Quantifiers and predicates help us translate everyday statements into precise logical expressions. We'll learn how to use universal and existential quantifiers to represent ideas like "all," "some," and "exactly one."

Understanding quantifiers is crucial for expressing complex relationships in logic. We'll explore how the order of quantifiers affects meaning and how to use multiple quantifiers to represent intricate statements accurately.

Translating Quantified Statements

Universal and Existential Conditionals

• Translate natural language statements into symbolic logic using quantifiers and predicates
• Universal conditional statements use the universal quantifier $\forall$ to express that a condition holds for all elements in a domain
  • $\forall x (P(x) \to Q(x))$ means "for all $x$, if $P(x)$ is true, then $Q(x)$ is true"
  • Example: "All cats are mammals" can be translated as $\forall x (\text{Cat}(x) \to \text{Mammal}(x))$
• Existential conditional statements use the existential quantifier $\exists$ to express that a condition holds for at least one element in a domain
  • $\exists x (P(x) \land Q(x))$ means "there exists an $x$ such that both $P(x)$ and $Q(x)$ are true"
  • Example: "Some birds can fly" can be translated as $\exists x (\text{Bird}(x) \land \text{CanFly}(x))$
• Conditional statements with quantifiers combine quantifiers and conditional operators to express complex relationships between predicates
  • Example: "If a number is even, then it is divisible by 2" can be translated as $\forall x (\text{Even}(x) \to \text{DivisibleBy2}(x))$

Scope and Order of Quantifiers

• The order of quantifiers in a statement affects its meaning and scope
  • The quantifier that appears first has a wider scope and binds the variable for the entire statement
  • Example: $\forall x \exists y (P(x,y))$ means "for every $x$, there exists a $y$ such that $P(x,y)$ is true"
  • Example: $\exists y \forall x (P(x,y))$ means "there exists a $y$ such that for every $x$, $P(x,y)$ is true"
• Changing the order of quantifiers can result in statements with different meanings
  • Example: $\forall x \exists y (x + y = 0)$ is true for real numbers, as for any $x$, there exists a $y$ ($-x$) such that their sum is 0
  • Example: $\exists y \forall x (x + y = 0)$ is false for real numbers, as there is no single $y$ that makes the sum 0 for all $x$

Multiple Quantifiers

Nested Quantifiers

• Statements can contain multiple quantifiers, creating nested structures
  • Example: $\forall x \exists y \forall z (P(x,y,z))$ means "for all $x$, there exists a $y$ such that for all $z$, $P(x,y,z)$ is true"
• The order of quantifiers determines the dependencies between variables
  • Example: $\forall x \exists y (x < y)$ means "for every $x$, there exists a $y$ greater than $x$"
  • Example: $\exists y \forall x (x < y)$ means "there exists a $y$ greater than all $x$"

Uniqueness Quantifier

• The uniqueness quantifier $\exists!$ expresses that there exists exactly one element satisfying a condition
  • $\exists! x (P(x))$ means "there exists a unique $x$ such that $P(x)$ is true"
  • Example: "Every person has exactly one biological mother" can be translated as $\forall x \exists! y (\text{Person}(x) \land \text{MotherOf}(y,x))$
• The uniqueness quantifier can be defined using the existential and universal quantifiers
  • $\exists! x (P(x)) \equiv \exists x (P(x) \land \forall y (P(y) \to y = x))$
  • This means "there exists an $x$ such that $P(x)$ is true, and for all $y$, if $P(y)$ is true, then $y$ is equal to $x$"

Quantifier Meanings

"At Least One" and "At Most One"

• The existential quantifier $\exists$ expresses the concept of "at least one"
  • $\exists x (P(x))$ means "there exists at least one $x$ such that $P(x)$ is true"
  • Example: "Some students are athletes" can be translated as $\exists x (\text{Student}(x) \land \text{Athlete}(x))$
• "At most one" can be expressed using the universal and existential quantifiers
  • $\forall x \forall y ((P(x) \land P(y)) \to x = y)$ means "for all $x$ and $y$, if both $P(x)$ and $P(y)$ are true, then $x$ and $y$ are the same"
  • Example: "At most one person can be the tallest in a group" can be translated as $\forall x \forall y ((\text{Person}(x) \land \text{Person}(y) \land \text{TallestInGroup}(x) \land \text{TallestInGroup}(y)) \to x = y)$

"Exactly One"

• "Exactly one" is equivalent to the uniqueness quantifier $\exists!$
  • $\exists! x (P(x))$ means "there exists exactly one $x$ such that $P(x)$ is true"
  • This can be expressed using the existential and universal quantifiers: $\exists x (P(x) \land \forall y (P(y) \to y = x))$
• "Exactly one" combines the concepts of "at least one" and "at most one"
  • Example: "Every country has exactly one capital city" can be translated as $\forall x (\text{Country}(x) \to \exists! y (\text{CapitalCity}(y) \land \text{IsCapitalOf}(y,x)))$
  • This means "for every country $x$, there exists a unique capital city $y$ such that $y$ is the capital of $x$"