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

## 1.

Formalise the following arguments as faithfully as you can as correct syntactic sequents.
(i) Aston, I fear, are going to be relegated. The problem is that to avoid this fate they must beat not only Sutton but also Weston. But they won't beat Weston, surely, unless Weston lose to Sutton – and even then they are not certain of beating them. But in any case Sutton will beat Weston only if they are the better side and, what is more, are at least as good as Aston. But in that case they won't lose to Aston.
(ii) Since John doesn't know very much about gardening, he will succeed in growing carrots only if Mary helps him. After all, to grow them to maturity he will have to get the seeds to germinate and will have to protect the plants from the carrot fly. And, though he might succeed in the former by good fortune, without Mary's help, he is bound to fail in the latter.
(iii) Without a change in conductor, there will not be any improvement in the orchestra's standards. A new conductor is unthinkable, though, with the present procedure for selecting orchestral managers; but replacing that involves a major administrative exercise, and any such exercise is bound to require an injection of new funds. If a new source of funding is required, then there will have to be a substantial grant from central government. Therefore a substantial governmental grant is essential if there is to be any improvement in orchestral standards.
(iv) Unless the Prime Minister intervenes, the Chancellor will cut taxes before the General Election. If he does, there will either be a run on the pound or the Government will increase the national debt. But the Prime Minister will not intervene. So, unless the Government increase the national debt, there will be a run on the pound.
(v) There will be no fight at the club tonight, because there will be one only if John and Henry are both at the club. But John will not be allowed into the club unless accompanied by Jenny, and if Jenny goes with John she will not go with Henry. I know, however, that Henry won't go unless he goes with Jenny.