A New Kind of Science: The NKS Forum - Propositional Equation Reasoning Systems

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A New Kind of Science: The NKS Forum (http://forum.wolframscience.com/index.php)
- Pure NKS (http://forum.wolframscience.com/forumdisplay.php?forumid=3)
-- Propositional Equation Reasoning Systems (http://forum.wolframscience.com/showthread.php?threadid=297)

Posted by Jon Awbrey on 04-15-2004 12:00 AM:

Propositional Equation Reasoning Systems

Note (August 2009). A newer version of this material is under development at this location.

PERS. Note 1

For eventual use on the Cactus Rules and Differential Logic threads
I will need to develop some elementary material on the formal calculi
that I describe as "propositional equation reasoning systems" (PERS).
This work follows up on Charles Sanders Peirce's "alpha graphs" and
George Spencer Brown's "calculus of indications" in 'Laws of Form'.

I will first give a reference version of the CSP-GSB axioms
and the first few theorems for the abstract calculus that is
formally knowable in a couple of senses as propositional logic.

The first order of business is to give the exact forms of the
axioms that I use, devolving from Peirce's "Logical Graphs" via
Spencer-Brown's 'Laws of Form' (LOF). In formal proofs, I will
use a variation of the annotation scheme from LOF to mark each
step of the proof according to which axiom, or "initial", is
being invoked to justify the corresponding step of syntactic
transformation, whether it applies to graphs or to strings.

The axioms are just four in number,
and they come in a couple of flavors:
the "arithmetic initials" I_1 and I_2,
and the "algebraic initials" J_1 and J_2.

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `( ) ( )` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Axiom I_1.` ` Distract <--- | ---> Condense ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` (( )) ` ` ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Axiom I_2.` ` ` Unfold <--- | ---> Refold ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a(a)` ` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Axiom J_1.` ` ` Insert <--- | ---> Delete ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `ab ` ac` ` ` ` ` ` ` b ` c ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` a @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `((ab)(ac)) ` ` = ` ` a((b)(c)) ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| Axiom J_2.` Distribute <--- | ---> Collect` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

Notice that all of the axioms in this set have the form of equations.
This means that all of the inference steps they allow are reversible.
In the proof annotation scheme below, I will use a double bar "====="
to mark this fact, but I may at times leave it to the reader to pick
which direction is the one required for applying the indicated axiom.

As you are undoubtedly aware, Peirce introduced these formal equations
at a level of abstraction that is one step higher than their customary
interpretations as propositional calculi, which two readings for logic
he called the "Entitative" and the "Existential" interpretations, and
so I will refer to the interpreted systems of graphs as "EnG" & "ExG",
respectively. The early CSP, as in his essay on "Qualitative Logic",
and also GSB, emphasized the EnG interpretation, while the later CSP
developed mostly the ExG interpretation. When it comes down to this
very primitive level of formal structure, I find it important to note
and I even tend to stress the formal significance of the circumstance
that this formal system is a "very abstract calculus" (VAC), devoid of
meaning in the usual logical sense. I cannot speak for how this works
out or what it might mean beyond this point, on the next levels of form.

Jon Awbrey

Posted by Jon Awbrey on 04-15-2004 02:48 AM:

Propositional Equation Reasoning Systems

PERS. Note 2

The first theorem is known as the "Double Negation Theorem" (DNT).

o-----------------------------------------------------------o
| C_1.` Double Negation Theorem ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` a ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ((a)) ` ` ` = ` ` ` ` a ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` Reflect <---- | ----> Reflect ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

The proof that follows it is derived from the one that was given
by George Spencer Brown in his book 'Laws of Form', and credited
to two of his students, John Dawes and D.A. Utting. This result
is annotated as "Consequence 1" (C_1) or as "Reflection" in LOF.

o-----------------------------------------------------------o
| C_1.` Double Negation Theorem.` Proof.` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< I2. Unfold "(())" >=========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` a o ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J1. Insert "(a)" >==========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `a o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` a o ` a o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J2. Distribute "((a))" >====o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` a o ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` `o` `a o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J1. Delete "(a)" >==========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J1. Insert "a" >============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` `o a` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o ` ` o a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J2. Collect "a" >===========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` `o a` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J1. Delete "((a))" >========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< I2. Refold "(())" >=========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< QED >=======================o

Jon Awbrey

Posted by Jon Awbrey on 04-15-2004 03:28 AM:

Propositional Equation Reasoning Systems

PERS. Note 3

One theorem of frequent use is what I like to call
the "Weed and Seed" Theorem. The proof is just an
inductive exercise, so I will let it go till later,
but it says that a label can be freely distributed
or freely erased (retracted or withdrawn) anywhere
in a subtree whose root is labeled with that label.
The second theorem on the list to be shown here is
actually the base case of this Weed & Seed theorem.
I talk about it under the LOF name of "Generation".

o-----------------------------------------------------------o
| C_2.` Generation Theorem` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` b o ` ` ` ` ` ` ` ` a o b ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` = ` ` ` a @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a(b)` ` ` ` = ` ` ` a(ab) ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` `Degenerate <---- | ----> Regenerate` ` ` ` ` ` |
o-----------------------------------------------------------o

Here is a proof of the Generation Theorem.

o-----------------------------------------------------------o
| C_2.` Generation Theorem. `Proof. ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` b o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< C1. Reflect "a(b)" >========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` b o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< I2. Unfold "(())" >=========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` b o ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` |/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J1. Insert "a" >============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` b o ` o a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o o a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` |/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J2. Collect "a" >===========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` b o ` o a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` |/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< C1. Reflect "a", "b" >======o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o b ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< QED >=======================o

Jon Awbrey

Posted by Jon Awbrey on 04-15-2004 03:48 AM:

Propositional Equation Reasoning Systems

PERS. Note 4

The third theorem to be proved here is one that GSB
annotates as "Integration", but I tend to regard it
as a matter of "Dominance or Recession" among forms.

o-----------------------------------------------------------o
| C_3.` Dominant Form Theorem ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a( )` ` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` `Remark <---- | ----> Recess ` ` ` ` ` ` ` `|
o-----------------------------------------------------------o

Here is a proof of the Dominant Form Theorem.

o-----------------------------------------------------------o
| C_3.` Dominant Form Theorem.` Proof.` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< C2. Regenerate "a" >========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< J1. Delete "a" >============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< QED >=======================o

If you scan the elementary steps that lead up to this point,
you will notice two distinct qualities of the proofs so far:

One brand of proof has that "falling off a log and rolling downhill"
sort of quality that is earnestly to be wished for but seldom to be
seen, at least, never so often as we'd wish.

The other kind, more kith o' death than kind, has a quality strained
past mercy, with a "how in the heck did anybody ever think of that?"
sort of subtlety that all too unfortunately rules the roost whenever
we begin to extend out practice to more and more compelling theories.

This is, to me, at least, a surprising observation,
and though I have no grand conclusion to draw from
it at the moment, it occurs to me that it might be
a useful measure to keep in mind as we essay forth.

Jon Awbrey

Posted by Jon Awbrey on 04-15-2004 12:36 PM:

Propositional Equation Reasoning Systems

PERS. Note 5

Now to take up a more interesting example,
here is the statement and a proof of the
Splendid Theorem from Leibniz that was
brought to my attention by John Sowa.

| If 'a' is 'b' and 'd' is 'c', then 'ad' will be 'bc'.
| This is a fine theorem, which is proved in this way:
|
| 'a' is 'b', therefore 'ad' is 'bd' (by what precedes),
|
| 'd' is 'c', therefore 'bd' is 'bc' (again by what precedes),
|
| 'ad' is 'bd', and 'bd' is 'bc', therefore 'ad' is 'bc'. Q.E.D.
|
| Leibniz, 'Logical Papers', page 41.
|
| Leibniz, G.W., 'Addenda to the Specimen of the Universal Calculus',
| pages 40-46 in Parkinson, G.H.R. (ed.), 'Leibniz: Logical Papers',
| Oxford University Press, London, UK, 1966. (Gerhardt, 7, p. 223).

o-----------------------------------------------------------o
| Praeclarum Theorema (Leibniz) ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` ` o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` a o ` o d ` ` o ad` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o---------o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` = ` ` ` ` ` ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| `((a(b))(d(c))((ad(bc)))) ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

And here's a neat proof of this nice theorem.

o-----------------------------------------------------------o
| Praeclarum Theorema (Leibniz).` Proof.` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` ` o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` a o ` o d ` ` o ad` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o---------o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< C1. Reflect "ad(bc)" >======o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` a o ` o d ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `ad o---------o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Weed "a", "d" >=============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `ad o---------o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< C1. Reflect "b", "c" >======o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `abcd o---------o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Weed "bc" >=================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `abcd o---------o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< C3. Recess "abcd" >=========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o---------o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< I2. Refold "(())" >=========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< QED >=======================o

Jon Awbrey

Posted by Jon Awbrey on 04-15-2004 04:01 PM:

Propositional Equation Reasoning Systems

PERS. Note 6

Let us now extend the CSP-GSB calculus in the following way:

The first extension is the "reflective extension of logical graphs",
or what I now refer to as the "cactus language", after its principal
graph-theoretic data structure. It is generated by generalizing the
negation operator "(_)" in a particular direction, treating "(_)" as
the "controlled", "moderated", or "reflective" negation operator of
order 1, and adding another such operator for each integer parameter
greater than 1. In sum, these operators are symbolized by bracketed
argument lists of the following shapes: "(_)", "(_,_)", "(_,_,_)",
and so on, where the number of slots is the order of the reflective
negation operator in question.

The formal rule of evaluation for a "k-lobe" or k-operator is:

o-----------------------------------------------------------o
| Evaluation Rule ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` x_1 `x_2` `...` x_k ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `o----o-...-o----o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `@` ` ` ` ` ` ` = ` ` ` ` ` ` `@` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ( x_1, x_2, ..., x_k )` ` = ` ` ` ` ` <space> ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` IF AND ONLY IF` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` |
| ` Just one of the x_1, x_2, ..., x_k` `=` `|` `=` `( )` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

The interpretation of these operators, read as assertions
about the values of their listed arguments, is as follows:

o-----------------------------------------------------------o
| Interpretation Rule ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` x_1 `x_2` `...` x_k ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o----o-...-o----o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| A "k-lobe operator" of the form "(x_1, ..., x_k)" ` ` ` ` |
| enjoys two commonly employed interpretations for` ` ` ` ` |
| propositional logic, in other words, two ways of` ` ` ` ` |
| taking it as an assertion about, or a constraint` ` ` ` ` |
| upon, the logical values of the listed arguments, ` ` ` ` |
| the mentioned variables x_j, for j = 1 through k. ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Existential Interpretation: ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `"Just one of the k arguments is not true." ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Entitative `Interpretation: ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `"Not just one of the k arguments is true." ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

Jon Awbrey

Posted by Jon Awbrey on 04-15-2004 04:48 PM:

Propositional Equation Reasoning Systems

PERS. Note 7

The Case Analysis-Synthesis Theorem (CAST)

The task at hand is to lay out what I think of as the pontoon bridge
between the model-theoretic and the proof-theoretic shores, and thus
between their diverging perspectives on logical procedure, even if I
can construct it at a point but so close to their common source that
it may not seem like it's worth the candle. Irregardless here it is.

The substance of this principle was known to Boole a sesquicentury ago,
tantamount to the "Boolean Expansion" that he uncovered while nameless.
So the only novelty here will rest in a certain manner of presentation,
in which I will prove the basic principle from the axioms given before.
One name for this rule is the "Case Analysis-Synthesis Theorem" (CAST).

The preparatory materials that we need are these:

I am going to revert to my customarily sloppy workshop manners
and refer to propositions and proposition expressions on rough
analogy with functions and function expressions, which implies
that a proposition will be regarded as the chief formal object
of discussion, enjoying many proposition expressions, formulas,
or sentences that express it, but worst of all I will probably
just go ahead and use any and all of these terms as loosely as
I see fit, taking a bit of extra care only when I see the need.

Let Q be a proposition with an unspecified, but context-fitting
number of variables, say, none, or x, or x_1, ..., x_k, as the
case may be. (More precisely, I should've said "sentence Q".)

