Joerg Endrullis and Lawrence S. Moss: Syllogistic Logic with "Most"

The classical syllogistic is the logical system whose sentences are of the form

All X are Y, Some X are Y, No X are Y

In a contemporary semantic account, these sentences are evaluated in a model by assigning a set [X] to each variable X and then using the evident truth definition. This logical system lies at the root of the western logical tradition. For this reason, modern logicians have occasionally looked back on it with an eye to its theoretical properties or to extending it in various ways.

This paper presents an extension of the syllogistic which includes sentences of the form

Most X are Y

Variables are interpreted by subsets of a given finite set, with the understanding that "Most X are Y" means that strictly more than half of the X are Y. We present a proof system which is strongly complete relative to the semantics: for every finite set S of sentences and every sentence s, s is derivable from S in our system if and only if every model of all sentences in S is again a model of s.

We also evaluate the complexity and present an algorithm to tell if a finite set S does indeed imply s.

The result is important because it shows that on top of syllogistic logic, one can add expressions which are not expressible in first-order logic and still have logical systems which are complete and highly manageable.