Uniqueness, Definability and Interpolation

This paper is meant to be a comment on Beth's definability theorem. In it we shall make the following points. Implicit definability as mentioned in Beth's theorem for first-order logic is a special case of a more general notion of uniqueness. If a is a nonlogical constant, Tat a set of sentences, cx* an additional constant of the same syntactical category as a and T7t* a copy of Tat with cx* instead of a, then for implicit definability of a in Ta, one has, in the case of predicate constants, to derive cx(x1,.. ., x,) *-* oc*(x1,.. ., x,) from T, u T1*, and similarly for constants of other syntactical categories. For uniqueness one considers sets of schemata S, and derivability from instances of Sat u Sac* in the language with both a and a*, thus allowing mixing of a and aX* not only in logical axioms and rules, but also in nonlogical assumptions. In the first case, but not necessarily in the second one, explicit definability follows. It is crucial for Beth's theorem that mixing of a and x* is allowed only inside logic, not outside. This topic will be treated in ?1. Let the structural part of logic be understood roughly in the sense of Gentzenstyle proof theory, i.e. as comprising only those rules which do not specifically involve logical constants. If we restrict mixing of a and x* to the structural part of logic which we shall specify precisely, we obtain a different notion of implicit definability for which we can demonstrate a general definability theorem, where a is not confined to the syntactical categories of nonlogical expressions of first-order logic. This definability theorem is a consequence of an equally general interpolation theorem. This topic will be treated in ??2, 3, and 4. Finally, in ?5 we shall show that under certain conditions, which in particular obtain in the case of implicit definability in the usual first-order case, the mixing of a and x* in logic can be reduced to their mixing in the structural part of logic, which makes Beth's definability theorem a consequence of our general definability theorem.