COMMENT ⊗   VALID 00010 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00003 00002	@Make[Report]
C00007 00003	@Chapter[What does it mean to "Understand" an Analogy]
C00009 00004	@Section["Analogical" Facts]
C00013 00005	@INCLUDE[NEW.MSS]
C00015 00006	@SubSection["New Analogical" Facts]
C00022 00007	@SubSection[Final Comments ]
C00026 00008	@Chapter[Notes]
C00027 00009	@BEGIN[Multiple]@Tag[Proofs]
C00046 00010	@BEGIN[Comment]	Outtakes
C00050 ENDMK
C⊗;
@Make[Report]
@Device[DOVER]
@LIBRARYFILE<SPECIALCHARACTERS>

@Comment< @Case{Device,	FILE "@Style(Endnotes)",
		PAGEDFILE "@Style(Endnotes)"} >

@Modify[Verbatim, Break Before]
@Modify[Description, Spread 0, Spacing 1.3, LeftMargin +10, Indent -10]
@Modify[Quotation, Indent 0]
@Modify[Itemize, Referenced <@#>]
@Modify[Enumerate, Referenced <@#@;#@:.@1@,@#.@a@,@#.@i>]
@Modify[Appendix, Numbered <@A.>, Referenced <@A>]
@Modify[Equation, FaceCode T]
@Comment{ Brian said this would work... Ha!
 @Modify[NoteCounter, Referenced <@#>] }
@Style[References=STD Alphabetic]
@DEFINE[Aside=NoteStyle, LeftMargin -16, Indent 0]
@DEFINE[SubAside=Quotation,Font SmallBodyFont,FaceCode I,Spacing 1,Spread 0.5]
@DEFINE[Subsubsection,Use HdX,Font TitleFont3,FaceCode I,Above .3inch,Centered]
@DEFINE[ENUM0=ENUMERATE, NumberFrom 0, SPREAD=0, Above=0.3 line, Below=0.1 line]
@DEFINE[ENUM1=ENUMERATE, SPREAD=0.1, Spacing=1, Above=0.3 line, Below=0.1 line]
@TEXTFORM[BEGENUM1 = 
	"@BEGIN<ENUMERATE, SPREAD=0.1, Spacing=1, Above=0.3 line, Below=0.1 line>"]
@TEXTFORM[ENDENUM1 = 
	"@END<ENUMERATE>"]
@DEFINE[ITEM1=Itemize, Spread=0.1, Spacing=1, Above=0.3 line, Below=0.1 line]
@DEFINE[DESC1=Description, Spread=0, Spacing=1, Above=0.3 line, Below=0.1 line]
@DEFINE[TEXT1 = TEXT, USE NoteStyle]
@Define[NoIndent,Continue Forced,Break Around]

@EQUATE[YY=R]

@Counter[EquationCounter, Numbered <(Facts @1)>, Referenced <(Facts @1)>,
IncrementedBy tag, INIT 0]

@Comment{ Now to keep from Breaking in front of chapters. }
@DEFINE(HdChap,Use HD1,PageBreak Off,Above .6inch)
@Counter(Chapter,TitleEnv HdChap,
	ContentsEnv tc1,Numbered [@1.],IncrementedBy Use,Referenced [@1],Announced) 
@Comment{How annoying!   Without this, the section numbers don't get reset!
	Note this is exactly what's in REPORT.MAK[scr,sys]. }
@Counter(Section,Within Chapter,TitleEnv HD2,ContentsEnv tc2,
	  Numbered [@#@:.@1],Referenced [@#@:.@1],IncrementedBy Use,Announced)

@Use(Bibliography =  "GENL.BIB[RDG,DBL]")
@Use(Bibliography =  "REPN.BIB[RDG,DBL]")
@Use(Bibliography = "META4.BIB[RDG,DBL]")


@Define[F1,FaceCode B,TabExport] 
@COMMENT<
@Specialfont(F1="ComputerModern10BI")
   @Define[Fancy,Font = F1,FaceCode B,TabExport] >
@Chapter[What does it mean to "Understand" an Analogy]

Before discussing how to communicate an analogy,
let us consider what it means to "understand" an analogy --
that is, how will one's corpus of facts will be alterred
as an analogy is ingested.
(We will later deal with mechanism by which this message 
(signal?)
can be conveyed.)

We view "analogy understanding" as a type of learning,
in that the hearer now knows something new.
@****** Must this be the case, or could it be otherwise?
Would it count if the only new fact is this analogical connection?*****@*
Furthermore, that addition fact must be "analogical",
another concept which remains to be defined.
We will address these two points in turn --
discussing first what it means for a fact to be considered analogical,
and then what it means to learn something new.