Strings and graphs sans labels are called "bare".
A bare terminal node, "o", is known as a "stone".
A bare terminal edge, "|", is known as a "stick".

Let the "replacement expression" of the form "Q[o/x]" denote
the proposition that results from Q by replacing every token
of the variable x with a blank, which is to say, erasing "x".

Let the "replacement expression" of the form "Q[|/x]" denote
the proposition that results from Q by replacing every token
of the variable x with a stick stemming from the site of "x".

In the case of a proposition Q, that is, an expression of it,
not having a token of the designated variable "x", let it be
stipulated that Q[o/x] = Q = Q[|/x].

I think that I am at long last ready to state the following:

o-----------------------------------------------------------o
| Case Analysis-Synthesis Theorem (CAST)` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `x` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `x` `|` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` Q[o/x] o---o Q[|/x] ` ` |
| ` ` ` ` ` ` `Q` ` ` ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` |
| ` ` ` ` ` ` `@` ` ` ` ` ` ` = ` ` ` ` ` ` `@` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `Q` ` ` ` ` ` ` = ` ( Q[o/x] x , Q[|/x] (x) ) |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

Jon Awbrey

Posted by Jon Awbrey on 04-15-2004 06:30 PM:

Propositional Equation Reasoning Systems

PERS. Note 8

CAST (cont.)

In order to think of tackling even the roughest sketch toward a proof
of this theorem, we need to add a number of axioms and axiom schemata.
Because I abandoned proof-theoretic purity somewhere in the middle of
grinding this calculus into computational form, I never got around to
finding the most elegant and minimal, or anything near a complete set
of axioms for the "cactus language", so what I list here are just the
slimmest rudiments of the hodge-podge of "rules of thumb" that I have
found over time to be necessary and useful in most working settings.
Some of these special "precepts" are probably provable from genuine
axioms, but I have yet to go looking for a more proper formulation.

o-----------------------------------------------------------o
| Precept L_1.` Indifference` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` a ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o---o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `(a, (a)) ` ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` Split <---- | ----> Merge ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

o-----------------------------------------------------------o
| Precept L_2.` Equality. `The Following Are Equivalent:` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` b ` ` ` ` ` ` ` a ` b ` ` ` ` ` ` ` a ` ` ` ` ` |
| ` ` ` ` ` o ` ` ` ` ` ` ` o---o ` ` ` ` ` ` ` o ` ` ` ` ` |
| ` ` ` a ` | ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` | ` b ` ` ` |
| ` ` ` o---o ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` o---o ` ` ` |
| ` ` ` `\ /` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` `\ /` ` ` ` |
| ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` = ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `(a, (b)) ` ` = ` ` ((a , b)) ` ` = ` ` ((a), b)` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

o-----------------------------------------------------------o
| Precept L_3.` Dispersion` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| For k > 1, the following equation holds:` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` y_1 ` `y_2` `...` ` y_k ` ` x y_1 `x y_2` `...` x y_k ` |
| ` `o------o-...-o------o` ` ` ` `o------o-...-o------o` ` |
| ` ` \ ` ` ` ` ` ` ` ` / ` ` ` ` ` \ ` ` ` ` ` ` ` ` / ` ` |
| ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` |
| ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` |
| ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` |
| ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` |
| ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` |
| ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` |
| ` ` ` ` ` `x @` ` ` ` ` ` ` = ` ` ` ` ` ` `@` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` x (y_1, ..., y_k) ` ` ` = ` ` (x y_1, ..., x y_k) ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` Distill ` ` <---- | ----> ` ` Disperse` ` ` ` ` |
o-----------------------------------------------------------o

To see why the Dispersion Rule holds, look at it this way:
If x is true, then the presence of "x" makes no difference
on either side of the equation, but if x is false, then both
sides of the equation are false.

Here is a proof sketch for the "Case Analysis-Synthesis Theorem" (CAST):

o-----------------------------------------------------------o
| Case Analysis-Synthesis Theorem.` Proof Sketch. ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `Q` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< L1. Split " " >=============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `x` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `x` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o---o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `Q @` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< L3. Disperse "Q" >==========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `x` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `x` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `Q o---o Q` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< C1. Reflect "x" >===========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `x` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `x` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `Q o---o Q[((x))/x] ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< C2. Weed "x", "(x)" >=======o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `x` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `x ` |` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` Q[o/x] o---o Q[|/x] ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< QES >=======================o

NB. QES = "Quod Erat Sketchiendum".

Jon Awbrey

Posted by Jon Awbrey on 04-15-2004 08:45 PM:

Propositional Equation Reasoning Systems

PERS. Note 9

CAST (concl.)

Some of the jobs that the CAST can be usefully put to work on
are proving propositional theorems and establishing equations
between propositions. Once again, let us turn to the example
of Leibniz's Praeclarum Theorema as a way of illustrating how.

o-----------------------------------------------------------o
| Praeclarum Theorema.` Proof by CAST.` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` ` o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` a o ` o d ` ` o ad` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o---------o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< CAST "a" >==================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `bc ` ` ` ` ` |
| ` ` b o ` o c ` o bc` ` ` ` ` ` ` b o ` o c ` o o ` ` ` ` |
| ` ` ` | ` | ` ` | ` ` ` ` ` ` ` ` ` | ` | ` ` |/` ` ` ` ` |
| ` ` ` o ` o d ` o d ` ` ` ` ` ` `o--o ` o d ` o d ` ` ` ` |
| ` ` ` `\ /` ` ` | ` ` ` ` ` ` ` ` ` `\ /` ` ` | ` ` ` ` ` |
| ` ` ` ` o-------o ` ` ` ` ` ` ` ` ` ` o-------o ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` a o-----------------------------o---o a ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Domination >================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` o bc` ` ` ` ` ` ` ` ` ` o c ` ` o ` ` ` ` |
| ` ` ` | ` | ` ` | ` ` ` ` ` ` ` ` ` ` ` | ` ` `/` ` ` ` ` |
| ` ` ` o ` o d ` o d ` ` ` ` ` ` `o--o ` o d ` o ` ` ` ` ` |
| ` ` ` `\ /` ` ` | ` ` ` ` ` ` ` ` ` `\ /` ` ` | ` ` ` ` ` |
| ` ` ` ` o-------o ` ` ` ` ` ` ` ` ` ` o-------o ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` a o-----------------------------o---o a ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Cancellation >==============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` o bc` ` ` ` ` ` ` ` ` ` o c ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` | ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` |
| ` ` ` o ` o d ` o d ` ` ` ` ` ` ` ` ` ` o d ` ` ` ` ` ` ` |
| ` ` ` `\ / ` ` `| ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` o-------o ` ` ` ` ` ` ` ` ` ` o-------o ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` a o-----------------------------o---o a ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Domination >================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o d ` o d ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o-------o ` ` ` ` ` ` ` ` ` ` o-------o ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` a o-----------------------------o---o a ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Cancellation >==============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o d ` o d ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o-------o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` a o-----------------------------o---o a ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< CAST "d" >==================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` b ` c ` `bc ` ` ` ` ` ` ` |
| ` ` b o ` o c ` o bc` ` ` ` ` ` o ` o o ` o o ` ` ` ` ` ` |
| ` ` ` | ` | ` ` | ` ` ` ` ` ` ` | ` |/` ` |/` ` ` ` ` ` ` |
| ` ` ` o ` o ` ` o ` ` ` ` ` ` ` o ` o ` ` o ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` | ` ` ` ` ` ` ` `\ /` ` ` | ` ` ` ` ` ` ` |
| ` ` ` ` o-------o ` ` ` ` ` ` ` ` o-------o ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` d o-------------------------o---o d ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `a o---o---o a` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Domination >================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` b ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` o bc` ` ` ` ` ` o ` ` o ` ` o ` ` ` ` ` ` |
| ` ` ` | ` | ` ` | ` ` ` ` ` ` ` | ` `/` ` `/` ` ` ` ` ` ` |
| ` ` ` o ` o ` ` o ` ` ` ` ` ` ` o ` o ` ` o ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` | ` ` ` ` ` ` ` `\ /` ` ` | ` ` ` ` ` ` ` |
| ` ` ` ` o-------o ` ` ` ` ` ` ` ` o-------o ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` d o-------------------------o---o d ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `a o---o---o a` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Cancellation >==============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` b ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` o bc` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` | ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o ` ` o ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` | ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o-------o ` ` ` ` ` ` ` ` o-------o ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` d o-------------------------o---o d ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `a o---o---o a` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Domination >================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o-------o ` ` ` ` ` ` ` ` o-------o ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` d o-------------------------o---o d ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `a o---o---o a` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Cancellation >==============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o-------o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` d o-------------------------o---o d ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `a o---o---o a` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< CAST "b" >==================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` |
| ` ` o ` o c ` o c ` ` ` ` ` ` o ` o c ` o c ` ` ` ` ` ` ` |
| ` ` | ` | ` ` | ` ` ` ` ` ` ` | ` | ` ` | ` ` ` ` ` ` ` ` |
| ` ` o ` o ` ` o ` ` ` ` ` ` ` o ` o ` ` o ` ` ` ` ` ` ` ` |
| ` ` `\ /` ` ` | ` ` ` ` ` ` ` `\ /` ` ` | ` ` ` ` ` ` ` ` |
| ` ` ` o-------o ` ` ` ` ` ` ` ` o-------o ` ` ` ` ` ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o-------------------------o---o b ` ` ` ` ` ` ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `d o---o---o d` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `a o---o---o a` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Cancellation >==============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` |
| ` ` ` ` o c ` o c ` ` ` ` ` ` ` ` o c ` o c ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` o ` ` ` ` ` ` ` o ` o ` ` o ` ` ` ` ` ` ` ` |
| ` ` ` `/` ` ` | ` ` ` ` ` ` ` `\ /` ` ` | ` ` ` ` ` ` ` ` |
| ` ` ` o-------o ` ` ` ` ` ` ` ` o-------o ` ` ` ` ` ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o-------------------------o---o b ` ` ` ` ` ` ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `d o---o---o d` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `a o---o---o a` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Domination >================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o c ` o c ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` o ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `/` ` ` | ` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o-------o ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o-------------------------o---o b ` ` ` ` ` ` ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `d o---o---o d` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `a o---o---o a` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Cancellation >==============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o c ` o c ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `/` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o-------o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o-------------------------o---o b ` ` ` ` ` ` ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `d o---o---o d` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `a o---o---o a` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< CAST "c" >==================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` ` o ` ` ` ` ` ` ` ` ` o ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` ` o ` ` ` ` ` ` ` ` ` o ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` `/` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` o-------o ` ` ` ` ` ` ` ` o-------o ` ` ` ` ` ` ` ` ` |
| ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` c o-------------------------o---o c ` ` ` ` ` ` ` ` ` ` |
| ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `b o---o---o b` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `d o---o---o d` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `a o---o---o a` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Cancellation >==============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` o ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` o-------o ` ` ` ` ` ` ` ` o-------o ` ` ` ` ` ` ` ` ` |
| ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` | ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` c o-------------------------o---o c ` ` ` ` ` ` ` ` ` ` |
| ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `b o---o---o b` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `d o---o---o d` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `a o---o---o a` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Cancellation >==============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `c o---o---o c` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `b o---o---o b` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `d o---o---o d` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `a o---o---o a` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< QED >=======================o

What we have harvested is the succulent equivalent of
a "disjunctive normal form" (DNF) for the proposition
with which we started. Remembering that a blank node
is the graphical equivalent of a logical value "true",
we can read this brand of DNF in the following manner:

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `c o---o---o c` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `b o---o---o b` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `d o---o---o d` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `a o---o---o a` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Either not 'a' and thus 'true'` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` Or ` ` 'a' and thus ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `Either not 'd' and thus 'true' ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `Or ` ` 'd' and thus` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` Either not 'b' and thus 'true'` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` Or ` ` 'b' and thus ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `Either not 'c' and thus 'true' ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `Or ` ` 'c' and thus true.` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

That is tantamount to saying that the proposition
being submitted for analysis is true in each case.
Ergo we are justly entitled to title it "Theorem".

