Consider the sentence below: “There is a country that borders both India and…
2018
Consider the sentence below:
“There is a country that borders both India and Nepal”
Which of the following represents the above sentence correctly?
- A.
\(∃𝑐 Country(𝑐)∧Border(𝑐,India)∧𝐵𝑜𝑟𝑑𝑒𝑟(𝑐,Nepal) \) - B.
\(∃𝑐 Country(𝑐)⇒[Border(𝑐,India)∧𝐵𝑜𝑟𝑑𝑒𝑟(𝑐,Nepal)] \) - C.
\([∃𝑐 Country(𝑐)]⇒[Border(𝑐,India)∧𝐵𝑜𝑟𝑑𝑒𝑟(𝑐,Nepal)] \) - D.
\(∃𝑐 Border(Country(𝑐),India∧Nepal)\)
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Correct answer: A
Correct formalization: ∃c Country(c) ∧ Border(c, India) ∧ Border(c, Nepal)
Explanation:
The existential quantifier ∃c asserts that there is at least one object c.
The conjunction Country(c) ∧ Border(c, India) ∧ Border(c, Nepal) requires that the same c is a country and that this same c borders both India and Nepal, which matches the English sentence.
Why other forms are incorrect:
A formula written with the existential quantifier as an antecedent of an implication, e.g. (∃c Country(c)) ⇒ (Border(c, India) ∧ Border(c, Nepal)), either leaves c in the consequent unbound or yields a conditional statement whose truth conditions are unrelated to "there exists a country that borders both". This does not force the same c to satisfy the border predicates.
A formula that places predicates where terms are expected, e.g. ∃c Border(Country(c), India ∧ Nepal), is syntactically incorrect: Country(c) is a predicate (not a term representing an entity) and India ∧ Nepal is a logical conjunction, not a single country term. The correct approach is to apply Border to an entity c and each country separately.
Summary: Use ∃c with the conjunction Country(c) ∧ Border(c, India) ∧ Border(c, Nepal) so that the same c is required to border both India and Nepal.
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