EQUIVALENCE (Direct and Contrary)

"Two things are identical if and only if what can be said of the one can be said salva veritate of the other" (Leibniz).

"It is self-evident that identity is not a relation between objects" (Wittgenstein, Tractatus 5.5301).

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Semantics

Two expressions are equivalent if they represent the same entity.


Syntax

(x ≡ y)  or  x≡y

Justification

Equivalence makes it easier for us to specify entities that are considered semantically indistinguishable, even if the form is different.


Examples

1. ⟨( x+y ≡ y+x )⟩ // the sum has the commutative property
   
   
2. ⟨( {x y} ≡ {y x} )⟩ // the conjunction has the commutative property

Remarks

• Equivalence is usually associated with generic expressions that specify properties.
  
  
• Two syntactically distinct expressions may be equivalent from a semantic point of view (when evaluated or interpreted). For example:
  
  ⟨( x^y^z ≡ (x^y)^z )⟩

⟨( (x★n x★m) ≡ x★(n+m) )⟩
  
  
• Equivalence strings can be defined, e.g., x≡y≡z, which is equivalent to (x≡y y≡z).

Axioms

1. ⟨( x ≡ x )⟩ // identity (every expression is equivalent to itself)
   
   
2. ⟨( (x ≡ y) ↔ (y ≡ x) )⟩ // commutativity
   
   
3. ⟨( (x ≡ y) → (y ≡ z) → (x ≡ z) )⟩ // transitivity
   
   
4. ⟨( (x = y) ↔ (x ≡ y) )⟩
   Equality and equivalence are equivalent at the conditional level (one implies the other).

Simplification of equivalent expressions

In an expression, all equivalent expressions are considered, always selecting the simplest, that is, the most compact. For example:

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

In this case, we consider the equivalent expressions

(a + b + a), (a + a + a + b) and (b + a + a), since the sum is commutative.


Equivalences with opposite concepts

It is possible to specify equivalences between contrary concepts. Examples:

1. (rich' ≡ poor) // the opposite of rich is equivalent to poor
   
   
2. (high' ≡ low) // the opposite of high is equivalent to low
   
   
3. (white' ≡ black) // the opposite of white is equivalent to black

And also between concepts affected by a numerical factor between 0 and 1:

(0.7*high ≡ 0.3*low)

In general, the following properties are satisfied:

1. ⟨( (x' ≡ y) ↔ (x ≡ y') )⟩
   
   That is, if, for example, (high' ≡ low), then (high ≡ low')
   
   
2. ⟨( (r*x ≡ (1−r)*x') )⟩
   For example, (0.7*high ≡ 0.3*low)

Contrary equivalence

As in the case of contrary substitution, the expression (x ≡' y) can be used as a condition. It is defined as follows:

⟨( (z ← (x ≡' y)) =: (z ←' (x ≡ y))) )⟩

For example,

(a ←(x ≡' y)) represents (a ←' (x ≡' y))

The contrary equivalence can also be used as a descriptive expression. For example,

(a ≡' b)
(a and b are not equivalent)