An algebraic study of Peterson's Intermediate Syllogisms

Peterson’s Intermediate Syllogisms, generalizing Aristotelian syllogisms by intermediate quantifiers ‘Many’, ‘Most’ and ‘Almost all’, are studied. It is demonstrated that, by associating certain values V, W and U on standard Lukasiewicz MV–algebra with the first and second premise and the conclusion, respectively, the validity of a corresponding intermediate syllogism is determined by a simple MV–algebra (in–)equation. Possible conservative extensions of Peterson’s system are discussed. Finally it is shown that Peterson’s bivalued intermediate syllogisms can be viewed as fuzzy theories in Pavelka’s fuzzy propositional logic, i.e. a fuzzy version of Peterson’s Intermediate Syllogisms is introduced.