As far as I know, a period is commonly used for an abbreviation (e.g.,
Inc.). But I can also see a slash in these cases:
Is there any grammar rule for an abbreviation? or is latter case an exception?
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There are a dozen or so "abbreviations" which employ the "/" character and which are generally recognized in US English. A few that come to mind:
(And there are no doubt hundreds if not thousands of such abbreviations used in individual disciplines such as medicine or electronics.)
Note that while some a most commonly lower case, others usually are upper case, and this may vary from one writer or context to the next.
The "/" character is also generally recognized as a shorthand for "or" separating a pair of options, as in "The person who insists on always using the properly-gendered personal pronoun will often find his/her efforts stymied by a lack of information."
Usually the "/" character is used without spaces between it and adjacent components, but in cases where one side is "exposed", such as "w/" for "with" (vs "without"), there should be a separating space.
This sort of abbreviation is used so rarely (and often in technical contexts) that it's difficult to define how "formal" it is.
Is there any grammar rule for an abbreviation?
Yes. Use an abbreviation only if your potential readers will immediately recognize and understand the abbreviation.
You want to communicate clearly and effectively with your reader. Therefore, you want to avoid any words, syntax, or abbreviations that will impede communication.
O/w if you fx employ abbreviations /s explaining what they denote, you're likely to perplex rather than promulgate precisely. (air?)
Translation: Otherwise, if you frequently use abbreviations without explaining what they mean, you are likely to confuse your reader rather than communicate clearly and effectively with him or her.
Note: 'air' = Am I right?, a rhetorical question I would not recommend using in any kind of formal or academic prose, whether you abbreviate it or not.