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| in Logic |
How do you show some conclusion of an argument cannot be derived in a complete system? Does one have to make the truth table to show that it is not valid and therefore, by definition it should be impossible for that conclusion to be derived?
June 24, 2010
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This question has essentially been answered, I think, as question 3211.
In brief: A truth-table is one way to show that a conclusion cannot be derived, but it is not by definition that it cannot be. This is a consequence of the soundness theorem, which states that every derivable formula is valid. Since the truth-table shows the formula is not valid, then it cannot be derived since, otherwise, it would be valid.
That said, a truth-table is not the only way to show that a formula cannot be derived. For one thing, there are also trees (or "tableaux"); for another, there are purely syntactic arguments that can be given to show, e.g., that a formula containing no more than one occurrence of any sentence letter cannot be derived. See question 3211 for a similar example.