HYPERSEMANTICS

The generic semantics of the "contrary" operator
"The contrary of a fact is falsehood, but the contrary of a profound truth may be another profound truth" (Niels Bohr).

"Nothing exists except in relation to its opposite" (Paul Twitchell).

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Opposite operator
Despite the fact that each primitive has a specific semantics, there is a hypersemantics, a common semantics referring to the contrary operators.

The contrary operator of an operator ⊥ is another operator that refers to the contrary concept to which ⊥ is associated.

1. If the operator is a constructor type, the opposite operator performs the opposite operation. For example:
   
   

   Concept	Contrary
   Add	Subtract
   Multiplying	Dividing
   Unite	Separate

   
   
2. If the operator is not constructive, the opposite operator has an opposite meaning, and that is a function of the operator. For example:
   
   

   Concept	Contrary
   Start	Finish
   Equal	Distinct
   Less	Not less
   Greater	Not greater

Remarks

• The "Contrary" operator is actually a meta-operator, that is, an operator acting on another operator.
  
  
• The "Contrary" operator is not a primitive. It is present in primitives either explicitly or implicitly.
  
  
• The "Contrary" operator is more generic than "logical negation". "Contrary" is universal. It applies to operators and to all kinds of expressions.

Contrary Constructive Monadic Operator
If ⊥ is a constructive monadic operator, the contravariant operator ⊥' satisfies the hyper-semantic property.

⟨( (x⊥)(⊥') = x )⟩

For example, ⟨( (x↓↑ = x )⟩

Calling (y = x⊥), we have conditional equivalence

⟨( (y = x⊥) ↔ (y(⊥') = x) )⟩

If the monadic operator can also be applied to the left, then the property is also true

⟨( (⊥')( ⊥x) = x )⟩

Calling (y = ⊥x), we have conditional equivalence

⟨( (y = ⊥x) ↔ ((⊥')y = x) )⟩


Contrary Constructive Dyadic Operator
When we have a dyadic expression, x⊥y, as is the case for exponentiation x^y (i.e., xy), two contractive operators are defined:

1. Right contrary operator.
   
   Symbolized by ⊥', it is the one that satisfies the hyper-semantic property
   
   ⟨( (x⊥y)(⊥')y = x )⟩
   
   Calling (z = x⊥y), we have conditional equivalence:
   
   ⟨( (z = x⊥y) ↔ (z(⊥')y = x) )⟩
   
   
2. Counterclockwise operator.
   
   Symbolized by '⊥, is the one that satisfies the hypersemantic property.
   
   ⟨( (x⊥y)('⊥)x = y )⟩
   
   Calling (z = x⊥y), we have conditional equivalence:
   
   ⟨( (z = x⊥y) ↔ (z('⊥)x = y) )⟩

Examples

Let's apply these concepts to the paradigmatic case of exponentiation, to see which are the opposite operators on the right and on the left: x^y (indicates xy).

1. Operator opposite to the right (^'):
   
   ((z^y)(^')y = z)
(x = z^y) // x = zy
(x(^')y = z) // y√x = z
   
   Therefore, x(^')y is y√x
   
   
2. Counterclockwise operator ('^):
   
   ((y^z)('^)y = z)
(x = y^z) // x = yz
(x('^)y = z) // logyx = z

Therefore, x('^)y is logyx

It is also worth considering conceptual recursion, i.e., the opposite operators to the right and left of ^' and of '^:

1. Opposite operator to the right of ^': (^')'
   
   Calling (⊥ = ^'), you have:
   
   ((z⊥y)( ⊥')y = z)

(x = z⊥y) // (x = z(^')y) x = y√z z = xy

(z = x(⊥')y) // z = xy
   
   Therefore,
   (x((^')')y = x^y) // xy
   
   
2. Opposite operator to the left of ^': '(^')
   
   Calling (⊥ = (^')), you have:
   
   ((y⊥z)('⊥)y = z)

(x = y⊥z) // (x = y(^')z) logzy = x

(z = x('⊥)y) // z = logxy
   
   Therefore,
   (x('(^'))y = y('^)x) // logxy
   
   
3. Opposite operator to the right of '^: ('^)'
   
   Calling (⊥ = ('^)), you have:
   
   ((z⊥y)( ⊥')y = z)

(x = z⊥y) // (x = z('^)y) logyz = x

x(⊥')y = z // z = yx
   
   Therefore,
   (x((('^)')y = y^x) // yx
   
   
4. Opposite operator to the left of '^: '('^)
   
   Calling (⊥ = ('^)), you have:
   
   (y⊥z)('⊥)y = z

(x = y⊥z) // (x = y('^)z) logzy = x zx = y

(x('⊥)y = z) // z = x√y
   
   Therefore,
   (x('('^))y = y(^')x) // x√y

These results are generalizable for any dyadic operator ⊥, the following hypersemantic properties being satisfied:

1. ⟨( x((⊥')')y = x⊥y )⟩
   
   
2. ⟨( x('('⊥))y = y(⊥')x )⟩
   
   
3. ⟨( x('(⊥'))y = y('⊥)x )⟩
   
   
4. ⟨( x(('⊥)')y = y⊥x )⟩

In case the operator is commutative, i.e., x⊥and ≡ y⊥x we have:

⟨( (z⊥y)( ⊥')y = z )⟩

⟨( x = z⊥y )⟩ ⟨( x(⊥')y = z )⟩

⟨( (y⊥z)('⊥)y = z )⟩

⟨( x = y⊥z )⟩ ⟨( x('⊥)y = z )⟩

Therefore,
⟨( x(⊥')y ≡ x('⊥)y )⟩ and (⊥' ≡ '⊥)

That is, the opposite operators on the left and on the right are equivalent.

Examples:

1. (a+b)(+')b // ev. a
(a+b)(+')a // ev. b
(a+b)('+)a // ev. b
(a+b)('+)b // ev. a
   
   
2. (a*b)(*')b // ev. a
(a*b)('*)a // ev. b

Counter Operator in Expressions
The "Contrary" operator can be used to define opposite values. For example,

(white' = black)
(black' = white)

Other examples:

1. If V and F represent the values "true" and "false", respectively, as these concepts are contrary to each other, we have: (V' = F) and (F' = V). In general,
   
   (f*V)' = (1-f)*V = f*F
   
   where f is a factor between 0 and 1. The case (V' = F) corresponds to f=1. If f=0.3, we have: (0.3*V)' = 0.7*V = 0.3*F.
   
   
2. If we have the opposite concepts high and low, the definitions are the same: (f*high)' = (1-f)*high = f*low

Properties

1. ⟨( (x'' = x) )⟩
2. ⟨( (x' = y) ↔ (y' = x) )⟩
3. ⟨( (f*x)' ≡ (1−f)*x )⟩
4. ⟨( (f*x)' ≡ f*x' )⟩