Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Digital systems, linear algebra, introduction to formal proof (resit) / 8 / a / ii / Solution

The dot after $\forall x$ gives it scope over the following implication: read the premise as $H=\forall x(P(x)\to Q(x))$. Choose a fresh arbitrary eigenvariable $u$. A natural deduction proof isAt line 6, $u$ is free in no undischarged assumption, so the universal quantifier introduction rule is valid.

The proposed variant is not derivable in a sound first-order logic. For a countermodel, use the two-element domain $\{0,1\}$, interpret both predicates $P,Q$ as $\{0\}$, and assign the free variable $x$ the value 0. Then $\forall x(P(x)\to Q(x))$ is true, as is $P(x)$ under the assignment, but $\forall xQ(x)$ is false at element 1. The premise's bound $x$ and the conclusion's free $x$ have different scopes.Attempting to generalize from $Q(x)$ in the variant would violate the eigenvariable condition, because $x$ remains free in the undischarged assumption $P(x)$.