Predicate Logic
PREDICATE LOGIC (also called first-order logic) extends propositional logic by adding VARIABLES and QUANTIFIERS. While propositional logic deals with whole sentences, predicate logic can describe properties of objects. It's the language used in formal mathematics, computer science, philosophy, and rigorous proofs.
New tools. PREDICATES: properties or relations. "Tall(x)" means "x is tall." "Loves(x, y)" means "x loves y." VARIABLES: x, y, z stand for objects. QUANTIFIERS: ∀ (FOR ALL — universal): ∀x Tall(x) means "everything is tall." ∃ (EXISTS — existential): ∃x Tall(x) means "something is tall." Combine: ∀x∃y Loves(x, y) means "everyone loves someone." Different from ∃y∀x Loves(x, y) which means "there is someone whom everyone loves" — note the order of quantifiers MATTERS.
Which English sentence does "∀x (Bird(x) → CanFly(x))" represent?
Why predicate logic matters. It's the language of mathematical PROOF. Russell, Frege, and Hilbert built it in the early 1900s. Computer science uses it for databases (SQL), AI knowledge bases, formal verification of programs. Philosophers use it to clarify arguments. In short: it's the most precise tool humans have for formal reasoning.
Translate Three
Translate to predicate logic: (1) "Some students are clever." (2) "All cats are mammals." (3) "Every student loves at least one teacher." Use predicates like Student(x), Clever(x), Cat(x), Mammal(x), Teacher(y), Loves(x, y).
Predicate logic is one of the most powerful tools in human history. It made modern mathematics rigorous and gave computer science its foundation. Worth the effort to learn.
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