Tutorial Contents Tutorial Three: Propositional Calculus: Tableaux - The Sentence Tableaux Rules - Testing Arguments for Validity - Sequents - Syntactic Sequents - Semantic Sequents - Soundness and Completeness - Correctness of Sequents and Validity of Arguments More Tutorials One Two Four Five Six Seven

### Correctness of sequents and validity of arguments

If one formalises an argument using the language of the propositional calculus, one ends up with something like this:

j1, j2, … So,  y  (where j1, j2, …, and y are formulae).

#### What is the relation between the validity of the argument and the correctness of the corresponding sequent?

The answer is that, if the corresponding sequent is correct, the argument must be valid. (As we saw when using the tableau method to test an argument for validity, if the tableau for the argument's counterexample set closes, the argument is valid.) But the converse does not hold. It is possible for an argument to be valid even though the corresponding sequent is incorrect. (As we saw when using the tableau method to test an argument for validity, we could not conclude that its counterexample set was consistent from the fact that the tableau did not close. We had to go back to the interpretation to see whether the simple unchecked formulae in an open branch were consistent.)