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Why can counting statements be expressed only with equality?

  • A.Without it, the domain of discourse would be infinite
  • B.Without it, distinct variables may denote the same object
  • C.Without it, quantifiers cannot be nested inside one another
  • D.Without it, no formula can refer to more than one object

Answer: B

Two existentially quantified variables may be satisfied by the same object unless their distinctness is asserted, so at least two requires an inequality. Nesting, reference and domain size are unaffected. That variables do not automatically denote different objects surprises most people learning the notation, and it is precisely why explicit inequalities litter formulas that count.

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