Abstract
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: (i) applying it to finite sublanguages and taking a limit; (ii) 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 Sigma _2-premiss sentences. I discuss implications of this failure for the first strategy and consequences for the entropy-limit conjecture.
Highlights
We can understand Inductive logic as a logical approach seeking to model thecertainty one ought attach to a conclusion of interest (ψ) givencertain premisses φ1, . . . , φk
Inductive logicians develop inductive entailment relations (e.g., [7]) and aim to determine the uncertainty attaching to the conclusion ψ, Y
Inductive logic was popularised by Carnap [2,3,4], his approach today lives on in the Pure Inductive Logic approach [9,10,22]
Summary
We can understand Inductive logic as a logical approach seeking to model the (un-)certainty one ought (or could) attach to a conclusion of interest (ψ) given (un-)certain premisses φ1, . . . , φk. While this first step is rather uncontroversial, the principle has a second part which applies, if there is more than one probability function satisfying all the premisses In this case, the principle compels rational agents to adopt one of these probability functions with maximum entropy ( the name of the principle). Possibly conflicting, ways to apply the Maximum Entropy Principle to predicate languages have been put forward It is not clear whether the resulting inductive logics agree. Both approaches agree (where they are both defined), this would point towards a unique consistent interpretation of the Maximum Entropy Principle on infinite domains providing further support for its application. In Theorem 2, I generalise this result to the existence of such sentences of all quantifier complexities greater than Σ2 In these cases, the maximal entropy function Pφ† does not exist [24, Section 2.2]. The Entropy-limit Conjecture even emerges strengthened, since it applies to fewer cases, and can fail in fewer cases
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