SUO: Re: Building Ontologies Through Signs And Inquiries (BOTSAI)
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Let us now illustrate the particulars that we find
in "The Case of the Missing Bus" by using the sort
of "propositional logic in a lattice" diagram that
I used to articulate the basic brands of inference,
in what now must seem like so many long notes past.
Let me recap the story as we know it so far in the syllogistic
or "propositional constraint reasoning" (PCR) style of picture.
Figure 1 sketchily summarizes the first phase of the reconstruction.
| A (A)
| o o
| \ /
| \ * * /
| \ /
| \ * * /
| \ /
| \ * * /
| \ /
| \ * * /
| \ /
| \ * * /
| \ /
| \ * * /
| \ /
| \ B o o (B) /
| \ /
| \ * * /
| \ /
| \ * * /
| \ /
| \ * * /
| \ /
| \ * * /
| \ /
| \ * * /
| \ /
| \ * * /
| \ /
| \*/
| o
| C
|
| Figure 1. Missing the Bus
|
| A = Arriving bus situations
| B = Best case situations
| C = Current situation
The point elements in these diagrams represent
the "propositions" that one is contemplating
with respect to a domain of objects, persons,
situations, and so on. Another option is to
treat them as the "terms" of the description:
Major term, Middle term, Minor term, and so on.
The line elements in these diagrams represent the "logical relations"
that are being considered between certain pairs of propositions, or else
the "premisses" that are being contemplated between various pairs of terms,
where roughly vertical lines indicate "implications", the antecedent lower
and the consequent higher, and where roughly horizontal placements indicate
relationship of "alteration" (change) or "alternation" (diversity), that is,
the situation among a number of alternatives, exclusive or inclusive, that
are available for one to change or to choose among.
It is my guess that something like this style of geometric figure
was used by Aristotle, and may have been a common sort of picture
at the time, at least, this is the impression that I get from the
way that he uses two different styles of language for indicating
the various sorts of logical relationships that are relevant to
the fundamental types of reasoning situation that he discusses.
For instance, Aristotle often uses the geometric label of the
line segment AB to indicate the premiss B => A. Of course,
this may just be a fluke of Greek grammar, or of its later
transcription.
There a convenient technical nomenclature that was added at a later date,
in which the various line elements depicting the premisses and relations
are customarily labeled as "Cases", "Facts", and "Rules", and I will use
this style of language rather freely to talk about the different roles
that different premisses may enjoy in the various forms of reasoning.
One other thing: I often use the following equivalent notations:
"(A)" = "~A" = "A'" = "Not A".
Among other things, this gives the following notational equality:
"(A (B))" = "A => B" = "Not A without B".
I hope that will be enough of a set-up to get this show on the road.
Data of the Situation:
1. Alternative Facts: (C (A)) versus ( C ((A))), that is, (C A).
2. Alternative Cases: (C (B)) versus ( C ((B))), that is, (C B).
3. Alternative Rules: (B (A)) versus ((B)((A))), that is, (A (B)).
We meet the surprising Fact : C => (A), depicted by the line segment (A)C.
The reason that this Fact is surprising is that we automatically expected
a different Fact, namely, C => A. And, assuming the current situation C,
which we always do -- since this whole intervention of C is just a gimmick
for supplying a pivot to our thought -- we were led moreover to expect A,
the arrival of the bus.
If we stop to think about it, we come to realize that there is
a middle term that we have been taking for granted, say "B",
the "benign" situation, the "best case" scenario (assuming
that the best case means catching the bus), or maybe just
the modal, normal, ordinary, or usual case, if you like
those terms better.
The name "reflection" seems to fit the process by which
we can become aware of the previously automatic, implicit,
and probably unconscious deduction that led to a current
expectation, the one that is subject to conflict with
a current observation, thereby generating a dilemma,
a problem, or a surprise.
Nota Bene. Actually, I use the word "problem" more specifically
to refer to a difference between an intention and an observation,
but that is another, yet related story.
In the process of reflecting on the "program" of a habitual deduction,
we become able to identify the intermediate and the middle terms that
go "into it", and at this point we become able to contemplate their
deliberate variation. In this way, we become able to pass from the
class of propositions that are schematized by "B" to one or two in
the class of propositions that are summarized by "~B", and thereby
guessing a new Case, for example, that the current situation has
the marks of a public holiday, C => H, where H => ~B, and so is
not beneficial for our immediate purposes, tedious as they are.
