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This paper proposes a new logic element model based on an Izhikevich (IZ) neuron and neural system that emulates two- and three-state logic behaviours. In a noise-free environment, with a periodic current of suitable amplitude and frequency, the IZ system is capable of performing logical AND and OR operations. Initially, a single IZ neuron demonstrates membrane dynamics in response to an input signal generated by combining two-state logic currents below the threshold. Subsequently, an IZ neural system model is introduced to enhance the reliability and resilience of the system. This model is characterised by electrical coupling with fast conduction and chemical coupling with a more adaptable structure. Each logic input independently influences each neuron within the system. Additionally, it has been observed that the reliability of the logic element is influenced by changes in synaptic strength, with a neural system lacking sufficient synaptic strength failing to generate logical output. Furthermore, the system displays a three-state logic behaviour under suitable forcing periodicity, thus enhancing the power efficiency of the logic element. The proposed IZ neuron and neural system are expected to significantly impact the development of brain-inspired logic elements.