Predikátlogika

Predikátlogika, also known as first-order logic or quantification theory, is a formal system used in logic and mathematics to represent statements about objects and their properties. It extends propositional logic by introducing predicates, which represent properties or relations, and quantifiers, which express the extent to which a predicate applies to a set of objects.

In predikátlogika, statements are broken down into terms, predicates, and quantifiers. Terms can be constants (specific objects), variables (placeholders for objects), or functions (which map objects to other objects). Predicates are symbols that can be true or false depending on the terms they are applied to. For example, "is_red(apple)" is a predicate that is true if the object 'apple' has the property of being red.

The core of predikátlogika lies in its quantifiers. The universal quantifier, denoted by '∀', means "for all" or "for every." For instance, ∀x (human(x) → mortal(x)) translates to "for all x, if x is human, then x is mortal." The existential quantifier, denoted by '∃', means "there exists" or "for some." For example, ∃x (cat(x) ∧ sleeps(x)) means "there exists an x such that x is a cat and x sleeps."

Predikátlogika provides a powerful framework for expressing complex logical relationships and is fundamental to fields like mathematics, computer science (especially artificial intelligence and database theory), and linguistics. Its ability to formalize reasoning about properties and relations makes it indispensable for precise and unambiguous communication in these disciplines.