Predicate Logic - Introduction and Terms

Question 1

What is the key difference between propositional and predicate logic?

Answer 1

Propositional logic deals with entire statements and their connectives, while predicate logic includes relations or properties of individuals and quantification over individuals.

Question 2

What is a predicate in predicate logic?

Answer 2

A predicate is a symbolic function that can be true or false depending on the values you assign to its variables.

Question 3

Define a term in the context of predicate logic.

Answer 3

A term refers to a constant, variable, or function in predicate logic that can stand for individual objects in the domain.

Question 4

What is a quantifier in predicate logic?

Answer 4

Quantifiers specify the scope of a statement in predicate logic. The two most common quantifiers are existential (∃) and universal (∀).

Question 5

What does the universal quantifier (∀) signify in predicate logic?

Answer 5

The universal quantifier (∀) signifies that the proposition holds for all instances within the domain of discourse.

Question 6

What does the existential quantifier (∃) signify in predicate logic?

Answer 6

The existential quantifier (∃) signifies that there exists at least one instance in the domain of discourse for which the proposition holds.

Question 7

What is the domain of discourse in predicate logic?

Answer 7

The domain of discourse, or universe of discourse, is the set of all entities over which variables can range.

Question 8

How do you symbolize “All humans are mortal” in predicate logic?

Answer 8

∀x (Human(x) → Mortal(x))

Question 9

How do you symbolize “Some birds can fly” in predicate logic?

Answer 9

∃x (Bird(x) ∧ CanFly(x))

Question 10

What is the difference between a bound variable and a free variable in predicate logic?

Answer 10

A bound variable is quantified within a statement, while a free variable is not quantified and can refer to any object in the domain.

Question 11

In predicate logic, what role do constants play?

Answer 11

Constants refer to specific, fixed entities in the domain of discourse.

Question 12

How can functions be represented in predicate logic?

Answer 12

Functions in predicate logic map individuals to individuals, often symbolized by expressions like f(x).

Question 13

What is an interpretation in predicate logic?

Answer 13

An interpretation assigns meaning to the symbols of the predicate logic language, typically mapping terms to objects, and predicates to relations in the domain.

Question 14

What does it mean for a predicate formula to be satisfiable?

Answer 14

A predicate formula is satisfiable if there exists an interpretation and assignment of variables that makes the formula true.

Question 15

How does predicate logic enhance the expressiveness compared to propositional logic?

Answer 15

Predicate logic allows for expressions about individual objects and their properties or relations, providing a more detailed and nuanced way to form logical statements.

Question 16

Can predicate logic express properties of individual objects?

Answer 16

Yes, predicate logic can express properties of individual objects using predicates and terms.

Question 17

What is the significance of specifying a domain of discourse in predicate logic?

Answer 17

Specifying a domain of discourse sets the context and range for what the variables, constants, and predicates refer to, making the logical statements meaningful.

Question 18

How do you represent “Every cat loves some dog” in predicate logic?

Answer 18

∀x (Cat(x) → ∃y (Dog(y) ∧ Loves(x, y)))

Question 19

In predicate logic, how would you express “There is a person who loves everyone”?

Answer 19

∃x (Person(x) ∧ ∀y (Person(y) → Loves(x, y)))

Question 20

What is the role of logical equivalence in predicate logic?

Answer 20

Logical equivalence in predicate logic shows that two statements are true in the same interpretations, maintaining consistency and allowing transformations and simplifications of logical expressions.