They are the definitions of 'another'

1. one more person or thing of the same type
2. DIFFERENT
a different one not the same thing, person etc, but a different one
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3. One man said, "Yesterday, I killed a big dragon with my kick."

4. Another man stepped up. "I have a talking dog," said the man.

[In that case, what is the meaning of 'another' in Sentence 4? Does 'another' in #4 mean #1 or #2 in meaning?]

In 4, it's a second man, but still a man. He's not a different type; he's just not the same man as in 3.

In sentence 4, the meaning of 'another' is closer to the definition of #1. It implies that there is an additional person, in this case, another man, who is also making a statement. In this context, 'another' is used to indicate that there is a second person, different from the first one mentioned, who is participating in the conversation or making a statement. So, the meaning of 'another' in sentence 4 aligns more with definition #1, which suggests adding one more person or thing of the same type.