If a set of assumed proofs have ⊥ as a consequence (the ⊥ sign is often dropped) then anything follows.

(One)

Similarly, if from the assumption that Δ is refuted we can conclude the statement that can never be refuted (⊤) then we can deduce any other statement from Δ. (remember ⊤ is the initial object in a co-Heyting algebra).

(Two)

Similarly the following rule shows that one must interpret the left side disjunctively.

(Three)

For if from a hypothetised refuted Δ one can refute 𝛽, then adding an arbitrary 𝛼 to Δ will not affect the refutation. This addition must be harmless. It explains also why the rule is called a weakening.