Jon Awbrey

Posted by Jon Awbrey on 04-16-2004 03:24 AM:

Propositional Equation Reasoning Systems

PERS. Note 10

Logic As Sign Transformation

We have been looking at various ways of transforming propositional
expressions, expressed in the parallel formats of character strings
and graphical structures, all the while preserving certain aspects of
their "meaning" -- and here I risk using that vaguest of all possible
words, but only as a promissory note, hopefully to be cached out in
a more meaningful species of currency as the discussion develops.

I cannot pretend to be acquainted with or to comprehend every form
of intension that others might find of interest in a given form of
expression, nor can I speak for every form of meaning that another
might find in a given form of syntax. The best that I can hope to
do is to specify what my object is in using these expressions, and
to say what aspects of their syntax are meant to serve this object,
lending these properties the interest I have in preserving them as
I put the expressions through the paces of their transformations.

On behalf of this object I have been spinning in the form
of this thread a developing example base of propositional
expressions, in the data structures of graphs and strings,
along with many examples of step-wise transformations on
these expressions that preserve something of significant
logical import, something that might be referred to as
their "logical equivalence class" (LEC), and that we
could as well call the "constraint information" or
the "denotative object" of the expression in view.

Jon Awbrey

Posted by Jon Awbrey on 04-16-2004 04:00 PM:

Propositional Equation Reasoning Systems

PERS. Note 11

Logic As Sign Transformation

To focus still more, let us return to that Splendid Theorem
noted by Leibniz, and let us look more carefully at the two
distinct ways of transforming its initial expression that
we just used to arrive at an equivalent expression, one
that made its tautologous character or its theorematic
nature as evident as it could be.

Just to remind you, here is the Splendid Theorem again:

o-----------------------------------------------------------o
| Praeclarum Theorema (PT)` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` b o ` o c ` ` o bc` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` a o ` o d ` ` o ad` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o---------o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` ` ` ` ` ` = ` ` ` ` ` ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| `((a(b))(d(c))((ad(bc)))) ` = ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

The first way of transforming the expression that appears
on the left hand side of the equation can be described as
"proof-theoretic" in character. That was given in Note 5.

PERS 5. http://forum.wolframscience.com/sho...tid=957#post957

The other way of transforming the expression that appears
on the left hand side of the equation can be described as
"model-theoretic" in character. That was given in Note 9.

PERS 9. http://forum.wolframscience.com/sho...tid=962#post962

What we have here amounts to a couple of different styles
of "communicational conduct", or "conductive communication",
if you prefer, that is to say, two sequences of signs of the
form e_1, e_2, ..., e_n, each one beginning with a problematic
expression and eventually ending with a clear expression of the
appropriate "logical equivalence class" (LEC) to which each and
every sign or expression in the sequence belongs.

Ordinarily, any orbit through a locus of signs can be taken
to reflect an underlying sign-process, a case of "semiosis".
So what we have here are two very special cases of semiosis,
and what we might just find it useful to contemplate is how
to characterize them as two species of a very general class.

We are starting to delve into some fairly picayune details
of a particular sign system, non-trivial enough in its own
right but still rather simple compared to the types of our
ultimate interest, and though I believe that this exercise
will be worth the effort in prospect of understanding more
complicated sign systems, I feel that I ought to say a few
words about the larger reasons for going through this work.

My broader interest lies in the theory of inquiry as a special
application or a special case of the theory of signs. Another
name for the theory of inquiry is "logic" and another name for
the theory of signs is "semiotics". So I might as well have
said that I am interested in logic as a special application
or a special case of semiotics. But what sort of a special
application? What sort of a special case? Well, I think
of logic as "formal semiotics" -- though, of course, I am
not the first to have said such a thing -- and by "formal"
we say, in our etymological way, that logic is concerned
with the "form", indeed, with the "animate beauty" and
the very "life force" of signs and sign actions. Yes,
perhaps that is far too Latin a way of understanding
logic, but it's all I've got.

Now, if you think about these things just a little more,
I know that you will find them just a little suspicious,
for what besides logic would I use to do this theory of
signs that I would apply to this theory of inquiry that
I'm also calling "logic"? But that is precisely one of
the things signified by the word "formal", for what I'd
be required to use would have to be some brand of logic,
that is, some sort of innate or inured skill at inquiry,
but a style of logic that is casual, catch-as-catch-can,
formative, incipient, inchoate, unformalized, a work in
progress, partially built into our natural language and
partially more primitive than our most artless language.
In so far as I use it more than mention it, mention it
more than describe it, and describe it more than fully
formalize it, then to that extent it must be consigned
to the realm of unformalized and unreflective logic,
where some say "there be oracles", but I don't know.

Still, one of the aims of formalizing what acts of reasoning
that we can is to draw them into an arena where we can examine
them more carefully, perhaps to get better at their performance
than we can unreflectively, and thus to live, to formalize again
another day. Formalization is not the be-all end-all of human
life, not by a long shot, but it has its uses on that behalf.

Jon Awbrey

Posted by Jon Awbrey on 04-16-2004 05:00 PM:

Propositional Equation Reasoning Systems

PERS. Note 12

Logic As Sign Transformation

This looks like a good place to pause and take stock.
The question arises: What is really going on here?
We have all these signs, but what is the object?

One object worth the candle is simply to study
a non-trivial example of a syntactic system,
simple in design but not entirely a toy,
just to see how these systems tick.

More than that, we would like to understand how sign systems
come to exist or can be placed in relation to object systems,
in the likes of which we possess some compelling independent
reason to take an interest.

What is the utility of setting up sets of strings and sets of graphs,
and sorting them according to their "semiotic equivalence class" (SEC)
based on this or that abstract notion of transformational equivalence?

Good questions.

I can only begin to address these questions in the present frame of work,
but I can't hope to answer them in anything like a satisfactory fashion.
Neverthless, I will not mind one bit if you keep them in mind as we go.

Jon Awbrey

Posted by Jon Awbrey on 04-16-2004 08:12 PM:

Propositional Equation Reasoning Systems

PERS. Note 13

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 ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `(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

Posted by Jon Awbrey on 04-17-2004 04:54 AM:

Propositional Equation Reasoning Systems

PERS. Note 14

Analysis of Contingent Expressions (cont.)

Let us now examine the propositional equation:

o-----------------------------------------------------------o
| Equation E_1` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` q r ` ` ` ` ` ` |
| ` ` ` ` `q o` `o r` ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` |
| ` ` ` ` ` `|` `|` ` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` |
| ` ` ` ` `p o` `o p` ` ` ` ` ` ` ` ` ` ` `p o` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` |
| ` ` ` ` ` ` `@` ` ` ` ` ` ` = ` ` ` ` ` ` `@` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` (p (q)) (p (r)) ` ` ` = ` ` ` ` `(p `(q r)) ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` [p=>q] & [p=>r] ` ` ` = ` ` ` ` `[p=>[q&r]] ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

There are three distinct ways that I can think of right off as to
how we might go about formally proving or systematically checking
the proposed equivalence, the evidence of whose truth we already
have before us clearly enough, and in a visually intuitive form,
from the venn diagrams that we examined above.

While we go through each of these ways let us keep one eye out
for the character and the conduct of each type of proceeding as
a semiotic process, that is, as an orbit, in this case discrete,
through a locus of signs, in this case propositional expressions,
and as it happens in this case, a sequence of transformations that
perseveres in the denotative objective of each expression, that is,
in the abstract proposition that it expresses, while it preserves
the informed constraint on the universe of discourse that gives
us one viable candidate for the informational content of each
expression in the interpretive chain of sign metamorphoses.

A "sign relation" L is a subset of a cartesian product O x S x I, where
O, S, I are sets known as the "object", "sign", and "interpretant sign"
domains, respectively. One writes "L c O x S x I", where the symbol "c"
is the Ascii symbol for the subset relation, "contained as a subset of".
Accordingly, a sign relation L consists of ordered triples of the form
<o, s, i>, where o, s, i belong to the domains O, S, I, respectively.

The "syntactic domain" of a sign relation L c O x S x I is
just the set-theoretic union S |_| I of its sign domain S
and its interpretant domain I.

It is not uncommon, especially in formal examples,
for the sign domain and the interpretant domain
to be equal as sets, in short, to have S = I.

Elsewhere I have discussed examples of sign relations that consist
of a finite set of triples of the form <o, s, i>, where o, s, i are
the "object", "sign", "interpretant sign", respectively, of what is
called the "sign triple" or the "elementary sign relation" <o, s, i>.

We will be taking a bit of a jump up from the finite case now,
since most of the examples of sign relations that interest us
in logic will have S and I being infinite sets, and usually O
will be infinite, too, in the long run, at least, although we
will frequently work up to the infinite object domains by way
of various series of finite approximations and gradual stages.

With that preamble behind us, let us turn to consider the case
of semiosis, or sign transformation process, that is generated
by our first proof of the propositional equation E_1.

o-----------------------------------------------------------o
| Equation E_1. `Proof 1. ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `q o` `o r` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `p o` `o p` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` (p (q)) (p (r)) ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Double Negation >===========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `q o` `o r` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `p o` `o p` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `(( (p (q)) (p (r)) ))` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Collection >================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `q o` `o r` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `p o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `(p ( ((q)) ((r)) ))` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Double Negation >===========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` q r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `p o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `(p (q r))` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< QED >=======================o

For some reason I always think of this
as the way that our DNA would prove it.

Jon Awbrey

Posted by Jon Awbrey on 04-17-2004 07:16 PM:

Propositional Equation Reasoning Systems

PERS. Note 15

Analysis of Contingent Expressions (cont.)

We are in the process of examining various proofs of the
propositional equation "(p (q))(p (r)) = (p (q r))", and
viewing them in the light of their character as semiotic
processes, in essence, as sign-theoretic transformations.

Here is a reminder of the equation in question:

o-----------------------------------------------------------o
| Equation E_1` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` q r ` ` ` ` ` ` |
| ` ` ` ` `q o` `o r` ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` |
| ` ` ` ` ` `|` `|` ` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` |
| ` ` ` ` `p o` `o p` ` ` ` ` ` ` ` ` ` ` `p o` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` |
| ` ` ` ` ` ` `@` ` ` ` ` ` ` = ` ` ` ` ` ` `@` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` (p (q)) (p (r)) ` ` ` = ` ` ` ` `(p `(q r)) ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` [p=>q] & [p=>r] ` ` ` = ` ` ` ` `[p=>[q&r]] ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

The second way of establishing the truth of this equation is
one that I see, rather loosely, as "model-theoretic", for no
better reason than the sense of its ending with a pattern of
expression, a variant of the "disjunctive normal form" (DNF),
that is commonly recognized to be the form that one extracts
from a truth table by pulling out the rows of the table that
evaluate to true and constructing the disjunctive expression
that sums up the senses of the corresponding interpretations.

In order to apply this "model-theoretic" method to an equation
between a couple of contingent expressions, one must transform
each expression into its associated DNF and then compare those
to see if they are equal. In the current setting, these DNF's
may indeed end up as identical expressions, but it is possible,
also, for them to turn out slightly off-kilter from each other,
and so the ultimate comparison may not be absolutely immediate.
The explanation of this is that, for the sake of computational
efficiency, it is useful to tailor the DNF that gets developed
as the output of a DNF algorithm to the particular form of the
propositional expression that is given as input.

o-----------------------------------------------------------o
| Equation E_1. `Proof 2, 1st Half. ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` q o ` o r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` | ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` p o ` o p ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< CAST "p" >==================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` q ` r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q o ` o r ` o o ` o o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` | ` | ` ` `\| ` |/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` o ` ` ` o ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\ /` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` p o-----------o---o p ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Domination >================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q o ` o r ` o ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` | ` | ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` o ` ` ` o ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\ /` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` p o-----------o---o p ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Cancellation >==============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q o ` o r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` | ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` p o-----------o---o p ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< CAST "q" >==================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o r ` ` o ` o r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` | ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o ` ` ` o ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\ /` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` q o-----------o---o q ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p o-------o---o p ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Cancellation >==============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o r ` ` ` ` o r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` ` ` o ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `/` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` q o-----------o---o q ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p o-------o---o p ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Domination >================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `/` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` q o-----------o---o q ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p o-------o---o p ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< CAST "r" >==================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `/` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` r o-----------o---o r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `\` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` / ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\`/` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q o-------o---o q ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p o-------o---o p ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Cancellation >==============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` r o-------o---o r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` / ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\ /` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q o-------o---o q ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p o-------o---o p ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< DNF >=======================o

What we have harvested is the succulent equivalent of
a "disjunctive normal form" (DNF) for the proposition
with which we started. Remembering that a blank node
is the graphical equivalent of a logical value "true",
we can read this brand of DNF in the following manner:

o-----------------------------------------------------------o
| DNF of "(p (q))(p (r))" ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` r o-------o---o r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` / ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\ /` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q o-------o---o q ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p o-------o---o p ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Either not 'p' and thus 'true'` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` Or ` ` 'p' and thus ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `Either not 'q' and thus 'false'` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `Or ` ` 'q' and thus` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` Either not 'r' and thus 'false' ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` Or ` ` 'r' and thus 'true'. ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

Sorry, the half-time show was cancelled by the censors.
But I'm guessing that the reader can probably finish off
the second half of the proof with a few scribbles on paper
faster than I can asciify it on my own, so at least there's
that entertainment to occupy the interval.

