First Order Logic Questions LOGIC FOR COMPSCI

What are the key additions that first-order logic brings to propositional logic?

First-order logic extends propositional logic by introducing predicates, which describe the properties of objects, functions that map objects to other objects, and quantifiers, which enable reasoning about multiple objects.

Explain the meaning and usage of the existential quantifier (∃).

Explain the meaning and usage of the existential quantifier (∃).

The existential quantifier (∃x. P(x)) means "there exists at least one x such that P(x) is true."

Explain the meaning and usage of the universal quantifier (∀).

The universal quantifier (∀x. P(x)) means "for all x, P(x) is true."

What is the significance of the scope of a variable in first-order logic?

The scope of a variable defines the part of the formula where the quantifier binding that variable is effective.

Explain the concept of bound and free variables in first-order logic.

A bound variable is a variable that is within the scope of a quantifier. A free variable is a variable that is not bound by any quantifier in the formula.

How does operator precedence work in first-order logic, especially concerning quantifiers and negation (¬)?

Quantifiers in first-order logic have precedence just below negation (¬). This means that in ∃x. P(x) ∧ Q(x), the intended parsing is often (∃x. P(x)) ∧ Q(x), which might not be what is desired if the intention is that x is quantified over the entire conjunction. Parentheses should be used to clarify scope, such as ∃x. (P(x) ∧ Q(x))

Describe how to translate the English statement "Some happy person wears a hat" into first-order logic using the predicates Happy(x) and WearingHat(x). Explain why the chosen logical connective is appropriate.

The statement "Some happy person wears a hat" translates to ∃x. (Happy(x) ∧ WearingHat(x)). The conjunction (∧) is used because we are stating that there exists at least one individual x who is both happy AND wears a hat.

Describe how to translate the English statement "Every happy person wears a hat" into first-order logic using the predicates Happy(x) and WearingHat(x). Explain why the chosen logical connective is appropriate

The statement "Every happy person wears a hat" translates to ∀x. (Happy(x) → WearingHat(x)). The implication (→) is used because we are stating that for all individuals x, IF x is happy, THEN x wears a hat.

State De Morgan's laws for quantified statements.

¬∀x. A is equivalent to ∃x.

¬∃x. A is equivalent to ∀x.

Explain how first-order logic can be used to express the idea that there is exactly one object with a certain property

To express that there is exactly one object with a property P(x), we can use the formula ∃x. (P(x) ∧ ∀y. (P(y) → y = x)). This states that there exists at least one x with property P, and for any y with property P, y must be the same as x.

First Order Logic

An extension of propositional logic that introduces predicates, functions,

and quantifiers to reason about objects and their properties.

Predicate

A symbol that describes a property or relationship among objects. It takes objects

as arguments and evaluates to true or false. (e.g., Loves(John, Mary), IsEven(x))

Function

A symbol that maps one or more objects to a single object. (e.g., fatherOf(John),

plus(2, 3))

Constant Symbol

A symbol that represents a specific object in the domain of discourse.

(e.g., John, Mary, 5)

Variable

A symbol that can refer to any object in the domain of discourse. (e.g., x, y, z).

Domain of Discourse

The set of all objects that the variables and quantifiers in a first-order

logic statement can refer to

Atomic Formula

The simplest form of a formula in first-order logic, consisting of a predicate

applied to terms (variables or constants). (e.g., P(x), Q(a, b)).

Logical Connectives

Symbols used to combine formulas, inherited from propositional logic

(e.g., ∧ (and), ∨ (or), ¬ (not), → (implies), ↔ (if and only if)).

Existential Quantifier

A symbol indicating that there exists at least one object in the

domain of discourse for which a given formula is true. (e.g., ∃x. P(x) means "There exists an

x such that P(x) is true")

Universal Quantifier

A symbol indicating that a given formula is true for all objects in

the domain of discourse. (e.g., ∀x. P(x) means "For all x, P(x) is true").

