L. & PASK, G., Eds, Progress in Cybernetics and Systems Research, vol. IX, pp. 379-386. Washington, D.C.: Hemisphere Publishing Company. STELZER, J. & KINGSLEY, E. (1975) Axiomatics as a paradigm in structuring subject matter. Instructional Science, 3, 383-450.
Analogy and axiomatics JOHN STELZER
501 Callahan Road, Roseburg, Oregon 97470, U.S.A. (Received 19 January 1981, and in revised form 12 April 1982) The concept of an analogy can be viewed from two perspectives. One active, being the process of drawing an analogy. The other passive, being the relationship of one thing being an analogy for another. The focus in this paper is on the latter meaning, or the static view of analogy. To be investigated are the types of analogies there are, not where they come from. Examples of analogies abound: LISP is analogous to SLIP; logic is analogous to Boolean algebra; electricity is analogous to hydraulics; home brewing is analogous to bread baking. The scope of this paper is further restricted to a consideration of analogies between subject matters. The subject matter may be logic, Boolean algebra, electricity, brewing, baking or some other. It need not be the complete subject matter but only some sub-part.
To be discussed are the types of analogies that might exist between subject matters. To this end, some m e t h o d of representing the subject matters of the knowledge spaces is required. The axiomatic method of representing subject matter has been investigated by Stelzer & Kingsley (1975). The analysis of axiomatic structures given below is an extension of this earlier work. The concept of an analogy will be viewed herein as a relationship that exists between subject matters themselves. The analogous relationships will be analysed in terms of functional relationships. Requiring analogies to be functional relationships is m o r e restrictive than, for example, the approach take by Gaines & Shaw (1982), who base their analysis on the notion of partial correspondences. The concept of an axiomatic theory is based on eight primitive concepts: L the underlying language in which the axiomatization proceeds, PE the set of primitive expressions representing primitive concepts, SE the set of secondary expressions representing defined concepts, PS the set of possible statements in the language L 1, an extension of L, TS the set of theoretical statements actually formulated in L 1 about the subject matter, Fxs the relationship "expression x is used to formulate statement s", A ~- s the relationship " s t a t e m e n t s is derived f r o m the set of statements A " and A ~ s the relationship "statement s can be derived from the set of statements ~' h where F __%_(PE t3 SE) x (PS), ~- G ~ (TS) x (TS), ___~ (PS) • (PS). 161 0020-7373/83/020161+ 13503.00/0
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DEFINITION 1. 91 = (L, PE, SE, PS, TS, F, ~-, ~) is an axiomatically structured theory for a subfect matter if and only if 91 satisfies the following 12 axioms. Axiom 1. PS is the set of well formed statements within some language L 1, which is a modification of the underlying language L, effected as follows. (i) Expressions appearing in PE or SE that do not appear in the lexicon of L are added to the lexicon of L; and transformational rules appropriate for these terms are added to the grammar for L. (ii) Transformational rules in the grammar for L for expressions appearing in either PE or SE, and in the lexicon of L, are modified as required to permit new transformational possibilities as appropriate. (iii) Step (ii) is carried out in such a way that nontheoretical uses of these expressions remain unchanged so that every statement expressible in L remains expressible in L 1. (iv) (Vx E P E w SE)(x E PS). That is, primitive and secondary expressions as members of the lexicon of L are not in and of themselves possible statements. Axiom 2. T S _ PS, TS is finite and the elements of TS are distinctly labelled so that different elements have different labels. Axiom 3. P E # 0 and PE is finite. Axiom 4. PE ~ SE = 0. That is, the same expression is not used to represent both a primitive and a defined concept. Axiom 5. T h e r e is a one-to-one function, h, from SE into TS. DEFINITION 2. For all x ESE, h(x) is referred to as the definition of x and is denoted bye. That is, h(x) =~. Axiom 6. (Vx E SE)Fxs That is, secondary expressions are used to formulate their own definitions. Axiom 7. (Vx ~ S E ) 7 (3A E ~ ( T S ) ) A ~- s That is, definitions are not proved. Axiom 8. There is no sequence of elements of SE, x a , . . . , x,(n >-1) such that the following hold: Fx 1J72, Fx2x3~
Fxn.~l. That is, there are no circular definition chains. Axiom 9. There is no sequence of subsets of TS, A I . . . . . An (n - 1) and elements of TS, Sl . . . . . s, such that the following hold:
s~EA2
A1 ~ s2, A2~-s3,
s, E A ,
A,~sl.
