Glossary

9.1 Universal and Existential Quantifiers

2 min read•august 7, 2024

Quantifiers are powerful tools in logic, allowing us to make sweeping statements about entire groups. Universal quantifiers () cover all elements, while existential quantifiers () assert the existence of at least one element satisfying a condition.

Understanding variables in quantified statements is crucial. Bound variables are tied to quantifiers, while free variables roam unconstrained. This distinction helps us grasp the scope and meaning of logical expressions in formal reasoning.

Quantifiers and Variables

Universal and Existential Quantifiers

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  • (∀) expresses that a is true elements in a given domain
  • (∃) expresses that a predicate is true for at least one element in a given domain
  • For all (∀) is used to represent the universal quantifier and is read as "for all" or "for every"
    • Example: P(x) is read as "for all x, P(x) is true"
  • (∃) is used to represent the existential quantifier and is read as "there exists" or "for some"
    • Example: is read as "there exists an x such that P(x) is true"

Variables in Quantified Statements

  • is a variable that is quantified by a quantifier within a statement
    • In the statement ∀x P(x), x is a bound variable because it is quantified by the universal quantifier ∀
  • is a variable that is not quantified by any quantifier within a statement
    • In the statement P(x) ∧ Q(y), both x and y are free variables because they are not quantified by any quantifier

Quantified Statements

Components of Quantified Statements

  • is a statement that involves one or more quantifiers and a predicate
    • Consists of a quantifier, a variable, and a predicate
    • Example: is a quantified statement with the universal quantifier ∀, variable x, and the predicate (P(x) → Q(x))
  • Predicate is a statement that contains one or more variables and becomes a proposition when the variables are assigned specific values
    • Example: "x is even" is a predicate, and it becomes a proposition when x is assigned a specific value, such as "4 is even"

Domain of Discourse

  • is the set of all possible values that the variables in a quantified statement can take
    • Specifies the context or universe in which the quantified statement is being evaluated
    • Example: In the statement "All students in this class have passed the exam," the domain of discourse is the set of all students in the specific class being referred to
  • The domain of discourse can be any non-empty set, such as the set of natural numbers, real numbers, or a specific set of objects
    • Example: In the statement "For every natural number n, n² + n + 41 is prime," the domain of discourse is the set of all natural numbers

Key Terms to Review (21)

