Past exam of the logic and set theory course of the University of Oxford / 2009 / A / 1 / e / Solution

The thinning rule, or weakening, isIt expresses that extra unused assumptions do not destroy a proof. In the context-sensitive predicate-calculus presentation it also allows a theorem first proved with a fresh generalization variable to be used later in a context containing that variable free. Simply replaying the same generalization under the larger context could violate its side condition. For instance a proof of $\forall x(x=x)$ from the empty context may generalize on $x$; it should remain available when an unused assumption $P(x)$ is added, even though that last generalization could not be replayed with $P(x)$ open.

In propositional calculus the axiom schemes and modus ponens have no analogous free-variable restriction. An induction on the original proof simply reuses each axiom, assumption and modus-ponens step under the enlarged assumptions, so weakening is admissible and need not be postulated separately. Context-sensitive Hilbert weakening concerns this proof presentation: systems already tracking only the assumptions on which each line depends can build admissible weakening into their treatment of predicate-calculus generalization as well.