Partition implication is defined by David P. Ellerman as follows,

$$

\mathrm{dit}(\pi_1 \implies \pi_2) := \mathrm{int}(\mathrm{dit}(\pi_1)^c\cup \mathrm{dit}(\pi_2))

$$

$$

\mathrm{dit}(\pi_1 \implies \pi_2) := \mathrm{int}(\mathrm{dit}(\pi_1)^c\cup \mathrm{dit}(\pi_2))

$$