A paired monophonic set M in a connected graph G is called a minimal a paired monophonic set if no proper subset of M is a paired monophonic set of G. The upper paired monophonic m_p^+ (G) of G is the maximum cardinality of a minimal paired monophonic set of G. Some general properties satisfied by this concept are studied. The paired monophonic number of some family of graphs is obtained. It is shown that for every pair of positive integers a and b with 4 ≤ a ≤ b, there exists a connected graph G such that m_p (G) = a and m_p^+ (G) = a + b, where m_p (G) is the paired monophonic number of G