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@Comment{ @Part<Examples, root "NAIVE.MSS[rdg,dbl]"> }

@Appendix<Smattering of Examples>
@Label<Examples>
This appendix elaborates the loose outline of categories of
analogies, given in Section @Ref<ExampleOverview>.
Each of the examples shown below is (for our purposes)
an instance of an analogy/metaphor/similarity.
(This list includes various linguistic tropes, such as metaphor and simile,
along with other forms of analogy.  
This reflects our view that all of these uses are applications of the 
same general analogizing process.)
Each particular clusters is designed to demonstrate some general application,
which is mentioned immediately before that set of examples.
Note that these groups are NOT designed to be disjoint from one another
-- many entries could easily fit in several categories.
Nor are they orthogonal --
some cases are in direct competition with others.
A more significant classification scheme,
which does in fact partition the space of analogies along certain defined axes,
was shown and used in the preceding Chapter @Ref<Categories>.)

@BEGIN{Enumerate}
@BEGIN<Multiple>@Tag<Comparison>
@B(Used for Comparisons)@*
@BEGIN(DESC1)

@i{Implicit question:} "Is A like B?" or "How is A like B?"

@i{Purpose:}@\The analogy establishes a connection between a pair of models,
designed to explain some fact(s) about B.

@i{Variables:}@\How explicit the comparison is;@*
the "size" of the compared model;@*
the number of features@Foot{
In this report the term "feature" refers to any proposition in which an
analogue participates. 
This is different from the more refined sense used 
by @Cite<Gentner> and @Cite<Hesse> --
where it refers only to unary predicates.}
mapped over.

@i{Notes:} @\The analogy is only used once in each case,
to establish a particular connection. 
After the analogizing link has been made,
and the analogy has been understood and exploited,
the analogy itself is usually thrown away.

@i{Examples:}
@END(DESC1)

@BEGIN(ITEM1)

@BEGIN(Multiple)
@i<Explicit Comparison>@*
@BEGIN(ITEM1)
How is electricity like fluid flow?@*
(A: They both satisfy the same equations.)

Should Law-Case#47 be a precedent for the current case?

Is this current case more like Case X or Case Y?@*
(@i{e.g.}, in terms of whether defendant should be acquitted or convicted.)

In what way is the Quaker song "Simple Gifts" similar to
Copland's "Appalachian Spring"?

Was Brittan's `A Young Person's Guide to the Orchestra' 
derived from Purcel's `Abdelezar'?

Which of Shakespeare's plays is most like "West Side Story"?
@END(ITEM1)
@END(Multiple)

@BEGIN<Multiple>
@i<Simile>@*
Here a word like "like" is used to indicate that this is a comparison.
(Omitting this "like" term leads to similarity metaphors,
which are mentioned in Example @Ref<SimMet>.
There the sentence omits not only the basis of the comparison,
but also the fact it is only a comparison.  
We also leave to later the idiom case, shown in Example @Ref(Idiom).)
@BEGIN(ITEM1)
John is like a bird.

Computers behave like people.

"West Side Story" is like "Romeo and Juliet".

"Love's Labor's Lost" is like "King Lear".
@END(ITEM1)
@END(Multiple)

@i(Only one feature mapped over)
@BEGIN(ITEM1)

"Learning at CalTech is like trying to sip water from a fire hydrant."@*
(in that a lot is forced into the student, under great pressure)

Consider almost any instance of a "that reminds me of" situation
-- where connection may be obscure to everyone (possibly) excluding the speaker.
In general the topic of the current conversation and the digression share
some single common feature.
For example, switching the conversation from Oscar Wilde to the
ancient Greek society.  
(Here, the key observation is that both were gay.)
@END(ITEM1)

@i<equation>@*
In this degenerate case of analogy, 
all the information needed for the comparison is made explicit.
@BEGIN(ITEM1)
John ate as many sun-flower seeds on June 24 as Polly parrot ate that day.

The nucleus is in the center of an atom, just as the sun is at the center of
the solar system.@*
(Notice this connection can be expanded, to cover the fact that electrons
are like planets, @i{etc.})
@END(ITEM1)

@END(ITEM1)
@END(Multiple)

@BEGIN<Multiple>@Tag<Prediction>
@B<Used for Prediction>@*
@BEGIN<DESC1>

@i{Implicit question:} Given that A is like B, and P(A), is P'(B) true?

