While NK constructs an actual proof (of the conclusion from the given assumptions), LK instead constructs a proof that proves the existence of a proof (of the conclusion from the assumptions).
For indirect calculi, the completeness property is expressed in terms of refutations since one establishes that Γ ⊨ α by showing the existence of a proof, not of α from Γ, but of ⊥ from Γ∪.
The reason is, of course, that the statement of the completeness of LK only has to assert, for each entailment relation, the existence of a proof tree but a reasoning program has the more difficult task of actually having to construct one.
Since the prosecutor was put on notice about the existence of exculpatory proof, didn't the prosecutor have an obligation to investigate the existence of the phone bill, or the exculpatory testimony from the motel staff, as well as any other favorable evidence supporting Mr. Fleming's claim of innocence?
Since his time, and possibly before, more elegant proofs have been given, and the mere existence of such proofs speaks about the beauty of the result.
Because the ancient mathematician is responsible for the existence of the proofs, he is arguably the dominant causal source of the descriptive information associated with the name 'Ibn Kahn'.
The main stages are simply stated: the opening of religious tradition to scholarly critique by Luther; the conclusive demonstration of the unworkability of existence proofs of God by Kant; the absorption of religion into secular philosophy by Hegel; and the announcement of what had become obvious by Nietzsche.
The problem solving process ends when one such proof is found, when the program is able to detect the non-existence of a proof, or when it simply runs out of resources.
After all, inability to imagine a proof is commonly due to a failure of ingenuity rather than the non-existence of a proof.
You might say that the existence of D12 is proof of the power of manly love.
The great American astronomer Carl Sagan, no less interested in alien intelligence, replied rapidly, if informally, that the existence of Arthur C Clarke was proof enough.
