DECLARATIVE EXPRESSIONS

"Language not only describes, but creates our reality" (Dr. Mario Alonso Puig).

"All language communicates itself" (Walter Benjamin).

"Meaning is the core of language" (Roger Schank).

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Definition

A declarative expression is an expression that establishes a relationship between expressions, whether or not those expressions have been previously defined. There are three types of declarative expressions:


Relations

One can use, for example, the following relations:

a<b a>b a≤b a≥b a=b a≠b
a≡b (a ≡' b)
a∈A a∉B A⊂B A⊃B A⊆B A⊇B A⊈B
a/b (a↓ = b) (a↓↓ = b)
a∈A A⊂B
(a b) {a b}

Examples:

1. a<b
   (it is declared that a is less than b, that this relation exists)
   (a<b → c) // ev. c
   
   
2. {a b c}
   (this relationship is declared to exist between a, b and c)
   (a↑ = b↑)→c // ev. c
   
   
3. a∈A
   (it is declared that a belongs to A, that this relation exists)
   
   The fact that it is specified that an element belongs to a set, does not imply that this element is added to the set. Declaration is not the same as construction.
   (a∈A → b) // ev. b
   
   
4. A⊂B
   (this relation is defined without having previously defined neither A nor B)
   (A⊂B → c) // ev. c
   
   
5. A∩B = {2 3}
   (we define A∩B without having previously defined neither A nor B)
   (A∩B)∪{1 2 3} // ev. {2 3}

Axioms

Axioms are declarative expressions. They are relations established a priori. For example,

1. ⟨( x+y ≡ y+x )⟩ // commutativity of sum.
   
   
2. ⟨( (x→y ∧ y→z) → (x→z) )⟩ // transitivity of condition.

Imaginary declarative expressions.

These are expressions that establish relationships that go against common sense, but that the language allows. Examples:

1. 1>3
(1>3 → c) // ev. c
   
   
2. a∈a
   (a belongs to itself).
   (a∈a → c) // ev. c
   
   
3. A⊂A
   (A is included in itself).
   (A⊂A → c) // ev. c
   
   
4. ⟨( n > n+1 )) // every natural number is greater than its successor.
(3>4 → c) // ev. c