Consider the sentence below. There is a country that borders both India and…

Correct formalisation: ∃c (Country(c) ∧ Borders(c, India) ∧ Borders(c, Pakistan))

• Key idea: There exists at least one object c such that c is a country and c shares a border with India and with Pakistan.

• Why this formula is correct: The existential quantifier binds the variable c and the conjunction requires the same c to satisfy Country(c), Borders(c, India), and Borders(c, Pakistan).

• Why the other forms fail:

  • The formula ∃c Country(c) ⇒ [Border (c, India) ∧ Border (c,Pakistan)] is interpreted as ∃c (Country(c) ⇒ ...). If c is not a country the implication is vacuously true, so a non-country could make the existential true. That does not enforce that a country exists that borders both India and Pakistan.

  • [∃c Country(c)] ⇒ [Border (c, India) ∧ Border (c,Pakistan)] is malformed because the variable c in the consequent is not bound by the existential (the quantifier is outside the implication). The consequence uses a free variable and so the formula does not properly express the intended meaning.

  • The expression ∃c Border (Country(c), India ∧ Pakistan) misuses predicate arguments: Country(c) is a predicate application not a term, and India ∧ Pakistan is not a single country term. The correct representation uses two separate border relations with c as the first argument.