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Thinning enlarges the assumptions without changing a conclusion. In a predicate-calculus presentation with context-sensitive generalization, a generalization valid before adding an unused assumption may not be replayable with that assumption open. An explicit weakening step nevertheless imports the already proved conclusion. In propositional calculus with only axiom schemes and modus ponens, weakening is admissible by replaying the proof. The need for a separately stated rule depends on how proof dependencies are represented.

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.

A logical derivation is a finite proof in the specified calculus, with conclusion $\Gamma ⊢\varphi$. Assumptions from $\Gamma$ and instances of A1–A7 are available; the rules are the modus ponens rule, universal generalization rule, and weakening.

In judgement form the universal generalization rule takes $\Delta ⊢\theta$ to $\Delta ⊢\forall x_i\theta$ only if $x_i$ is not free in any undischarged assumption in $\Delta$. Weakening permits $\Delta ⊢\theta$ to become $\Delta \cup \Sigma ⊢\theta$. All instances of A4 and A5 must meet their stated substitution and free variable conditions. Derivability means existence of such a finite proof with assumptions contained in $\Gamma$. Equivalently one tracks the finite assumption set used by each proof line.

A common-context sequent version of the disjunction elimination rule isThe second and third premises record exactly the two boxed subproofs of the natural deduction presentation. Their additional case assumptions do not occur in the conclusion: that is assumption discharge. The surrounding context $\Gamma$ remains throughout. A formulation with different surrounding contexts is equivalent after retaining their union by weakening.