=================================================================================

In machine learning, logical statements refer to the use of logical conditions or rules to make decisions or predictions based on input data. These statements are often used in decision-making processes within algorithms to determine the outcome or classification of a given instance. Logical statements can take various forms, including simple if-else conditions or more complex logical rules. These statements are designed to capture patterns or relationships in the data, allowing the machine learning model to generalize and make predictions on new, unseen data. 

While logical statements and mathematical concepts are distinct, machine learning models can incorporate mathematical operations and properties, such as the distributive property, as part of their internal computations.  

Some examples of logical statements are:

Example i:

Given the statements: 

P∨R (P or R) is true. 

 ¬P (not P) is true. 

Conclusion: 

 R is true.

Example ii:

Given the statements: 

P ∨ R  ∨ G₁ ∨ G₂ ∨ ... ∨ G_n is true. 

 ¬P is true. 

Conclusion: 

R  ∨ G₁ ∨ G₂ ∨ ... ∨ G_n is true.

Example iii:

Given the statements:

A∨B is true.

¬A∨C is true.

Conclusion:

B∨C is true.

Example iv:

De Morgan's Laws: 

  ¬(P ∧ Q) ≡ (¬P) ∨ (¬Q)

  ¬(P ∨ Q) ≡ (¬P) ∧ (¬Q)

Example v:  

Distributive Property: 

 A ∧ (B ∨ C) ≡ (A ∧ B) ∨ (A ∧ C) 

A ∨ (B ∧ C) ≡ (A ∨ B) ∧ (A ∨ C)

Example vi:  ("→" represents implication)

The contrapositive of P → Q is ¬Q → ¬P. 

Example vii: 

Disjunction of literals is a logical concept that is often encountered in the context of Boolean logic and propositional calculus:

A | B | A OR B 

-------------- 

T | T | T 

T | F | T 

F | T | T 

F | F | F  

Example viii:  

Conjunctive normal form (CNF) is a standard representation of logical formulas used in propositional logic. A logical formula is in CNF if it is a conjunction (AND) of one or more clauses, where each clause is a disjunction (OR) of literals. In other words, CNF is a way of expressing a logical formula as a product of sums. The general form of a CNF formula is: 

Example viii: 

Tautology: A statement that is always true, regardless of the truth values of its variables. 

Example ix:  

Contradiction: A statement that is always false, regardless of the truth values of its variables.

Example x: Conversion to CNF

Eliminate biconditionals:

Turn (A ↔ B) into (A → B) ∧ (B → A).

Eliminate implications:

Turn (A → B) into ¬A ∨ B.

Move ¬ inwards using De Morgan's Laws:

E.g. turn ¬(A ∧ B) into  ¬A ∨ ¬B.

Turn (A ∨ B) → C into  ¬(A ∨ B) ∨ C, after eliminate implication, into (¬A ∨ ¬B) ∨ C, De Morgan's Law, into (¬A ∨ C) ∧ (¬B ∨ C), distributive law.

Example xi: Resolution. 

============================================