[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 55

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Arisbe List, Cybernetics List,

Analysis of Contingent Propositions

For all of the reasons mentioned above, and for the sake of
a more compact illustration of the in and outs of a typical
"propositional equation reasoning system" (PERS), let's now
take up a much simpler example of a contingent proposition:

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` q o ` o r ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` | ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` p o ` o p ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` O ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `(p (q)) (p (r))` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

For the sake of simplicity in discussing this example,
I will revert to the "existential interpretation" (ExG)
of logical graphs and their corresponding parse strings.

Under ExG the expression "(p (q))(p (r))" interprets as
the vernacular expression "p=>q & p=>r", so this is the
reading that we'll want to keep in mind for the present.

Where brevity is required, and it occasionally is, we may invoke
the propositional expression "(p (q))(p (r))" under the name "f"
by making use of the following definition:  "f = (p (q))(p (r))".

Since the expression "(p (q))(p (r))" involves just three variables,
it may be worth the trouble to draw a venn diagram of the situation.
There are in fact a couple of different ways to execute the picture.

Figure 1 indicates the points of the universe of discourse X
for which the proposition f : X -> B has the value 1 (= true).
In this "paint by numbers" style of picture, one simply paints
over the cells of a generic template for the universe X, going
according to some previously adopted convention, for instance:
Let the cells that get the value 0 under f remain untinted, and
let the cells that get the value 1 under f be painted or shaded.
In doing this, it may be good to remind ourselves that the value
of the picture as a whole is not in the "paints", in other words,
the 0, 1 in B, but in the pattern of regions that they indicate.
NB.  In this Ascii version, I use [```] for 0 and [^^^] for 1.

o-----------------------------------------------------------o
| ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^o-------------o^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ / ` ` ` ` ` ` ` \ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ ^ ^/` ` ` ` ` ` ` ` `\^ ^ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ ^ / ` ` ` ` ` ` ` ` ` \ ^ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ ^/` ` ` ` ` ` ` ` ` ` `\^ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ / ` ` ` ` ` ` ` ` ` ` ` \ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^o` ` ` ` ` ` ` ` ` ` ` ` `o^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^|` ` ` ` ` ` ` ` ` ` ` ` `|^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^|` ` ` ` ` ` P ` ` ` ` ` `|^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^|` ` ` ` ` ` ` ` ` ` ` ` `|^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^|` ` ` ` ` ` ` ` ` ` ` ` `|^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^|` ` ` ` ` ` ` ` ` ` ` ` `|^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ o--o----------o ` o----------o--o ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^/^ ^ \ ` ` ` ` `\`/` ` ` ` ` / ^ ^\^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ / ^ ^ ^\` ` ` ` ` o ` ` ` ` `/^ ^ ^ \ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^/^ ^ ^ ^ \ ` ` ` `/^\` ` ` ` / ^ ^ ^ ^\^ ^ ^ ^ ^ |
| ^ ^ ^ ^ / ^ ^ ^ ^ ^\` ` ` / ^ \ ` ` `/^ ^ ^ ^ ^ \ ^ ^ ^ ^ |
| ^ ^ ^ ^/^ ^ ^ ^ ^ ^ \ ` `/^ ^ ^\` ` / ^ ^ ^ ^ ^ ^\^ ^ ^ ^ |
| ^ ^ ^ o ^ ^ ^ ^ ^ ^ ^o--o-------o--o^ ^ ^ ^ ^ ^ ^ o ^ ^ ^ |
| ^ ^ ^ | ^ ^ ^ ^ ^ ^ ^ ^ | ^ ^ ^ | ^ ^ ^ ^ ^ ^ ^ ^ | ^ ^ ^ |
| ^ ^ ^ | ^ ^ ^ ^ ^ ^ ^ ^ | ^ ^ ^ | ^ ^ ^ ^ ^ ^ ^ ^ | ^ ^ ^ |
| ^ ^ ^ | ^ ^ ^ ^ ^ ^ ^ ^ | ^ ^ ^ | ^ ^ ^ ^ ^ ^ ^ ^ | ^ ^ ^ |
| ^ ^ ^ | ^ ^ ^ Q ^ ^ ^ ^ | ^ ^ ^ | ^ ^ ^ ^ R ^ ^ ^ | ^ ^ ^ |
| ^ ^ ^ | ^ ^ ^ ^ ^ ^ ^ ^ | ^ ^ ^ | ^ ^ ^ ^ ^ ^ ^ ^ | ^ ^ ^ |
| ^ ^ ^ o ^ ^ ^ ^ ^ ^ ^ ^ o ^ ^ ^ o ^ ^ ^ ^ ^ ^ ^ ^ o ^ ^ ^ |
| ^ ^ ^ ^\^ ^ ^ ^ ^ ^ ^ ^ ^\^ ^ ^/^ ^ ^ ^ ^ ^ ^ ^ ^/^ ^ ^ ^ |
| ^ ^ ^ ^ \ ^ ^ ^ ^ ^ ^ ^ ^ \ ^ / ^ ^ ^ ^ ^ ^ ^ ^ / ^ ^ ^ ^ |
| ^ ^ ^ ^ ^\^ ^ ^ ^ ^ ^ ^ ^ ^\^/^ ^ ^ ^ ^ ^ ^ ^ ^/^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ \ ^ ^ ^ ^ ^ ^ ^ ^ o ^ ^ ^ ^ ^ ^ ^ ^ / ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^\^ ^ ^ ^ ^ ^ ^ ^/^\^ ^ ^ ^ ^ ^ ^ ^/^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ o-------------o ^ o-------------o ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ |
| ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ |
o-----------------------------------------------------------o
Figure 1.  Venn Diagram for (p (q))(p (r))

