This is from the Oxford Learner's Dictionary.
based on, or holding, the belief that everyone is equal and should have the same rights and opportunities
I wonder why it is necessary to put a comma after 'holding'.
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I don't think it is required, but it just makes the sentence easier to understand. Keep in mind that the 2 commas work together to offset the "or holding" phrase; it would be a mistake to only have one or the other.
It emphasizes that "based on" and "holding" are two parallel phrases that both point to the noun "belief." It still works without the commas, but it's less clear.
It would also be correct to write the sentence without any commas: “Based on or holding the belief ....” The reason the authors didn’t write it that was is probably that the sentence becomes much harder to parse without the commas. Which words in that sentence would the “or” refer to? We have to read far into the sentence to rule out the possibility that it will say something like, “Based on or holding the belief that everyone is equal implies the conclusion that ...,” where the clause parallel to “based on [the conclusion]” would be, “holding [that] the belief that everyone is equal implies [the conclusion].”
The purpose of the commas is to remove this ambiguity. If there’s a comma before the phrase “or holding,” however, there needs to also be one afterward. The function of the comma here is to set off “or holding” as parenthetical phrase, parallel to “based on.” It’s pretty similar to writing, “Based on (or holding) the belief that ....”
This is in a dictionary definition - already pointed out that is a different normal usage than sentence grammar. In sentences, a comma often indicates a pause, or slight pause, that would/should/or could indicate a separation in the flow of a sentence. Commas are part of the rules and structure of grammar, which helps to define the written and spoken parts of language.
As to the pause, it separates portions of the flow such as phrases or clauses within the flow of the piece as a whole. There are fairly exacting rules for commas' use and placement, but these may vary by type of writing, convention, or even by personal style or preference.
As the basis of deciding whether a comma has been used correctly (after considering the above) try reading the sentence out loud using the established cues and context, then see if the pause or slight pause a comma should indicate sounds correct. Consider it is the writer's ear that established a comma's placement, in the written language, so it may be different in a novel, or a non-fiction piece, or a newspaper/journalistic story. Other types of writing may have vastly different rulesets: code, databases, telegrams (txt msgs R th mdrn equiv) and communiqués for instance.
Now we're back to where and why the sentence was written: as a definition in a dictionary. There the styles and preferences are very likely dictated and fairly rigidly used for the sake of clarity and axactness.