Jon Awbrey

Posted by Jon Awbrey on 04-17-2004 08:42 PM:

Propositional Equation Reasoning Systems

PERS. Note 16

Analysis of Contingent Expressions (cont.)

We are still in the middle of contemplating a particular example
of a propositional equation, namely, "(p (q))(p (r)) = (p (q r))",
and we are still considering the second of three formal methods that
I intend to illustrate in the process of thrice-over establishing it.

o-----------------------------------------------------------o
| Equation E_1` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` q r ` ` ` ` ` ` |
| ` ` ` ` `q o` `o r` ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` |
| ` ` ` ` ` `|` `|` ` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` |
| ` ` ` ` `p o` `o p` ` ` ` ` ` ` ` ` ` ` `p o` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` |
| ` ` ` ` ` ` `@` ` ` ` ` ` ` = ` ` ` ` ` ` `@` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` (p (q)) (p (r)) ` ` ` = ` ` ` ` `(p `(q r)) ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` [p=>q] & [p=>r] ` ` ` = ` ` ` ` `[p=>[q&r]] ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

I know that it must seem tedious, but I probably ought to go ahead
and carry out the second half of this analogically "model-theoretic"
strategy, just so that we will have the security of this concrete and
shared experience on which to fall back at every later point in what
may quickly become a rather abstruse discussion. Here then is the
rest of the necessary chain of equations:

o-----------------------------------------------------------o
| Equation E_1. `Proof 2, 2nd Half. ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `q r` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` p o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< CAST "p" >==================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `q r` ` ` q r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` o ` ` o o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` | ` ` `\| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` o ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p o-------o---o p ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Domination >================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `q r` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` o ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` | ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` o ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p o-------o---o p ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Cancellation >==============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `q r` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p o-------o---o p ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< CAST "q" >==================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o r ` ` o r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q o-------o---o q ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p o-------o---o p ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Domination >================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o r ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q o-------o---o q ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p o-------o---o p ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Cancellation >==============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q o-------o---o q ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p o-------o---o p ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< CAST "r" >==================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` r o-------o---o r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` / ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\ /` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q o-------o---o q ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p o-------o---o p ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< Cancellation >==============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` r o-------o---o r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` / ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\ /` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q o-------o---o q ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` p o-------o---o p ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o=============================< DNF >=======================o

This is not only a logically equivalent DNF,
but exactly the same DNF expression that we
obtained before, so we have established the
given equation "(p (q))(p (r)) = (p (q r))".

Incidentally, one may wish to note that this DNF
expression quickly folds into the following form:

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` pqr o-------o---o p ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` (p q r, (p))` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

This can be read to say "either p q r, or not p",
which gives us yet another expression equivalent
to the sentences "(p (q))(p (r))" and "(p (q r))".

Still another way of writing the same thing would be like so:

o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` p o-------o pqr ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ((p , p q r)) ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

In other words, "p is equivalent to p and q and r".

Jon Awbrey

Posted by Jon Awbrey on 04-18-2004 03:12 AM:

Propositional Equation Reasoning Systems

PERS. Note 17

Analysis of Contingent Expressions (cont.)

Let's pause to refresh ourselves with a few morsels of lemmas bread.
One lemma that I can see just far enough ahead to see our imminent
need of is the principle that I canonize as the "Emptiness Rule".
It says that a bare lobe expression like "(... , ...)", with any
number of places for arguments but nothing but blanks as filler,
is logically tantamount to the proto-typical expression of its
type, namely, the constant expression "()" that ExG interprets
as denoting the logical value "false". To depict the rule in
graphical form, we have the continuing sequence of equations:

o-----------------------------------------------------------o
| Emptiness Rule` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` o---o ` ` ` o-o-o ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `\ /` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` = ` ` @ ` ` = ` ` @ ` ` = ` `...` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

Yet another rule that we'll need is the following:

o-----------------------------------------------------------o
| Indistinctness Rule ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` a ` a ` ` ` a a a ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` o---o ` ` ` o-o-o ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `\ /` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` = ` ` @ ` ` = ` ` @ ` ` = ` `...` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

This one is easy enough to derive from rules that are
already known, but call it the "Indistinctness Rule"
just on behalf of ready reference and employment.

Finally, let me introduce a rule-of-thumb that is a bit
more suited to routine computation, and that will serve
to replace the indistinctness rule in many of the cases
where we actually have to call on it. This is actually
just a special case of the evaluation rule listed above:

o-----------------------------------------------------------o
| Evaluation Rule ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` | `x_2` `... x_k` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` o---o-...-o---o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` x_2 ... x_k ` ` ` ` |
| ` ` ` ` ` ` `@` ` ` ` ` ` ` = ` ` ` ` ` ` `@` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ((), x_2, ..., x_k) ` ` = ` ` ` ` x_2 ... x_k ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` `Setup` ` ` <---- | ----> ` ` `Spike` ` ` ` ` ` |
o-----------------------------------------------------------o

Jon Awbrey

Posted by Jon Awbrey on 04-18-2004 06:28 PM:

Propositional Equation Reasoning Systems

PERS. Note 18

Analysis of Contingent Expressions (concl.)

To continue with the beating of this still kicking horse that is
known as the propositional equation "(p (q))(p (r)) = (p (q r))",
let's now take up the "third way" that I mentioned for examining
propositional equations, but I believe that you will be relieved
to know that it is literally a third way only at the very outset,
almost immediately breaking up according to whether one proceeds
by way of the more "routine" model-theoretic path or else by way
of the more "strategic" proof-theoretic path. I think that I'll
take the low road today.

Let's convert the equation between propositions:

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

into the corresponding equational proposition:

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

If you're like me, you'd rather see it in pictures:

o-----------------------------------------------------------o
| Equation E_1, Written as an Equation in Cactus Language ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `q o` `o r` `q o r` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `|` `|` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `p o` `o p` `p o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o---------o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` `\ /` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `(( `(p (q))` (p (r)) ` , ` (p` (q r))` ))` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` [[p=>q] & [p=>r]] `<=>` [p=>[q&r]]` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o

We may now interrogate the alleged equation for the third time,
working by way of the "case analysis-synthesis theorem" (CAST).

o-----------------------------------------------------------o
| Equation E_1. `Proof 3. ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` q o ` o r ` q o r ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` | ` | ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` p o ` o p ` p o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\ /` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o---------o ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\ ` ` ` /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \` ` `/ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\ ` /` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o==================================< CAST "p" >=============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `q` `r` ` q r ` `q` `r` ` qr` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` `o` ` `o` ` `o o o o` `o o` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `|` `|` ` `|` ` `|/ `|/ ` `|/ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` `o` ` `o` ` `o` `o` ` `o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ / ` ` `|` ` ` \ / ` ` `|` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o-------o` ` ` `o-------o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` / ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` `/` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ / ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `p o---------------o---o p` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o==================================< Domination >===========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `q` `r` ` q r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` `o` ` `o` ` ` `o` `o` ` `o` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `|` `|` ` `|` ` ` / ` / ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` `o` ` `o` ` `o` `o` ` `o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ / ` ` `|` ` ` \ / ` ` `|` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o-------o` ` ` `o-------o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` / ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` `/` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ / ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `p o---------------o---o p` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o==================================< Cancellation >=========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `q` `r` ` q r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `|` `|` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ / ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o-------o` ` ` `o-------o` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` / ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` `/` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ / ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `p o---------------o---o p` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o==================================< Emptiness >============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `q` `r` ` q r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `|` `|` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ / ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o-------o` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` / ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` `/` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ / ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `p o---------------o---o p` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o==================================< Cancellation >=========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `q` `r` ` q r ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `|` `|` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ / ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o-------o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `p o---------------o---o p` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o==================================< CAST "q" >=============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `r` ` `r` ` `|` `r` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `o` `o` ` `o` ` `o` `o` ` `o r` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `|` `|` ` `|` ` `|` `|` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `o` `o` ` `o` ` `o` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` \ / ` ` `|` ` ` \ / ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `o-------o` ` ` `o-------o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` ` / ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` `/` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ / ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `q o---------------o---o q` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `p o-------o---o p` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o==================================< Domination >===========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `r` ` `r` ` `|` `r` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `o` `o` ` `o` ` `o` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `|` `|` ` `|` ` `|` `|` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `o` `o` ` `o` ` `o` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` \ / ` ` `|` ` ` \ / ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `o-------o` ` ` `o-------o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` ` / ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` `/` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ / ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `q o---------------o---o q` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `p o-------o---o p` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o==================================< Cancellation >=========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `r` ` `r` ` ` ` `r` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` ` `o` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `|` ` `|` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` ` `o` ` `o` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` / ` ` `|` ` ` \ / ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `o-------o` ` ` `o-------o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` ` / ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` `/` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ / ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `q o---------------o---o q` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `p o-------o---o p` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o==================================< Domination >===========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `r` ` `r` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `|` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` ` `o` ` `o` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` / ` ` `|` ` ` \ ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `o-------o` ` ` `o-------o` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` ` / ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` `/` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ / ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `q o---------------o---o q` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `p o-------o---o p` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o==================================< Spike >================o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `r` ` `r` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `|` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` / ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `o-------o` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` ` / ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` `/` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ / ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `q o---------------o---o q` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `p o-------o---o p` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o==================================< Cancellation >=========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `r` ` `r` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `|` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` / ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `o-------o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `q o---------------o---o q` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `p o-------o---o p` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o==================================< CAST "r" >=============o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `|` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `o` ` `o` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `|` ` `|` ` ` ` `|` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `o` ` `o` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` / ` ` `|` ` ` ` / ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `o-------o` ` ` `o-------o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` \ ` ` / ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `\` `/` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` \ / ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `o` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `r o---------------o---o r` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `q o-------o---o q` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `p o-------o---o p` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o==================================< Cancellation >=========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `o-------o` ` ` `o-------o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` \ ` ` / ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `\` `/` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` \ / ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `o` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `r o---------------o---o r` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `q o-------o---o q` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `p o-------o---o p` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o==================================< Emptiness & Spike >====o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `o` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `o` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `|` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `r o---------------o---o r` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `q o-------o---o q` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `p o-------o---o p` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o==================================< Cancellation >=========o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `r o-------o---o r` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `q o-------o---o q` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `p o-------o---o p` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o==================================< QED >==================o

And that, of course, is the DNF of a theorem.

Jon Awbrey

Posted by Jon Awbrey on 04-19-2004 12:34 AM:

Propositional Equation Reasoning Systems

PERS. Note 19

Proof as Semiosis

We have been looking at several different ways of proving
one particular example of a propositional equation, and
along the way we have been exemplifying the species of
sign transforming process that is commonly known as
a "proof", more specifically, an equational proof
of the propositional equation at issue.

Let us now draw out these semiotic features of
the business of proof and place them in relief.

Our syntactic domain S contains an infinite number of signs
or expressions, which we may choose to view in either their
text or their graphic forms, glossing over for now the many
details of their parsicular correspondence.

