Feste, attempting to impress Orsino with some complex word-play, observes ‘if your four negatives make your two affirmatives ...’ (TN V.i.20). This supposed rule endeared itself greatly to prescriptive grammarians in later centuries, who tried to impose it on the whole of English, despite the fact that the strict mathematical logic was used only in a few formal styles of expression. They were largely successful. It is now said to be ‘bad grammar’ to use a double negative within a single clause in standard English unless one is being logically precise.
In Shakespearean English we often find sequences of negative words whose sole function is to intensify the negative meaning of an utterance.
  (This is how the rule continues to operate in regional dialects today where, for example, I’ve not got any money is strengthened by I’ve not got no money, and further strengthened by I’ve not got no money neither.) A range of negative forms is involved, such as no, not, none, neither, never, nowhere, and nothing, as well as words with negative implication, such as deny. When Richard accuses Queen Elizabeth of Hastings’ imprisonment, he uses two negatives to make his point, ‘You may deny that you were not the mean / Of my Lord Hastings’ late imprisonment’ (R3 I.iii.89). There is no ‘cancelling out’ of meanings here, nor in any of the examples below.


AYL I.ii.16 You know my father hath no child but I, nor none is like to have
CE IV.ii.17 I cannot nor I will not hold me still
Ham V.i.131 [Hamlet] What man dost thou dig it for? [First Clown] For no man, sir. [Hamlet] What woman then? [First Clown] For none neither.
Ham III.i.164 Nor what he spake ... / Was not like madness
MW IV.ii.149 [Ford] he’s not here I seek for. [Page] No, nor nowhere else but in your brain
Tem III.ii.18 [Stephano] We’ll not run ... [Trinculo] Nor go neither; but you’ll lie like dogs, and yet say nothing neither


AYL I.ii.26 love no man in good earnest, nor no further in sport neither
LLL V.i.144 [Holofernes] Thou hast spoken no word all this while. [Dull] Nor understood none neither, sir
TN III.i.156 [Viola, of her heart] And that no woman has, nor never none / Shall mistress be of it, save I alone