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Quantum analogues of the (classical) Logical Neural Networks (LNN) models are proposed in (q-LNN for short). We shall here further develop and investigate the q-LNN composed of the quantum analogue of the Probabilistic Logic Node (PLN) and the multiple-valued PLN (MPLN) variations, dubbed q-PLN and q-MPLN respectively. Besides a clearer mathematical description, we present a computationally efficient and simply described quantum learning algorithm in contrast to what has been proposed to the quantum weighted version.