👁️‍🗨️Formal Logic I Unit 9 – Quantifiers and Their Scope

What Are Quantifiers?

• Quantifiers are logical symbols used to specify the quantity or extent of a statement in predicate logic
• Allow for making general statements about collections of objects without referring to specific individuals
• Quantifiers bind variables within a formula, determining the range of values the variable can take
• Two main types of quantifiers: universal quantifier (∀) and existential quantifier (∃)
• Quantifiers are essential for expressing complex logical statements and reasoning about properties of sets
• Enable the formalization of natural language statements involving words like "all", "some", "none", and "every"
• Quantifiers introduce variables into a logical formula, which can then be used to represent arbitrary objects

Types of Quantifiers

• Universal quantifier (∀): Expresses that a property holds for all elements in a given domain
  • Denoted by the symbol ∀, read as "for all" or "for every"
  • Example: ∀x P(x) means "for all x, P(x) is true"
• Existential quantifier (∃): Asserts that there exists at least one element in the domain for which a property holds
  • Denoted by the symbol ∃, read as "there exists" or "for some"
  • Example: ∃x P(x) means "there exists an x such that P(x) is true"
• Uniqueness quantifier (∃!): Asserts the existence of exactly one element in the domain satisfying a property
  • Denoted by the symbol ∃!, read as "there exists a unique"
  • Can be expressed using a combination of universal and existential quantifiers
• Empty quantifier (∀x ∈ ∅): Vacuously true statement, as it quantifies over an empty set
• Bounded quantifiers: Restrict the domain of quantification to a specific set or range of values

Scope of Quantifiers

• Scope refers to the portion of a logical formula that a quantifier applies to or has jurisdiction over
• Determined by the placement of parentheses or the order of quantifiers in a formula
• Variables bound by a quantifier can only be used within its scope
• Helps avoid ambiguity and ensures the correct interpretation of the quantified statement
• Nested quantifiers: When one quantifier appears within the scope of another
  • The order of quantifiers affects the meaning of the statement
  • Example: ∀x ∃y P(x, y) is different from ∃y ∀x P(x, y)
• Free variables: Variables not bound by any quantifier within a formula
  • Can lead to ambiguity or undefined behavior

Quantifier Rules and Notation

• Quantifier elimination: Process of removing quantifiers from a formula while preserving its meaning
  • Existential instantiation: Replacing ∃x P(x) with P(c), where c is a new constant
  • Universal instantiation: Replacing ∀x P(x) with P(t), where t is any term
• Quantifier negation: Negating a quantified statement changes the quantifier and the formula
  • ¬∀x P(x) is equivalent to ∃x ¬P(x)
  • ¬∃x P(x) is equivalent to ∀x ¬P(x)
• Quantifier distributivity: Quantifiers distribute over logical connectives in specific ways
  • ∀x (P(x) ∧ Q(x)) is equivalent to (∀x P(x)) ∧ (∀x Q(x))
  • ∃x (P(x) ∨ Q(x)) is equivalent to (∃x P(x)) ∨ (∃x Q(x))
• Quantifier ordering: The order of quantifiers matters when they are nested
  • ∀x ∃y P(x, y) means "for every x, there exists a y such that P(x, y) holds"
  • ∃y ∀x P(x, y) means "there exists a y such that for every x, P(x, y) holds"

Translating Sentences with Quantifiers

• Identify the domain of discourse: The set of objects the sentence is referring to
• Recognize the main predicate or property being discussed in the sentence
• Determine the quantifier(s) needed to express the sentence accurately
  • "All" or "every" typically indicate the universal quantifier (∀)
  • "Some", "at least one", or "there exists" suggest the existential quantifier (∃)
• Introduce variables to represent the objects in the domain
• Construct the logical formula using the identified quantifiers, variables, and predicates
• Ensure that the scope of the quantifiers is correctly represented using parentheses
• Example: "Every student in the class has a favorite subject" can be translated as ∀x (Student(x) → ∃y (Subject(y) ∧ FavoriteOf(y, x)))

Nested Quantifiers

• Nested quantifiers occur when one quantifier appears within the scope of another
• The order of the quantifiers affects the meaning of the statement
• Nested quantifiers can express complex relationships between objects in the domain
• Example: ∀x ∃y P(x, y) means "for every x, there exists a y such that P(x, y) holds"
  • The choice of y can depend on the value of x
  • Different x values may be associated with different y values
• Example: ∃y ∀x P(x, y) means "there exists a y such that for every x, P(x, y) holds"
  • The same y value must work for all possible x values
  • The choice of y is independent of the value of x
• Nested quantifiers can involve multiple variables and predicates, creating intricate logical statements

Common Mistakes and Pitfalls

• Confusing the order of quantifiers: ∀x ∃y P(x, y) is not equivalent to ∃y ∀x P(x, y)
• Misplacing parentheses: Incorrect scope can change the meaning of the statement
  • Example: ∀x (P(x) → Q(x)) is different from (∀x P(x)) → Q(x)
• Using free variables: All variables should be properly bound by quantifiers
• Mixing up the domain of discourse: Ensure that the quantifiers and variables refer to the intended set of objects
• Neglecting the existential import: The existential quantifier assumes that the domain is non-empty
  • If the domain is empty, statements like ∀x P(x) are vacuously true
• Overcomplicating the translation: Aim for a clear and concise representation of the sentence
• Ignoring the context: The meaning of a statement can depend on the specific context or field of study

Practice Problems and Applications

• Translate natural language sentences into predicate logic using quantifiers
  • "Every prime number greater than 2 is odd"
  • "There exists a solution to the equation x^2 + 1 = 0"
• Determine the truth value of quantified statements given a specific domain
  • Let the domain be the set of integers. Is the statement ∀x ∃y (x + y = 0) true or false?
• Negate quantified statements and simplify the result
  • Negate the statement ∀x (P(x) → ∃y Q(x, y)) and simplify it
• Analyze the logical equivalence of quantified formulas
  • Are the formulas ∀x (P(x) ∧ Q(x)) and (∀x P(x)) ∧ (∀x Q(x)) logically equivalent?
• Apply quantifiers to real-world problems and situations
  • In a database of employees and their salaries, express the statement "Every employee earns more than $50,000 per year"
  • In a social network, represent the statement "Each person has at least one friend who has more friends than they do"