  The paper investigates from a proof-theoretic perspective various
non-contractive logical systems circumventing logical and semantic paradoxes.
Until recently, such systems only displayed additive quantifiers (Gri\v{s}in,
Cantini). Systems with multiplicative quantifers have also been proposed in the
2010s (Zardini), but they turned out to be inconsistent with the naive rules
for truth or comprehension. We start by presenting a first-order system for
disquotational truth with additive quantifiers and we compare it with
Gri\v{s}in set theory. We then analyze the reasons behind the inconsistency
phenomenon affecting multiplicative quantifers: after interpreting the
exponentials in affine logic as vacuous quantifiers, we show how such a logic
can be simulated within a truth-free fragment of a system with multiplicative
quantifiers. Finally, we prove that the logic of these multiplicative
quantifiers (but without disquotational truth) is consistent, by showing that
an infinitary version of the cut rule can be eliminated. This paves the way to
a syntactic approach to the proof theory of infinitary logic with infinite
sequents.