@i{Purpose:}@\The analogy is used to conjecture properties of one object,
based on facts about another.

@i{Variables:}@\The analogues can be described in great detail,
or quite sparsely
(including just those few facts which need to be carried over);@*
the size of the models (varying from a single situation or example,
to a complete known, general model).

@i{Notes:}@\In these situations it is often useful to keep the analogy,
as it may be a source of future conjectures/facts about B;@*
in general the new assertions about B will be merely plausible conjectures,
rather than guaranteed true facts;@*
in some (rare) cases the analogy may work both ways -- one might notice
new facts about A based on facts about B.
(@Cite<Interfield> makes this point.)

@i{Examples:}
@END(DESC1)

@BEGENUM1<>

@i(Large and Comprehensive)@*
@TAG(MM)
Here the models involved are large.
Such analogies can be used to define an entire scholarly pursuit.
@BEGIN(ITEM1)
The atom is like a miniature solar system.@*
@i{I.e.} electrons revolve about a heavy nucleus, like the planets about the sun.

Electricity is like water flow.@*
@i{I.e.} batteries can be compared to dammed resouviors.

Quantum mechanical particles behave as elements in an abstract group.@*
@i{I.e.} One can predict the energy for new particles based on certain relations
to known members of this QM family.

The genetics concept of a "gene" shares many properties with the
biochemical concept of chromosomes.@*
@i{I.e.} they come in pairs, one from each parent.

The `Solar metaphor' theory used to explain the creation of myths.@Foot{
This theory held that people would use anecdotes to describe the movement
of the sun in the heavens --
phrasing its movements in terms of the
life story of a particular god or hero.
This view was popular @i(ca.) 1870-1900.  See @Cite<SolarMetaphor>.}@*
@i{I.e.} Hercules would peak at the "noon" of his life, then descend.

Physiologically, people are like electronic circuits.@*
@i{I.e.} the many loops in our endocrin system would behave much like
feedback circuit.
(In particular, expect a certain type of problem to arise if ever the loop
becomes open.)

Cognitively, people are like computers.@*
@i{I.e.}, we have both short and long term memories, (with different properties,)
interrupts, etc. (see the list of borrowed terms, shown below in
Example @Ref{Borrow}.)
@END(ITEM1)

@i(Moderate in Size - Interfield)@*
These comparisons help to refine (but not define) a domain.
Here the "other" analogue is outside the current domain.
@BEGIN(ITEM1)

"How is a Knowledge Representation language like a piano?"@*
@i{I.e.} both are used as instruments, need to be tuned, @i{etc}.

"Metaphor is like a Solar Eclipse"@*
(see @Cite<Ortony1>, p169)

Music is like poetry.@*
@i{I.e.} both are art forms, designed for esthetics, and based on symmetry, ...

Music (as sequences) are like math sequences.@*
@i{I.e.} there is a regularity in both -- leading to a predictable behaviour, ...

Doing research is like climbing to the top of a mountain.@*
@i{I.e} take any path which leads to the top, attending only to things on the
"critical path."
Hence everything to the side, or behind you, should be ignored.

Doing research is like constructing a building.@*
@i{I.e.} each and every layer from the foundation up should be solid.
This means it may be worth the time now
to fix known flaws on any underlayer, as they will only be harder to
reach and correct later.

Consider three models for a text editor:@Foot{
This example is taken directly from @Cite<R&N>.}
@BEGIN(ITEM1)
Secretary@*
@Comment{ Pro: either will follow your instructions in
performing the tasks you prescribe.
(This includes "taking dictation", etc.).@*
Con: The "still in append bug" -- an editor can't distinguish commands from text.}

card file@*
@Comment{(That is, the text file being editted is like a file of cards.)@*
Pro: Permits insertion of text into arbitrary places; multiple files; ...@*
Con: There can be other sub-units (not just cards);
nothing like a global Substitute, ...}

tape recorder@*
@Comment{(That is, the file is like the tape, on the recorder.)@*
Pro: Handles overwriting.  Insertions are like splicing, multiple tapes, ...@*
Con: Still no global finds/substitute commands, ...}
@END(ITEM1)

HPP Executive Council is like a council of barons.@*
@i{I.e.} each has absolute say within his own project,
except in cases when such decisions influence the rest of the empire.

