Background

We have a motivational poster in our office that says:

None of us is as smart as all of us.

I think that it's grammatically incorrect, and here is my reasoning:

• All of the tigers have spots.
• All of us are here.
• None of us are dead yet.

The three examples all sound correct when using the plural "are", rather than "is".

Question

Unfortunately, some of my coworkers disagree with me. They believe that the quote is correct when it uses the word "is". So my question is this:

Is the motivational quote grammatically correct or should it say "all"?

Solution 1:

Semantically, none is neither singular nor plural. It's less than one and much less than many.
So its subject agreement is entirely arbitrary. Plus, negatives are noted for their funny grammar.

Positive quantifiers are either singular in verb agreement, like each and every, or plural, like all. None can be either, depending on context.

• Every boy is playing today.
• Every one of the boys is playing today.
• Each boy is playing today.
• Each of the boys is playing today.

• All of the boys are playing today.

• None of them are playing today.

• None of them is armed.

If it quantifies a plural noun phrase, as in none of us, then it's equivalent to the negation of a universal quantifier in many cases.

For instance,

• None of us are going to the party.

is equivalent to

• All of us are not going to the party.

which is ambiguous; that's one reason not to say it this way.
instead of

• All of us are staying away from the party.

which is unequivocally plural.

As for the motivational slogan

• None of us is as smart as all of us (note, no full stop)

It's intended to make you think, and it seems to have succeeded.
If you like, you can take it as short for

• None of us is as smart as all of us together are.

which is meant to contrast individual intelligence with groupthink,
and therefore is unequivocally singular.

tl;dr It'd be grammatical with are, but it'd also be a less efficient slogan.

Solution 2:

According to Oxford Online Dictionaries, either is correct:

It is sometimes held that none can only take a singular verb, never a plural verb: none of them is coming tonight rather than none of them are coming tonight. There is little justification, historical or grammatical, for this view. None is descended from Old English nān meaning ‘not one’ and has been used for around a thousand years with both a singular and a plural verb, depending on the context and the emphasis needed

In this case, however, I believe "is" is more appropriate to stress the meaning that no individual is as smart as the collective all of us.

Solution 3:

As almost everybody else here mentions (not "mention" :D), none comes from not one, so grammatically, it should be used as a singular (it baffles me how some people conclude the opposite from the same fact), similar to the usage of every:

Usage Note: Every is representative of a group of English words and expressions that are singular in form but felt to be plural in sense. The class includes noun phrases introduced by every, any, and certain uses of some. These expressions invariably take a singular verb; we say Every car has (not have) been tested, Anyone is (not are) liable to fall ill

However, as with many other linguistic corruptions, usage of none with the plural has become an acceptable part of the language due to its long usage, as mentioned in the Usage Note of the American Heritage Dictionary:

Usage Note: It is widely asserted that none is equivalent to no one, and hence requires a singular verb and singular pronoun: None of the prisoners was given his soup. It is true that none is etymologically derived from the Old English word ān, "one," but the word has been used as both a singular and a plural since the ninth century. The plural usage appears in the King James Bible ("All the drinking vessels of king Solomon were of gold ... none were of silver") as well as the works of canonical writers like Shakespeare, John Dryden, and Edmund Burke. It is widespread in the works of respectable writers today. Of course, the singular usage is perfectly acceptable. Choosing between singular or plural is thus more of a stylistic matter than a grammatical one.

Having said all that, to get to your specific question, the sentence None of us is as smart as all of us is absolutely correct.

Using are would also be understood and correct given the above. But as the American Heritage Dictionary says, choice between the two is more a matter of style (which is subjective), and in my opinion, the correct stylistic choice has been made in this case: the sentence is meant to say and emphasize that "no single individual is as smart as society as a whole (when we pool our intellectual resources)". But again, this is more a matter of opinion.

Solution 4:

One IS smarter. The not just negates the "one is smarter" i.e. not "one is smarter". The not applies to the whole sentence, not just the word one. It's still one is smarter and therefore it's still none is smarter.

Solution 5:

None of us are as smart as all of us. 
Zero of us are as smart as all of us. 

This version of the sentence and its obvious paraphrasing, if strictly interpreted, are the exact opposite of motivational.  They imply that everyone thinking together is no better than no one thinking at all -- so why make the effort? 

None of us is as smart as all of us. 
No one of us is as smart as all of us. 

This version and its common interpretation express the intended motivating sentiment.  Whether right or wrong, it claims that no single person in the group can outperform the whole group working together -- so we should use our combined intelligence. 

If you've ever dealt with design by committee, you might find the version not cast in the singular to be closer to truthful, however far from motivational.