The difference between "few" and "a few"

It's one of the things that many non-native speakers of English get wrong: the distinction between few and a few. For example, let's take the following pair of sentences:

At the start of the 2014 World Cup, there were few signs that the final would include Germany.

At the start of the 2014 World Cup, there were a few signs that the final would include Germany.

What's the difference? While most native English speakers would immediately recognize the difference, to many English-language learners that I have encountered in the past, there is no discernible distinction. This could get them into a bit of a pickle in writing.

The simple explanation of the difference is that few means "almost zero" and is generally of a negating connotation, while in contrast a few means "a small amount, but definitely not zero" and is generally of a positive/additive connotation. Therefore, the above sentences would mean something like:

At the start of the 2014 World Cup, there were almost no signs that the final would include Germany. (i.e., It was unimaginable that Germany would make it to the final.)

At the start of the 2014 World Cup, there were some signs that the final would include Germany. (i.e., It was possible to think that Germany would make it to the final.)

As to why this distinction exists, it's partly because few is an adjective, while a few is a noun.

As an adjective, it implies that the expected value is vanishingly small. Therefore, if we initially had a high expectation that Brazil would reach the finals, the possibility that it wouldn't was approaching zero.

In contrast, as a noun, it implies that the actual amount was more than zero. Therefore, if we initially had an expectation that Germany would fail to go to the finals, our perception of that possibility would have been that it was greater than zero.

In reality, the number in question (e.g., the number of signs that Brazil wouldn't reach the finals) could be known or determined. The difference is how you interpret that number. For example, if we stated that the odds for making the finals of the 2014 World Cup stated on July 5, 2014 were:

Odds for World Cup semi-finals:

Brazil wins 80.09 %
Germany wins 19.91 %

Argentina wins 61.60 %
Netherlands wins 38.40 % 

It means that - on July 5th - there was an expected 19.91% chance that Germany would win their semi-final match to enter the finals against the winner of Argentina vs. Netherlands. This is almost a 1-in-5 chance that they would make it. And here is the crux in how our interpretations of the same number would change the meaning of the sentence.

If we had interpreted that 1-in-5 as a vanishingly small chance, then we'd use, "...there were few signs that the final would include Germany."

However, if we had interpreted that 1-in-5 as a distinct (if small) possibility, we'd use, "...there were a few signs that the final would include Germany."

... and that's basically it. Simple, no?

One additional note: The meanings for few and a few remain consistent when the thing your talking about is stated as a negative. For example:

There were few signs of Brazil not reaching the final.

Here, the direct implication is, "There were almost no signs of Brazil not reaching the final." After doing the trick in English of transforming two negatives into a positive, we have the implicitly understood interpretation of this statement as something like, "There were many signs that Brazil would reach the final." The use of a few leads to a contrasting sentence, similar to the examples using Germany:

There were a few signs of Brazil not reaching the final.

Here, the direct implication is, "There were some signs of Brazil not reaching the final," which is almost the direct opposite of the implication of the previous statement.

So, in short:

• If you want to emphasize the near-zero quantity of something, use few.
• If you want to emphasize the not-at-all-zero quantity of something, use a few.

Addendum: Of course, there is also the use of the few. However, as with most uses of the to convert an adjective into a noun (e.g., the elderly, the hungry, the poor, the tired, the wealthy, the powerful, etc.), it is referring to a distinct and defined (either implicitly or explicitly) group, usually of people. Thus we understand the recruiting phrase used by the US Marine Corps, "The few, the proud, the Marines," as referring to an implicit exclusivity that is associated with the culture of the US Marine Corps. Similarly, given the interchangeability of pronouns for articles, we can understand the following use of we few from the St. Crispin's Day speech in Henry V:

From this day to the ending of the world,
But we in it shall be remembered-
We few, we happy few, we band of brothers;

... and an excuse to post this lovely rendition of the St. Crispin's Day speech: