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C00007 00003	∂Discussion with MRG 21-Oct-82
C00009 00004	Outline
C00010 00005	What is an analogy? ... and what is it used for?
C00015 00006	<<<here  -what of this should be save?>>>
C00019 00007	It is still possible that, somehow, both analogues together decide whether
C00024 00008	//Below is a note 
C00031 00009	So now that we know that [**] is insufficient, what now?
C00033 00010	Argument must be different -- that there are several ...
C00034 00011	Then why are theory's used, not objects?
C00035 00012	Situation: When looking for someone who is musical.
C00036 ENDMK
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∂Discussion with MRG 21-Oct-82
(his comments on my paper)
1. Form:
   First: intro
	a motivation -- why is analogy needed?
	this paper describes a formalism, not the USE of an analogy, or ...
   Second -- lead into 5 place predicate, beginning with intuitions
	which need to be honed
   Then a demonstration of the special cases: how mapping and commonality
	
∂Thoughts based on conversation w/DBL, on 25/Oct/82

The system can hold different levels of facts -- 
depth measured wrt causal reasoning.
Analogies can be drawn at any these levels; and then "checked" by
seeing if the encompassing "causal system" can explain this...
This can produce confirmation, or invalidation (i.e. show something
is laughable.)

Outline

Intro/Motivation
   [Why analogy] Analogies used all the time
   [Why paper] No one has formalized what is going on
   [What is paper] NOT a sketch of how to use an analogy;
	instead a description of what it is -- i.e. not code, ...

>2 args
   At least 2 - the analogues
   Must be something else - the context, or "answer"(mapping, theory, ...)

What are analogues
   [Other args are function of this -- e.g. mapping over SOMETHING]
   Cannot be objects themselves, ... rather ... a description of it
   --- leads to reformulation

What is an analogy? ... and what is it used for?

The idea is to derive a proposition of the form
Analogous(A B ...);
and later to use this to derive some fact, like
f(A B)
[usually this is just f'(B); which may have considered f''(A) --
i.e. Analogous(A B ...)
	f''(A)
-------
f'(B)
-------
-------

What must go into this analogy relation?  
Certainly a pair of things corresponding to the two analogues.
Anything else; or is analogy a two place predicate?
Note that we do indeed say that 
"Viral meningitis is like Bacterial meningitis",
or "Electrical system is like water flow";
which sounds a lot like.
	Analogous(VM BM).

<<<Note here - yes, that proposition does encode our statement, but what now?
To get any result, ... 

In and of itself, this is meaningless.
Let's now consider what we want to do with it -- what type of conclusions
we can draw ... >>>

Consider cases where A is like B in more than one way --
"Bill is physically like George" vs
"Bill's brawn resembles George's brain".
In both cases we're comparing Bill to George, but we would expect
to reach quite different conclusions in these two situations.
(ex: Bill's strength would correspond to George's strength in the first
case, but to George's IQ in the second.)

<<<BEGIN SideIssues>>>
Two issues come up here:

First, perhaps this is really a linguistic confusion;
we should have been comparing "Bill'sPhysique" to "George'sPhysique" in the first
case, and to "George'sMind" in the second.
That is, the whole problem was in going from the English statement to the 
actual statement of analogy --
Analogous(Bill'sPhysique George'sPhysique) and
Analogous(Bill'sPhysique George'sMind), respectively.
Perhaps, but consider now the analogy that 
"Bill is heavy, as George is tall".  
Is the analogy here 
Analogous(Bill'sPhysique-Mass George'sPhysique-Height), to distinquish
this analogy from comparing their respective weights, which would be stated
as Analogous(Bill'sPhysique-Mass George'sPhysique-Mass).
And if you'll permit that, what about ...
Where does it end?

Second, why was it this binary Analogous(A B) was so seductive?
Why did it seem sufficient?
In general,
(1) there is a single prefered perspective in which to view the
analogues, which well forces the analogy itself.
(2) in the rare case when there are more than one way to consider A and B,
usually one of pair of these perspectives makes any sense (the others being
funny.)  Again, no problem.
(3) in many of the remaining cases, when A1 and A2 (perspectives of A)
match to B, these matchings are independent ... the "mappings" can be merged.
(4) even of the remaining problematic situations,
where A and B can correspond in different and INCOMPATIBLE ways,
there is usually some specific context which dictates which perspective
should be considered -- and again, the other perspectives seem silly.
---
It is only the final case, when there is no strong context, or where this
context does NOT further disambiguate, that we have problems.

<<<END SideIssues>>>

To consider a case where this makes a real difference --
Washington / Lincoln...
<<<here  -what of this should be save?>>>

But now what can we do?
Well, knowing that BM is a disease, we probably would like to conclude
that VM is a disease.

Silly conjecture:
	Analogous(A B) => Disease(A)

Obviously false. (From Analogous(ES WF), we should *NOT* conclude Disease(WF).)