Sl E A1
That is, there are no circular proof chains. Axiom 10. (Vs E TS)(VAE ~(TS))(A ~ s ~ A ~ s). That is, if a theoretical statement is proved, it can be proved. Axiom 11. ( V x E S E ) ( V s E P S ) [ F x s ~ ( 3 s l ~ P S ) ( V w ~ P E u S E ) (fwsl~--~w~x & (Fws v FwY)) & {~} ~ s<--~Sl]. That is, the definition of any secondary expression,
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x, is formulated in such a way that any possible statement, s, formulated in terms of x is provably equivalent to another possible statement, sl, that is formulated in terms of expressions other than x, and that appear in s or ~; the equivalence proof using only s (Eliminability). Axiom 12. ( V x E S E ) ( V s E P S ) [ ~ F x s ~ ( V A E ~ ( T S ) ) ( A w { s & (V~slEA)-n FXSl ~ A ~ s)]. That is, the definition of any secondary expression, x, is formulated in such a way that any possible statement, s, not formulated in terms of x and derivable from ~ and a set of theoretical statements, A, no members of which are formulated in terms of x, remains derivable from A only (Non-creativity). DEFINITION 3. The set of definitions, theorems and axioms for an A S T , (i)
Def(91) ={s ~ TS: (3x ~ S E ) s =s
(ii)
Th(91) --{s c TS: ( 3 A ~ ~ ( T S ) ) A ~- s},
(iii)
Ax(')l) = TS ~ [ D E F ( g ) u TH(91)].
We can show in an AST, that SE is finite and that Def (91), Th (91), Ax(91) partition TS. Stelzer & Kingsley (1975) investigate the notion of dependency in an AST. A matrix method that is deterministic is developed that can be used to identify all dependency relationships. It can also be proved that all secondary expressions representing defined concepts can be eliminated from any AST for a subject matter. In essence, axioms 1-12 satisfy our expectations concerning the results that should be forthcoming concerning ASTs. They provide an adequate characterization of the concept on an AST. Below we assume throughout that 91 = (L, PE, SE, PS, TS, F, ~-, ~), 91" = (L*, PE*, SE*, PS*, TS*, F, ~*, ~) and 91"* = (L**, PE**, SE**, PS**, TS**, F, ~**, %) are all ASTs. It is not necessary to differentiate between F, F*, F** and ~, ~*, ~** since the context of use suffices to determine which AST is being referenced. DEFINITION 4. For all expressions t, t* in P E w PE* u S E & SE*, t, t* have the same logical structure if and only if (i)
t, t* are both individual names, or
(ii)
t, t* are both names of sets, or
(iii)
t, t* are both n-place predicates.
Besides comparing the logical structure of primitive and secondary expressions, it is necessary to be able to discuss how primary and secondary expressions function syntactically. Two expressions t, t* are said to function syntactically the same way (or have the same syntactic functions) if and only if for all well formed statements s such that Fts, if every occurrence of t in s is replaced by t*, then the resulting statement is also well formed. Alternatively, it might be said that t and t* are in the same transformational categories (noun phrase, verb phrase, etc.), or that transformational rules applicable to t apply as well to t* and, conversely.
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DEFINITION 5. The set of expressions used to formulate the axioms for an AST. Ax exp (9/) = {t ~ PE u SE: (3s r Ax (9/))Fts}. As will be seen, an analogy results when certain axioms for one subject matter are re-interpreted in another subject matter. To do so means that the primitive and secondary expressions appearing in the axioms selected from one AST are replaced by primitive and secondary expressions from the analogous AST. The replacement of expressions must be done in such a way that every replacement expression has the same logical structure as the expression it replaces. The statement that results from the replacement of expressions must be comparable with selected axioms and theorems in the analogous subject matter in specified ways based on the notion of implication. More precisely, suppose 9/* is intended to be an analogy for 9/. On the one hand we must be able to describe how much of 9/'s structure carries over to 9/*. Three possible categories describing three general cases of relative structural characteristics will be distinguished. On the other hand, we must be able to compare the relative strengths of statements analogous to the axioms of 9/. That is, we must be able to describe how strong (derivationally) 9/'s axioms are in their analogous form. T h r e e possible categories describing relative derivational characteristics will be defined. A nine-fold description of the concept of analogy will result from these definitions. The defintion of the concept of analogy that provides the most general category describing relative structural characteristics will be given first. DEFINITION 6. A n analogy for an A S T 9/ is an A S T 9/* where L = L*, and a pair
of functions f, g satisfying the following four conditions. (i) f i s a one-to-one function from a non-empty subset. A , of A x (9/) into Ax(9/*) u Th(9/*). (ii) g is a one-to-one function from B ={t~PEuSE:
(3s~A)Fts}
intoPE*uSE*.