Here are some of the expressions that
we find salient enough to single out
and confer an epithetic nickname on:

| e_0 = "()"
|
| e_1 = " "
|
| e_2 = "(p (q))(p (r))"
|
| e_3 = "(p (q r))"
|
| e_4 = "(p q r, (p))"
|
| e_5 = "(( (p (q))(p (r)) , (p (q r)) ))"

Under ExG we have the following interpretations:

| e_0 expresses the logical constant "false"
|
| e_1 expresses the logical constant "true"
|
| e_2 says "not p without q, and not p without r"
|
| e_3 says "not p without q and r"
|
| e_4 says "p and q and r, or else not p"
|
| e_5 says that e_2 and e_3 say the same thing

We took up the Equation E_1 that read as follows:

(p (q))(p (r)) = (p (q r)).

Each of our proofs is a finite sequence of signs,
and thus, for a finite integer n, takes the form:

s_1, s_2, s_3, ..., s_n.

Proof 1 proceeded by the "straightforward approach",
starting with e_2 as s_1 and ending with e_3 as s_n,
That is, it commenced from the sign "(p (q))(p (r))"
and ended up at the sign "(p (q r))" by legal moves.

Proof 2 lit on by "burning the candle at both ends",
changing e_2 into a normal form that reduced to e_4,
changing e_3 into a normal form that reduced to e_4,
in this way tethering e_2 and e_3 to a common stake.
In more detail, one route went from "(p (q))(p (r))"
to "(p q r, (p))", and another went from "(p (q r))"
to "(p q r, (p))", thus equating the two departures.

Proof 3 took the path of reflection, expressing the
met-equation between e_2 and e_3 in the naturalized
equation e_5, then taking e_5 as s_1 and exchanging
it by dint of value preserving steps for e_1 as s_n.
Thus we went from "(( (p (q))(p (r)) , (p (q r)) ))"
to the blank expression that ExG recognizes as true.

Review:

Pf 1. PERS 14. http://forum.wolframscience.com/sho...tid=973#post973
Pf 2a. PERS 15. http://forum.wolframscience.com/sho...tid=976#post976
Pf 2b. PERS 16. http://forum.wolframscience.com/sho...tid=977#post977
Pf 3. PERS 18. http://forum.wolframscience.com/sho...tid=988#post988

Jon Awbrey

Posted by Jon Awbrey on 04-19-2004 06:08 PM:

Propositional Equation Reasoning Systems

PERS. Note 20

Computation and Inference as Semiosis

Equational reasoning, as distinguished from implicational reasoning,
is well-evolved in mathematics today but grievously short-schrifted
in contemporary logic textbooks. Consequently, it may be advisable
for me to draw out and place in relief some of the more distinctive
characters of equational inference that may have passed beneath the
notice of a casual reading of these notes.

By way of a very preliminary orientation, let us consider the
distinction between an "information maintaining process" (IMP)
and an "information reducing process" (IRP). To conform with
prudent practice, let's make our first acquaintance with this
difference in the medium of some concrete and simple examples.

Example 1. Modus Ponens

IRP Version:

p => q
p
-------
q

IMP Version:

p => q
p
=======
p q

I think it is probably best if
I let that one sink in a while.

Text added later. Questions from a reader's private message
tell me that I ought to supply some exposition at this point.

A display of the form:

Expression_1
Expression_2
-------------
Expression_3

is used to state a "rule of inference" (ROI).
The expressions above the line of inference
are called "premisses" and the expression
below the line is called a "conclusion",
(also "outcome", "result", or "upshot").

If the ROI in question is succinct enough, one may write it
in-line, as "Premiss_1, Premiss_2 |- Conclusion", where the
symbol "|-" is called the "(proof-theoretic) turnstile".

Either way, one reads such a ROI in the following manner:
"From Expression_1 and Expression_2, infer Expression_3".

Looking to our first Example, the ROI that is classically
known as "modus ponens" says the following: If one has
that p implies q, and one has that p is true, then one
has a "way of putting it forward" that q is true.

Modus ponens is an "illative" or "implicational" rule.
Passage through its turnstile incurs the toll of some
information loss, and thus from a fact of q alone one
cannot infer with any degree of certainty that p => q
and p are the reasons why q might indeed happen to be.

Further considerations along these lines may lead us to appreciate
the difference between "implicational rules of inference" (IROI's)
and "equational rules of inference" (EROI's), the latter indicated
by an "equational line of inference" (ELOI) or 2-way turnstile ||-.

Jon Awbrey

Posted by Jon Awbrey on 04-20-2004 03:20 AM:

Propositional Equation Reasoning Systems

PERS. Note 21

Variations on a Theme of Transitivity

The next Example is extremely important, and for reasons
that reach well beyond the level of propositional calculus
as it is ordinarily conceived. But it's slightly tricky to
get all of the details right, so it will be worth taking the
trouble to look at it from several different angles and as it
appears in diverse frames, genres, or styles of representation.

In discussing this Example, it is convenient to observe that
the implication relation that is ordinarily indicated by the
propositional form x => y is equivalent to an order relation
x =< y on boolean values, where 0 is taken to be less than 1.

Example 2. Transitivity

IRP Version:

p =< q
q =< r
--------
p =< r

IMP Version:

p =< q
q =< r
=============
p =< q =< r

In stating the IMP analogue of transitivity,
I have taken advantage of a common idiom in
the use of order relation symbols, one that
represents their logical conjunction by way
of a catenated syntax. Thus, "p =< q =< r"
means that p =< q and that q =< r, and this
claims more than the 2-adic relation p =< r.

To study the differences between these two versions of
transitivity within what is locally a familiar setting,
let's view the propositional forms involved as if they
were elementary cellular automaton rules, resulting in
the following Table.

Table 21. Composite and Compiled Order Relations
o---------o------------o-----------------o----------------o-------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` `| L_5 ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` `| ` ` ` ` ` ` |
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` `| Order ` ` ` |
o---------o------------o-----------------o----------------o-------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` `| ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` `| ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` `| ` ` ` ` ` ` |
o---------o------------o-----------------o----------------o-------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` `| ` ` ` ` ` ` |
| q_207 ` | q_11001111 | 1 1 0 0 1 1 1 1 | (p` `(q)) ` ` `| p =< q` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` `| ` ` ` ` ` ` |
| q_187 ` | q_10111011 | 1 0 1 1 1 0 1 1 | ` ` `(q ` (r)) | q =< r` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` `| ` ` ` ` ` ` |
| q_175 ` | q_10101111 | 1 0 1 0 1 1 1 1 | (p` ` ` ` (r)) | p =< r` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` `| ` ` ` ` ` ` |
| q_139 ` | q_10001011 | 1 0 0 0 1 0 1 1 | (p (q))(q (r)) | p =< q =< r |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` `| ` ` ` ` ` ` |
o---------o------------o-----------------o----------------o-------------o

Jon Awbrey

Posted by Jon Awbrey on 04-20-2004 03:00 PM:

Propositional Equation Reasoning Systems

PERS. Note 22

Transitivity (cont.)

Taking up another angle of incidence by way of extra perspective,
let us now reflect on the venn diagrams of our four propositions.

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
q_207. (p (q))

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
q_187. (q (r))

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
q_175. (p (r))

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
q_139. (p (q))(q (r))

Among other things, these images make it visually obvious that the
constraint on the three boolean variables p, q, r that we indicate
by asserting either of the forms "(p (q))(q (r))" or "p =< q =< r"
is one that implies a constraint on the two boolean variables p, r
that we indicate by either of the forms "(p (r))" or "p =< r", but
that it imposes additional constraints on these variables that are
not captured by the illative conclusion.

Jon Awbrey

Posted by Jon Awbrey on 04-21-2004 02:16 AM:

Propositional Equation Reasoning Systems

PERS. Note 23

Transitivity (cont.)

One way to view a proposition f : B^k -> B is to
consider its "fiber of truth", f^(-1)(1) c B^k,
and to regard it as a k-adic relation L c B^k.

By way of general definition, the "fiber" of a function
f : X -> Y at a given value y of its co-domain Y is the
"antecedent" (pre-image or inverse image) of y under f.
This is a subset, possibly empty, of the domain X, and
it gets rather barbarously asciified as f^(-1)(y) c X.

In particular, if f is a proposition f : X -> B, then we think
of f^(-1)(1) c X as the subset of X that is "indicated" by the
proposition f. Whenever we "assert" a proposition f : X -> B,
we are saying that what it indicates is all that happens to be
the case in the relevant universe of discourse X. Because the
special case of the fiber of truth is used so often in logical
contexts, I will sometimes use the notation [|f|] = f^(-1)(1).

Using this panoply of notions and notations, we may treat the
fiber of truth of each proposition f : B^3 -> B as if it were
a relational data table of the shape {<p, q, r>} c B^3, where
the <p, q, r> are bit vectors indicated by the proposition f.

Thus we obtain the following four relational data tables
for the propositions that we are looking at in Example 2.

[| q_207 |] = [| p =< q |]
o---------o---------o---------o
|` ` p ` `|` ` q ` `|` ` r ` `|
o---------o---------o---------o
|` ` 0 ` ` ` ` 0 ` ` ` ` 0 ` `|
|` ` 0 ` ` ` ` 0 ` ` ` ` 1 ` `|
|` ` 0 ` ` ` ` 1 ` ` ` ` 0 ` `|
|` ` 0 ` ` ` ` 1 ` ` ` ` 1 ` `|
|` ` 1 ` ` ` ` 1 ` ` ` ` 0 ` `|
|` ` 1 ` ` ` ` 1 ` ` ` ` 1 ` `|
o-----------------------------o

[| q_187 |] = [| q =< r |]
o---------o---------o---------o
|` ` p ` `|` ` q ` `|` ` r ` `|
o---------o---------o---------o
|` ` 0 ` ` ` ` 0 ` ` ` ` 0 ` `|
|` ` 0 ` ` ` ` 0 ` ` ` ` 1 ` `|
|` ` 0 ` ` ` ` 1 ` ` ` ` 1 ` `|
|` ` 1 ` ` ` ` 0 ` ` ` ` 0 ` `|
|` ` 1 ` ` ` ` 0 ` ` ` ` 1 ` `|
|` ` 1 ` ` ` ` 1 ` ` ` ` 1 ` `|
o-----------------------------o

[| q_175 |] = [| p =< r |]
o---------o---------o---------o
|` ` p ` `|` ` q ` `|` ` r ` `|
o---------o---------o---------o
|` ` 0 ` ` ` ` 0 ` ` ` ` 0 ` `|
|` ` 0 ` ` ` ` 0 ` ` ` ` 1 ` `|
|` ` 0 ` ` ` ` 1 ` ` ` ` 0 ` `|
|` ` 0 ` ` ` ` 1 ` ` ` ` 1 ` `|
|` ` 1 ` ` ` ` 0 ` ` ` ` 1 ` `|
|` ` 1 ` ` ` ` 1 ` ` ` ` 1 ` `|
o-----------------------------o

[| q_139 |] = [| p =< q =< r |]
o---------o---------o---------o
|` ` p ` `|` ` q ` `|` ` r ` `|
o---------o---------o---------o
|` ` 0 ` ` ` ` 0 ` ` ` ` 0 ` `|
|` ` 0 ` ` ` ` 0 ` ` ` ` 1 ` `|
|` ` 0 ` ` ` ` 1 ` ` ` ` 1 ` `|
|` ` 1 ` ` ` ` 1 ` ` ` ` 1 ` `|
o-----------------------------o

Jon Awbrey

Posted by Jon Awbrey on 04-21-2004 04:40 PM:

Propositional Equation Reasoning Systems

PERS. Note 24

Transitivity (cont.)

In the medium of these unassuming examples, we begin to see
the activities of logical inference and methodical inquiry
as "information clarifying operations" (ICO's).

Allow me to clarify.

First, we drew a distinction between information maintaining
and information reducing processes and we noted the related
distinction between equational and implicational inferences.
I will use the acronyms EROI and IROI, respectively, for the
equational and implicational analogues of the various rules
of inference.

For example, we considered the brands of "information fusion"
that are involved in a couple of standard rules of inference,
taken in both their equational and their illative variants.

In particular, let us assume that we begin from a state of uncertainty
about the universe of discourse X = B^3 that is standardly represented
by a uniform distribution u : X -> B such that u(x) = 1 for all x in X,
in short, by the constant proposition 1 : X -> B. This amounts to the
"maximum entropy sign state" (MESS). As a measure of uncertainty, let
us use either the multiplicative measure given by the cardinality of X,
commonly notated as |X|, or else the additive measure given by log |X|.
In this frame we have |X| = 8 and log |X| = 3, to wit, 3 bits of doubt.

