Language Invariance and Spectrum Exchangeability in Inductive Logic

Abstract

A sufficient condition is given for a probability function in Inductive Logic (with relations of all arities) satisfying spectrum exchangeability to additionally satisfy Language Invariance. This condition is shown to also be necessary in the case of homogeneous probability functions.

A survey of some recent results on Spectrum Exchangeability in Polyadic Inductive Logic

“…The good news, see Landes (2009) and Vencovská (2006), 9 is that in the presence of Ex, JPSP implies Sx, tying in with the unary situation where JSP implies Ax. However, the arguably bad news is that within this formulation JPSP proves to be just too demanding.…”

“…Unfortunately the more general the result becomes the worse the notation. For that reason we shall just give two examples of such 'conformity', from which the wider picture may be glimpsed [for more details see Landes (2009)]. …”

The Theory of Spectrum Exchangeability

“…6 We are now ready to prove the converse to our earlier observation that for w a t-heterogeneous probability function, if ζ j t |= θ( a) then w(θ( a)) = 1.…”

“…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.…”

“…Let δ > 0 be fixed and let θ ∈ SL. By a result of Landes [6,Theorem 12] (or see [15,Lemma 34.4]), for t ∈ N + and anyp…”