  We study the first-order axiomatisability of finite semiring interpretations
or, equivalently, the question whether elementary equivalence and isomorphism
coincide for valuations of atomic facts over a finite universe into a
commutative semiring. Contrary to the classical case of Boolean semantics,
where every finite structure can obviously be axiomatised up to isomorphism by
a first-order sentence, the situation in semiring semantics is rather
different, and strongly depends on the underlying semiring. We prove that for a
number of important semirings, including min-max semirings, and the semirings
of positive Boolean expressions, there exist finite semiring interpretations
that are elementarily equivalent but not isomorphic. The same is true for the
polynomial semirings that are universal for the classes of absorptive,
idempotent, and fully idempotent semirings, respectively. On the other side, we
prove that for other, practically relevant, semirings such as the Viterby
semiring, the tropical semiring, the natural semiring and the universal
polynomial semiring N[X], all finite semiring interpretations are first-order
axiomatisable (and thus elementary equivalence implies isomorphism), although
some of the axiomatisations that we exhibit use an infinite set of axioms.
