We'll start with the basics.
Starting a Sentence
You capitalize the first word in each sentence. Examples are:
Lisa and Jenny walked down the street.
A bunny hopped up to Lisa.
"Hello, little bunny," said Lisa.
You capitalize the names of people, as shown above. To go with this, you also capitalize titles of people:
Mrs. Jane Smith headed into the church.
Father William met her at the door.
Doctor Smith stood waiting within.
NOTE: The name God is always capitalized when it refers to a singular God. So "God damnit" is capitalized. The only time God is not capitalized is if you're using it as generic noun, such as in "a woodland god would watch over this gentle forest". So it's the difference between "a god of many gods" vs "The God in a religion". It's sort of like the difference between "the electric bill was paid" (bill being a generic noun) and "my boyfriend Bill called me on the phone" (a specific entity).
You capitalize the proper names of places. So for example:
Paul Revere went to Boston, Massachusetts.
Donald Trump is well known in New York City.
It could be fun to kayak on the Amazon River.
Mark was born in Argentina.
This also refers to names that designate a region. So the East Coast is capitalized, but "driving east on Route 290" is not capitalized. You can visit relatives in the South to drink mint juleps, but you look south to watch for migrating hummingbirds.
Days and Months
You capitalize days of week and month. So for example:
The full moon will fall on a Friday this month.
I'm looking forward to my vacation in July.
Titles are only capitalized if they are part of a person's direct addressed name. So you would capitalize President George Washington, but you would not capitalize "Bill Smith is president of Soulmate Pharmaceuticals." Director Lisa Shea came to the party. However, Lisa Shea, our production director, was available for questions.
If you are talking about the generic idea of "France has an army" there is no capitalization there. However, if you are specifically referring to an entity, like the Army of the United States, or the United States Marine Corps, those get capital letters. Those are proper nouns of named groups. It's the same thing as referring to the Salvation Army or the New England Patriots. Those are all proper nouns of groups.
If you are referring to a specific situation, you want to capitalize it. Marie Antoinette ran into problems during the French Revolution. The party was held on the Fourth of July. If your timeframe is more generic, you leave it lowercase. The first telegraph message was sent during the nineteenth century. I love photographing butterflies in the summer.
There seems to be no consensus on A.M. vs a.m. - you'll see both rules vociferously defended. Figure out who your target audience is and do the best you can. Just stay consistent.
Courses and School
If you are naming a proper noun of a specific course, capitalize it. But if it's a general field of study, do not capitalize it. So, "I was taking Psychology 101 in the spring and my teacher was unintelligible." But "I am majoring in psychology, with a minor in literature."
These cover the main uses of capitalization. Let me know if you have any questions!
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