@END(ITEM1)

@i(Moderate in Size - Intrafield)@*
Here we compare two objects from the same domain.
@BEGIN(ITEM1)
A circle is like a sphere.@*
@i{I.e.} expect @G(p) to be involved in the formula for surface area.@*
(This is just one instance of the 2D to 3D system of analogies.)

Recursion is like iteration.@*
@i{I.e.} you need to worry about "boundary conditions" in both cases.
@Comment{Both reduce a problem to a simple case}

Abstraction is like simplification/analogy...@*
@i{I.e.} in all cases one has to consider a simpler form of the problem...
@END(ITEM1)

@i<Instance to Instance>@*
@TAG(II)
Here the "models" are at the level of a single instance.
(This may be considered the standard case of analogy.
In many cases the novel object is a specific situation 
(usually the current focus).
When the vehicle (@i<i.e.> the already known analogue) is also specific,
this is considered "learning from example".)
@BEGIN(ITEM1)
The EMACS command M-F is just like C-F,
except it applies to words rather than single characters.

Patient#75 today is similar to Patient#75 a week ago, but ...

LegalCase#92 is like LegalCase#53, except ...

This "group" proof is just like that "ring" one.@*
(@i<c.f.> @Cite<Kling>.)

Knowing that Fred is like John in that both are people,
we can assume that Fred has two arms because John does.
@END(ITEM1)
@ENDENUM1<>
@END(Multiple)

@BEGIN<Multiple>@Tag<FindOrGen>
@B<Find or Produce an Analogue>@*
@BEGIN(DESC1)

@i{Implicit question:} Find or create a B which is like A in some manner.

@i{Purpose:}@\Construction of a new object with certain desired properties
(using the analogical mapping as an intentional definition).

@i{Variables:}@\Whether the new analogue already exists or must be "CONSed up";@*
how constrained the "answer" is 
(whether there is a single possible answer, many allowed answers, or none);@*
the new analogue may be simply acceptable, as opposed to optimal;@*
the nature of the search space from which to find a suitable B,
(in particular how constrained it is)@*

@i{Notes:}@\The analogy itself may then be further refined,
based on the analogue found.

@i{Examples:}
@END(DESC1)

@BEGENUM1<>
@BEGIN(Multiple)@Tag(ProblemXForm)
@i<Problem Transformation>@*
A major application of this is problem transformation:
Given problem T, find a similar, (but easier to solve) problem S.
(That is the analogue finding part.
Solving T involves solving S,
and then mapping the result back to the T situation.)@*
We may consider solving a problem as travelling from a starting position, T,
to a destination, the solution, @G(t).
Problem transformation suggests taking the circuitous route
from T up to S, over to @G(s), 
(@i{i.e.}, solve the problem in that space,)
and then map down to the answer @G(t).
Picturally,@Foot{
The author found that diagrams like the one above can confuse as well as clarify.
Similar pictures are often used to describe a
slightly different problem solving strategy,
in which one solves a considerably different problem first,
then uses this result to (indirectly) help solve the original problem.
For example, @Cite[FOL] talks about "hopping to the meta-level"
to decide how to solve a problem.  
It is important to realize that, unlike the problem transformation case,
this new S problem has relatively little to do with the original
T problem --
it is here in a different language, and involves totally different types of
objects.}
@BEGIN(Example)
S     @G(s)
@K<SmallCircle>  @K<DoubleRightArrow> @K<SmallCircle>
@K<DoubleUpArrow>     @K<DoubleDownArrow>
@K<SmallCircle>     @K<SmallCircle>
T     @G(t)
@END(Example)
Here we are considering the task of finding that analogous problem
-- finding the S from T.
(This can be viewed as finding a proportional analogies:
When all is said and done, we have 
@QUOTATION[T:S :: @G<t>:@G<s>).]

@BEGIN(ITEM1)
The goal, @G(t), is a computer program which performs task T.
The approach is to first find a similar task, S,
for which a program, @G(s) has already been written.
That code is then modified appropriately, in the @G(s) to @G(t) step.
(See @Cite<Evolution> or @Cite<M&Ua>.)
@Cite<RBrown> addresses the same problem,
but uses an abstraction of the program (called its plan),
rather than the code itself, for the transformation.

