I'm currently a postgraduate studying philosophy of mind. I have studied logic, but only at undergraduate level, and it was over 30 years ago. So I apologise for the probable naivety of this question.
I'm going to take the liberty of quoting from Melvin Bragg's email newsletter about a radio programme on logic hosted by him, broadcast yesterday on the UK station BBC Radio 4. In a post-programme discussion between the participants,

A C Grayling, usually more slicingly exact than a Gillette razor, defended vagueness in the following terms: if he rushed into a bedroom at two o’clock in the morning and shouted “Fire!” this could open a wide range of possibilities for any logical person.  Was he talking about a candle being lit downstairs?  Was he talking about something that had happened in another place?  But we mere mortals would be vague enough to understand that the house was ablaze and rush out.  Something to be said for vagueness then. (Melvyn Bragg's In Our Time Newsletter - Logic - 21/10/2010)

This seems very wrong to me. Given the context, the shouted word "Fire!" seems really quite precise. And if context is entirely ignored, then any human communication whatsoever is vague to the point of utter meaninglessness. What this illustration demonstrates is not the power of vagueness, but that of context.

But then AC Grayling is quite a "big name". Or maybe Melvin got it wrong. If not, what am I missing? What's the relationship between vagueness and context?

(Link to the programme on BBC iPlayer.)