This paper studies proof systems for the logics of super-strict implication $$\textsf{ST2}$$
ST
2
–$$\textsf{ST5}$$
ST
5
, which correspond to C.I. Lewis’ systems $$\textsf{S2}$$
S
2
–$$\textsf{S5}$$
S
5
freed of paradoxes of strict implication. First, Hilbert-style axiomatic systems are introduced and shown to be sound and complete by simulating $$\textsf{STn}$$
STn
in $$\textsf{Sn}$$
Sn
and backsimulating $$\textsf{Sn}$$
Sn
in $$\textsf{STn}$$
STn
, respectively (for $${\textsf{n}} =2, \ldots , 5$$
n
=
2
,
…
,
5
). Next, $$\textsf{G3}$$
G
3
-style labelled sequent calculi are investigated. It is shown that these calculi have the good structural properties that are distinctive of $$\textsf{G3}$$
G
3
-style calculi, that they are sound and complete, and it is shown that the proof search for $$\mathsf {G3.ST2}$$
G
3
.
ST
2
is terminating and therefore the logic is decidable.