I'm trying to explain to someone why there's no article in this sentence:
I'm always looking for inspiration
What is the rule that applies here? And is there a recorded set of similar words that are normally used without articles?
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As pointed out by medica, inspiration is an uncountable or mass noun. The relevant portion of the link is:
In English, mass nouns are characterized by the fact that they cannot be directly modified by a numeral without specifying a unit of measurement, and that they cannot combine with an indefinite article (a or an). Thus, the mass noun "water" is quantified as "20 litres of water" while the count noun "chair" is quantified as "20 chairs". However, both mass and count nouns can be quantified in relative terms without unit specification (e.g., "much water," "so many chairs").
In general, intangible things (concepts, ideas, emotions, thoughts etc., such as inspiration) are mass nouns.
Also, any noun that can refer to multiples without being pluralized (such as water in the wiki example and candy in the comments) may be used as a mass noun. You can say
Let justice roll down like waters and righteousness like a mighty stream.
But you don't need to pluralize waters to make a valid sentence.
Similarly, you can say "I want a bunch of candies" or "I want a bunch of candy" and mean "many pieces of candy" in both cases.
Finally, you actually can say "I'm always looking for an inspiration." and have it still make sense under the same 4th meaning: a thing or person that inspires. The connotation would change slightly. Without the article, the sentence brings to mind finding inspiration from intangibles, like beauty or art. With an article, the connotation is a more for a concrete person, place or event (which could be personified as a muse: a source of inspiration; especially : a guiding genius).