(iii) For all t in B, t and g(t) have the same logical structure and have the same syntactic function. (iv) For all s in A and for all t in B, Fts +-~Fg(t)f(s). Condition (i) of Definition 6 asserts that a subset of the set of axioms for 9/2[is mapped by f into a subset of the union of the axioms and theorems for 9/*. The subset of Ax(9/) on which f is defined is assumed to be non-empty to ensure that the analogy to 9/* is non-trivial. Thus, f associates some axioms for 9/with unique statements in 9/* that are known to be true of 9/*'s subject matter. T o say that f is one-to-one means that unique axioms in 9 / a r e associated with unique statements in 9/*. It can be shown that the set B in condition (ii) of Definition 6 is a subset of A x exp (9/) Thus, g is a mapping from a subset of Ax exp (9/) into PE* u SE*. The function g associates each element in this subset of Ax exp (9/) with a unique element in PE* u SE* that has the same logical structure [condition (iii), Definition 6]. Finally, condition (iv) asserts that an expression is used to formulate an axiom in A just in case the image under g of the expression is used to formulate the image of the axiom in ~* under f. The notion of analogy can be represented diagramatically as shown in Fig. 1. In Figure 1, 9/ and 9/* are represented by closed spaces. It is important to realize that
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FIG. 1. ~*, f and g provide an analogy for 9.1.
an analogy is a triple consisting of an AST, 9.t*, and the two functions f and g. That is, one AST, 9~*, can serve as an analogy for another, 9~, in various ways depending upon the definitions of f and g. Below, we distinguish three types of analogy. The AST 9~* with two specific functions f, g cannot be more than one type of analogy. But by redefining f and g, while keeping 9~* constant, more than one type of analogy can be generated. Thus, the structure of 9~* alone does not classify uniquely the kind of analogy. When classifying analogies the functions f and g must be taken into consideration also. Below ~ ( f ) , ~ ( g ) and Y~(f), ~ ( g ) denote the domain and range of f and g, respectively. DEFINITION 7. Let ~*, f, g be an analogy for 9~. (i) 9~*, f, g is partial analogy if and only if ~ ( f ) = Ax (9~). (ii) 9~*, f, g is a complete analogy if and only if ~ ( f ) =Ax(9~). (iii) 9~*, f, g, is an exact analogy if and only if 9.1", f, g is a complete analogy and (f) = A x (9~*). That is, in a partial analogy some of the axioms do not find analogous interpretations. In a complete analogy all axioms carry over to the analogous subject matter. In an exact analogy all axioms are reinterpreted as axioms in the analogous domain. Figure 1 illustrates a partial analogy while Figs 2 and 3 illustrate complete and exact analogies, respectively. ~1"
Ax(,tl)
f
y
FIG. 2. ~*, f, g provide a complete analogyfor 9~.
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J. STELZER
f
Axexp('~)
Axexp(~ * }
FIG. 3. 9~*,f, g provide an exact analogy for 9A.
Proposition I (i) If 92*, f, g are a partial analogy for 92, then
~ ( g ) ~ Ax exp (92). (ii) If 92", f, g are a complete analogy for 92, then ~ ( g ) = A x exp (92). (iii) If 92*, f, g are an exact analogy for 92, then ~ ( g ) = A x exp(92)
and
~ ( g ) = A x exp(92*).
