Which of the following predicate logic formulae/formula is/are CORRECT…
2025
Which of the following predicate logic formulae/formula is/are CORRECT representation(s) of the statement: “Everyone has exactly one mother”?
The meanings of the predicates used are:
• \(𝑚𝑜𝑡ℎ𝑒𝑟(𝑦, 𝑥): 𝑦\) is the mother of \(x\)
• \(𝑛𝑜𝑡𝑒𝑞(𝑥, 𝑦): 𝑥\) and \(y\) are not equal
- A.
\(∀𝑥∃𝑦∃𝑧(𝑚𝑜𝑡ℎ𝑒𝑟(𝑦, 𝑥) ∧ ¬𝑚𝑜𝑡ℎ𝑒𝑟(𝑧, 𝑥))\) - B.
\(∀𝑥∃𝑦[𝑚𝑜𝑡ℎ𝑒𝑟(𝑦, 𝑥) ∧ ∀𝑧(𝑛𝑜𝑡𝑒𝑞(𝑧, 𝑦) → ¬𝑚𝑜𝑡ℎ𝑒𝑟(𝑧, 𝑥))]\) - C.
\(∀𝑥∀𝑦[𝑚𝑜𝑡ℎ𝑒𝑟(𝑦, 𝑥) → ∃𝑧(𝑚𝑜𝑡ℎ𝑒𝑟(𝑧, 𝑥) ∧ ¬𝑛𝑜𝑡𝑒𝑞(𝑧, 𝑦))]\) - D.
\(∀𝑥∃𝑦[𝑚𝑜𝑡ℎ𝑒𝑟(𝑦, 𝑥) ∧ ¬∃𝑧(𝑛𝑜𝑡𝑒𝑞(𝑧, 𝑦) ∧ 𝑚𝑜𝑡ℎ𝑒𝑟(𝑧, 𝑥))]\)
Attempted by 73 students.
Show answer & explanation
Correct answer: B, D
Correct formalizations:
∀x ∃y [ mother(y,x) ∧ ∀z ( mother(z,x) → ¬noteq(z,y) ) ]
∀x ∃y [ mother(y,x) ∧ ¬∃z ( noteq(z,y) ∧ mother(z,x) ) ]
Explanation:
Existence: The ∀x ∃y mother(y,x) part ensures every person x has at least one mother.
Uniqueness (first form): ∀z (mother(z,x) → ¬noteq(z,y)) means any mother z of x must not be different from y, i.e. z = y.
Uniqueness (second form): ¬∃z (noteq(z,y) ∧ mother(z,x)) states there is no different individual z who is also a mother of x.
Why the other given formulas fail:
The formula ∀x ∃y ∃z (mother(y,x) ∧ ¬mother(z,x)) only requires there be some non-mother for each x in addition to at least one mother; it does not rule out multiple distinct mothers, so it does not express "exactly one."
The formula ∀x ∀y [mother(y,x) → ∃z (mother(z,x) ∧ ¬noteq(z,y))] is essentially tautological about any given mother y (it asserts there is a mother equal to y) and does not guarantee every x has a mother nor that there is only one mother, so it fails to capture the intended meaning.