Category theory gives us the duality between subsets and quotient sets (= partitions = equivalence relations). Certain ‘classical’ theories are based on the subset or subobject side of the duality: Boolean logic of subsets (usually presented as “propositional logic”) and the Birkhoff-von-Neumann quantum logic of subspaces (of a separable Hilbert space). Hence there are two dual theories: 1) Since partitions are dual to subsets, there is a dual logic of partitions, and 2) for vector spaces, direct-sum decompositions are the partitional dual to subspaces, so there is a quantum logic of direct-sum decompositions.
The quantitative version of Boolean subset logic is finite discrete probability theory, and the quantitative version of partition logic is 3) the logical theory of information—where the Shannon notions of simple, compound, conditional, and mutual entropies are derived by a uniform requantifying transformation from the corresponding natural logical notions of entropy. And the standard quantum information theory notion of von Neumann entropy can be arrived at from Shannon entropy by substituting density matrices for probability distributions and traces for sums. Similarly, the new 4) quantum logical entropy is arrived at from the notion of logical entropy by the same substitutions. Those are the four new partition-related theories that will be sketched in the talk.
References (downloadable from www.ellerman.org) :
1) Ellerman, David. 2010. “The Logic of Partitions: Introduction to the Dual of the Logic of Subsets.” Review of Symbolic Logic 3 (2 June): 287–350., or Ellerman, David. 2014. “An Introduction to Partition Logic.” Logic Journal of the IGPL 22 (1): 94–125. https://doi.org/10.1093/jigpal/jzt036. See also Brendan Fong’s MIT Category Theory Seminar talk on partition logic at: https://www.youtube.com/watch?v=5I7v9mvOC2E
2) Ellerman, David. 2018. “The Quantum Logic of Direct-Sum Decompositions: The Dual to the Quantum Logic of Subspaces.” Logic Journal of the IGPL 26 (1 (January)): 1–13. https://doi.org/10.1093/jigpal/jzx026.
3) Ellerman, David. 2017. “Logical Information Theory: New Foundations for Information Theory.” Logic Journal of the IGPL 25 (5 Oct.): 806–35. https://doi.org/10.1093/jigpal/jzx022.
4) Ellerman, David. 2018. “Logical Entropy: Introduction to Classical and Quantum Logical Information Theory.” Entropy 20 (9): Article ID 679. https://doi.org/10.3390/e20090679.