(iv) 92", f, g provide an exact analogy for 92 if and only if 92, f - l , g-1 provide an exact analogy for 92*. (v) If 92", f, g provide an exact analogy for 92 and if 92"*, fx, gl provide an exact analogy for 92*, then 92"*, f o f 1 and g o gl provide an exact analogy for 92. A result similar to (iv), above, is not forthcoming for analogy, complete analogy, or partial analogy. While f-~ and g-~ have the appropriate structure (one-to-one) in these cases, there is no guarantee that ~ ( f - i ) c h x (~[:~). Indeed, it is entirely possible for ~ ( f - x ) = ~(f)__-Th (92*). It is assumed here that 92", f and g provide either an analogy, a complete analogy, or a partial analogy for 92. Now consider (v), above, where 92", f, g provide an exact analogy for 92; and 92"*, fx, g~ provide an exact analogy for 92*. Then 92** along with the composite functions f o f~ and g o gl provide an exact analogy for 92. That is, the composite functions f o f~ and g o g l provide the requisite mappings from 92 to 92**. Composites of one-to-one functions are known to be one-to-one and so f o f~ and g o gl satisfy conditions (i), (ii) and (iii) of Definition 6. For condition (iv) we have both Fts ~->Fg(t)f(s)
and Fg(t)f(s) ~->Fgl(g(t))fl(s)).
Thus, Fts +->Fg o gl(t)f ~f l(s).
Finally ~([ oA) = ~(f) = Ax(~)
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and
~(f
oA) = ~ ( A ) = A x (92"*)
and, thus, Definition 7 (iii) is satisfied as required. As with (iv), a result similar to (v) is not forthcoming for analogy, complete analogy, or partial analogy. As before, the reason is that we do not know in these cases that (f) _ Ax (92*). Thus we are not assured that the composite function f o jq is defined. That is, for f o f~ to be defined, it must be the case that
(f) -~ ~ (A). But, we only know that (fl) c_ Ax (92*) and we cannot guarantee that
(f) =_Ax (92*) holds. We can establish the following proposition. Proposition 2 (i) If 92", f, g provide an exact analogy for 92, and 92"*, fl, gl provide a complete analogy for 92", then 92**, f ~ g ~ gl provide a complete analogy for 92. (ii) If 92*, f, g provide an exact analogy for 92, and 92"*, fl, gl provide an analogy (or a partial analogy) for 92", then 92"*, f ~ g ~ provide an analogy (or partial analogy) for 92. The difference between (i) and (ii) in Proposition 2 is quite simple. In (i), every element in Ax(92) is mapped into Ax(92**)uTh (92**) by f ~ In (ii), only a subset of Ax(92) is mapped into this set. That is, in (ii), at best, (fi) ___Ax (92*).
Thus, only those elements, s, in Ax (92) such that
f(s) ~ ~ ( A ) have images in Ax (92**)w Th (92**). Similar remarks apply to Ax exp (92) as well. This completes the discussion of the purely structural characteristics of analogies. Now turn to the problem of comparing the derivational strengths of analogies. Let 92", f, g provide an analogy for 92. Suppose s is both in Ax (92) and in ~ ( f ) . This means that s has an image under f,f(s), in Ax(92*)v0Th (92*). We will refer to f(s) as the analogous interpretation (in 92*) of s. It is of interest to compare all s in ~ ( f ) to their analogous interpretations in 92*. Three types of analogies can be distinguished in this way: proper, approximate, and enveloping. One preliminary definition is required. DEFINITION 8. Let 92", f, g be an analogy for 92. For all s E ~ ( f ) , Subs(s/g(t), t) denotes the result of substituting in statement s, g(t) for every occurrence of every term t in ~(g).
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J. STELZER
From Definition 8 it follows that if ~*, f, g is an analogy for 9~ and if s a ~ ( f ) , then: (i) (ii)
Subs (s/g(t), t) ~ PS* and Subs (Subs (s/g(t), t)/g-l(g(t)), g(t)) -- s.
