Question

Consider the following predicate / First-Order logic sentence: ∀a(∃b(X(a,b)∧Y(b))⇒[∀c(¬Z(c,a))]) Assume that X,Y, and Z are predicates. Domains for variables are:

Dₐ={A1, A2, A3}
Dᵦ={B1, B2}
D={C1,C2,C3,C4}
​Now convert the sentence above to its CNF form (show and explain all steps):

Answer 1

The process of converting a first-order logic sentence to CNF involves logical equivalence transformations, standardization of quantifiers, and a series of steps to maintain the structure and meaning of the original predicates and logical form.

Step-by-step explanation:

Converting a predicate / first-order logic sentence to its Conjunctive Normal Form (CNF) involves several steps that ensure the structure of the initial argument is maintained. The sentence ∀a(∃b(X(a,b)∧Y(b))⇒[∀c(¬Z(c,a))]) presents a universal quantifier followed by an implication, with nested existential and universal quantifiers combined with logical connectives like conjunction and negation inside the implication.

To begin the conversion, apply logical equivalences: distribute the negation using De Morgan's laws and the implications using the implication equivalence (p ⇒ q ≡ ¬p ∨ q). Quantifiers also need to be standardized, and Skolem functions might be introduced to eliminate existential quantifiers. This process ensures that the predicates maintain their meaning (descriptive terms or concepts) and the logical form of valid deductive inferences such as disjunctive syllogism and modus ponens. After these transformations, the statement can be further simplified into a combination of disjunctions and conjunctions that adhere to CNF's strictures.