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

(One)

Δ⊢⊥Δ⊢α(Weak-R)

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)

Δ⊣⊤Δ⊣α(Weak-R)

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

(Three)

Δ⊣βΔ,α⊣β(Weak-L)

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.

Notes:

With great respect, Henry Story, born 29th July 1967 - died 5th September 2023

From Mathematics StackExchange, Henry Story, April 2020:

Examples of co-implication (a.k.a co-exponential)

Link to publishing experiments