LOGICAL OPERATIONS (Negation, Conjunction and Disjunction)
"Logic is a deep grammar of the rational" (Patrick K. Bastable).
"Logic was born as an attempt to mechanize the intellective processes of reasoning" (Douglas Hofstadter).
"Logic is concerned only with the pure form of thought" (Kant).
Introduction
The logical operations Negation, Conjunction and Disjunction are derived operations. They all derive from primitives. The only "pure" logical operation is the primitive Condition.
Logical operations in MENTAL apply to all types of expressions. The result of the logical operations Negation, Conjunction and Disjunction is always an existential value (θ or α).
Logical negation
Semantics and Syntax
The logical negation of an expression x is the opposite of its existential value (θ or α).
Definition
⟨( ¬x = (x?)' )⟩ or
⟨( ¬x = (α ←' x? → θ) )⟩
Examples
¬3 // ev. θ
¬(3=4) // ev. α
Logical Conjunction and Disjunction
Semantics and Syntax
The logical conjunction (symbolized by ∧) of two expressions is θ if the existential value of one of them is θ, and α otherwise.
The logical disjunction (symbolized by ∨) of two expressions is α if the existential value of one of them is α, and θ otherwise.
Both operations are dual and are analogous to the connectives of classical logic, where θ plays the role of "false" and α the role of "true". Both existential values are contrary to each other: (α' = θ) and (θ' = α).
Existential tables, analogous to the truth tables of classical logic, are:
x
y
x∧y
x∨y
α
α
α
α
α
θ
θ
α
θ
α
θ
α
θ
θ
θ
θ
Definitions
⟨( x∧y = (x? ← y?) )⟩ // Conjunction
⟨( x∧y = (α ← x ← y) )⟩ // Equivalent definition
⟨( x∨y = {x? y?}↓ )⟩ // Disjunction
The duality is expressed as follows: (∨' = ∧) and, therefore, (∨' = ∧).
Examples
3∧4 // ev. α
3∨4 // ev. α
(4<3)∧(3<4) // ev. θ
(4<3)∨(3<4) // ev. α
(x° = 7) (y° = θ)
(x∧y) // ev. θ
(x∨y) // ev. α
Properties
All the known properties of classical logic are fulfilled, among them, De Morgan's laws, the commutative laws, the distributive laws. All these laws are dual. One property is obtained from another by interchanging the logical operators of conjunction and disjunction.
