Acceptance and approximation : studies on probabilistic stability, acceptance rules on atomless spaces, and algebraic logics of approximation
Keywords: Philosophy.

This dissertation comprises two parts. The first part studies acceptance rules on Bayesian statistical models and atomless probability spaces. We show that, for a very wide class of probabilistic learning problems, Leitgeb's probabilistic stability rule yields a notion of acceptance that either fails to be conjunctive (accepted hypotheses are not closed under finite conjunctions) or is trivial (only hypotheses with probability one are accepted). These results apply to most canonical Bayesian models involving exchangeable random variables, such as parametric models with Dirichlet priors over discrete distributions (and, in particular, to every confirmation function in Carnap's gamma-delta continuum). These results exhibit a serious tension for the stability rule: in Bayesian statistical models, important properties of priors that are conducive to inductive learning—Bayesian consistency, as well as certain symmetries in the agent's probability assignments—act against conjunctive acceptance. We then investigate impossibility theorems for acceptance rules on atomless spaces. We first strengthen a theorem by Smith, which generalises a form of the Lottery Paradox to atomless probability spaces, revealing an important conflict between certain invariance constraints on rational acceptance and logical closure requirements on accepted hypotheses. We introduce a refined class of acceptance rules, under which acceptance is sensitive to the structure of the evidence available (for instance, the acceptance of a hypothesis can depend on the structure of the observable data in the given experimental setup). While these rules escape Smith's result, we prove that, on learning problems based on standard Borel probability spaces, sufficiently invariant acceptance rules are either (1) not conjunctive, (2) inconsistent, or (3) trivial in a learning-theoretic (topological) sense. These results raise a number of questions about the context-sensitivity of acceptance, the aggregation of uncertain information in highly symmetric probability models, and the relationship between acceptance rules and statistical testing procedures. In the second part, we study the modal logics of approximation operators on Boolean algebras. We axiomatise the logics of approximation operators generated by arbitrary sets, sublattices, complete sublattices and subalgebras. For each class of substructures, we provide a representation theorem which characterises the corresponding class of closure-kernel algebras, and prove a completeness theorem, identifying the corresponding bimodal logic. Our investigations show that these approximation semantics allow to recover a natural family of modal systems: the fusions S4+S4 and EMNT4+EMNT4, the temporal logic S4t, as well as S4 and S5 are all shown to be logics of approximation operators. By varying the structural constraints imposed on the set D which generates the approximation operators, we obtain a natural family of modal logics. The logic of approximation operators generated by arbitrary subsets is the subnormal logic EMNT4+EMNT4. We characterise the corresponding class of algebras: that is, algebras with closure and kernel operators representable as approximation operators of the same underlying set. We show that representable algebras do not constitute an algebraic variety. We then prove that the complete logic of sublattice-generated closures and kernels is the fusion S4+S4. Our completeness proof, which uses a result by van Benthem et al. on vertical-horizontal bitopologies on the rational plane, relies on showing that the interior-closure algebras of pairwise zero-dimensional bitopological spaces correspond to sublattice-generated closure-kernel algebras. We note, as a corollary, that S4+S4 is the bimodal logic of pairwise zero-dimensional bitopological spaces. When D is a complete sublattice, we obtain exactly the temporal logic S4t, exploiting a Tarski-like duality between complete sublattices and pairs of closure and kernel operators. When D is a subalgebra, the two modalities collapse into one as they become each other's duals, and we obtain monomodal S5 (for complete subalgebras) and S4 (for subalgebras in general)