COS3751
Assignment 03
Unique No: 717989
Due 15 August 2025
,Question 1
Vocabulary Provided
• Customer(p1, p2): Person p1 is a customer of person p2
• Boss(p1, p2): Person p1 is a boss of person p2
• Doctor(p): Person p is a doctor
• Surgeon(p): Person p is a surgeon
• Lawyer(p): Person p is a lawyer
• Actor(p): Person p is an actor
• Emily, Joe: Constants denoting individuals
First-Order Logic Translations and Explanations
(a) Emily is either a surgeon or a lawyer (but not both).
This is a classic exclusive disjunction (XOR), meaning one of the two conditions must be
true, but not both.
Let Surgeon(Emily) denote that Emily is a surgeon, and Lawyer(Emily) that she is a
lawyer.
(Surgeon(Emily) ∨ Lawyer(Emily)) ∧ ¬(Surgeon(Emily) ∧ Lawyer(Emily))
Explanation: This expression ensures that Emily satisfies exactly one of the two predi-
cates, by combining inclusive disjunction with a negated conjunction.
1
, (b) All surgeons are doctors.
This is a universal implication stating that being a surgeon implies being a doctor:
∀p (Surgeon(p) → Doctor(p))
Explanation: The standard form ∀x(A(x) → B(x)) expresses subclass relationships,
ensuring that all surgeons are doctors.
(c) Joe does not have a lawyer (i.e., he is not the customer of any lawyer).
We are told Joe is not a customer of any lawyer. Two equivalent logical formulations are:
Existential negation form:
¬∃p (Lawyer(p) ∧ Customer(Joe, p))
Universal implication form (preferred):
∀p (Lawyer(p) → ¬Customer(Joe, p))
Explanation: The second form reflects the natural language phrasing “Joe is not a cus-
tomer of any lawyer” and is clearer in expressing universal negation.
(d) There exists a lawyer all of whose customers are doctors.
This is an existential statement containing a nested universal:
∃p (Lawyer(p) ∧ ∀q (Customer(q, p) → Doctor(q)))
Explanation: This asserts that there is at least one lawyer such that everyone who is
their customer is also a doctor.
2
Assignment 03
Unique No: 717989
Due 15 August 2025
,Question 1
Vocabulary Provided
• Customer(p1, p2): Person p1 is a customer of person p2
• Boss(p1, p2): Person p1 is a boss of person p2
• Doctor(p): Person p is a doctor
• Surgeon(p): Person p is a surgeon
• Lawyer(p): Person p is a lawyer
• Actor(p): Person p is an actor
• Emily, Joe: Constants denoting individuals
First-Order Logic Translations and Explanations
(a) Emily is either a surgeon or a lawyer (but not both).
This is a classic exclusive disjunction (XOR), meaning one of the two conditions must be
true, but not both.
Let Surgeon(Emily) denote that Emily is a surgeon, and Lawyer(Emily) that she is a
lawyer.
(Surgeon(Emily) ∨ Lawyer(Emily)) ∧ ¬(Surgeon(Emily) ∧ Lawyer(Emily))
Explanation: This expression ensures that Emily satisfies exactly one of the two predi-
cates, by combining inclusive disjunction with a negated conjunction.
1
, (b) All surgeons are doctors.
This is a universal implication stating that being a surgeon implies being a doctor:
∀p (Surgeon(p) → Doctor(p))
Explanation: The standard form ∀x(A(x) → B(x)) expresses subclass relationships,
ensuring that all surgeons are doctors.
(c) Joe does not have a lawyer (i.e., he is not the customer of any lawyer).
We are told Joe is not a customer of any lawyer. Two equivalent logical formulations are:
Existential negation form:
¬∃p (Lawyer(p) ∧ Customer(Joe, p))
Universal implication form (preferred):
∀p (Lawyer(p) → ¬Customer(Joe, p))
Explanation: The second form reflects the natural language phrasing “Joe is not a cus-
tomer of any lawyer” and is clearer in expressing universal negation.
(d) There exists a lawyer all of whose customers are doctors.
This is an existential statement containing a nested universal:
∃p (Lawyer(p) ∧ ∀q (Customer(q, p) → Doctor(q)))
Explanation: This asserts that there is at least one lawyer such that everyone who is
their customer is also a doctor.
2
