Quantities, State Spaces, and Laws

Abstract

It is well-known that first-order languages can contain, in addition to monadic and polyadic predicates, n-place function symbols for any n. Still, many of the central concepts introduced in Chapter 2 (Q-predicates, constituents etc.) are primarily applicable to formal languages containing qualitative predicates. To be sure, Carnap’s notion of a monadic conceptual system leaves room for attribute spaces which are based upon some quantity (cf. Chapter 2.2). However, Carnap wished to avoid uncountable families of predicates, since that would lead to an uncountable number of Q-predicates as well. Therefore, he assumed that the value space of a real-valued quantity is partitioned into at most a denumerably infinite number of intervals. A more direct and more elegant way of treating quantitative concepts and laws — the so-called state space conception of theories — is given in this chapter. We shall see that the idea of a quantitative state space is in fact obtainable as a limiting case of Carnapian discrete conceptual systems.