We consider two formalizations of the notion of irrelevance as a rationality principle within the framework of (Carnapian) Inductive Logic: Johnson's Sufficientness Principle, JSP, which is classically important because it leads to Carnap's influential Continuum of Inductive Methods and the recently proposed Weak Irrelevance Principle, WIP. We give a complete characterization of the language invariant probability functions satisfying WIP which generalizes the Nix-Paris Continuum. We argue that the derivation of two very disparate families of inductive methods from alternative perceptions of (...) 'irrelevance' is an indication that this notion is imperfectly understood at present. (shrink)

The application of the maximum entropy principle to determine probabilities on finite domains is well-understood. Its application to infinite domains still lacks a well-studied comprehensive approach. There are two different strategies for applying the maximum entropy principle on first-order predicate languages: applying it to finite sublanguages and taking a limit; comparing finite entropies of probability functions defined on the language as a whole. The entropy-limit conjecture roughly says that these two strategies result in the same probabilities. While the conjecture is (...) known to hold for monadic languages as well as for premiss sentences containing only existential or only universal quantifiers, its status for premiss sentences of greater quantifier complexity is, in general, unknown. I here show that the first approach fails to provide a sensible answer for some \-premiss sentences. I discuss implications of this failure for the first strategy and consequences for the entropy-limit conjecture. (shrink)

The application of the maximum entropy principle to determine probabilities on finite domains is well-understood. Its application to infinite domains still lacks a well-studied comprehensive approach. There are two different strategies for applying the maximum entropy principle on first-order predicate languages: applying it to finite sublanguages and taking a limit; comparing finite entropies of probability functions defined on the language as a whole. The entropy-limit conjecture roughly says that these two strategies result in the same probabilities. While the conjecture is (...) known to hold for monadic languages as well as for premiss sentences containing only existential or only universal quantifiers, its status for premiss sentences of greater quantifier complexity is, in general, unknown. I here show that the first approach fails to provide a sensible answer for some \-premiss sentences. I discuss implications of this failure for the first strategy and consequences for the entropy-limit conjecture. (shrink)

We give a unified account of some results in the development of Polyadic Inductive Logic in the last decade with particular reference to the Principle of Spectrum Exchangeability, its consequences for Instantial Relevance, Language Invariance and Johnson's Sufficientness Principle, and the corresponding de Finetti style representation theorems.

Spectrum Exchangeability, Sx, is an irrelevance principle of Pure Inductive Logic, and arguably the most natural extension of Atom Exchangeability to polyadic languages. It has been shown1that all probability functions which satisfy Sx are comprised of a mixture of two essential types of probability functions; heterogeneous and homogeneous functions. We determine the theory of Spectrum Exchangeability, which for a fixed languageLis the set of sentences ofLwhich must be assigned probability 1 by every probability function satisfying Sx, by examining separately the (...) theories of heterogeneity and homogeneity. We find that the theory of Sx is equal to the theory of finite structures, i.e., those sentences true in all finite structures forL, and it emerges that Sx is inconsistent with the principle of Super-Regularity. As a further consequence we are able to characterize those probability functions which satisfy Sx and the Finite Values Property. (shrink)