I left off last time at the point where you were just beginning to
contemplate the possibility that your current situation might fall
under the case description of a public holiday, thereby explaining
the absence of the expected bus, and a hypothesis which, if true,
would reduce your affective sense of surprise at the accustomed
bus not being there at the place-time that you were accustomed
to observe it.
Now, if you're like me, you might eventually think to look up,
and then to look around your surrounding neighborhood, to see
if you can observe any further evidence or any other naturally
occurring signs that might bear on your new hypothesis one way
or another.
This, of course, brings us to the deductive phase of our present inquiry.
And, equally of course, our immedately present phase of deduction must
be distinguished from all of those previous deductions, not to mention
their Promethean and Epimethean (fore and aft) bracketings by all of
those previous bits of abductive and inductive reasoning that went
into making up what were no doubt many previous cycles, and a vast
host of parallel cycles, and a countless array of epicycles on
our deference to an inquiry that may be indefinitely deferred.
Well, after that importunate word from our spontaneity,
I think that it is due time to get back to our story.
We have all been waiting for this bus long enough!
[This essay was written just after Easter 2000.]
If I had been walking on a residential street hereabouts,
through most of last week, when this "missing of the bus"
caper was alleged to have happened, I could have looked up
and looked around and seen all the gaily colored balloons,
the flapping ribbons, and the many other festive decorations
that were decked out on many of the houses and the trees by
all of the neighborhood parents who were throwing together
to treat their collective broods to an Easter Egg Hunt.
So that would have served to confirm the hypothesis of
a holiday, and perhaps it may have even altered my sense
of what was "best", "benign", "beneficial" -- trudging off
on my accustomed way, in pursuit of my habitual goals, or
stopping to enjoy the signs of another custom, and even
to follow them -- but that's another story altogether!
Anyway, it behooves me to try and size up the present moment of inquiry.
Let me unfold the map again and make a few additional notations upon it.
| A D (A)
| o o o
| \ /
| \ * * * /
| \ /
| \ * * * /
| \ /
| \ * * * /
| \ /
| \ * * * /
| \ /
| \ * * * /
| \ /
| \ * * * /
| \ /
| \ B o * o (B) /
| \ /
| \ * * * /
| \ /
| \ * o H /
| \ /
| \ * * /
| \ /
| \ * * /
| \ /
| \ * * /
| \ /
| \ * * /
| \ /
| \*/
| o
| C
|
| Figure 2. Missing the Bus, Again
|
| A = Arriving bus situations,
| B = Best case situations,
| C = Current situation,
| D = Decoration situations,
| H = Holiday situations.
I think that this pretty graphically says what I've been striving to say
in the last thousand words or so, and I am tempted to leave it at that,
but temptations to desist, you will have observed, are the sorts of
temptations I can easily resist! So let me attempt to sum it up
all over again, this time once again in schematic symbols and
in rather more verbose but slightly more descriptive phrases.
Abduction of a Case:
Fact: C => (A), In the current situation, the bus is not arriving.
Rule: H => (A), If it is a holiday, the bus would not be arriving.
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Case: C => H , Perhaps the current situation is a holiday.
The validity of this abduction as a form of reasoning, in the only way
that its particular form of non-demonstrative inference can be said to
be valid, depends on the validity of the corresponding deduction, from
the Case : C => H and the Rule : H => ~A to the Fact : C => ~A. So it
needs to be remembered that the utility of this deduction, which only
concludes what has already been observed, is that it succeeds in its
aim to reduce the surprise of that observation.
Deduction of a Fact:
Case: C => H , In the current situation, it is a holiday.
Rule: H => D , If it is a holiday, there will be decorations.
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Fact: C => D , In the current situation, there will be decorations.
The inductive phase, in this situation, consists of looking up and testing
whether the prediction comes true. I have been studying for few years now,
and still remain a bit puzzled, as to how exactly this sense of induction
fits in logically, if it does at all, with the other meaning of induction,
namely, of a non-demonstrative inference from a Case and a Fact to a Rule.
Jon Awbrey
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