THE CALCULUS OF DISTINCTIONS
TABLE OF CONTENTS
Calculus of Distinctions
Non-Numerical Arithmetic
The Algebra of Distinctions
Types of
Dinstinctions
Logical Interpretations of the Calculus
The symbols and expressions of the calculus of distinctions may be interpreted and applied to any expression dealing with conceptual, perceptual, and existential distinctions.
This facility was briefly demonstrated when we mapped the consequences of the primary expressions of the non-numerical arithmetic (Reference: the section on non-numerical arithmetic, where we obtained the set of standard logic tables). With the algebra of distinctions established, we may now expand the logical interpretation of the calculus to include algebraic expressions. We begin by recognizing the logical content or interpretation of some of the basic symbolic expressions of the calculus3. (see in Table 1).
A great number of logical consequences may be derived from the initial equations of the algebra of distinctions. Many such algebraic relations also have interesting and meaningful logical interpretations which prove useful in the application of the calculus (see Table 2).
These symbolic relations may be easily demonstrated by substitution and simplification. Their use will speed up algebraic transformations in applications to problems of logic and mathematical physics by facilitating the interpretation or simplification of a problem.
One additional operational rule is worth bringing up at this time:
Equivalent expressions at odd and even depths within a complex expression may be cancelled.
This may be stated because of the structure of the algebra of distinctions. The value of this rule lies in the fact that its use may result in the almost immediate reduction of a set of statements to the logical conclusion or conclusions which may be drawn from them. Compared to the step-by-step procedure of the standard methods of symbolic logic, the calculus is clearly more efficient.
An example will serve to illustrate the point. Consider the following set of statements:
1. A physicist (Physicist A) developed a method for measuring the speed of light coming from distant stars.
2. Another physicist (Albert Einstein) developed a theory (the theory of relativity) which states that all light, from whatever source, may exhibit only one speed, c, to any observer.
3. A number of physicists, (Physicists B through Z) made experimental observations confirming the theory of relativity.
4. Among the experimental observations of the group of Physicists B-Z, were several measurements of the speed of light from distant stars.
In order to apply symbolic logic to these statements, we first define the essential elements of the statements as follows.
Let a represent the results of Physicist A's measurements, while b represents the theory of relativity, c represents the singular speed of light predicted by the theory of relativity, d represents the results of Physicists B-Z's observations, and e represents the speed of light from distant stars.
Assuming that the four statements are true and interpreting them as statements in standard symbolic logic, we have:
Step 1.
which may be regrouped as
Step 2.
and reduced by substitution of
and
into Step 2 to get
Step 3.
thus the conclusion is which translates to the conclusion that Physicist A's method for measuring the speed of light from distant stars will yield the speed c.
Analysis of the same set of statements with the calculus of distinctions is accomplished by translating the four statements into the calculus using the logical equivalents of Table 2:
| STATEMENT | LOGICAL EQUIVALENTS |
| 1 |  |
| 2 |  |
| 3 |  |
| 4 |  |
Combining the logical equivalents into one expression in accordance with Table 1, we have:
Applying the rule of cancellation of equivalent expressions in even and odd depths, we have:
i.e., Physicist A's method will yield the speed c.
This is, of course, a very simple example. However, it demonstrates the efficiency of the method. The rule applies equally well to more complex expressions with many separate distinctions and sub-partitions. For example, a complex expression like
may be quickly simplified by the rule to:
 |
= aa |
by Ex. 3 |
| = a | by Ex. 2 |
Simplification of a given complex expression may often be accomplished by more than one path. With practice comes greater familiarity with various theorems, consequences and methods, and greater facility at seeing the most efficient route to simplification or, in some cases, appropriate transformations to meaningful patterns or results. For example, an even quicker route to simplification of the expression used in the example above would be:
|
|
by even-odd depth rule for c |
| = a | by reduction (See Table 2) |
One final note before we turn to applications of the theory and the calculus to questions of classical and modern physics: The calculus of distinctions presents a very effective method of testing hypotheses and theories. In the examples above, the statements were assumed to be valid. If, in the transformation of expressions obtained by the translation of verbal or quantitative statements into the calculus of distinctions, a clearly invalid expression is derived, two or more of the statements, assuming they were properly translated, must be in conflict, revealing a logical error in the theory or hypothesis as a whole.