So maybe,
	∀A,B,F. Analogous(A B) & F(A) => F(B)
[these F may really be λ things.
 also, they may have to be sufficiently general to encompass both A and B, ...]

Hence, given Analogous(VM BM) and Disease(BM), we correctly conclude
Disease(VM).

But still problematic: what of CausedBy(BM Bacteria)?
Clearly not CausedBy(VM Bacteria) -- by def'n it is caused by a Virus.

That's the first nail.  Still might be some solution, by claiming that,
inherent to the analogues A and B are a set of "transferable facts".

Perhaps it is due to but one of the analogues.
ie. 
	∀A,B,F. Analogous(A B) & F(A) 
		& PredicatedOKedBy(A F)
		=> F(B)

This too fails.  Consider the "a spoon is like a fork" analogy,
[note: the "the" above is inappropriate -- we'll later see that "a" should have
been used.]
Hence we would need
	PredicatedOKedBy(Spoon SilverWare)	[*]

to see that a fork is also a piece of silverware -- hence
	Analogous(Spoon Fork) & SilverWare(Spoon) 
		& PredicatedOKedBy(Spoon SilverWare)
		=> SilverWare(Fork)
	
But now we want to assert that "a spoon is like a shovel",
in that either can be used for scooping.
Given
	PredicatedOKedBy(Spoon SilverWare)
we could, using [*] above, deduce that 
	SilverWare(Shovel)
-- WRONG!

Reciprocally we can show that
	∀A,B,F. Analogous(A B) & F(A) 
		& PredicatedOKedBy(B F)
		=> F(B)
(i.e. where the other analogue gets to confirm the predicate) doesn't work.

It is still possible that, somehow, both analogues together decide whether
some property should be transfered over -- i.e. that
PredicatedOKedBy is a tertiary relation, of A, B and F.

Hence
	∀A,B,F. Analogous(A B) & F(A) 		[**]
		& PredicatedOKedBy(A B F)
		=> F(B).

While there is no "logical counterexample" to this formalism,
it violates one of the basic assumptions we all make about analogies:
namely, that the analogy itself may depend on the context or the situation,
and there is no way to describing this in this formalism.
(Reworded, they should be considered when deciding whether to accept a 
given predicate.)
In one case one predicate F1 may make sense,
while in another, F2 seems apt;
the kicker is that F1 and F2 are incompatible, in the sense that they
cannot both hold at once -- i.e.
	F1(B) & F2(B) & (other facts re: B) => False.

Consider the proportional analogy question:
What is the best ? for
	"Washington : 1 :: Lincoln : ?"?
[cite whoever first mentioned this]
If we regard Washington and Lincoln as presidents, and observe that
Washington was first president, the obvious answer is ?=9,
as Lincoln was the ninth president.

If we let F1(x) mean we consider only facts about x @i{qua} president,
then F1(Washington) [can be shown to] imply that the (relevant) relation
connecting Washington to 1 is "cardinality of presidency".
Now how can we use the formalism above (i.e. [**]) to derive that F1(Lincoln),
which would, in turn imply that ?=9?
This can be deduced when
PredicatedOKedBy(Washington Lincoln F1), and F1(A);
both of which seem reasonable.

Consider now the F2 predicate, where
F2(x) means the only facts we will consider about x deal with x as part of
a US bill.
F2(Washington) would mean that Washington's relation to 1 is in terms of the
fact that Washington's portrait appears on a 1 dollar bill.
It does seem reasonable to assert that 
PredicatedOKedBy(Washington Lincoln F2), and F2(A),
which would mean that ?=5, as Lincoln appears on 5 dollar bills.

So which is the answer, ?=9 or ?=5?
In the above notation, it depends on whether F1(B) or F2(B) holds.
That is, [**] does NOT lead to the contradiction F1(B) & F2(B).
But it also does NOT permit us to state that F1 is apt in some situations,
while F2 should be considered under others.
The point here is that we do want to be able to reach either conclusion,
each under its own set of circumstances.

----
*** Tangent ***
Given PredicatedOKedBy(Washington Lincoln F1) and
PredicatedOKedBy(Washington Lincoln F2); now what?
We have, in some sense, too much.
Now whenever F1(Washington) we cannot help but get F1(Lincoln).
- so?
//Below is a note 
-- a proof of why no logical contradiction is possible...
<here>

Unfortunately, there is no "rigorous demonstration" of why
this definition is erroneous -- that is, it does not lead to ...
We can, in fact, even prove that no such proof is possible.
<here>
First the intuition:
the analogy may depend on the context or the situation,
which should be considered when deciding whether to accept a given predicate.
In one case one predicate F1 may make sense,
(in that PredicatedOKedBy(A B F1) seems reasonable, and F1(A) can hold),
while in another, F2 seems apt;
the kicker is that F1 and F2 are incompatible, in the sense that they
cannot both hold at once -- i.e.
	F1(B) & F2(B) & (other facts re: B) => False.

