Quantifiers are powerful tools in first-order logic, allowing us to express statements about all or some elements in a domain. They come in two flavors: universal () for "all" and existential () for "some."
Negating quantifiers flips their meaning, turning "all" into "some not" and "some" into "none." This property is crucial for understanding logical relationships and constructing valid arguments in first-order logic.
Quantifiers
Universal and Existential Quantifiers
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∀ expresses that a formula holds for all values of a variable
Formally, ∀xP(x) means that P(x) is true for every possible value of x in the domain
Can be read as "for all", "for every", or "for each" (∀x∈N,x≥0 reads "for all natural numbers x, x is greater than or equal to 0")
∃ expresses that a formula holds for at least one value of a variable
Formally, ∃xP(x) means that there exists at least one value of x in the domain for which P(x) is true
Can be read as "there exists", "for some", or "there is at least one" (∃x∈R,x2=4 reads "there exists a real number x such that x squared equals 4")
Quantifier Negation
Negation of a universally quantified formula ∀xP(x) is equivalent to an existentially quantified negation of the formula ∃x¬P(x)
¬(∀xP(x))≡∃x¬P(x) (reads "not for all x, P(x)" is equivalent to "there exists an x such that not P(x)")
Example: ¬(∀x∈N,x>0)≡∃x∈N,x≤0 (reads "not all natural numbers are greater than 0" is equivalent to "there exists a natural number less than or equal to 0")
Negation of an existentially quantified formula ∃xP(x) is equivalent to a universally quantified negation of the formula ∀x¬P(x)
¬(∃xP(x))≡∀x¬P(x) (reads "there does not exist an x such that P(x)" is equivalent to "for all x, not P(x)")
Example: ¬(∃x∈Z,x2=2)≡∀x∈Z,x2=2 (reads "there does not exist an integer x such that x squared equals 2" is equivalent to "for all integers x, x squared does not equal 2")
Quantifier Transformations
Quantifier Distribution and Elimination
Quantifier distribution refers to the rules for distributing quantifiers over logical connectives
∀x(P(x)∧Q(x))≡(∀xP(x))∧(∀xQ(x)) (universal quantifier distributes over conjunction)
∃x(P(x)∨Q(x))≡(∃xP(x))∨(∃xQ(x)) (existential quantifier distributes over disjunction)
However, ∀x(P(x)∨Q(x))≡(∀xP(x))∨(∀xQ(x)) and ∃x(P(x)∧Q(x))≡(∃xP(x))∧(∃xQ(x))
Quantifier elimination is the process of removing quantifiers from a formula while preserving its satisfiability
One approach is to replace the quantified variable with a constant that satisfies the formula, if such a constant exists
Example: ∃x(x>0∧∀y(y>x→y>1)) can be simplified to ∃x(x>0∧x<1) by eliminating the universal quantifier
Prenex Normal Form and Skolemization
Prenex normal form is a formula where all quantifiers appear at the beginning, followed by a quantifier-free matrix
Any formula can be converted to prenex normal form by applying quantifier distribution and renaming variables to avoid name clashes
Example: ∀x∃y(P(x)∨Q(y)) is in prenex normal form, while (∀xP(x))∨(∃yQ(y)) is not
Skolemization is a process of eliminating existential quantifiers from a formula in prenex normal form
Each existentially quantified variable is replaced by a Skolem function that depends on the universally quantified variables preceding it
The resulting formula is satisfiable if and only if the original formula is satisfiable
Example: ∀x∃yP(x,y) can be Skolemized to ∀xP(x,f(x)), where f is a Skolem function