Consider the following problem from @Cite<Polya1>:@*
Given two points lying in one of the half planes defined by a line,
find the minimal path joining these point, subject to the constraint
that that path must go touch the line.@*
(Solution: Rather than solve the problem given, T,
solve instead the S problem,
which is derived from T by
reflecting one of the points about the line, and then asking for the shortest
path between the points which intersects that line.
The solution to this problem (@G<s>) is trivial 
-- it is just the straight line connecting those points.
This solution, @G(T), is readily mapped back to the original problem,
and implies the familiar "angle of incidence equals angle of reflection" rule.)

The goal, @G(t), is to show that some task, T, is NP-complete.
The standard approach is to find some similar task, S, 
which is known to be NP-Complete.
(Hence @G(s), the proof that S in NP-Complete, is trivial.)
The T @K{DoubleRightArrow} S
and @G(s) @K{DoubleRightArrow} @G(t) maps involve demonstrating that
T can be simulated in S --
@i<i.e.>, that T can be transformed into S in polynomial time, and vice versa.

@END(ITEM1)
@END(Multiple)

@BEGIN(Multiple)@i{Other Cases}@*
@BEGIN(ITEM1)
Has anyone previously designed an IC chip with a similar function to
these specifications?

This laboratory technique successfully cloned rat-insulin.
Can it be modified to clone human-insulin?
Can variants of it be used for varied molecular genetic experiments?

Write a composition of a particular musical theme.
@END(ITEM1)
@END(Multiple)
@ENDENUM1<>
@END(Multiple)

@BEGIN(Multiple)@Tag(ProblemRestate)
@B<Problem Restatement>@*
@BEGIN(DESC1)

@i{Implicit question:} Find an alternative representation of problem A 
(which renders the problem easier to solve).@Foot{
It has been claimed that many of the significant advances science has made
have been via restructuring of known problems into tractable forms
(? Polya, Kuhn ?).}

@i{Purpose:}@\To solve problem A.@*
(One might also derive new insights into
a large range of similar problems as a side effect.)

@i{Variables:}@\This new language may be totally 
different from the original descriptive
language, or may just involve small embellishments,@*
the transformation may be information preserving, or may abstract
out certain facts.¬

@i{Notes:}@\This whole class may be considered a subclass of 
Example @Ref(FindOrGen) above --
in that we are trying to find an analogue, along some dimension, to a given
problem.
It seems sufficiently important to deserve its own category though.@*@*
This is the critical step in problem reformulation.@*@*
@i(N.B.) we are still solving the same problem, out in the real world --
but now we have a better, more apt, handle on that situation.
(This is NOT true in the previous problem transformation case above,
Example @Ref(ProblemXForm)
-- there, a different problem was solved.)@*@*
In the "correct" representation,
it is easy to see the differences and similarities connecting the two analogues.
Ideally, the differences can be seen as simply substituting one value of
a parameter for another.@*@*
Many of these points are explicated in Appendix @Ref<Reform>.

@i{Examples:}
@END(DESC1)

@BEGIN(ITEM1)
@i(Fixed Algorithmic Transformations)
@BEGIN(ITEM1)
Transposing a musical piece into a different key.

Switching from a rectangular to a spherical coordinate system.

Writing a FORTRAN program in PASCAL, or LISP.

Use a bit vector representation for a set of elements, rather than a linked list.

Our visual system does a lot of work transforming the initial retinal image
into higher level, useful information.
@i{E.g.}, we readily recognize a friend,
independent of his current position and orientation relative to our eyes 
-- the received retinal images is
automatically normalized (or canonicalized).@Foot{
The work of @Cite<HubelWeisel> shows that much of this is learned --
as opposed to hardwired from birth.}
@END(ITEM1)

@i<Heuristic Methods>
@BEGIN(ITEM1)

@BEGIN(Multiple)
"Visualizing" a problem --
@i{e.g.}, usings drawings or charts rather than equations.@Foot{
This was discussed in @Cite<Simon-Repn> -- where it was claimed that experts
used apt representations, which let the answer be "read off" the diagram.}
Examples -
@BEGIN(ITEM1)
the 2D reformulation of problem transformation case,
shown in Example @Ref<ProblemXForm> above.

the standard representation of a fraction as a piece of a pie.@*
(@Cite<R&N> discuss how useful this representation is
when teaching students how to add and subtract fractions.)
@END(ITEM1)
@END(Multiple)

Playing a musical piece with a different instrument.@*
(This requires understanding nuances of each instrument's style,
partially discussed in Example @Ref<Style>.)

