Past exam of the logic and set theory course of the University of Oxford / 2010 / A / 1 / d

Use structural induction on formulas. Every term denotes the unique element. Every atomic predicate is true by hypothesis, and every equality is true because its two terms denote that same element.

If both immediate subformulas of an logical implication are true, their logical implication is true. A universal or existential quantification ranges over the singleton domain; its matrix is true for the only possible value by induction, so either quantified formula is true. These are all the formation cases in the reduced language. Hence

No. The full language can express $\neg (x_0\dot{=}{x}_0)$, which is false in the singleton structure from part (i). Every formula of the reduced language is true in that structure under every assignment in a first-order structure. Therefore no formula of the reduced language can be logically equivalent to this negated equality.

This is a singleton truth-preservation obstruction to expressive completeness. Adding both quantifiers to logical implication does not repair the failure: on this particular nonempty singleton model, quantification cannot create a false formula from true atomic formulas.