There are a number of standard ways in mathematics and statistics
for talking about "the subset W of the domain X that gets painted
with the value z by the indicator function f : X -> B".  The subset
W c X is called the "antecedent", the "fiber", the "inverse image",
the "level set", or the "pre-image" in X of z under f, and is defined
as W = (f^(-1))(z).  Here, f^(-1) is called the "converse relation" or
the "inverse relation" -- it is not in general an inverse function --
corresponding to the function f.  Whenever possible in simple examples,
I will use lower case letters for functions f : X -> B, and I will try
to employ capital letters for subsets of X, if possible, in such a way
that F will be the fiber of 1 under f, in other words, F = (f^(-1))(1).

The easiest way to see the sense of the venn diagram is
to notice that the expression "(p (q))", read as "p=>q",
can also be read as "not p without q".  Its assertion
effectively excludes any tincture of truth from the
region of P that lies outside the rule of Q.

Likewise for the expression "(p (r))", read as "p=>r",
and also readable as "not p without r".  Asserting it
effectively excludes any tincture of truth from the
region of P that lies outside the rule of R.

Figure 2 shows the other standard way of drawing a venn diagram
for such a proposition.  In this "punctured soap film" style of
picture -- others may elect to give it the more dignified title
of a "logical quotient topology" or some such thing -- one goes
on from the previous picture to collapse the fiber of 0 under X
down to the point of vanishing utterly from the realm of active
contemplation, thereby arriving at a degenre of picture like so:

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` o-------------o ` o-------------o ` ` ` ` ` ` |
| ` ` ` ` ` `/` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` `\` ` ` ` ` ` |
| ` ` ` ` ` / ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` \ ` ` ` ` ` |
| ` ` ` ` `/` ` ` ` ` ` ` ` `/`\` ` ` ` ` ` ` ` `\` ` ` ` ` |
| ` ` ` ` / ` ` ` ` ` ` ` ` / P \ ` ` ` ` ` ` ` ` \ ` ` ` ` |
| ` ` ` `/` ` ` ` ` ` ` ` `/` ` `\` ` ` ` ` ` ` ` `\` ` ` ` |
| ` ` ` o ` ` ` ` ` ` ` ` o-------o ` ` ` ` ` ` ` ` o ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` | ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` | ` ` ` |
| ` ` ` | ` ` ` ` Q ` ` ` | ` ` ` | ` ` ` R ` ` ` ` | ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` | ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` | ` ` ` |
| ` ` ` o ` ` ` ` ` ` ` ` o ` ` ` o ` ` ` ` ` ` ` ` o ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` `\ ` ` /` ` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` / ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` `/` ` ` ` ` |
| ` ` ` ` ` \ ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` / ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` `/`\` ` ` ` ` ` ` `/` ` ` ` ` ` |
| ` ` ` ` ` ` o-------------o ` o-------------o ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
Figure 2.  Venn Diagram for (p (q r))

This diagram indicates that the region where p is true is
wholly contained in the region where both q and r are true.
Since only the regions that are painted true in the previous
figure show up at all in this one, it is no longer necessary
to distinguish the fiber of 1 under f by means of any stipple.

In sum, it is immediately obvious from the venn diagram that
in drawing a representation of the propositional expression:

   (p (q))(p (r)),

in other words,

   [p => q] & [p => r],

we are also looking at a picture of:

   (p (q r)),

in other words,

   p => [q & r].

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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