We herein demonstrate a brain-inspired synaptic device using a poly(vinylidene fluoride) and trifluoroethylene (PVDF-TrFE)/silicon nanowire (Si NW) based ferroelectric field effect transistor (FeFET). The PVDF-TrFE/Si NW FeFET structure achieves reliable synaptic plasticity such as symmetrical potentiation and depression, thanks to the reversible dynamics of the PVDF-TrFE permanent dipole moment. The calculated asymmetric ratio of potentiation and depression is as low as 0.41 at the optimized bias condition, indicating a symmetrical synaptic plasticity behavior. Pattern recognition accuracy based on the actual synaptic plasticity data of the synaptic device can be estimated via the CrossSim simulation software. Our simulation result reveals a high pattern recognition accuracy of 85.1%, showing a potential feasibility for neuromorphic systems. Furthermore, the inverter-in-synapse transistor consisting of the Si NW FeFET synapse and resistor connected in series is able to provide energy-efficient logic circuits. A total noise margin [(NMH + NML)/VDD] of 41.6% is achieved, and the power consumption [Ps = VDD(ID,L + ID,H)/2] of the logic-in-synapse transistor is evaluated to be 0.6 µW per logic gate. This study would shed light on the way toward a brain-inspired neuromorphic computing system based on the FeFET synapse device.