Let us now consider the various rules of inference for transitivity
in the light of their performance as information-developing actions.

Transitive Law (IROI)

p =< q
q =< r
--------
p =< r

By itself, the information p =< q would reduce
our uncertainty from log 8 bits to log 6 bits.

By itself, the information q =< r would reduce
our uncertainty from log 8 bits to log 6 bits.

By itself, the information p =< r would reduce
our uncertainty from log 8 bits to log 6 bits.

In this situation, the application of the IROI for transitivity
to the information p =< q and the information q =< r to get the
information p =< r does not increase the measure of information
beyond what any one of the three propositions has independently
of the other two. In a sense, then, this IROI operates only to
move the information around without changing its measure in the
slightest bit.

Transitive Law (EROI)

p =< q
q =< r
=============
p =< q =< r

The contents and the measures of information
that are associated with the propositions
p =< q and q =< r are the same as before.

On its own, the information p =< q =< r would
reduce our uncertainty from log(8) = 3 bits
to log(4) = 2 bits, a reduction of 1 bit.

These are just some of the initial observations that can be
made about the dimensions of information and uncertainty in
the conduct of logical inference, and there are many issues
to be taken up as we get to the thick of it. In particular,
we are taking propositions far too literally at the outset,
reading their spots at face value, as it were, without yet
considering their species character as fallible signs.

All in good time ...

Jon Awbrey

Posted by Jon Awbrey on 04-21-2004 08:48 PM:

Propositional Equation Reasoning Systems

PERS. Note 25

For ease of reference during the rest of this discussion,
let us refer to the propositional form f : B^3 -> B such
that f<p, q, r> = q_139 <p, q, r> = (p (q))(q (r)) as
the "syllogism mapping", written as syll : B^3 -> B,
and let us refer to the fiber syll^(-1)(1) c B^3 as
the "syllogism relation", written as Syll c B^3.
Table 25-a shows Syll as a relational dataset.

Table 25-a. Syllogism Relation
o---------o---------o---------o
|` ` p ` `|` ` q ` `|` ` r ` `|
o---------o---------o---------o
|` ` 0 ` ` ` ` 0 ` ` ` ` 0 ` `|
|` ` 0 ` ` ` ` 0 ` ` ` ` 1 ` `|
|` ` 0 ` ` ` ` 1 ` ` ` ` 1 ` `|
|` ` 1 ` ` ` ` 1 ` ` ` ` 1 ` `|
o-----------------------------o

One of the first questions that a certain type of person,
in this case me, tends to ask about a 3-adic relation,
in this case Syll, is whether it is "determined by"
its 2-adic projections. I will illustrate what
this means in the present case.

Table 25-b repeats the relation Syll in the first column,
listing its 3-tuples in bit-string form, followed by the
2-adic or "planar" projections of Syll in the next three
columns. For instance, Syll_pq is the 2-adic projection
of Syll on the pq "plane" that is arrived at by deleting
the r column and counting each 2-tuple that results just
one time. Likewise, Syll_pr is obtained by deleting the
q column and Syll_qr is derived by deleting the p column,
ignoring whatever duplicate pairs may result. The final
row of the right three columns gives the propositions of
the form f : B^2 -> B that indicate the 2-adic relations
that result from these projections.

Table 25-b. Dyadic Projections of the Syllogism Relation
o-------------o-------------o-------------o-------------o
| ` `Syll ` ` | ` Syll_pq ` | ` Syll_pr ` | ` Syll_qr ` |
o-------------o-------------o-------------o-------------o
| ` ` 000 ` ` | ` ` 00 ` ` `| ` ` 00 ` ` `| ` ` 00 ` ` `|
| ` ` 001 ` ` | ` ` 00 ` ` `| ` ` 01 ` ` `| ` ` 01 ` ` `|
| ` ` 011 ` ` | ` ` 01 ` ` `| ` ` 01 ` ` `| ` ` 11 ` ` `|
| ` ` 111 ` ` | ` ` 11 ` ` `| ` ` 11 ` ` `| ` ` 11 ` ` `|
o-------------o-------------o-------------o-------------o
| p =< q =< r | ` (p (q)) ` | ` (p (r)) ` | ` (q (r)) ` |
o-------------o-------------o-------------o-------------o

I have an appointment with another sort of table ...

Jon Awbrey

Posted by Jon Awbrey on 04-22-2004 03:10 PM:

Propositional Equation Reasoning Systems

PERS. Note 26

Let us make the simple observation that taking a projection,
in our framework, deleting a column from a relational table,
is like taking a derivative in differential calculus. What
it means is that our attempt to return to the integral from
whence the derivative was derived will in general encounter
an indefinite variation on account of the circumstance that
real information may have been destroyed by the derivation.

One will find that some relations can be reconstructed from
various types of derivatives and projections, others cannot.
The reconstuctible relations are said to be "reducible" to
the types of reductive data in question, while the others
are said to be "irreducible" with respect to those means.

The analogies between derivation, differentiation, implication,
projection, and others sorts of information reducing operation
will undergo extensive development in the remainder and sequel
of the present discussion.

Jon Awbrey

Posted by Jon Awbrey on 04-30-2004 08:32 PM:

Propositional Equation Reasoning Systems

PERS. Note 27

We were in the middle of discussing the relationships
between information preserving rules of inference and
information destroying rules of inference -- folks of
a 3-basket philosophical bent will no doubt be asking
"And what of information creating rules of inference?"
but there I must wait for some signs of enlightenment,
desiring not to tread on the rules of that succession.

The contrast between the information destroying and the
information preserving versions of the transitive rule of
inference led us to examine the relationships among several
boolean functions, namely, those that qualify locally as the
elementary cellular automata rules q_139, q_175, q_187, q_207.

The function q_139 : B^3 -> B and its fiber [| q_139 |] c B^3
appeared to be key to many structures in this setting, and so
I singled them out under the new names of syll : B^3 -> B and
Syll c B^3, respectively.

Managing the conceptual complexity of our considerations at this
juncture put us in need of some conceptual tools that I broke off
to develop in my notes on "Reductions Among Relations". The main
items that we need right away from that thread are the definitions
of relational projections and their inverses, the tacit extensions.

But the more I survey the problem setting the more it looks like we
need better ways to bring our visual intuitions to play on the scene,
and so I want next to lay out some visual schemata that are designed
to facilitate that.

Jon Awbrey

Posted by Jon Awbrey on 05-01-2004 03:24 AM:

Propositional Equation Reasoning Systems

PERS. Note 28

Figure 28-a shows the familiar picture of a boolean 3-cube,
wherein the points of B^3 are coordinated as bit strings of
length three. Looking at the functions f : B^3 -> B and the
relations L c B^3 on this pattern, one views the construction
of either type of object as a matter of coloring the nodes of
the 3-cube with choices from a pair of colors that stipulate
which points are in the relation L = [|f|] and which points
are out of it. Bowing to common convention, we may use the
color "1" for points that are "in" a given relation and the
color "0" for points that are "out" of that same relation.
However, it will be more convenient here to indicate the
former case by writing the coordinates in the place of
the node and to indicate the latter case by plotting
the point as an unlabeled node "o".

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` 111 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` /|\ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `/`|`\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / `|` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` `|` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` / ` `|` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` ` `|` ` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` / ` ` `|` ` ` \ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` 110 ` ` 101 ` ` 011 ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|\ ` ` / \ ` ` /|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|`\` `/` `\` `/`|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` \ / ` ` \ / `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` `\` ` ` `/` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` / \ ` ` / \ `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|`/` `\` `/` `\`|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|/ ` ` \ / ` ` \|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` 100 ` ` 010 ` ` 001 ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` `|` ` ` / ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` `|` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` `|` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `|` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ `|` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\`|`/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \|/ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` 000 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 28-a. Boolean 3-Cube B^3

Table 28-b shows the 3-adic relation Syll c B^3 again,
and Figure 28-c shows it plotted on a 3-cube template.

Table 28-b. Syll c B^3
o-----------------------o
| ` p ` ` ` q ` ` ` r ` |
o-----------------------o
| ` 0 ` ` ` 0 ` ` ` 0 ` |
| ` 0 ` ` ` 0 ` ` ` 1 ` |
| ` 0 ` ` ` 1 ` ` ` 1 ` |
| ` 1 ` ` ` 1 ` ` ` 1 ` |
o-----------------------o

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` 111 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` /|\ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `/`|`\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / `|` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` `|` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` / ` `|` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` ` `|` ` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` / ` ` `|` ` ` \ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` `o` ` ` 011 ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|\ ` ` / \ ` ` /|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|`\` `/` `\` `/`|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` \ / ` ` \ / `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` `\` ` ` `/` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` / \ ` ` / \ `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|`/` `\` `/` `\`|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|/ ` ` \ / ` ` \|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` `o` ` ` 001 ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` `|` ` ` / ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` `|` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` `|` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `|` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ `|` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\`|`/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \|/ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` 000 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 28-c. Triadic Relation Syll c B^3

Jon Awbrey

Posted by Jon Awbrey on 05-01-2004 01:32 PM:

Propositional Equation Reasoning Systems

PERS. Note 29

We return once more to the plane projections of Syll c B^3.

Table 29-a. Syll c B^3
o-----------------------o
| ` p ` ` ` q ` ` ` r ` |
o-----------------------o
| ` 0 ` ` ` 0 ` ` ` 0 ` |
| ` 0 ` ` ` 0 ` ` ` 1 ` |
| ` 0 ` ` ` 1 ` ` ` 1 ` |
| ` 1 ` ` ` 1 ` ` ` 1 ` |
o-----------------------o

Table 29-b. Dyadic Projections of Syll
o-----------o o-----------o o-----------o
| `Syll_12` | | `Syll_13` | | `Syll_23` |
o-----------o o-----------o o-----------o
| ` p ` q ` | | ` p ` r ` | | ` q ` r ` |
o-----------o o-----------o o-----------o
| ` 0 ` 0 ` | | ` 0 ` 0 ` | | ` 0 ` 0 ` |
| ` 0 ` 1 ` | | ` 0 ` 1 ` | | ` 0 ` 1 ` |
| ` 1 ` 1 ` | | ` 1 ` 1 ` | | ` 1 ` 1 ` |
o-----------o o-----------o o-----------o
| `(p (q))` | | `(p (r))` | | `(q (r))` |
o-----------o o-----------o o-----------o

In showing the 2-adic projections of a 3-adic relation L c B^3,
I will translate the coordinates of the points in each relation
to the plane of the projection, there dotting out with a dot "."
the bit of the bit string that is out of place on that plane.

Figure 29-c shows Syll and its three 2-adic projections:

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` 111 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` /|\ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `/`|`\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / `|` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` `|` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` / ` `|` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` ` `|` ` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` / ` ` `|` ` ` \ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` `o` ` ` 011 ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|\ ` ` / \ ` ` /|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|`\` `/` `\` `/`|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` \ / ` ` \ / `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` `\` ` ` `/` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` / \ ` ` / \ `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|`/` `\` `/` `\`|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|/ ` ` \ / ` ` \|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` `o` ` ` 001 ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` `|` ` ` / ` ` ` ` ` ` ` ` |
| ` ` 11. ` ` ` ` `\` ` `|` ` `/` ` ` ` ` .11 ` ` |
| ` ` `|\ ` ` ` ` ` \ ` `|` ` / ` ` ` ` ` /|` ` ` |
| ` ` `|`\` ` ` ` ` `\` `|` `/` ` ` ` ` `/`|` ` ` |
| ` ` `|` \ ` ` ` ` ` \ `|` / ` ` ` ` ` / `|` ` ` |
| ` ` `|` `\` ` ` ` ` `\`|`/` ` ` ` ` `/` `|` ` ` |
| ` ` `|` ` \ ` ` ` ` ` \|/ ` ` ` ` ` / ` `|` ` ` |
| ` ` `|` ` `\` ` ` ` ` 000 ` ` ` ` `/` ` `|` ` ` |
| ` ` `|` ` ` \ ` ` ` ` ` ` ` ` ` ` / ` ` `|` ` ` |
| ` ` `o` ` ` 01. ` ` ` ` ` ` ` ` `o` ` ` .01 ` ` |
| ` ` ` \ ` ` `|` ` ` ` ` ` ` ` ` `|` ` ` / ` ` ` |
| ` ` ` `\` ` `|` ` ` ` ` ` ` ` ` `|` ` `/` ` ` ` |
| ` ` ` ` \ ` `|` ` ` ` ` ` ` ` ` `|` ` / ` ` ` ` |
| ` ` ` ` `\` `|` ` ` ` 1.1 ` ` ` `|` `/` ` ` ` ` |
| ` ` ` ` ` \ `|` ` ` ` / \ ` ` ` `|` / ` ` ` ` ` |
| ` ` ` ` ` `\`|` ` ` `/` `\` ` ` `|`/` ` ` ` ` ` |
| ` ` ` ` ` ` \|` ` ` / ` ` \ ` ` `|/ ` ` ` ` ` ` |
| ` ` ` ` ` ` 00. ` `/` ` ` `\` ` .00 ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` / ` ` ` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` ` ` ` ` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` / ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` 0.1 ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` 0.0 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 29-c. Syll c B^3 and its Dyadic Projections

Jon Awbrey

Posted by Jon Awbrey on 05-01-2004 05:00 PM:

Propositional Equation Reasoning Systems

PERS. Note 30

We now compute the tacit extensions of the 2-adic projections
of Syll, alias q_139, and this makes manifest its relationship
to the other functions and fibers, namely, q_175, q_187, q207.

