INTERROGATION

"What is a question?" (Wittgenstein. Philosophical Investigations).

"Every question is an invitation to look in a certain direction" (Dr. Mario Alonso Puig).

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Semantics

The interrogation of an expression x is its existential value: θ (non-existence) or α (existence). These values are used by MENTAL instead of the logical values "true" and "false".

The meta-expressions θ and α are contrary to each other: (θ' = α), (α' = θ), as with the traditional logical values:

(V' = F), (F' = V).

Syntax

x? // interrogation of x

Definition

⟨( x? = (α ← x →' θ) )⟩

Justification

This operation allows you to obtain the existential values of expressions for later use in other expressions, mainly as conditions or types of expressions.


Examples

1. x=7
(x=7)? // ev. α
(x=4)? // ev. θ
(x<4)? // ev. θ
(x>5)? // ev. α
(x≥5)? // ev. α
   
   
2. z=(x y)
(z=(x 3))? // ev. θ
(z=(x y))? // ev. α
(z=(x α))? // ev. α
(z=(α α))? // ev. α
(z=(x αº))? // ev. θ (symbol α is taken into account)
   
   
3. C={a b c}
(a∈C)? // ev. α (a belongs to C)
(d∈C)? // ev. θ (d does not belong to C)
(α∈C)? // ev. α (C contains some element, C is not empty)

Properties

1. ⟨( (x = x)? = α )⟩ // every expression is equal to itself
   
   
2. ⟨( (x ≠ x)? = θ )⟩ // there is no expression that is not equal to itself
   
   
3. ⟨( (x > x)? = θ )⟩
   
   
4. ⟨( (x < x)? = θ )⟩
   
   
5. ⟨( (x ≥ x)? = α )⟩
   
   
6. ⟨( (x ≤ x)? = α )⟩
   
   
7. ⟨( x?? = x? )⟩
   
   
8. (θ? = θ)
   
   
9. (α? = α)
   
   
10. ⟨( x←y ≡ x←(y?) )⟩
    
    
11. ⟨( (x ←' y) ≡ (x←((y?))') )⟩
    
    
12. ⟨( (x? ← y?) ≡ (x ← y)? )⟩
    
    
13. ⟨( (x? ∧ y?)' ≡ ((x?)' ∨ (y?)')') )⟩
    
    
14. ⟨( {x? ∨ y?}' ≡ ((x?)' ∧ (y?)')') )⟩
    
    This property and the previous one are analogous to De Morgan's laws of Boolean logic.