Predicate Logic

An extension of Propositional Logic that uses variables and quantifiers to represent and analyse Statement.


Is a Statement that includes a variable.

  • P(x)P(x): "x is a prime number"

A predicate becomes a proposition when the variable are substituted for values.

  • P(2)P(2): "2 is a prime number" (True)


Quantifiers describe how many of a thing there are.

Universal Quantifier

Symbol: \forall

Means "For all" or "Every".


x,P(x)\forall x, P(x): "For every x, x is a prime number"

Existential Quantifier

Symbol: \exists

Means "There exists" or "Some".


x,P(x)\exists x, P(x): "There exists an x such that x is a prime number"

DeMorgan's Laws for negating quantifiers

∼[(∀x)P(x)] ≡ (∃ x)[∼P(x)] ∼[(∀x)P(x)] ≡ (∀x)[∼P(x)]