[Inquiry] Re: Examples Of Inquiry

[Jon Awbrey]

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EOI.  Note 12

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Cathy Legg's "Missing The Bus" Example (cont.)

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 8 sketchily summarizes the first phase of the reconstruction.

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| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `A` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` (A) ` |
| ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` |
| ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * / ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * ` / ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` |
| ` ` ` ` ` \ ` ` * ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` * ` ` / ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` ` * ` ` ` ` ` ` ` ` ` ` ` ` * ` ` ` / ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` B o ` ` ` ` ` ` ` ` o (B) ` ` / ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` ` `*` ` ` ` ` ` ` `*` ` ` ` / ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` * ` ` ` ` ` ` * ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` `*` ` ` ` ` `*` ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ ` ` * ` ` ` ` * ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` `*` ` ` `*` ` / ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` * ` ` * ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` \ `*` `* `/ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` \ * * / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| A `=` Arriving bus situations ` \*/ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| B `=` Best case situations` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| C `=` Current situation ` ` ` ` `C` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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Figure 8.  Cathy Legg's "Missing the Bus" Example

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:

   Alternative Facts:  (C (A))  versus  ( C ((A))),  that is,  (C  A).

   Alternative Cases:  (C (B))  versus  ( C ((B))),  that is,  (C  B).

   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.

Jon Awbrey

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