Just to continue.

Lots of arguments do not allege to be valid. The premisses are supposed to provide some
support for the conclusion, but not conclusive support. Deductive arguments are those
that allege to be valid. Proofs are a proper subset of deductive arguments.

What is required for a proof? It must be valid, it must be sound (valid with true premisses),
but this isn't enough. 'The sky is blue; so the sky is blue' is sound but no proof, because
it is circular. It couldn't be used to demonstrate the conclusion for someone who doubted
it.

So we might say that a proof is a sound, non-circular argument--but this is insufficient too.

I might give a sound non-circular argument that we don't know is sound, because
we don't know the premiss is true. That won't prove the conclusion, since
for all we know the premiss is false.

So, supposing the last presidential election was going to be close.
"Obama will win. He is a Dem; So a Dem will win.'

'McCain will win. He is a Rep; So a Rep will win.'

As one or the other will win, and the second premisses are both true, and both are valid,
one of these is sound but we don't know which, so neither is a proof.

So I suggest that a proof is a sound, non-circular argument where we are in a position
to know it is sound.