  We develop a proof-theoretic semantics -- in particular, a base-extension
semantics -- for multi-agent S5 modal logic (and hence also for the usual
unindexed S5). Following the inferentialist interpretation of logic, this gives
us a semantics in which validity is based on proof rather than truth. In
base-extension semantics, the validity of formulae is generated by provability
in a `base' of atomic rules and an inductive definition of the validity of the
connectives. Base-extension semantics for many interesting logics has been
explored by several authors and, in particular, a base-extension semantics for
the modal logics K, KT, K4, and S4 has been developed by the present authors.
Here, we give a base-extension semantics for multi-agent S5 with $\square_a$,
for an agent a, as our primary operators, framed as the knowledge operator K_a.
Similarly to Kripke semantics, we make use of relational structure between
bases, allowing us to establish a correspondence between certain bases and
worlds. We use this to establish the appropriate soundness and completeness
results. We conclude by discussing how this semantics can be extended to
Dynamic Epistemic Logics (DEL) starting with Public Announcement Logic (PAL).