Transforming a musical work to a different style;
@i{e.g.}, baroque rather than classical.@*
(Again, this issues discussed in Example @Ref<Style> are relevant.)

Any of the transformations discussed in 
@Cite<Amarel>'s  Missionary & Cannibal reformulations.

Adding an object which was NOT stated in the problem itself.
(@i{E.g.}, adding a
new point during a geometric proof, or dropping a perpendicular.)
This issue was addressed by @Cite<RBrown>,
where this ability was considered the critical step required to solve many problem;
its absense could severely limit a program's performance.)

The use of additional slots or relations to describe some object.
(The value in the new slots may be defned in terms of the values of existing slots.)

This is closely tied with the change of representation issues discussed in 
@Cite<ArchMRS>.
The task defined there is to find a (possibly new) representation for
a given problem,
in which a certain set of queries can be solved efficiently.
(Much of the Analysis of Algorithms work has a similar theme.
See @Cite<AHU>.)

@END(ITEM1)
@END(ITEM1)
@END(Multiple)

@BEGIN(Multiple)@Tag(Literal)
@B<Literary uses to conjure images>@*
usually by exploiting cute, usually serendipitous coincidences.@*
@BEGIN(DESC1)

@i{Implicit question:} The goal is to describe some feature, P(x), of a certain
situation, B.
Here it is usually achieved by finding a commonly known object, A,
which has a pronounced salient feature corresponding to this P.

@i{Purpose:}@\Suggesting that P(B) holds.  (Using the notation defined above.)

@i{Variables:}@\Whether the analogy must be constructed or is already present
in the minds of the viewer;@*
how strained the comparison is.

@i{Notes:}@\This case is very similar to the 
Comparison case, Example @Ref(Comparison) above.
The major difference is the explicitness of the comparision 
(here it usually goes unmentioned)
and its "unlikeliness" 
(as this usually joins terms from quite different fields.)@*@*
This can be used for both similarity and proportional metaphors.@*@*
The salient features of the vehicle must be fairly obvious.
This makes the transference straightforward.
(Stated another way,
if the particular property to be mapped over is not obvious,
the effectiveness of this metaphor will be lost.
(This point is also discussed in @Cite<Gentner>, p.45-47.)

@i{Examples:}
@END(DESC1)

@BEGENUM1<>

@BEGIN(Multiple)
@Tag<SimMet>
@i<Similarity Metaphor>@*
@BEGIN(ITEM1)
People are birds.

She's a packrat.

He was dynamite!
@END(ITEM1)
@END(Multiple)

@BEGIN(Multiple)
@Tag(Idiom)
@i<Idiom>@*
Some "universally known" fact, associated with the idiom,
is transfered to the referent.
(Again, only one feature is mapped over.)
@BEGIN(ITEM1)
"... his Achilles tendon ..."@*
@i{i.e.}, that is his single penetrable part.

"... hanging like a sword of Damocles ..."@*
@i{i.e.}, temporarily safe, but in a very tense situation.

"... like a fiddler on the roof..."@*
@i{i.e.}, this considers day to day existence as a delicate operation,
in precarious situation.
@END(ITEM1)
@END(Multiple)

@BEGIN(Multiple)
@Tag(LitMet)
@i<Literary Metaphor>@*
Each of these cases will "compose" an analogy in the viewer's mind as it is read.
(The prior cases all exploited an already known analogue,
whose salient features were already well understood.)

@BEGIN(ITEM1)
"Now is the winter of our discontent;@*
Made bright by the sun of Bolingbrook" - or something like that
@Cite<Shakespeare>, @i(Henry II?), ? ?.

"Oh, what a rogue and peasant slave am I"@*
(@i(Hamlet), ? ?)

"It is the east, and Juliet is the sun"@*
(@i(Romeo and Juliet), ? ?)
@END(ITEM1)
@END(Multiple)

@ENDENUM1<>
@END(Multiple)

@BEGIN(Multiple)@Tag(Proportion)
@B<Proportional Metaphors>@*
@BEGIN(DESC1)

@i{Implicit question:} Find @i{y} which is to @i{B} as @i{x} is to @i{A}.