Table 30-a. Syll c B^3
o-----------------------o
| ` p ` ` ` q ` ` ` r ` |
o-----------------------o
| ` 0 ` ` ` 0 ` ` ` 0 ` |
| ` 0 ` ` ` 0 ` ` ` 1 ` |
| ` 0 ` ` ` 1 ` ` ` 1 ` |
| ` 1 ` ` ` 1 ` ` ` 1 ` |
o-----------------------o

Table 30-b. Dyadic Projections of Syll
o-----------o o-----------o o-----------o
| `Syll_12` | | `Syll_13` | | `Syll_23` |
o-----------o o-----------o o-----------o
| ` p ` q ` | | ` p ` r ` | | ` q ` r ` |
o-----------o o-----------o o-----------o
| ` 0 ` 0 ` | | ` 0 ` 0 ` | | ` 0 ` 0 ` |
| ` 0 ` 1 ` | | ` 0 ` 1 ` | | ` 0 ` 1 ` |
| ` 1 ` 1 ` | | ` 1 ` 1 ` | | ` 1 ` 1 ` |
o-----------o o-----------o o-----------o
| `(p (q))` | | `(p (r))` | | `(q (r))` |
o-----------o o-----------o o-----------o

Table 30-c. Tacit Extensions of Projections of Syll
o---------------o o---------------o o---------------o
| `TE(Syll_12)` | | `TE(Syll_13)` | | `TE(Syll_23)` |
o---------------o o---------------o o---------------o
| ` p ` q ` r ` | | ` p ` q ` r ` | | ` p ` q ` r ` |
o---------------o o---------------o o---------------o
| ` 0 ` 0 ` 0 ` | | ` 0 ` 0 ` 0 ` | | ` 0 ` 0 ` 0 ` |
| ` 0 ` 0 ` 1 ` | | ` 0 ` 1 ` 0 ` | | ` 1 ` 0 ` 0 ` |
| ` 0 ` 1 ` 0 ` | | ` 0 ` 0 ` 1 ` | | ` 0 ` 0 ` 1 ` |
| ` 0 ` 1 ` 1 ` | | ` 0 ` 1 ` 1 ` | | ` 1 ` 0 ` 1 ` |
| ` 1 ` 1 ` 0 ` | | ` 1 ` 0 ` 1 ` | | ` 0 ` 1 ` 1 ` |
| ` 1 ` 1 ` 1 ` | | ` 1 ` 1 ` 1 ` | | ` 1 ` 1 ` 1 ` |
o---------------o o---------------o o---------------o
| [| (p (q)) |] | | [| (p (r)) |] | | [| (q (r)) |] |
o---------------o o---------------o o---------------o
| [| `q_207` |] | | [| `q_175` |] | | [| `q_187` |] |
o---------------o o---------------o o---------------o

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` 111 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` /|\ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `/`|`\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / `|` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` `|` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` / ` `|` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` ` `|` ` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` / ` ` `|` ` ` \ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` 110 ` ` `o` ` ` 011 ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|\ ` ` / \ ` ` /|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|`\` `/` `\` `/`|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` \ / ` ` \ / `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` `\` ` ` `/` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` / \ ` ` / \ `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|`/` `\` `/` `\`|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|/ ` ` \ / ` ` \|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` 010 ` ` 001 ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` `|` ` ` / ` ` ` ` ` ` ` ` |
| ` ` 11. ` ` ` ` `\` ` `|` ` `/` ` ` ` ` ` ` ` ` |
| ` ` `|\ ` ` ` ` ` \ ` `|` ` / ` ` ` ` ` ` ` ` ` |
| ` ` `|`\` ` ` ` ` `\` `|` `/` ` ` ` ` ` ` ` ` ` |
| ` ` `|` \ ` ` ` ` ` \ `|` / ` ` ` ` ` ` ` ` ` ` |
| ` ` `|` `\` ` ` ` ` `\`|`/` ` ` ` ` ` ` ` ` ` ` |
| ` ` `|` ` \ ` ` ` ` ` \|/ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `|` ` `\` ` ` ` ` 000 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `|` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `o` ` ` 01. ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` \ ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` `\` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` \ ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `\` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` \ `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `\`|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` \|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` 00. ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 30-d. Tacit Extension TE_12_3 (Syll_12)

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` 111 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` /|\ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `/`|`\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / `|` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` `|` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` / ` `|` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` ` `|` ` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` / ` ` `|` ` ` \ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` 101 ` ` 011 ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|\ ` ` / \ ` ` /|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|`\` `/` `\` `/`|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` \ / ` ` \ / `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` `\` ` ` `/` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` / \ ` ` / \ `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|`/` `\` `/` `\`|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|/ ` ` \ / ` ` \|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` 010 ` ` 001 ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` `|` ` ` / ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` `|` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` `|` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `|` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ `|` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\`|`/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \|/ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` 000 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` 1.1 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` / \ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `/` `\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / ` ` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` ` ` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` / ` ` ` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` ` ` ` ` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` / ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` 0.1 ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` 0.0 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 30-e. Tacit Extension TE_13_2 (Syll_13)

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` 111 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` /|\ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `/`|`\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / `|` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` `|` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` / ` `|` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` ` `|` ` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` / ` ` `|` ` ` \ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` 101 ` ` 011 ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|\ ` ` / \ ` ` /|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|`\` `/` `\` `/`|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` \ / ` ` \ / `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` `\` ` ` `/` `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|` / \ ` ` / \ `|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|`/` `\` `/` `\`|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `|/ ` ` \ / ` ` \|` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` 100 ` ` `o` ` ` 001 ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` `|` ` ` / ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` `|` ` `/` ` ` ` ` .11 ` ` |
| ` ` ` ` ` ` ` ` ` \ ` `|` ` / ` ` ` ` ` /|` ` ` |
| ` ` ` ` ` ` ` ` ` `\` `|` `/` ` ` ` ` `/`|` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ `|` / ` ` ` ` ` / `|` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\`|`/` ` ` ` ` `/` `|` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \|/ ` ` ` ` ` / ` `|` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` 000 ` ` ` ` `/` ` `|` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` `|` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `o` ` ` .01 ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` / ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|` ` `/` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|` ` / ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|` `/` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|` / ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|`/` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|/ ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` .00 ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 30-f. Tacit Extension TE_23_1 (Syll_23)

Jon Awbrey

Posted by Jon Awbrey on 05-02-2004 03:10 AM:

Propositional Equation Reasoning Systems

PERS. Note 31

The reader may wish to contemplate Figure 31
and use it to verify the following two facts:

Syll = TE(Syll_12) |^| TE(Syll_23)

Syll_13 = Syll_12 o Syll_23

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `*` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` /|\ ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `/`|`\` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` / `|` \ ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `/` `|` `\` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` / ` `|` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` ` `|` ` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` / ` ` `|` ` ` \ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` `o` ` ` `*` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` /|\ ` ` / \ ` ` /|\ ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `/`|`\` `/` `\` `/`|`\` ` ` ` ` ` ` |
| ` ` ` ` ` ` / `|` \ / ` ` \ / `|` \ ` ` ` ` ` ` |
| ` ` ` ` ` `/` `|` `\` ` ` `/` `|` `\` ` ` ` ` ` |
| ` ` ` ` ` / ` `|` / \ ` ` / \ `|` ` \ ` ` ` ` ` |
| ` ` ` ` `/` ` `|`/` `\` `/` `\`|` ` `\` ` ` ` ` |
| ` ` ` ` / ` ` `|/ ` ` \ / ` ` \|` ` ` \ ` ` ` ` |
| ` ` ` `/` ` ` `o` ` ` `o` ` ` `*` ` ` `\` ` ` ` |
| ` ` ` / ` ` ` ` \ ` ` /|` ` ` / \ ` ` ` \ ` ` ` |
| ` ` `*` ` ` ` ` `\` `/`|` ` `/` `\` ` ` `*` ` ` |
| ` ` `|\ ` ` ` ` ` \ / `|` ` / ` ` \ ` ` /|` ` ` |
| ` ` `|`\` ` ` ` ` `/` `|` `/` ` ` `\` `/`|` ` ` |
| ` ` `|` \ ` ` ` ` / \ `|` / ` ` ` ` \ / `|` ` ` |
| ` ` `|` `\` ` ` `/` `\`|`/` ` ` ` ` `/` `|` ` ` |
| ` ` `|` ` \ ` ` / ` ` \|/ ` ` ` ` ` / \ `|` ` ` |
| ` ` `|` ` `\` `/` ` ` `*` ` ` ` ` `/` `\`|` ` ` |
| ` ` `|` ` ` \ / ` ` ` / \ ` ` ` ` / ` ` \|` ` ` |
| ` ` `o` ` ` `*` ` ` `/` `\` ` ` `o` ` ` `*` ` ` |
| ` ` ` \ ` ` `|` ` ` / ` ` \ ` ` `|` ` ` / ` ` ` |
| ` ` ` `\` ` `|` ` `/` ` ` `\` ` `|` ` `/` ` ` ` |
| ` ` ` ` \ ` `|` ` / ` ` ` ` \ ` `|` ` / ` ` ` ` |
| ` ` ` ` `\` `|` `/` ` `*` ` `\` `|` `/` ` ` ` ` |
| ` ` ` ` ` \ `|` / ` ` / \ ` ` \ `|` / ` ` ` ` ` |
| ` ` ` ` ` `\`|`/` ` `/` `\` ` `\`|`/` ` ` ` ` ` |
| ` ` ` ` ` ` \|/ ` ` / ` ` \ ` ` \|/ ` ` ` ` ` ` |
| ` ` ` ` ` ` `*` ` `/` ` ` `\` ` `*` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` / ` ` ` ` \ ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `/` ` ` ` ` `\` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` / ` ` ` ` ` ` \ ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `o` ` ` ` ` ` ` `*` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `\` ` ` ` ` `/` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `*` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 31. Syll = TE(Syll_12) |^| TE(Syll_23)

Jon Awbrey

Posted by Jon Awbrey on 05-03-2004 07:10 PM:

Propositional Equation Reasoning Systems

PERS. Note 32

I don't know about you, but I am still puzzled by all of this stuff,
that is to say, by the entanglements of composition and projection
and their relationship to the information processing properties
of logical inference rules. What I lack is a single picture
that could show me all of the pieces and make the pattern
of their informational relationships clear.

In accord with my experimental way, I will stick with the case
of transitive inference until I have pinned it down thoroughly,
but of course the real interest is much more general than that.

At first or maybe second sight, the relationships seem easy enough
to write out. Figure 32 shows how the various logical expressions
are related to each other: The expressions "(p (q))" and "(q (r))"
are conjoined in a purely syntactic fashion -- much in the way that
one might compile a theory from axioms without knowing what either
the theory or the axioms were about -- and the best way to sum up
the state of information implicit in taking them together is just
the expression "(p (q)) (q (r))" that would the canonical result
of an equational or reversible rule of inference. From that
equational inference, one might arrive at the implicational
inference "(p (r))" by the most conventional implication.

