Complex Individuals and Multigrade Relations
Adam Morton · Noûs · 1975
Goodman and Leonard pointed out in 1940 that by using the calculus of individuals one could give formalizations within first-order logic of many idioms involving what they called "multigrade relations".These are relations such as 'are brothers', 'are compatriots', or 'built the bridge', which do not take any fixed number of arguments.One can say 'a and b are compatriots' or 'a and b and . . .and z are compatriots'.The purpose of this paper is to show that the reverse is also true; I give a formal account of multigrade relations and some related idioms, and show that there is a natural translation of the vocabulary of the calculus of individuals into the notation I provide which takes all the theorems of that calculus to valid sentences of the formalism.'I. PLURAL SUBJECTS The subject of a predicate such as 'live together' may be a string of names, such as 'Adam and Milly and Stephen', or a plural noun phrase such as 'the Mortons', or a string of plural noun phrases such as 'the Mortons and the MacDougals' or 'the Mortons and the MacDougals and some of the Hanrahans'.'The Mortons' is not shorthand for 'Adam and Milly and Stephen' even if these are all and only the Mortons, for in using 'the Mortons' one leaves open who are Mortons and how many they are.Instead, the force of a sentence such as 'The Mortons live together' is to say something like 'there are some people, pi, P2,..., and pi is a Morton and so is P2 and so are all the others, and p' and P2 and . . .live together'.Notice that these idioms involve nopresupposition that the subject covers only finitely many individuals; a multigrade relation can relate infinitely many relata.Notice also that multigrade relations can