@i{Purpose:}@\(Rapid) transference of facts/assertion to @i{y} based on
either @i{x} itself or its role, with respect to @i{A}.

@i{Variables:}@\How constrained the possible values of @i{y} are,
(in simple, artificial cases, there are only a few known values allowed for @i{y}.
In general, there will be no additional constraints imposed on that new object.)@*
How explicit the @i{A} term is in the problem statement;@*
whether the speaker realizes he is using a metaphor;@*
whether @i{x} is relevant, or its role with respect to @i{A}.@*

@i{Notes:}@\This can be used either to specify an object, @i{y},
or to communicate some additional properties of this @i{y}.
There can be a large coherent collection of terms borrowed over 
-- see especially the Example @Ref(Lak) case below.

@i{Examples:}
@END(DESC1)

@BEGENUM1<>
@BEGIN(Multiple)
@Tag<ExpPropor>
@i<Explicit question>
@BEGIN(ITEM1)
"a ewe's calf"@*
@i{I.e.} find @i{?} such that
@i{Cow : Calf :: Ewe : ?}@*
(note that cow was implicit)@*
@i{?} = Lamb

"the phonemes of music"@*
@i{I.e.} find @i{?} such that  
@i{Language : Phoneme :: Music : ?}@*
(note that Language was implicit)@*
@i{?} = Interval, Timbre, or ...

Who is the first lady of England?@*
@i{I.e.}, find @i{?} such that
@i{US : First-lady :: England : ?}@*
(note that US was implicit)@*
@i{?} = Lady Diane, Mr Thatcher, ...

<Any of the geometric puzzles @Cite<Evans> examined.>@*
Here everything is explicit, and the range of accepted answers is limited.

@END(ITEM1)
@END(Multiple)

@i<Extending a term from its "native" domain to a foreign one.>@*
@Tag(Borrow)
Here a familiar term is used in an inappropriate context;
and its "metaphoric" meaning is extended (loosened)
by figuring how some facts about that known object 
could be transfered into this other field.
@BEGIN(ITEM1)
Consider "@i(tree)" --
Originally a botanical term, it is now used to refer to almost any
instance of a hierarchy.
There are family trees (Geneology),
language trees (Linguistics), 
the data structure "tree" (Computer Science)
[Notice it often brings along the terms @i(root, leaves, branchiness).
However it consistently leaves behind 
its subspecies, color and bark.]

Terms like @i{Anatomy, Physiology} and @i{Diagnosis}
once applied only to the study of people.
They are now been applied to arbitrary physical systems (such as computers) as well.

@i(Channels, bugs, swapped out, processing) went the other way 
-- originally describing only a particular device (a computer),
they can now be used to describe people.

Notice that almost any use of scare quotes means the embedded term is really
metaphoric.  Consider `@i("native")' above.
@END(ITEM1)

@i<Antiquation>@*
@Tag(Anteq)
Many terms are no longer even regarded as a metaphor 
-- @i{i.e.}, these meanings of the terms have been totally assimilated into the
language.
@BEGIN(ITEM1)
the @i{leg} of a chair

the @i{head} of a pin
@END(ITEM1)

@BEGIN(Multiple)
@Tag(Lak)
@i<Sub-conscious use of metaphors>@*
In the above "standard" above case of borrowed terms,
(Example @Ref(Borrow),)
the user is still aware that this usage is metaphoric.
Here, like the Example @Ref(Anteq) case, 
the speaker does not even realize that he is using a metaphor.
Other than that, these cases are quite similar to the Example @Ref(Borrow) case.
In particular,
such systems of metaphors are largely
consistent, systematic, and coherent.
(This observation was taken from @Cite[Lakoff].
See Note @Ref<Lakoffette> in Appendix @Ref<Misc>.)

@BEGIN(ITEM1)
"he was feeling down"

"time is money"

"argument is war"

"ideas are playthings"
@END(ITEM1)
@END(Multiple)

@BEGIN(Multiple)
@i(Mathematics)@*
Mathematicians often extend familiar functions to apply to new domains:
The @K{PlusSign} operator was originally meaningful only over reals,
but now this same symbol,
and much of its associated semantics 
(given below,)
have been borrowed by other (mathematical sub)fields 
-- such as fields, matrices, transfinite ordinals, sets, @i{et cetera}.
The times operator, @K{Times}, has been similarly extended.
There is an underlying unity to all uses of a given symbol.
For example
in all cases @K{PlusSign} retains its traditional properties:
its arity is 2,
it is commutative and associative,
there is a zero element (another metaphor),
domain elements have a unique inverse,
@K{PlusSign} distributes over the multiplication operation 
(when multiplication is defined),
and it is often used to define that multiplication operation.
If there is a binary non-commutative operator, it is usually @K{Times} 
-- consider non-abelian groups, matrices, or cross product.

