  The probabilistic symbol is the right-hand side derivative of the
characteristic functions corresponding to the one-dimensional marginals of a
stochastic process. This object, as long as the derivative exists, provides
crucial information concerning the stochastic process. For a L\'evy process,
one obtains the characteristic exponent while the symbol of a (rich) Feller
process coincides with the classical symbol which is well known from the theory
of pseudodifferential operators. Leaving these classes behind, the most general
class of processes for which the symbol still exists are L\'evy-type processes.
It has been an open question, whether further generalizations are possible
within the framework of Markov processes. We answer this question in the
present article: within the class of Hunt semimartingales, L\'evy-type
processes are exactly those for which the probabilistic symbol exists. Leaving
quasi-continuity behind, one can construct processes admitting a symbol.
However, we show, that the applicability of the symbol might be lost for these
processes. Surprisingly, in our proofs the upper and lower Dini derivatives
corresponding to certain singular functions play an important r\^ole.
