The effects of logarithmic singularities in rescattering processes are investigated. The reaction πN-->ππN is considered, but treated purely as an S-wave, spinless model. A particular triangle graph is analyzed in detail; it contains as an intermediate state the (3,3) nucleon isobar I, which is described as a spinless particle of complex mass. The graph is calculated from a dispersion relation as a function of the mass s of the two pions in the final state, for low values of the over-all c.m. system energy W. The relation is then analytically continued in W. For a narrow range in W, an enhancement of the square of the amplitude is found near s=4 (the pion mass is unity). The analogous enhancement also appears in the W channel near W=I+1, for a small range of s only, near s=4. The prominence of the effect depends on the width of I, being closely connected with the nearness to the physical region of one of the two logarithmic singularities (anomalous thresholds) of the graph: this distance increases sharply with the isobar width. The positions of the singularities are interpreted as the phase-space limits for the simultaneous production of states with mass s and I. The conclusion is that such a "double excitation" process leads to an enhancement of the triangle amplitude only if, in general, s and I fall in certain narrow ranges. The implications of this result for models of the higher resonances in the elastic channel (πN-->πN) is briefly discussed.