A Note on Binary Inductive Logic

Abstract

We consider the problem of induction over languages containing binary relations and outline a way of interpreting and constructing a class of probability functions on the sentences of such a language. Some principles of inductive reasoning satisfied by these probability functions are discussed, leading in turn to a representation theorem for a more general class of probability functions satisfying these principles.

Language Invariance and Spectrum Exchangeability in Inductive Logic

“…, a r ) and S(Θ) = S (so S = r) let N L (S, T ) be the number of state descriptions Φ extending Θ with S(Φ) = T . By results in [16], [17], [13] this does not depend on the particular Θ with S(Θ) = S which is chosen.…”

“…By a straightforward generalization of the result in [17] (where just two colors were considered) u p L satisfies Ex and Sx.…”

“…In common with recent developments in Inductive Logic, see for example [17] (and [1], [2], [3] for the classical approach), we shall work within a first order predicate language L with finitely many relation symbols, countably many constants a 1 , a 2 , a 3 , . .…”

“…A number of such principles have been suggested, see for example [1], [2], [3], [8], [10], [13], [15], [16], [17], [19], the most basic of which asserts that w should not treat the constants a i differently, equivalently:…”

The Theory of Spectrum Exchangeability

“…It is shown in [6], [8], [9], [10], [15] that any probability function on SL which satisfies Sx may be expressed as a convex sum of probability functions of two basic types: heterogeneous and homogeneous functions, defined as follows.…”