This article is centred on two themes. The first is the extension of Zadeh’s basic Z-numbers into a tool for level-2 Computing With Words (CWW) and consequently subjective natural language understanding. We describe an algorithm and new operators (leading to complex or spectral Z-numbers), use them to simulate differential diagnosis, and highlight the inherent strengths and challenges of the Z-numbers. The second theme deals with the design of a, Minsky’s Society of Mind based, natural language comprehending machine-mind architecture. We enumerate its macro-components (function modules and memory units) and illustrate its working mechanism through simulation of metaphor understanding; validating system outputs against human-comprehension responses. The framework uses the aforementioned new Z-number paradigm to precisiate knowledge-frames. The completeness of the conceptualized architecture is analyzed through its coverage of mind-layers (Minsky) and cerebral cortex regions. The research described here draws from multiple disciplines and seeks to contribute to the cognitive-systems design initiatives for man-machine symbiosis. © Springer International Publishing Switzerland 2015