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Symmetry Principles in Polyadic Inductive Logic
[Thesis]. Manchester, UK: The University of Manchester; 2016.
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Abstract
We investigate principles of rationality based on symmetry in Polyadic Pure Inductive Logic. The aim of Pure Inductive Logic (PIL) is to determine how to assign probabilities to sentences of a language being true in some structure on the basis of rational considerations. This thesis centres on principles arising from instances of symmetry for sentences of first-order polyadic languages.We begin with the recently introduced Permutation Invariance Principle (PIP), and find that it is determined by a finite number of permutations on a finite set of formulae. We test the consistency of PIP with established principles of the subject and show, in particular, that it is consistent with Super Regularity. We then investigate the relationship between PIP and the two main polyadic principles thus far, Spectrum Exchangeability and Language Invariance, and discover there are close connections. In addition, we define the key notion of polyadic atoms as the building blocks of polyadic languages. We explore polyadic generalisations of the unary principle of Atom Exchangeability and prove that PIP is a natural extension of Atom Exchangeability to polyadic languages.In the second half of the thesis we investigate polyadic approaches to the unary version of Constant Exchangeability as invariance under signatures. We first provide a theory built on polyadic atoms (for binary and then general languages). We introduce the notion of a signature for non-unary languages, and principles of invariance under signatures, independence, and instantial relevance for this context, as well as a binary representation theorem. We then develop a second approach to these concepts using elements as alternative building blocks for polyadic languages.Finally, we introduce the concepts of homomorphisms and degenerate probability functions in Pure Inductive Logic. We examine which of the established principles of PIL are preserved by these notions, and present a method for reducing probability functions on general polyadic languages to functions on binary languages.