Abstract
Building on previous work [4, 5] that bridged Formal Learning Theory and Dynamic Epistemic Logic in a topological setting, we introduce a Dynamic Logic for Learning Theory (DLLT), extending Subset Space Logics [9, 17] with dynamic observation modalities \([o]\varphi \), as well as with a learning operator 
, which encodes the learner’s conjecture after observing a finite sequence of data 
. We completely axiomatise DLLT, study its expressivity and use it to characterise various notions of knowledge, belief, and learning.
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- 1.
From probabilistic and statistical formalisms based on Bayesian reasoning, Popper-style measures of corroboration, through default and non-monotonic logics, Carnap-style ‘inductive logic’, to AGM-style rational belief revision and theory change.
- 2.
‘The Logic of Reliable Inquiry’ is the title of a classic text in FLT-based epistemology [16].
- 3.
We will return to it, with complete definitions, later in the paper. Our DLLT is interpreted over such frames.
- 4.
In the tautological information state X, the learner believes P iff \(\mathbb {L}(X)\subseteq P\).
- 5.
This topology is T1 iff for every two distinct points \(x\not =y\) there exist an observation \(O\in \mathscr {O}\) with \(x\in O\) and \(y\not \in O\).
- 6.
The observational topology is T0 iff points can be distinguished by observations; i.e. if x and y satisfy the same observable properties in \(\mathscr {O}\), then \(x=y\). Obviously, T0 is a minimally necessary condition for any kind of learnability of the real world from observations.
- 7.
A set is locally closed if it is the intersection of a closed and an open set.
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Acknowledgements
The research of Nina Gierasimczuk is supported by an Innovational Research Incentives Scheme Veni grant 275-20-043, Netherlands Organisation for Scientific Research (NWO) and by the OPUS grant 2015/19/B/HS1/03292, National Science Centre Poland (NCN). Aybüke Özgün acknowledges financial support from European Research Council grant EPS 313360.
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Baltag, A., Gierasimczuk, N., Özgün, A., Vargas Sandoval, A.L., Smets, S. (2018). A Dynamic Logic for Learning Theory. In: Madeira, A., Benevides, M. (eds) Dynamic Logic. New Trends and Applications. DALI 2017. Lecture Notes in Computer Science(), vol 10669. Springer, Cham. https://doi.org/10.1007/978-3-319-73579-5_3
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