Predicate Logic - Syntax and Semantics Flashcards
By the end of this deck, learners will be adept at constructing and recognizing well-formed formulas (WFFs) in predicate logic. They will thoroughly understand the use and implications of universal and existential quantifiers and will be able to accurately determine the scope and binding relationships of variables within logical expressions.
What characterizes a well-formed formula (WFF) in predicate logic?
A WFF in predicate logic is a syntactically correct expression composed of predicates, quantifiers, logical connectives, variables, and constants following the rules of predicate logic.
What symbol is used for the universal quantifier in predicate logic?
The universal quantifier is symbolized by ∀.
What symbol represents the existential quantifier in predicate logic?
The existential quantifier is symbolized by ∃.
How do you interpret the universal quantifier in a logical expression?
The universal quantifier (∀) indicates that the statement following it is true for every element in the domain of discourse.
What does the existential quantifier indicate in a predicate logic expression?
The existential quantifier (∃) indicates that there exists at least one element in the domain of discourse for which the statement following it is true.
Define the scope of a quantifier in predicate logic.
The scope of a quantifier refers to the part of the logical expression over which the quantifier has authority, usually indicated by the portion of the formula following the quantifier.
What does it mean for a variable to be bound in predicate logic?
A variable is bound if it falls within the scope of a quantifier, which specifies its value or range of values.
What is a free variable in predicate logic?
A free variable is not bound by any quantifier and, therefore, can refer to any element within the domain of discourse.
How can a well-formed formula be negated in predicate logic?
A WFF can be negated by placing the negation symbol (¬) in front of the formula, altering its truth value.
In predicate logic, how do you express “All dogs are mammals”?
∀x (Dog(x) → Mammal(x))
How would you symbolize “Some cats are black” in predicate logic?
∃x (Cat(x) ∧ Black(x))
What is the significance of parentheses in predicate logic?
Parentheses are used to clarify the structure and order of operations within logical expressions, especially to delineate the scope of quantifiers and logical connectives.
How does the universal quantifier relate to negation?
The negation of a universally quantified statement is logically equivalent to an existentially quantified statement of the negated predicate: ¬∀x P(x) is equivalent to ∃x ¬P(x).
How is the existential quantifier affected by negation?
The negation of an existentially quantified statement is equivalent to a universally quantified statement of the negated predicate: ¬∃x P(x) is equivalent to ∀x ¬P(x).
What is the difference between using a universal quantifier and an existential quantifier in terms of commitment to the domain of discourse?
The universal quantifier asserts that a property holds for all elements in the domain, while the existential quantifier asserts the property holds for at least one element.
