  This paper presents a proof of reciprocity laws for the Parshin symbol and
for two new local symbols, defined here, which we call 4-function local
symbols. The reciprocity laws for the Parshin symbol are proven using a new
method - via iterated integrals. The usefulness of this method is shown by two
facts - first, by establishing new local symbols - the 4-function local symbols
and their reciprocity laws and, second, by providing refinements of the Parshin
symbol in terms of bi-local symbols, each of which satisfies a reciprocity law.
The K-theoretic variant of the first 4-function local symbol is defined in the
Appendix. It differs by a sign from the one defined via iterated integrals.
Both the sign and the K-theoretic variant of the 4-function local symbol
satisfy reciprocity laws.