DEFINITION 9. Let 9~*, f, g be an analogy for 91. (i) 9~*, f, g is a proper analogy if and only if for all s ~ ~ ( f )
PS* ~ Subs (s/g(t), t),~--~f(s). (ii) 9X*, f, g is an approximate analogy if and only if for all s ~ ~ ( f )
PS* ~ Subs (s/g(t), t)-->f(s). (iii) 9.I*, f, g is an enveloping analogy if and only if for all s ~ @ (f)
PS* ~ f(s) ~ Subs (s/g(t), t). In an approximate analogy the images are weaker than the substitution instances and in an enveloping analogy the image statements are stronger. It should be clear that Definitions 7 and 9 are combinable yielding the concepts of: proper partial analogies, proper complete analogies, proper exact analogies, approximate partial analogies, approximate complete analogies, approximate exact analogies, enveloping partial analogies, enveloping complete analogies and enveloping exact analogies. In Proposition 1 (iv) an elementary result was given concerning exact analogies based on properties of inverses of one-to-one functions. Proposition l(v) and Proposition 2(i) and (ii) give results based on composites of functions. Due to the definition given for the concept Subs (s/g(t), t), and due to the fact that the definitions of proper, approximate and enveloping analogies are based on the notion of implication; further results along these lines are forthcoming. That is, based on elementary properties of inverses, composites and implication, reflexive and transitive properties of analogies can be established. The following three tables summarize these results. The first table simply summarizes elementary results that follow immediately from the basic TABLE 1 ASSUME 96*, f, g is the following type analogy for 96
THEN 96", f, g is also the following type analogy for 96.I
exact proper exact approximate exact enveloping exact proper proper partial proper complete proper exact proper proper partial proper complete proper exact
complete proper complete approximate complete enveloping complete approximate approximate partial approximate complete approximate exact enveloping enveloping partial enveloping complete enveloping exact
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ANALOGY AND AXIOMATICS TABLE 2 ASSUME 92", f, g is the following type analogy for 92
THEN 92, f-l, g-1 is the following type analogy for 92*
exact proper exact enveloping exact approximate exact
exact proper exact approximate exact enveloping exact
TABLE 3 ASSUME 92*, f, g is the following type analogy for 92
ASSUME 92**, fl, g~ is the following type analogy for 92*
THEN 92"*, f ~ g ~ gl is the following type analogy for 92
exact exact exact proper exact proper exact proper exact proper exact proper exact proper exact approximate exact approximate exact approximate exact approximate exact approximate exact approximate exact proper exact proper exact proper exact enveloping exact enveloping exact enveloping exact enveloping exact enveloping exact enveloping exact exact proper exact proper exact proper exact approximate exact approximate exact enveloping exact enveloping exact
exact complete partial proper exact proper complete proper partial approximate exact approximate complete approximate partial proper exact proper complete proper partial approximate exact approximate complete approximate partial enveloping exact enveloping complete enveloping partial proper exact proper complete proper partial enveloping exact enveloping complete enveloping partial analogy proper analogy approximate analogy enveloping analogy proper analogy approximate analogy proper analogy enveloping analogy
exact complete partial proper exact proper complete proper partial approximate exact approximate complete approximate partial approximate exact approximate complete approximate partial approximate exact approximate complete approximate partial enveloping exact enveloping complete enveloping partial enveloping exact enveloping complete enveloping partial enveloping exact enveloping complete enveloping partial analogy proper analogy approximate analogy enveloping analogy approximate analogy approximate analogy enveloping analogy enveloping analogy
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definitions. Notice also that the results given in Proposition l(iv) and Proposition 2(i) and (ii) are included for completeness. Table 3 can be interpreted as follows. If an analogy of the type in column 1 is followed by an analogy of the type in column 2, then the resultant (composite) type is the type listed in column 3. For example, the last line means an enveloping exact analogy (from 92 to 92*) followed by a proper partial analogy (92* to 92**) results in an enveloping partial analogy (92 to 92**). Notice that the only exact analogies that appear, appear on the left in this schema. Notice also that in combining analogies in this way, approximate analogies never follow enveloping analogies; nor do enveloping analogies follow approximate analogies. A close look at the definitions of these concepts show that this must be the case due to the implication that may hold between elements. That is, from sl implies s2 (s2 approximates sl) and from s3 implies s2 (s3 envelops s2) we cannot infer any relationship between sl and s3. Similarly, knowing that s2 implies sl (s2 envelops s~) and s2 implies s3 (s3 approximates s2) we can infer nothing about the relationship between sx and sa. It is important to realize that analogies have been characterized solely in terms of axioms and subject-matter-related expressions appearing in axioms. Analogies are independent of definitions and theorems concerning subject matters. This is appropriate as is now suggested. Since all definitions are eliminable from an AST, it can be asserted that given the primitive expressions, all required definitions can be generated. That is, given some type of an analogy, we can proceed to define, within the analogy itself, concepts that are analogous to interesting concepts within the original subject matter. More specifically, g maps a set of expressions from 92 into 92*. Any definition in 92 definable only with expressions in the domain of g will have an analogous definition formulated solely with expressions in the range of g. Thus, having the analogies for (some) axioms and their expressions can result in analogies for other definitions in the subject matter. By way of an example of some of these concepts consider the analogy between elementary electronics and hydraulics. A simple circuit with a battery and resistor, both suitably grounded, is analogous to having a pond with a pump circulating water through a pipe containing a restriction. The restriction may be a reduction from a larger to a smaller pipe in the same direction as the water flow. Voltage is analogous to the potential for water flow as established by the pump: water pressure. Furthermore, current is analogous to water flow in the pipe: water flow. Resistance is analogous to a restriction impeding water flow. Finally, grounding of the ends of the circuit is analogous to having the pump pump water out of the pond through the pipe back into the pond. This situation is depicted graphically in Fig. 4. 92 will be the electronics AST and 92* will be the hydraulics AST. We being to specify 92 and 92* as follows. L = English, PE = {E, I, R, series circuit, parallel circuit}, SE = {total resistance (RT)}, L* = English,
171
ANALOGY AND AXIOMATICS Flow --->
Restriction
f--.-~ 'VVV~A~
FIG. 4. Simple circuit/pond pumping analogy.