//
Note that this task is not impossible:
these F1 and F2 much simply consider other parameters 
(i.e. the situation, somehow encoding).
//

-----
I tried here to reach some logical contradiction --
ie trying to "prove" anything silly about B, or to deduce False.
My method was to find some pair A, B which are analogous in two ways.
//Consider RDG and MRG, where each is both 
(i) associated with Stanford CS dept, and
    [hence if one wanted to contact someone in the CS department, one might
    reason that RDG would know him, based on the fact that MRG knew him, and
    RDG was also a co-Stanford CS dep't member.]
(ii) a recorder player.
    [hence if one had a music-related question, and knew that MRG could answer it,
    perhaps RDG could as well, as they both were recorder-ers.]

This means we would want
  PredicatedOKedBy(MRG RDG InSCS)		[3a]
and
  PredicatedOKedBy(MRG RDG Recorder-er)		[3b]
//

The (unattainable) goal here is to somehow show that these two
propositions lead to some contradiction -- either that they, by
themselves, are inconsistent; or that, when using the 2nd order fact (**),
they derive a pair of incompatible compatible solutions.

Case i) Given that we have no axioms about this PredicatedOKedBy relation,
there is no way we could assert that [3a] and [3b] are inconsistent.

[Side note: it is easy to derive that various boolean connectives in the third
argument "distribute" outside --
	PredicatedOKedBy(x y F1&F2) ≡	
	PredicatedOKedBy(x y F1) & PredicatedOKedBy(x y F2),
based on [*]; but so what?]

Cae ii) How could we deduce that F1(B) and F2(B) are contradictory, when
they both must be true -- otherwise this whole claim falls down.

<<Before realizing this, I played with

Derive first F1(B), and then F2(B),
which are somehow contradicatory -- i.e. 
	F1(B) & F2(B) & (other facts re: B) => False.
Note that we must have had F1(A) and F2(A) though, to have derived these on B.

I thought that we could have some additional facts about B which lead to this
contradiction. E.g. Suppose 
	F1(x) ≡ [x has >2 elements], and 
	F2(x) ≡ [x has <4 elements],
Then if A has 3 elements, the F1(A) and F2(A).
Now suppose we know that B has an EVEN number of elements.

NO NO NO
There is a model B -- which must have some number of elements...
I loose...
>>>

This means we had to use some other approach.
Perhaps this F must be meta-level, or otherwise complicated.
For example, let's take the Washington is like Lincoln analogy.

Let F1(x) be true if "the (most important) number associated with x is
the cardinality of his presidency".

Thus F1(Washington) => "the relation of Washington to 1 means that Washington
is the first president".
Using [**], and the premises that
Analogous(Washington Lincoln) & F1(Washington)
	& PredicatedOKedBy(Washington Lincoln F1),
we see that F1(Lincoln) -- which means that the answer to the proportional analogy
"Washington : 1 :: Lincoln : ?" must be ?=9.

Another person may assume that F2(Washington) held, where
F2(x) iff "the (most important) number associated with x is
the value of the US bill which contain x's likeness".

It seems reasonable to assume that PredicatedOKedBy(Washington Lincoln F2) would
hold as well, which means, given
F2(Washington), that F2(Lincoln).
This contradicts the above result, as
"Washington : 1 :: Lincoln : ?" now means ?=5.

So now that we know that [**] is insufficient, what now?
How can we "spice" it up to handle such cases?
We need to somehow incorporate something IN ADDITION TO THE ANALOGUES THEMSELVES,
somewhere in [**].
The obvious place is the ANALOGOUS relation.
Thinking of Washington and Lincoln as presidents is different from
thinking of them as portraits on various US bills --
these are TWO different analogies, and should be noted as such.

//
To those of you who noted that it would be "mathematically adequate" to enhance
the PredicatedOKedBy relation,
to let it consider the current situation:
Realize we will eventually "incorporate" the effects of this relation in the
ANALOGOUS relation anyway, so this distinction is only transitory.

Besides, why let it have all the fun?
//

Thus far all we have established is that something else is needed.
But what?

Something about the type of analogy; or the way of considering the analogues,
...
perhaps a mapping.
We'll return to that later.
<here>
Argument must be different -- that there are several ...

Fred is more like a door than a window.

confirmation:
Consider tasks like "What is like x?"
Usually that is retort when that analogy not found: but I thought ...


----

Then why are theory's used, not objects?
Q: Is analogy a property of the object, or of the representation?

[People claim not to have an analogy if ... ]

----
consider analogy between two vectors: if they each have a zero coordinate.
But given ANY pair of vectors, we can transform cartesian system into an 
"equivalent" one in which this is true - so they are analogous.
Situation: When looking for someone who is musical.
Knowing that Mike is like Jeanne in that both play a recorder ...

We can intepret this as meaning that there is some binary relation, R1,
such that R1(Washington, 1) and
R1(Lincoln, ?), for some value of ?.