1. Being tired, he went to bed early.

2. Tired, he went to bed early.

3. Being strong, he could lift the box.
4. Strong, he could lift the box.

5. Being left along, the baby began to cry.
6. Left alone, the baby began to cry.

7. It being fine, he went out for a walk.

8. It fine, he went out for a walk.
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Which one is better? Do we have to use 'being' or omit 'bing'? Are they all grammatical <in written English>?
Do native speakers us those sentences in casual speech?

OK sentences:

1, 3, 5 (but with "alone" not "along"), and 6

Not OK sentences:
2, 4, 7, and 8

Since it was a fine evening, he want out for a walk.
Because he was strong, he could lift the box.
Because he was tired, he went to bed early.

They are not all grammatical in written english; However, 6 is grammatical.

natives do use these sentences in casual speech.

(I corrected one spelling error.)

1. Being tired, he went to bed early.
2. Tired, he went to bed early.

3. Being strong, he could lift the box.
4. Strong, he could lift the box.

5. Being left alone, the baby began to cry.
6. Left alone, the baby began to cry.

7. It being fine, he went out for a walk.

8. It fine, he went out for a walk.

Good sentences:

1, 3, 5, and 6

Incorrect sentences:
2, 4, 7, and 8

All of the sentences you provided are grammatically correct in written English. However, there are some differences in terms of style and usage.

Sentence 1: "Being tired, he went to bed early."
Sentence 2: "Tired, he went to bed early."

Both sentences have the same meaning, but they have different stylistic impacts. Sentence 1, with the use of "being," emphasizes the state of being tired as a temporary condition that affected his decision to go to bed early. It adds a slightly more descriptive and explanatory tone to the sentence. Sentence 2, on the other hand, conveys the same information more concisely by omitting "being." This style is preferred in more straightforward, concise writing.

Sentences 3 and 4 follow the same pattern as sentences 1 and 2. "Being strong, he could lift the box" emphasizes the temporary condition of being strong, while "Strong, he could lift the box" presents the information more directly.

Sentence 5: "Being left alone, the baby began to cry."
Sentence 6: "Left alone, the baby began to cry."

Both sentences are correct, but again, they have different stylistic impacts. Sentence 5, with "being," emphasizes the action or situation as a temporary condition. It suggests that once the baby is no longer left alone, the crying may stop. Sentence 6, without "being," presents the action or situation more directly, without implying the temporary nature of it.

Sentence 7: "It being fine, he went out for a walk."

This sentence is grammatically correct, but it is less commonly used in modern English. The construction "it being" is somewhat dated and may sound formal or literary. In casual speech, it is more common to say "Since it was fine, he went out for a walk" or "Because it was fine, he went out for a walk."

Sentence 8: "It fine, he went out for a walk."

This sentence is not grammatically correct in standard English. The phrase "It fine" lacks a subject or verb and can be seen as incomplete. To make it grammatical, you could say "It was fine, so he went out for a walk." or "As it was fine, he went out for a walk."

In casual speech, native speakers might use different variations of these sentences depending on their style, personal preferences, and regional dialects. But overall, the use of "being" or omitting it is a matter of style and preference, and both are grammatically acceptable.