'No man or woman have' or 'no man or woman has'? The first didn't sound right to me, but you know how it is, once you question something, you're no longer sure of the answer. I had to look it up to be certain.
    If I hadn't had access to a copy of The Oxford Manual of Style, what could I have done to avoid making an embarrassing mistake? There are numerous grammar and spelling websites, of course, but you need to be careful that you use the right country's English e.g. not US English when you want UK and vice versa. Writers' forums (fora?) are another option, but these have their drawbacks too. How can you be sure that those answering your questions are better informed than you? And what do you do if you get conflicting advice?
    My motto in such instances is: if in doubt, rewrite. change the words around or use a simpler sentence construction. It's usually a good idea to detach subordinate clauses and parenthetical statements if you can. You don't want your style to be monotonous, but unless you're Henry James, generally the less convoluted your sentences the better. If you're unsure whether your style is too complex, try translating them into another language. German is a good one to try. If you end up with a pile of verbs at the end of your sentences, it's time to split them up. Seriously, try reading your work aloud and take a look at the comma count. If you have a proliferation of punctuation, or stumble over the pauses, you could have a problem.
    As to 'no man or woman have', the verb should be in agreement with the subject. In this case a man or a woman. 'Or' indicates that there is only one at a time, therefore the verb should be singular i.e. 'has'. If the phrase had been 'no man and woman' then the verb should be 'have'. Simple!

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