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The concept of provability is important because it suggests how we can automate the determination of logical entailment
Starting from a set of premises Δ, we enumerate conclusions from this set
If a sentence φ appears, then it is provable from Δ and is, therefore, a logical consequence
If the negation of φ appears, then ¬φ is a logical consequence of Δ and φ is not logically entailed (unless Δ is inconsistent)
Note that it is possible that neither φ nor ¬φ will appear
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