- 'Nothing is a puppy'
- 'A puppy is nothing'
Can these two all mean there's no puppy?
No. These don't that there's no puppy. What they mean depends a little on context, and neither expression is common.
The first means the opposite of "Something is a puppy", or "Puppies don't exist". People aren't likely to say that because it's not true. Moreover the "don't exist" form is easier. So people say "Ghosts don't exist" instead of "Nothing is a ghost".
The second means "A puppy costs nothing, or is worth nothing". It is probably used hyperbolically.
I got my girlfriend a puppy for her birthday.
Hey, a puppy is nothing! I got my girlfriend a lion cub!
As you see it is not a common thing to say!
More common could be:
Nothing is important to me.
My ex-wife is nothing. I only love you!
There are other possible meanings, in context.
The short answer is no. 'Nothing is a puppy' does not mean 'A puppy is nothing'.
The word "nothing" in the first sentence and the word "nothing" in the second sentence mean two different things.
"Nothing is a puppy": The word "nothing" here is a pronoun and it means "Not anything; no single thing."
A puppy is nothing: The word "nothing" here is an adjective and it means "of no value".