∀: The symbol ∀ represents the universal quantifier in logic, indicating that a statement applies to all elements within a particular domain. This concept is essential for expressing general truths and plays a crucial role in understanding predicates and translating categorical propositions into formal logic.
∀x: The symbol ∀x, known as the universal quantifier, indicates that a statement applies to all elements within a specific domain. It asserts that for every individual 'x' in that domain, the property or predicate following the quantifier holds true. This concept is crucial in logic and mathematics, as it allows for generalizations and forms the basis of logical arguments and proofs.
∀x (p(x) → q(x)): The expression ∀x (p(x) → q(x)) is a universal quantification that states 'for all x, if p(x) is true, then q(x) is also true'. This logical statement connects two predicates, p and q, and indicates a conditional relationship between them. It showcases the concept of universal quantifiers in formal logic, where the truth of the overall statement depends on the truth of p(x) leading to q(x) for every element x in the domain.
∃: The symbol ∃ represents the existential quantifier in logic, which asserts that there exists at least one element in a specified domain that satisfies a given property. It connects closely to the notion of predicates and is essential for expressing statements about existence within various logical frameworks.
∃x: The symbol ∃x represents the existential quantifier in logic, which asserts that there exists at least one element in a particular domain that satisfies a given property or condition. This concept is crucial for making statements about the existence of certain objects or individuals within a set and is used to express propositions that may not apply universally, distinguishing it from universal quantification.
∃x p(x): The notation ∃x p(x) is used in formal logic to represent the existential quantifier, meaning 'there exists an x such that p(x) is true.' This expression asserts that at least one element in a particular domain satisfies the property p. Understanding this concept is essential for distinguishing between universal and existential claims, where universal claims assert that a property holds for all elements, while existential claims only require at least one instance.
Bound Variable: A bound variable is a variable that is quantified and thus has its value restricted by a quantifier in a logical expression. This restriction allows the variable to take on values from a specified domain, making it dependent on the quantifier's scope. Understanding how bound variables function is essential for interpreting predicates, evaluating the impact of quantifiers, and analyzing the relationships within logical statements.
Domain of Discourse: The domain of discourse refers to the specific set of objects or elements that a particular quantifier, function, or logical statement is concerned with. This concept is crucial because it defines the boundaries within which variables and quantifiers operate, impacting the truth value of quantified statements and the interpretation of logical expressions.
Dual Quantifiers: Dual quantifiers refer to the relationship between universal and existential quantifiers in formal logic, highlighting how the truth of statements can interchangeably affect one another. Essentially, the universal quantifier asserts that a property holds for all elements in a domain, while the existential quantifier states that there exists at least one element for which the property is true. Understanding dual quantifiers helps in translating complex logical statements and understanding their implications.
Existential Quantifier: The existential quantifier is a logical symbol used to express that there exists at least one element in a particular domain for which a given predicate holds true. This concept is crucial for expressing statements involving existence and is represented by the symbol $$\exists$$, often translated as 'there exists' or 'for some'.
For All: The term 'for all' is used in logic to indicate a universal quantifier, which asserts that a statement applies to every member of a particular set or category. This concept is crucial for making generalizations in formal arguments and can be represented symbolically as '∀', which denotes that the proposition holds true without exception across the entire domain of discourse.
Free variable: A free variable is a variable in a logical expression that is not bound by a quantifier and can take on any value from its domain. It is crucial for understanding how predicates operate within statements, especially when distinguishing between the elements of an expression and the scope of variables. Free variables allow us to express generality and create more complex logical structures without being limited by specific quantification.
Illicit conversion: Illicit conversion refers to the improper inference made when the premises of a categorical syllogism are used to draw conclusions that are not logically valid. This concept highlights the dangers of assuming that the relationship between subjects can be reversed without proper justification, especially when dealing with universal and existential quantifiers and the nuances of multiple quantification. Understanding illicit conversion is essential for grasping the rules governing logical relationships and avoiding invalid conclusions in reasoning.
Predicate: A predicate is a statement or expression that asserts something about a subject, often involving properties or relations. It typically contains a verb and can be understood as a function that assigns truth values based on the subject it is linked to. This concept plays a crucial role in understanding how statements are formed, especially when dealing with quantified expressions and logical reasoning.
Quantified statement: A quantified statement is a type of logical assertion that uses quantifiers to express the extent to which a predicate applies to a subject. These statements can specify whether a property holds for all or some members of a given set, often using universal quantifiers like 'for all' (denoted as ∀) and existential quantifiers like 'there exists' (denoted as ∃). Understanding quantified statements is essential for formal reasoning and logical deduction.
Quantifier Exchange: Quantifier exchange refers to the logical principle that allows for the swapping of the order of universal and existential quantifiers in certain statements without altering their truth value. This principle is significant because it enables more flexible interpretations and transformations of logical expressions, particularly in relation to universal quantification and existential quantification.
Quantifier negation: Quantifier negation is a logical principle that describes how the negation of quantified statements affects their meaning. Specifically, it states that negating a universally quantified statement transforms it into an existentially quantified statement and vice versa. This principle highlights the relationship between universal quantifiers (like 'for all') and existential quantifiers (like 'there exists'), which is crucial for understanding logical expressions and reasoning.
Quantifier Shift: Quantifier shift refers to the change in the position of quantifiers in logical expressions, altering the meaning or truth conditions of those expressions. This concept is particularly important when dealing with universal and existential quantifiers, as switching their order can lead to different interpretations of a statement, impacting how we understand relationships between elements within a logical framework.
Rules of Inference: Rules of inference are logical principles that outline the valid steps we can take to derive conclusions from premises in a logical argument. They serve as the foundation for deductive reasoning, enabling us to establish new truths based on previously accepted statements. Understanding these rules is essential for working with quantifiers, allowing for precise reasoning about universally and existentially quantified statements.
There exists: The phrase 'there exists' is used in logic to indicate the existence of at least one element within a particular set or domain that satisfies a given property or condition. This phrase is often denoted by the existential quantifier '∃', which asserts that for some element in the domain, a specific statement holds true. Understanding this term is crucial for interpreting statements involving existence and for distinguishing between universal and existential claims.
Universal Quantifier: The universal quantifier is a symbol used in logic and mathematics to indicate that a statement applies to all members of a specified set. It is commonly represented by the symbol '∀', and its role is crucial in expressing generalizations and universal truths in logical expressions.
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