Recently, Logic Explained Networks (LENs) have been proposed as explainable-by-design neural models providing logic explanations for their predictions. However, these models have only been applied to vision and tabular data, and they mostly favour the generation of global explanations, while local ones tend to be noisy and verbose. For these reasons, we propose LEN^p, improving local explanations by perturbing input words, and we test it on text classification. Our results show that (i) LEN^p provides better local explanations than LIME in terms of sensitivity and faithfulness, and (ii) its logic explanations are more useful and user-friendly than the feature scoring provided by LIME as attested by a human survey.