COS3751
ASSIGNMENT 03
Assignment No: 717989
Due 15 August 2025
,Question 1: First-Order Logic Translations
Vocabulary:
• Customer(p1, p2): Predicate — person p1 is a customer of person p2.
• Boss(p1, p2): Predicate — person p1 is a boss of person p2.
• Doctor(p): Predicate — person p is a doctor.
• Surgeon(p): Predicate — person p is a surgeon.
• Lawyer(p): Predicate — person p is a lawyer.
• Actor(p): Predicate — person p is an actor.
• Emily, Joe: Constants denoting individuals.
Translations to First-Order Logic:
(a) Emily is either a surgeon or a lawyer (but not both).
This is an exclusive disjunction — one of the two roles holds for Emily, but not both
simultaneously.
(Surgeon(Emily) ∨ Lawyer(Emily)) ∧ ¬(Surgeon(Emily) ∧ Lawyer(Emily))
Explanation: The disjunction ensures Emily has at least one role, while the negation
excludes the possibility of both roles, implementing a logical XOR.
(b) All surgeons are doctors.
This is a universal implication: for all individuals, being a surgeon implies being a doctor.
∀p (Surgeon(p) → Doctor(p))
Explanation: The formula applies to all people p, enforcing that the set of surgeons is a
subset of doctors.
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, (c) Joe does not have a lawyer.
Joe is not the customer of any individual who is a lawyer.
∀p (Lawyer(p) → ¬Customer(Joe, p))
Alternative (equivalent) form:
¬∃p (Lawyer(p) ∧ Customer(Joe, p))
Explanation: Both forms are logically equivalent. The universal formulation aligns more
closely with the phrasing “not the customer of any lawyer.”
(d) There exists a lawyer all of whose customers are doctors.
This asserts the existence of at least one lawyer whose every customer is a doctor.
∃p (Lawyer(p) ∧ ∀q (Customer(q, p) → Doctor(q)))
Explanation: The inner universal quantifier ensures that all customers q of the lawyer
p are doctors. If the lawyer has no customers, the statement holds vacuously, which is
standard in first-order logic.
(e) Every surgeon has a lawyer.
Each surgeon is a customer of at least one lawyer.
∀p (Surgeon(p) → ∃q (Lawyer(q) ∧ Customer(p, q)))
Explanation: The existential quantifier inside the implication ensures that for each sur-
geon p, a corresponding lawyer q exists whom they are a customer of. The quantifier
order is essential to reflect the correct logical relationship.
Summary of Logical Translations:
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ASSIGNMENT 03
Assignment No: 717989
Due 15 August 2025
,Question 1: First-Order Logic Translations
Vocabulary:
• Customer(p1, p2): Predicate — person p1 is a customer of person p2.
• Boss(p1, p2): Predicate — person p1 is a boss of person p2.
• Doctor(p): Predicate — person p is a doctor.
• Surgeon(p): Predicate — person p is a surgeon.
• Lawyer(p): Predicate — person p is a lawyer.
• Actor(p): Predicate — person p is an actor.
• Emily, Joe: Constants denoting individuals.
Translations to First-Order Logic:
(a) Emily is either a surgeon or a lawyer (but not both).
This is an exclusive disjunction — one of the two roles holds for Emily, but not both
simultaneously.
(Surgeon(Emily) ∨ Lawyer(Emily)) ∧ ¬(Surgeon(Emily) ∧ Lawyer(Emily))
Explanation: The disjunction ensures Emily has at least one role, while the negation
excludes the possibility of both roles, implementing a logical XOR.
(b) All surgeons are doctors.
This is a universal implication: for all individuals, being a surgeon implies being a doctor.
∀p (Surgeon(p) → Doctor(p))
Explanation: The formula applies to all people p, enforcing that the set of surgeons is a
subset of doctors.
1
, (c) Joe does not have a lawyer.
Joe is not the customer of any individual who is a lawyer.
∀p (Lawyer(p) → ¬Customer(Joe, p))
Alternative (equivalent) form:
¬∃p (Lawyer(p) ∧ Customer(Joe, p))
Explanation: Both forms are logically equivalent. The universal formulation aligns more
closely with the phrasing “not the customer of any lawyer.”
(d) There exists a lawyer all of whose customers are doctors.
This asserts the existence of at least one lawyer whose every customer is a doctor.
∃p (Lawyer(p) ∧ ∀q (Customer(q, p) → Doctor(q)))
Explanation: The inner universal quantifier ensures that all customers q of the lawyer
p are doctors. If the lawyer has no customers, the statement holds vacuously, which is
standard in first-order logic.
(e) Every surgeon has a lawyer.
Each surgeon is a customer of at least one lawyer.
∀p (Surgeon(p) → ∃q (Lawyer(q) ∧ Customer(p, q)))
Explanation: The existential quantifier inside the implication ensures that for each sur-
geon p, a corresponding lawyer q exists whom they are a customer of. The quantifier
order is essential to reflect the correct logical relationship.
Summary of Logical Translations:
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