Cognitive Computing has become a catchphrase in the technology world, with the promiseof new smart services offered by industry giants like IBM and Google. Recent technological advances in Artificial Intelligence (AI) have thrown into the public sphere some old questions about the relationship between machine computation and human intelligence. While much of the industry and media hype suggests that many traditional challenges have been overcome, we show examples from language processing which demonstrate that present day Cognitive Computing still struggles with fundamental, long-standing problems with AI. An alternative conceptualization of artificial intelligence is presented, following Licklider’s lead in adopting man-computer symbiosis as a metaphor for designing software systems that enhance human cognitive performance. A survey of existing proposals based on this view suggests that a distinction can be made between weak and strong versions of symbiosis. We propose a Strong Cognitive Symbiosis which dictates an interdependence rather than simply cooperation between human and machine functioning, and show two systems under development where the symbiotic relationship benefits both actors in achieving the task outcome.