**USING THE CALCULUS OF DIMENSIONAL DISTINCTIONS**

^{©}**Edward R. Close, December 21, 2017**

**Here are the Initial Equations of the calculus:**

The symbol indicating a distinction (upside-down L) represents a distinction of any kind. The blank on the right-hand side of equation 2.) denotes no distinction.

*

__Note__: The same logic symbols used by G. Spencer Brown in__Laws of Form__are used here to avoid confusion when the reader refers to Brown’s work. **A and B in this table represent algebraic variables or functions that may take on any value allowed in the CoDD.
CoDD
expressions often involve nested symbols up to seven to nine levels deep, or
more, conveying the geometric structure of the calculus and of existential
reality. Three-dimensional representations would be more realistic, and three-dimensional
representations can be developed for use in videos or slide presentations, but distinctions
of four or more dimensions can only be represented by projections onto
two-dimensional media, and are generally incomprehensible to anyone who has not
spent a lot of time studying them. Two-dimensional nested expressions are used because
they are relatively easily drawn on sheets of paper or a blackboard.

**A Demonstration of Calculation:**

Assume
that a hypothesis about some aspect the reality we experience can be expressed
in terms of the CoDD by the expression H consisting of existential distinctions:

These calculations reveal that H is equal to a non-distinction. Therefore, the hypothesis represented by expression H is false.

**Application of the CoDD to a Simple Example:**

Consider
the following 3 statements:

1. 1. All
cars produced by the Ford Motor Company before 1927 were black.

2. 2. John
has two antique Ford cars, and they are both black.

3. 3. Therefore,
John’s antique cars were manufactured before 1927.

Those
familiar with automotive history know that most cars produced before 1927 were
black. Henry Ford famously said: “A customer can have a car painted any color he
wants, as long as it’s black!” And everyone familiar with cars knows that the Ford
Motor Company has also produced many black cars since 1927, and that there are black
antique cars other than Fords. But suppose you didn’t know anything about cars other
than what is given in the first two statements; how would you proceed? Even
though this is a simple example, comparing the analysis using symbolic logic
with the CoDD analysis is instructive.

Let
the symbol P represent cars manufactured prior to 1927; let V represent John’s vintage
cars; and let B represent black cars. Translating the statements from English
to symbolic logic and the CoDD, we have:

Looking
at the symbolic logic approach first, we see that the first two statements are
given as true, and their truth in conjunction has the following logical
consequence:

(P

**ϵ**B)**․**(V**ϵ**B) ⸧ (P**․**V)**ϵ**B, which has two possible, mutually exclusive consequences:
[(P

**․**V)**ϵ**B] ⸧ (P**․**V) and [(P**․**V)**ϵ**B]**~**(P**․**V). Thus, the third statement is not supported by the information given in the first two. It can be either True or False, therefore, while statements**1**and**2**are true, statement**3**is indeterminate; it may be either true or false, we just don’t know based on the information given.
Now, let’s look at the
CoDD analysis. We start with the CoDD representation of statement #3 as our
hypothesis: The groups of vehicles that P, B and V represent are existential.
Therefore, they can be replaced by symbols of distinction,
and we have:

This
calculation is performed by applying initial equation #2 three times in the
first step, three times in the second step, and once in the last step. Since
this CoDD expression of statement #3 reduces to, and is therefore equivalent to
non-distinction, the statement is false. Statement #1 and #2 taken together, do
not imply statement #3.

The CoDD is a powerful tool because it allows one to check the logical validity of statements and hypotheses of any kind very quickly once it is translated into the CoDD notation. With practice, it is much more direct and easier than conventional symbolic logic. The simplification steps for even a very complex CoDD expression can be done visually.

Edward R. Close December 21, 2017

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ReplyDeleteOh.. it's been a long time since I've here and I almost missed this one!

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