@Section["Analogical" Facts]

The (informal) specifications below will serve to define 
@Comment[? constrain ?]
the sort of analogies we will (and will not) consider.
@BEGIN[Itemize]
An analogy connects a pair of things --
which may be objects, ideas, relations, or what-have-you.
(This eliminates "familial resemblences",
but still includes both similarity and proportional analogies.)

The purpose of an analogy is to enhance the hearer's understanding
of one of the analogues,
which somehow corresponds to some known features of the other.
(Note @Ref[NonAnalogy] lists a few situations which appear analogy-like,
which are NOT analogies.)
@END[Itemize]

As our vocabulary implies, we are taking the model of analogy
as an communication act, where the speaker is trying to convey a
body of facts about an object, @T<A>, to a hearer 
and does so by claiming that this @T<A> is like @T<B>,
in some manner @T<C>.@Foot{
We will later discuss various enhancements which spring from this theme:
when both the speaker and the hearer are the same individual;
times when that "manner C" can be left implicit (not explicitly stated);
the relevence of the speaker's model of hearer, and vice versa;
etc.}

Given these constraints, we can ask
what it means for two things to be analogous.
We intuitively think this means they are, in some manner, similar.
This intuition is trivial to formalize:
We consider @T<A> and @T<B>
@***** Should I use the same symbols (fonts) for the symbols
as for their referents?  *****@*
to be analogous if there is some assertion which is true of both of them --
that is, if there is some formula, @T<@g[f](x)>,
such that both @T<@g[f][A]> and @T<@g[f][B]> are true.

(There are, of course, rival views of analogy.
Some would claim that @T[A] and @T[B] are analogous if they each satisfy
the same partial theory.
Others view analogy as a mapping,
which sends various parts of one analogue into parts of the other.
We will deal with these views later.)

@****** Anything else I should say here? *****@*
@INCLUDE[NEW.MSS]

At the heart of analogy understanding is the notion of newness --
after this communication the hearer now knows something about one
of the analogues which he didn't know before.
This section will provide a general definition of "new",
applicable to any learning experience,
not just those related to analogies.
The next section will use this description,
combined with the specification of analogical facts given in the prior section,
to describe what it means to be new in an analogical sense.
@SubSection["New Analogical" Facts]

Now that we know what it means to be "new" in general,
and what it means to be an analogical connection,
we can ask what it means to be new in an analogical sense.
This subsection will discuss this issue --
to define what it means to understand an analogy.

Recall now that the purpose of analogy is educational
-- that is, our mission is to find sentence @g[s] such that
@T<New(Th,A,@g[s])>.
We must also find some way that @T<A> is similar to @T<B> --
i.e. some formula @T<@g[f](x)> such that
(after understanding the analogy)
@T<@g[f][B]> and @T<@g[f][A]> are each provable from @T[Th'].

Is it enough to find some formula @g[f] such that
@T<@g[f][B]> is in @T[Th], and
@T<@g[f][A]> is independent of @T[Th] -- i.e.
@BEGIN[Equation]
@Tag[Defn-f]
(@g[f][B] @K[MemberOf] Th) @K[And] (@g[f][A] @K[NotMemberOf] Th) @K[And] @~
(@K[Not]@g[f][A] @K[NotMemberOf] Th)?
@END[Equation]

The answer, it turns out, is yes.
We have to prove, first, that this corresponds to an analogy connection,
and second that this would be a new fact about @T[A].
The first is trivial:
@g[f] is the property 
(well, formula)
which both @T[A] and @T[B] share.
The second part is more difficult.

By assumption @T<@g[f][A]> is independent of @T[Th];
it remains only to show that there is some constructable constant,
@T[R], such that
@T<@F1{I}@-[Th'][R] @K[ProperSubSet] @F1{I}@-[Th][R]>, where
@T<Th' @K[Gets] T @K[Union] @K[LeftBrace] @g[f][A] @K[RightBrace]>.

@****** Now what?  Redo everything below! *****@*

The easiest way of showing this is by constructing the model of @T<Th'>
which has an interpretation of @T<A> which cannot be amoung 
@T<@F1{I}@-[Th'][A]>.
We simply construct a Henken model of @T<Th'>,
(technically the model corresponding that theory...,)
and complete it with @T<@K[Not]@g[f][A]>.
@****** Is this correct, and legal? *****@*
Clearly this interpretation of @T<A> will not be among the
interpretations of @T<Th'>, @T<@F1{I}@-[Th']>, QED.

The statement that @T<@K[Not](Th @K[RightTurnstile] @g[f][A])>,
eliminates the cases when @T<@g[f][A]> is trivial (e.g. a tautology);
and it seems that any statement that non-trivially involves @T<A>
must say something new about @T<A>.

