When people combine concepts these are often characterisedas “hybrid”, “impossible”, or “humorous”. However, when simplyconsidering them in terms of extensional logic, the novel conceptsunderstood as a conjunctive concept will often lack meaning having anempty extension (consider “a tooth that is a chair”, “a pet flower”,etc.). Still, people use different strategies to produce new non-emptyconcepts: additive or integrative combination of features, alignment offeatures, instantiation, etc. All these strategies involve the ability to dealwith conflicting attributes and the creation of new (combinations of)properties. We here consider in particular the case where a Head concepthas superior ‘asymmetric’ control over steering the resulting conceptcombination (or hybridisation) with a Modifier concept. Specifically, wepropose a dialogical approach to concept combination and discuss animplementation based on axiom weakening, which models the cognitiveand logical mechanics of this asymmetric form of hybridisation.