P E * = {pr, fl, re, series pipe, parallel pipe}, SE* = {total resistance (ReT)}. T h e function g will be: g(E) = pr (pressure), g(I) = fl (flow), g (R) = re (resistance), g(RT) = reT (total resistance), g(series circuit) = series pipe, g (parallel circuit) = parallel pipe. TS in electronics contains four statements. sl: E = I x R , s2: In a series circuit Ra- = R1 + R2, s3: In a parallel circuit RT is p r o p o r t i o n a l to R1 x R2 RI• s4: In a parallel circuit RT = R1 + R----~" F u r t h e r m o r e , TS is partitioned by:
Ax(9~)={sl, s2, s3} and In hydraulics TS* has four statements. s*:pr=flxre, s* : In a series pipe reT = re1 + re2, s* : In a parallel pipe reT = re~ x re2 re1 9 re2'
Th(9.I)={s4}.
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J. STELZER s4* : In a parallel pipe reT is proportional to re1 x re2, Ax(92*)=
* s*}, {sl,* s~,
Th(92*) = {s* }. Successive analogy types are generated by varying f. (1) Partial analogy: no image under f for s3:
/(Sl)
= sL
f(s~) = s L
(2) Complete analogy. Add to f above f(s3) = s L
This is not an exact analogy since $4 ~ A x (92").
(3) Exact analogy: Change the image of s3: f(s3) = s L
(4) Proper analogy. Both (1) and (2), above, are proper analogies: (1) is a proper partial analogy and (2) is a proper complete analogy; (3) is not a proper analogy. Change Ax (92*) and define f as follows to get a proper exact analogy: Ax(92*) ={s*, s*2, s 4*lJ', f(Sl) = S*,
f(s2) = S~,
f(s3) = S*.
(5) Approximate analogy. Define Ax (92), Ax (92*) and f as follows: Ax (92) = {sl, s2, $4}, Ax(92*) = { s * , s*2, s 3.1I, f ( s l ) = S~,
f ( s 4 ) = 84~.
This yields an approximate partial analogy. Extending f to include f(s~) = s*
results in an approximate complete analogy. Change Ax (92*) so that Ax (92") = (s*, s*, s*} and leave f as defined to get an approximate exact analogy. (6) Enveloping analogy. Restore Ax (92) so that A x (92) = {sl, sz, s3}.
Define f as follows: f(Sl) = s*,
f(s~) = s*,
to get an enveloping partial analogy. To make this enveloping analogy complete, add: f(s~) = s*.
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Restore Ax (~*) so that Ax (gA*)={s*, s*2, s .13s and the analogy becomes an enveloping exact analogy. Research performed under contract (DAHC19-74-C-0066) to the U.S. Army Research Institute for the Behavioral and Social Sciences; Organization and Systems Laboratory, Dr Joseph Zeidner, Director; while the author was a Senior Scientist at the Human Resources Research Organization, 1974/75. The author thanks Dr Leon Nawrocki and Dr Joe Ward of the former Institute, and Dr Robert Seidel of the latter for their support.
References GAINES, B. R. & SHAW, M. L. G. (1982). Analysing analogy. In TRAPPL, R., RICCIARDI, L. & PASK, G., Eds, Progress in Cybernetics and Systems Research, vol. IX, pp. 379-386. Washington, D.C.: Hemisphere Publishing Company. STELZER, J. & KINGSLEY, E. (1975) Axiomatics as a paradigm in structuring subject matter. Instructional Science, 3, 383-450.