<<Are there other examples of non-linguistic applications?>>
@END(Multiple)

@ENDENUM1<>
@END(Multiple)

@BEGIN(Multiple)@Tag(Nary)
@B<Familial resemblance>@*
@BEGIN(DESC1)

@i{Implicit Question}: Given a set {A@-<i>}, find how they are all similar.

@i{Purpose:}@\This is used as an imprecise, but widely-used classification
method.

@i{Variables:}@\Presence of "negative examples" to curtail size of commonality,@*
existence of single shared common feature.

@i{Notes:}@\This is a form of "rough classification".@*
There need not be a single pairwise commonality -- see @Cite<Wittgenstein>'s
description of familial resemblances.@*
People seem to use this categorizing scheme very often -- but often have
a very difficult time justifying it.@*
It is also true that some categorizing is considerably
simpler than others.
For example, Bach's work seem quite easy for anyone 
(even musical novices like the author) to discern;
whereas few other composers do have so distinctive a "signature".
(The same might be said for larger categories 
-- @i{e.g.}, in the Baroque style.)@Foot{
As a digression, one might ask why this is true 
-- why are some artists easier to recognize that others?
Perhaps identifiability comes from some distinguishing mark,
or from being a larger delta from the others?@*
<<Other thoughts?>>}

@i{Examples:}
@END(DESC1)

@BEGENUM1<>
@BEGIN(Multiple)
@Tag(ExplainNary)
@i<(Justification of a) Classification>@*
@BEGIN(ITEM1)
Solve puzzle #N (say 12) from @Cite<Bongard>.

What do all Bach works have in common?

What do all @Cite[Bongard] problems have in common?

Consider different recordings of a given musical piece 
-- by various conductors with different orchestra,
performed with different instances of a given set of instruments,
performed with different sets of instruments,
with different styles...
Why do we feel they are all the same piece?

Explain why the various senses of "game" are denoted with the same word.
(Likewise the different senses of "analogy" -- see Section @Ref(Senses).)

Retinal images of the same object.@*
Here we regard all (reasonable, non-degenerate) retinal images
corresponding to the same 3D object as analogous to one another.
(Of course we are not conscious of this process, having
rather powerful, parallel hardware to do much of the work.)@Foot{
The moral here is that in the correct representation,
analogous things will be be "seen" as being identical. 
(See Appendix @Ref<Reform>.)
Indeed neurophysiologists have (seriously) talked about a "Fred" neuron,
which is triggered by the sight of your friend Fred.}

What do all character "a"s have in common?@*
(@i{C.f.} preface to @Cite<Inversions>.)

Is there any common element to this collection of
(alledged) examples of familial resemblances?
@END(ITEM1)
@END(Multiple)

@BEGIN(Multiple)
@Tag(Style)
@i<Issues of Style>
@BEGIN(ITEM1)
Which of the following contemporary interpretations of Shakespeare's "King Lear"
is closest to what the original Elizabethian audience would have seen?

Generate the correct 16@+{th} character ("p") for an alphabet whose first five
entries are printed as "@u{abcde}".
(Again @i{c.f.} preface to @Cite<Inversions>.)

Write a musical piece on a particular theme in a Mozart-ish style.@*
(Or in a Renaissance style, or for a classical violinist.)
@END(ITEM1)
@END(Multiple)

@BEGIN(Multiple)
@Tag(FuzzyClass)
@i<(Pseudo-)Class Membership>@*
Given a class {A@-<i>}, should a new object, B,
be considered a member of this (fuzzily-defined) group?
@BEGIN(ITEM1)
Was this work really composed by Bach?

Is this an authentic Rodin sculpture?

Did Shakespeare or Bacon really write that play?
@END(ITEM1)
@END(Multiple)
@ENDENUM1<>
@END(Multiple)

@END{Enumerate}