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q ` ` ` ` | ` ` ` ` | ` ` ` ` r ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` p o ` ` ` ` | ` ` ` ` | ` ` ` q o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `(p (q))` ` ` | ` ` ` ` | ` ` `(q (r))` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_207 ` ` ` | ` ` ` ` | ` ` ` q_187 ` ` ` |
o---------o---------o ` ` ` ` o---------o---------o
` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` `
` ` ` ` ` ` \ ` ` ` Conjunction ` ` ` / ` ` ` ` ` `
` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` `
` ` ` ` ` ` ` v ` ` ` ` ` ` ` ` ` ` v ` ` ` ` ` ` `
` ` ` ` ` ` ` `o-------------------o` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` q ` r ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` o ` o ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` | ` | ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` p o ` o q ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` `\`/` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` @ ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `o-------------------o` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| `(p (q)) (q (r))` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `o-------------------o` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` q_139 ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `o---------o---------o` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` Implication ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` `v` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` `o---------o---------o` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` r ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` o ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` | ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` p o ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` | ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` @ ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `o-------------------o` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` `(p (r))` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `o-------------------o` ` ` ` ` ` ` `
` ` ` ` ` ` ` `| ` ` ` q_175 ` ` ` |` ` ` ` ` ` ` `
` ` ` ` ` ` ` `o-------------------o` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
Figure 32. Expressive Aspects of Transitive Inference

Most of the customary names for this type of process
have turned out to have misleading connotations, and
so I will experiment with calling it the "expressive"
aspect of the various rules for transitive inference,
simply to emphasize the fact that rules can be given
for it that operate solely on signs and expressions,
without necessarily needing to look at the objects
that are denoted by these signs and expressions.

Jon Awbrey

Posted by Jon Awbrey on 05-04-2004 03:32 PM:

Propositional Equation Reasoning Systems

PERS. Note 33

In the way of many experiments, the word "expressive"
does not seem to work for what I wanted to say here,
since we too often use it to suggest something that
expresses an object or a purpose, and I wanted it
to imply what is purely a matter of expression,
shorn of consideration for anything objective.
Aside from coining a word like "ennotative",
some other options would be "connotative",
"hermeneutic", "semiotic", "syntactic" --
each of which works in some range of
interpretation but fails in others.
Trial 2. Let's try "formulaic".

Despite how simple the formulaic aspects of transitive inference
might appear on the surface, there are problems that wait for us
just beneath the syntactic surface, as we quickly discover if we
turn to considering the kinds of objects, abstract and concrete,
that these formulas are meant to denote, and all the more so if
we try to do this in a context of computational implementations,
where the "interpreters" to be addressed take nothing on faith.
Thus we engage the "denotative semantics" or the "model theory"
of these extremely simple programs that we call "propositions".

Figure 33 is an attempt to outline the model-theoretic relationships
that are involved in our study of transitive inference. In it I use
a number of abbreviated notations:

I use the forms X:Y:Z and x:y:z as alternative notations for the
cartesian product X x Y x Z and the tuple <x, y, z>, respectively.
In situations where we have products like X:Y:Z with X = Y = Z = B,
and relations like L c X:Y, M c X:Z, N c Y:Z, I will use forms like
L c B:B:~, M c B:~:B, N c ~:B:B to remind us that we are considering
particular ways of situating L, M, N within the product space X:Y:Z.

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` 0:0:0 ` ` ` | ` ` ` ` | ` ` ` 0:0:0 ` ` ` |
| ` ` ` 0:0:1 ` ` ` | ` ` ` ` | ` ` ` 0:0:1 ` ` ` |
| ` ` ` 0:1:0 ` ` ` | ` ` ` ` | ` ` ` 0:1:1 ` ` ` |
| ` ` ` 0:1:1 ` ` ` | ` ` ` ` | ` ` ` 1:0:0 ` ` ` |
| ` ` ` 1:1:0 ` ` ` | ` ` ` ` | ` ` ` 1:0:1 ` ` ` |
| ` ` ` 1:1:1 ` ` ` | ` ` ` ` | ` ` ` 1:1:1 ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
|TE(Syll_12) c B:B:B| ` ` ` ` |TE(Syll_23) c B:B:B|
o-------------------o ` ` ` ` o-------------------o
| ` `[| q_207 |]` ` | ` ` ` ` | ` `[| q_187 |]` ` |
o----o---------o----o ` ` ` ` o----o---------o----o
` ` `^` ` ` ` ` \ ` ` ` ` ` ` ` ` / ` ` ` ` `^` ` `
` ` `|` ` ` ` ` `\`Intersection `/` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` `v` ` ` ` ` `v` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` 0:0:0 ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` 0:0:1 ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` 0:1:1 ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` 1:1:1 ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` `Syll c B:B:B ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` `[| q_139 |]` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o---------o---------o` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` ` Projection` ` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` ` ` ` `v` ` ` ` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` `o---------o---------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` `0:0` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` `0:1` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` `1:1` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| `Syll_13 c B:~:B` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` [| (p (r)) |] ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o----o---------o----o` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` `^` ` ` ` ` `^` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` / ` ` ` ` ` ` \ ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` `/` Composition `\` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` / ` ` ` ` ` ` ` ` \ ` ` ` ` `|` ` `
o----o---------o----o ` ` ` ` o----o---------o----o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `0:0` ` ` ` | ` ` ` ` | ` ` ` `0:0` ` ` ` |
| ` ` ` `0:1` ` ` ` | ` ` ` ` | ` ` ` `0:1` ` ` ` |
| ` ` ` `1:1` ` ` ` | ` ` ` ` | ` ` ` `1:1` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| `Syll_12 c B:B:~` | ` ` ` ` | `Syll_23 c ~:B:B` |
o-------------------o ` ` ` ` o-------------------o
| ` [| (p (q)) |] ` | ` ` ` ` | ` [| (q (r)) |] ` |
o---------o---------o ` ` ` ` o---------o---------o

Figure 33. Denotative Aspects of Transitive Inference

Jon Awbrey

Posted by Jon Awbrey on 05-04-2004 07:14 PM:

Propositional Equation Reasoning Systems

PERS. Note 34

I think that I've stared at this problem long enough
that I can draw a few conclusions about its etiology.

A piece of syntax like "(p (q))" or "p => q" is an
abstract description, and abstraction is a process
that loses information about the objects described.
So when we go to reverse the abstraction, as we do
when we look for models of that description, there
is a degree of indefiniteness that comes into play.

For example, the proposition (p (q)) is typically assigned the
functional type B^2 -> B, but that is only its canonical or its
minimal abstract type. No sooner do we use it in a context that
invokes additional variables, as we do when we next consider the
proposition (q (r)), than its type is tacitly adjusted to fit the
new context, for instance, acquiring the extended type B^3 -> B.
This is one of those things that most people eventually learn to
do without blinking an eye, that is to say, unreflectively, and
this is precisely what makes the same facility so much trouble
to implement properly in computational form.

Both the fibering operation, that takes us from the function
(p (q)) to the relation [| (p (q)) |], and the tacit extension
operation, that takes us from the relation [| (p (q)) |] c B:B
to the relation [| q_207 |] c B:B:B have this same character of
abstraction-undoing or modelling operations that require us to
re-interpret the same pieces of syntax under different types.
This accounts for a large part of the apparent ambiguities.

Jon Awbrey

Posted by Jon Awbrey on 05-05-2004 04:02 AM:

Propositional Equation Reasoning Systems

PERS. Note 35

Up till now I've concentrated mostly on the abstract types
of domains and propositions, things like B^k and B^k -> B,
respectively. This is a little like trying to do physics
all in dimensionless quantities without keeping track of
the qualitative physical units. So much abstraction has
its obvious limits, not to mention its hidden dangers.

To remedy this situation I will start to introduce the
concrete types of domains and propositions, once again
as they pertain to our current collection of examples.

With visions of venn diagrams in the back of my mind,
I have been using the lower case letters p, q, r for
the basic propositions of abstract type B^3 -> B and
the upper case letters P, Q, R for the basic regions
of the universe of discourse where p, q, r hold true,
respectively. The set of signs !X! = {"p", "q", "r"}
is the "alphabet" for the universe of discourse that
we notate as X% = [!X!] = [p, q, r], already getting
sloppy about quotation marks to single out the signs.
The universe X% is composed of two layers of objects,
the space of positions X = <|p, q, r|> = {<p, q, r>}
and the space of propositions X^ = (X -> B). Let us
take /P/ = {(p), p}, /Q/ = {(q), q}, /R/ = {(r), r},
in effect, three sets of two abstract signs each, as
giving the qualitative dimensions of the universe X%.

Given this framework, we can say that the concrete type
of the space X is /P/ x /Q/ x /R/ ~=~ B^3 and that the
concrete type of each proposition in X^ = (X -> B) is
/P/ x /Q/ x /R/ -> B. Given the length of the type
markers, we will often omit the cartesian product
symbols and write just /P//Q//R/.

An abstract reference to a point of X is a triple in B^3.
A concrete reference to a point of X is a conjunction of
signs from the dimensions /P/, /Q/, /R/, picking exactly
one sign from each dimension.

To illustrate the use of concrete coordinates for points
and concrete types for spaces and propositions, Figure 35
translates the contents of Figure 33 into the new language.

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` (p)(q)(r) ` ` | ` ` ` ` | ` ` (p)(q)(r) ` ` |
| ` ` (p)(q)`r` ` ` | ` ` ` ` | ` ` (p)(q)`r` ` ` |
| ` ` (p) q (r) ` ` | ` ` ` ` | ` ` (p) q `r` ` ` |
| ` ` (p) q `r` ` ` | ` ` ` ` | ` ` `p`(q)(r) ` ` |
| ` ` `p` q (r) ` ` | ` ` ` ` | ` ` `p`(q)`r` ` ` |
| ` ` `p` q `r` ` ` | ` ` ` ` | ` ` `p` q `r` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
|TE(Syll_12) c B:B:B| ` ` ` ` |TE(Syll_23) c B:B:B|
o-------------------o ` ` ` ` o-------------------o
| ` `[| q_207 |]` ` | ` ` ` ` | ` `[| q_187 |]` ` |
o----o---------o----o ` ` ` ` o----o---------o----o
` ` `^` ` ` ` ` \ ` ` ` ` ` ` ` ` / ` ` ` ` `^` ` `
` ` `|` ` ` ` ` `\`Intersection `/` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` \ ` ` ` ` ` ` / ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` `v` ` ` ` ` `v` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` (p)(q)(r) ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` (p)(q)`r` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` (p) q `r` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` `p` q `r` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| Syll c /P//Q//R/` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` `[| q_139 |]` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o---------o---------o` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` ` Projection` ` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` ` ` ` `v` ` ` ` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` `o---------o---------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` `(p) (r)` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` `(p)` r ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` p ` r ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` ` ` ` ` ` ` ` ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| Syll_13 c /P//R/` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o-------------------o` ` ` ` `|` ` `
` ` `|` ` ` ` `| ` [| (p (r)) |] ` |` ` ` ` `|` ` `
` ` `|` ` ` ` `o----o---------o----o` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` `^` ` ` ` ` `^` ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` ` / ` ` ` ` ` ` \ ` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` `/` Composition `\` ` ` ` ` `|` ` `
` ` `|` ` ` ` ` / ` ` ` ` ` ` ` ` \ ` ` ` ` `|` ` `
o----o---------o----o ` ` ` ` o----o---------o----o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` `(p) (q)` ` ` | ` ` ` ` | ` ` `(q) (r)` ` ` |
| ` ` `(p)` q ` ` ` | ` ` ` ` | ` ` `(q)` r ` ` ` |
| ` ` ` p ` q ` ` ` | ` ` ` ` | ` ` ` q ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| Syll_12 c /P//Q/` | ` ` ` ` | Syll_23 c /Q//R/` |
o-------------------o ` ` ` ` o-------------------o
| ` [| (p (q)) |] ` | ` ` ` ` | ` [| (q (r)) |] ` |
o---------o---------o ` ` ` ` o---------o---------o

Figure 35. Denotative Aspects of Transitive Inference

Jon Awbrey

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