Four-wave mixing (FWM) effect in metal-insulator-metal (MIM) waveguide filled with a Kerr nonlinear medium is investigated, followed by the study about polarization-dependent FWM in such nanoplasmonic structures due to the local enhancement effect of surface plasmons (SP) and birefringent characteristics. When the pumps with the polarization parallel to each other are launched into the waveguide with 250-nm thickness, the FWM efficiency reaches -33 dB. However, when the pumps are perpendicularly polarized, idler waves can not be obtained. For the waveguide with 50-nm thickness, when the pumps are both TM polarized, the FWM efficiency reaches -30 dB. While under other polarization states of pumps, idler waves can not be observed due to the cut off of TE modes. Such FWM-based mechanisms are utilized to construct all-optical logical gates.