MENTAL, AN OPERATIONAL AND DESCRIPTIVE LANGUAGE

"Knowledge is language that names and describes" (Wittgenstein).

"One must describe, rather than explain. Every explanation is a hypothesis" (Wittgenstein).

"To give the essence of description means to give the essence of all description, hence the essence of the world" (Wittgenstein, Tractatus 4.711).

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Types of expressions

There are three types of expressions:

• Operational.
  An operational expression is one that deals with one or more operations and produces a result. That result may be a descriptive expression.
  
  
• Descriptive.
  A descriptive expression is one that represents several expressions, which may themselves be descriptive or operational. A descriptive expression may or may not be parameterized.
  
  
• Mixed.
  They are expressions that combine the operational and the descriptive.

Examples of operative expressions

• Arithmetic expressions.
  
  (x = a+a) // ev. (x = 2*a)

(x = a*a) // ev. (x = a^2)
  
  
• Substitution expressions.
  
  (a b c)/(a=3) // ev. (3 b c)

{a 2*a 3*a}/(a=3) // ev. (3 6 9)
  
  
• Operations with groups (sets or sequences).
  
  (1 2)∪(3 4) // ev. (1 2 3 4) ev. 1234

{a b}∪{c d} // ev. {a b c d}

Examples of descriptive expressions

• Parameterized generic expressions.
  
  ⟨( f(x y) = (x+y x*y) )⟩ // describes a function
  
  
• Set.
  
  {⟨( n ← n>5 )⟩} // represents the set of natural numbers greater than 5
  
  
• Distribution.
  
  ([[a b] [1 2 3]]) // represents (a 1 a 2 a 3 b 1 b 2 b 3)
  
  
• Range.
  
  1…100 // represents the open expression 1 2 3 hellip; 100

1…10 // represents the sequence (1 2 3 … 10)

(1 3 … 13) // represents the open expression 1 3 5 7 9 11 13
  
  
• Repetition.
  
  12☆5 // represents the open expression 12 12 12 12 12 12 12

12★5 // represents the sequence (12 12 12 12 12 12 12)
  
  
• Mixed descriptive expressions.
  
  ([[a b] [1…3]]) // represents a (a 1 a 2 a 3 b 1 b 2 b 3)

[[a b]☆3] // represents a [[a b] [a b] [a b] [a b]< b>], which in turn represents a a a a a a b a b a a a b b b b a a a a a b a b a a b b b

Examples of mixed descriptive expressions

• Function defined by particularizing a function.
  
  ⟨( f(x y) = (x+y x*y) )⟩

⟨( g(x) = f(x 3) )⟩ // evaluates as ⟨( g(x) = (x+3 x*3) )⟩