That is an advantage of knowing that @T<Th @K[RightTurnstile] @g[f][B]> --
clearly all the symbols of @g[f] must be in this @T<@F1[L]@-[Th]>.
@SubSection[Final Comments ]
We present some closing comments
before leaving this topic.
Many of these will motivate the next topic of this paper,
which addresses the issue of what goes into the statement of an analogy.

@BEGIN[Itemize]
As should be obvious,
we are attempting to be descriptive here, as opposed to prescriptive.
That is, we have yet to say anything about how to generate such a @g[f],
given @T[Th], @T[B] and @T[A].
Some hints 
will be given later in this paper;
a fully descriptive procedure is for a different paper.

A related point is that this description 
only addresses what constitutes a legal analogy,
but says nothing about what it means to be a good (or even acceptable) analogy.
Again a subsequent chapter will (at least) hint at some comparative criteria.

Everything above will still work, even if we do not start
with an explicit symbol for the analogue in question --
it might be some implicit ... (e.g. the value of a function, or ...)
It just gets trickier -- having first to reify ...
There are many ways around this issue; but it has to be mentioned.
@****** This still has to be shown *****@*
@END[Itemize]

@Chapter[Notes]

@BEGIN[Enumerate]

@BEGIN[Multiple]@Tag[NonAnalogy]
It is worth mentioning various cases of NON-analogy:
(If we allowed these to count as analogy, the above premises would not
be sufficient.)

@BEGIN[Itemize]
It might be to tell the hearer that you, the speaker,
just happen to also know this something.

It might be the basis of induction --
from
@BEGIN[Equation]
MRG went to good school,
@K[And] RDG is like MRG (in terms of education)
@END[Equation]
deduce 
@BEGIN[Equation]
all stanford CS people well educated.
@END[Equation]
This is induction, NOT analogy.
@END[Itemize]
@END[Multiple]

@BEGIN[Multiple]@Tag[Proofs]
This chapter includes many (non)examples of analogies.
This note will confirm that this model of analogy does 
"behave" correctly on these cases.

@SubHeading[Case 1: Father(Arnold)]
<<do this>>

@SubHeading[Case 2: Floogle(A)]
Here we add in the sentence @T<Floogle(A)>,
where @T<Floogle> is not in @F1[L].
(The only essential feature is that nothing in @T<Th> relates to
this @T<Floogle> relation.
Otherwise, we would need to resort to some type of extensible signature table,
and other hairy messes.)
On introducing this relation, we now have a factor of @T<2@+[n]> more models
of @T<T>, where @T<n> is the number of constants (? constructable elements ?) --
depending on whether @T<Floogle(x)> is true for that element @T<x>, or not.

While the number of interpretations, @T<@F1[I]@-[Th']>, will clearly increase,
the number of instantiations of @T<A>, @T<@F1[I]@-[Th'][A]> will not.
There is nothing in @T<A>'s newly noted @T<Floogle>@i<hood> which can
eliminate any members of that set of interpretations.
Hence we claim that
@T<@K[Not]New(Th, A,"Floogle(A)")>;
i.e. the sentence @T<Floogle(A)>,
which clearly deals with @T<A>,
says nothing new about @T<A>.
(Realize that this was the case above when we added in @T<Analogous(A,B,...)>,
when we had no rules which joined this @T<Analogous> relation (symbol?)
to anything then in @T<Th>.)

@SubHeading[Case 3: A = C]
<<do this>>

@SubHeading[Case 3': Only One Member]
<<do this>>
@END[Multiple]

@END[Enumerate]
@BEGIN[Comment]	Outtakes

analogicistic@Foot{
This is like "historistic", versus "historical".
The term "analogistic" refers to those matters
which deal with the topic of analogy, but which are not, themselves,
instances of an analogy.
This leaves "analogical" to refer to those instances of analogies.}

It is also possible that different symbols, in different instantiations,
may refer to the same real world object.
@****** Is this important? *****@*
For example, the symbols @T<MRG> and @T<MRG'> may refer to the real world objects
(whose canonical referent are) 
@T<Prof Micheal R. Genesereth> and @T<Dr Milton R. Grinberg>, respectively,
in one model,
but oppositely in another.

The flaw with this argument is simple:
@T<@F1[I]@-[Th'] @K[ProperSubset] @F1[I]@-[Th]>,
does not imply that
@T<@F1[I]@-[Th'][A] @K[ProperSubset] @F1[I]@-[Th][A]>.
There may, for example, be another interpretation, @T<I@-[k]>,
which remained in @T<@F1[I]@-[Th']>, and
which mapped @T<A> onto this same element --
i.e. @T<I@-[j][A] @K[Eq] I@-[k][A]>, meaning that
@T<@F1[I]@-[Th'][A]> might still equal @T<@F1[I]@-[Th][A]>.

@END[Comment]