 Solution

ID: past-exam-of-the-logic-and-set-theory-course-of-the-university-of-oxford/2010/a/1/b/ii/solution

False. Choose a variable

x_{j}

different from

x_{i}

and take

ϕ = \forall x_{j} P_{0}^{(1)} (x_{j}) .

The variable

x_{i}

does not occur at all, so in particular it is not free. Yet the formula has one universal-quantifier symbol and zero occurrences of

x_{i}

. The hypothesis on free occurrences of one variable does not control quantifiers binding other variables.

 New to topics? Read the docs here!