Predicate Logic & Quantifiers

From Propositions to Predicates

Predicate logic (first-order logic) extends propositional logic with predicates — statements about objects — and quantifiers that range over domains.

A predicate $P(x)$ is a statement about $x$ that becomes a proposition when $x$ is given a value. The two quantifiers are:

$\forall x \in D,\ P(x) \quad \text{(universal: P holds for every x in D)}$ $\exists x \in D,\ P(x) \quad \text{(existential: P holds for at least one x in D)}$

Negation of quantifiers (De Morgan for quantifiers): $\neg(\forall x,\ P(x)) \equiv \exists x,\ \neg P(x)$ $\neg(\exists x,\ P(x)) \equiv \forall x,\ \neg P(x)$

def for_all(domain, predicate):
    return all(predicate(x) for x in domain)

def exists(domain, predicate):
    return any(predicate(x) for x in domain)

# Every number in {1..5} is positive
print(for_all(range(1, 6), lambda x: x > 0))  # True
# Some number in {1..5} equals 3
print(exists(range(1, 6), lambda x: x == 3))  # True

Nested Quantifiers

Order matters: $\forall x\, \exists y,\ P(x,y)$ differs from $\exists y\, \forall x,\ P(x,y)$.

Your Task

Implement for_all(domain, predicate) and exists(domain, predicate) using Python's built-in all and any.