Conventions set by earlier grammarians set the logic here, to the extent there is much logic.
As John Lawler points out in a comment to this question, the grammatical number of a negative is arbitrarily set. As pronouns, neither and none are neither singular nor plural. Neither of two or none of many is zero, not many (plural) and not one (singular). These pronouns are used with either singular or plural verbs because those are the two simple options available in English for number.
Which one to use is arbitrary on the whole. Yes, you can probably find situations where a singular verb or plural verb sound right to a fluent ear, and you can try to reason from context (Grammar Girl provides an example of thinking through whether the pronoun none refers to a singular or plural antecedent), but in my experience, only convention dictates an absolute difference. Thus, without a clear logic, language users are instead guided by their own conventions or whims. The Oxford English Dictionary, for instance, has many examples of both singular and plural usages with neither.
B. pron.
- Not the one nor the other (of two).
a. Standing alone or with of.
[...]
(a) With singular or unmarked agreement.
[examples from 1300 onward omitted for brevity]
- 1923 World's Work May 563/2 He..expected that he had been arrested in mistake for me. Unfortunately neither of us was justified in our hopes of misprision.
- 1970 G. F. Newman Sir, You Bastard iii. 111 Both the detectives' names and ranks were correct; neither was upped to DCS.
- 2001 N. Jones Rough Guide Trav. Health ii. 399 There are several suitable drugs to eradicate the worms, such as tiabendazole..and albendazole.., neither of which should be taken in pregnancy.
(b) With plural verb or anaphoric plural pronoun.
Use of plural agreement with neither is frequently criticized by grammarians in the latter part of the 18th cent.
[examples from 1550 onward omitted for brevity]
- 1886 R. C. Moberly Probl. & Princ. (1904) x. 314 They are co-ordinate,..they neither override the other.
- 1902 Philos. Rev. 11 140 At this stage of development it is nonsense to talk about the self or the non-self, egoism or altruism, the individual or the external world, for neither exist as far as the child is concerned.
- 1991 Daily Tel. 5 Jan. 30/7 In the girls' singles, neither of the top two seeds were seriously challenged.
The note in B.1.a.(b) is interesting - it was the 18th century grammarians that insisted on treating neither as singular. In many cases, it was convention - and not logic - that would dictate the choice of neither is over neither are.
Incidentally, none also has a history of flexible usage, with singular (A.2.a) and plural (A.2.b) examples. While A.2.b notes that the plural usage is now more common, "the singular being expressed by no one," it is not too hard to find errant "none is" or "none ... [singular verb]" in the wild.
So, at least in this case, convention sets the motivation for using none and neither, and especially whether you want to be judged negatively by someone who holds dear to convention.