Aristotle's syllogistic is extended to include denumerably many quantifiers such as 'more than 2/3' and 'exactly 2/3.' Syntactic and semantic decision procedures determine the validity, or invalidity, of syllogisms with any finite number of premises. One of the syntactic procedures uses a natural deduction account of deducibility, which is sound and complete. The semantics for the system is non-classical since sentences may be assigned a value other than true or false. Results about symmetric systems are given. And reasons are given (...) for claiming that syllogistic validity is relevant validity. (shrink)