1. He left New York for Chicago.

2. He started New York for Chicago.
3. He left for Chicago.
4. He started for Chicago.

(Are they all grammatical?)

1 and 3 are best.

2. is incorrect.

4. sounds is if he started for Chicago, but didn't arrive there.

Out of the four sentences you provided, two are grammatically correct while the other two have slight errors in them. Let's examine each sentence individually:

1. "He left New York for Chicago." - This sentence is grammatically correct. It conveys the idea that someone departed from New York with the destination being Chicago.

2. "He started New York for Chicago." - This sentence is not grammatically correct. The appropriate preposition to use in this context is "from" rather than "for." Therefore, it should be corrected to: "He started from New York for Chicago."

3. "He left for Chicago." - This sentence is grammatically correct. Although it doesn't specify the starting point, it implies that someone left (from an undisclosed location) with the intention of going to Chicago.

4. "He started for Chicago." - This sentence is not grammatically incorrect, but it may sound slightly incomplete and ambiguous. Adding additional context or information would help to clarify the meaning. For example, you could say: "He started his journey for Chicago from New York."

In summary, sentences 1 and 3 are grammatically correct, while sentences 2 and 4 need minor corrections to be grammatically accurate.