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Commonsense reasoning has long been considered as one of the holy grails of artificial intelligence. Most of the recent progress in the field has been achieved by novel machine learning algorithms for natural language processing. However, without incorporating logical reasoning, these algorithms remain arguably shallow. With some notable exceptions, developers of practical automated logic-based reasoners have mostly avoided focusing on the problem. The paper argues that the methods and algorithms used by existing automated reasoners for classical first-order logic can be extended towards commonsense reasoning. Instead of devising new specialized logics we propose a framework of extensions to the mainstream resolution-based search methods to make these capable of performing search tasks for practical commonsense reasoning with reasonable efficiency. The proposed extensions mostly rely on operating on ordinary proof trees and are devised to handle commonsense knowledge bases containing inconsistencies, default rules, taxonomies, topics, relevance, confidence and similarity measures. We claim that machine learning is best suited for the construction of commonsense knowledge bases while the extended logic-based methods would be well-suited for actually answering queries from these knowledge bases.