Scope of a Quantifier

The part of a formula where the quantifier has influence over the

variable it introduces

Bound Variable

An occurrence of a variable that is within the scope of a quantifier that

introduces that variable

Free Variable

An occurrence of a variable that is not bound by any quantifier in the formula

Sentence/Closed Formula

A formula in first-order logic that contains no free variables (variables without a quantifier). Its truth value is independent of any specific assignment of values to variables

Term

An expression that refers to an object in the domain of discourse. Terms can be

variables, constants, or the result of applying a function symbol to other terms

Operator Precedence

The rules that determine the order in which logical operators and

quantifiers are evaluated in a formula (quantifiers have precedence below ¬).

De Morgan’s Law for Quantified Statements

Equivalences that describe how negation

interacts with quantifiers: ¬∀x. A ≡ ∃x. ¬A and ¬∃x. A ≡ ∀x. ¬A.

Restricted Quantifier

A shorthand notation for quantification over a specific set, often

written as ∀x ∈ S. P(x) or ∃x ∈ S. P(x)

Uniqueness Quantifier (∃!)

A notation (though not a basic symbol) used to express that

there exists exactly one object with a certain property. It can be formally defined using the

basic quantifiers and equality.

Interpretation

In model theory, an interpretation assigns meaning to the symbols of a first-

order logic language in terms of a domain and functions and relations on that domain

Model

An interpretation under which a given set of formulas (a theory) are all true

Logical Axiom

A formula that is considered to be universally true in all interpretations.

Inference Rule

A rule that allows the derivation of new formulas from a given set of

formulas

Theorem

A formula that can be derived from a set of axioms using valid inference rules

Decidable Theory

A theory for which there exists an algorithm that can determine, for any

given closed formula, whether it is a theorem of that theory or not

Discuss the fundamental differences and relationships between propositional logic and first-

order logic. How do the additions in first-order logic enhance its expressive power? Provide

examples to illustrate these differences and enhancements

Propositional logic involves simple statements that can be either true or false, while first-order logic extends this by incorporating quantifiers and predicates, allowing for more complex expressions about objects and their relationships. For example, in first-order logic, one can express statements like "All humans are mortal" using quantifiers, which is not possible in propositional logic.

Explain the roles and significance of predicates, functions, and quantifiers in first-order logic.

How do these components work together to represent complex statements about objects and

their relationships?

Predicates define properties or relations among objects, functions map objects to other objects, and quantifiers express the extent of a statement (universal or existential). Together, they allow for rich and nuanced representations of logical relationships.

Analyze the importance of variable scope and binding in first-order logic. Discuss potential

ambiguities that can arise from improper handling of scope and provide strategies for writing

clear and unambiguous first-order logic formulas

Variable scope refers to the context in which a variable is defined and can be referenced, while binding relates to the assignment of values to those variables. Improper management of scope can lead to ambiguities, such as unintended variable capture, where a variable's reference is altered by a surrounding quantifier. To avoid these issues, it's crucial to use clear quantifier notation and limit the use of variables to specific scopes.

Compare and contrast the uses and interpretations of the existential quantifier (∃) and the

universal quantifier (∀). Provide examples of how each quantifier is used to translate different

types of English statements into first-order logic, paying attention to the choice of logical

connectives.

Existential quantifier (∃) asserts the existence of at least one object satisfying a property, while universal quantifier (∀) states that all objects satisfy a property. For example, "There exists a human who is a philosopher" can be expressed as ∃x (Human(x) ∧ Philosopher(x)), whereas "All humans are mortal" is expressed as ∀x (Human(x) → Mortal(x)).

Discuss the concept of negating quantified statements in first-order logic, focusing on De

Morgan's laws for quantifiers and their relationship to negating propositional connectives.

Illustrate with examples how to effectively negate complex first-order logic formulas and

translate the results back into English

Morgan's laws for quantifiers state that negating a universally quantified statement results in an existential quantifier, and vice versa. For example, the negation of "All x (P(x))" is equivalent to "There exists an x such that not P(x)". This concept allows for the systematic handling of negation in complex first-order logic formulas, ensuring proper translations into English.