The Logic of Partitions: Introduction to the Dual of the Logic of Subsets

Ellerman, David

doi:10.2139/ssrn.1368796

Cited by 30 publications

(71 citation statements)

References 22 publications

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“…The existence of a finite countermodel for any non-tautology in subset logic is trivial but the corresponding finite model property for partition logic is also an open question. The existing correctness and completeness proofs for partition logic [9] use semantic tableaus (adapted to deal with partitions rather than propositions). But a Hilbert-style axiom system for partition tautologies together with a proof of completeness is not currently known (all to the author's knowledge).…”

Section: Discussionmentioning

confidence: 99%

“…The point was to show that the 15 functions, and thus all their further combinations, could be defined in terms of the four primitive operations of join, meet, implication and nand. 9 The fourteen non-zero operations occur in natural pairs: ⇒ and , ⇐ and , ≡ and ≡, ∨ and ∨, and ∧ and | in addition to σ and ¬σ , and τ and ¬τ . Except in the case of the join ∨ (and, of course, σ and τ ), the second operation in the pair is not the negation of the first.…”

Section: Distributing Interior Across Intersections Gives Partition Cnfmentioning

confidence: 99%

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An introduction to partition logic

Ellerman

2013

Logic Journal of IGPL

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Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (non-empty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality-which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence the idea arises of a dual logic of partitions. That dual logic is described here. Partition logic is at the same mathematical level as subset logic since models for both are constructed from (partitions on or subsets of) arbitrary unstructured sets with no ordering relations, compatibility or accessibility relations, or topologies on the sets. Just as Boole developed logical finite probability theory as a quantitative treatment of subset logic, applying the analogous mathematical steps to partition logic yields a logical notion of entropy so that information theory can be refounded on partition logic. But the biggest application is that when partition logic and the accompanying logical information theory are 'lifted' to complex vector spaces, then the mathematical framework of quantum mechanics (QM) is obtained. Partition logic models the indefiniteness of QM while subset logic models the definiteness of classical physics. Hence partition logic may provide the backstory so the old idea of 'objective indefiniteness' in QM can be fleshed out to a full interpretation of quantum mechanics. In that case, QM will be the 'killer application' of partition logic.

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Section: Discussionmentioning

confidence: 99%

Section: Distributing Interior Across Intersections Gives Partition Cnfmentioning

confidence: 99%

An introduction to partition logic

Ellerman

2013

Logic Journal of IGPL

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“…In partition logic on sets ( [4], [5]), the set partition operations (e.g., join, meet, and implication) on the partitions on a given universe set U can be represented as subset operations on certain subsets of U U , i.e., on certain binary relations on U . For a set partition = fB 1 ; :::; B m g on U , a distinction or dit of is an ordered pair (u; u 0 ) 2 U U of elements in distinct blocks of .…”

Section: Exploiting Duality In Quantum Partition Logicmentioning

confidence: 99%

“…The simplest is that the join of partitions which corresponds to the union of ditsets: for set partitions and on U , dit ( _ ) = dit ( ) [ dit ( ). For the meet and implication operations, we need to use the re ‡exive-symmetric-transitive closure operation on subsets of U U where for S U U , the RST-closure cl (S) is the equivalence relation that is the intersection of all the equivalence relations containing S. 4 Then the interior, int (S), is the complement of the closure of the complement, i.e., int (S) = cl (S c ) c (where () c is the set complement operation). Then the other operations on partition relations isomorphic to the partition operations are:…”

Section: Exploiting Duality In Quantum Partition Logicmentioning

confidence: 99%

“…It was then a theorem, not a de…nition as in most modern logic texts, that it was su¢ cient for validity to consider only the "propositional" special case (truth-table validity) where the universe was, in e¤ect, a one element set (with subsets symbolized as 0 and 1 or F and T ). One advantage in going "back to Boole" and considering the logic of subsets (instead of only the propositional special case) is that the concept of a subset has a (category-theoretic) dual in the concept of a quotient set, equivalence relation, or partition-and hence the development of the logic of partitions ( [4]; [5]). 1 "Subsets of a universe set"linearize to "subspaces of a vector space"and thus the usual quantum logic can also be seen as being about subspaces (e.g., the closed subspaces of a Hilbert space)-or the associated propositions about a vector being in the subspaces (just as the variables in the Boolean logic of subsets can also be interpreted in the usual way about propositions, e.g., that a generic element is in a subset).…”

Section: Introductionmentioning

confidence: 99%

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The quantum logic of direct-sumdecompositions: the dual to the quantum logic of subspaces

Ellerman

2017

Logic Journal of the IGPL

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Since the pioneering work of Birkho¤ and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the "quantum logic" of subspaces of a general vector space-which is then specialized to the closed subspaces of a Hilbert space. But there is a "dual" progression. The set notion of a partition (or quotient set or equivalence relation) is dual (in a categorytheoretic sense) to the notion of a subset. Hence the Boolean logic of subsets has a dual logic of partitions. Then the dual progression is from that logic of set partitions to the quantum logic of direct-sum decompositions (i.e., the vector space version of a set partition) of a general vector space-which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the quantum logic of direct-sum decompositions to express measurement by any self-adjoint operators. The quantum logic of direct-sum decompositions is dual to the usual quantum logic of subspaces in the same sense that the logic of partitions is dual to the usual Boolean logic of subsets.

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Deductive and Analogical Reasoning on a Semantically Embedded Knowledge Graph

Summers-Stay

2017

Artificial General Intelligence

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Representing knowledge as high-dimensional vectors in a continuous semantic vector space can help overcome the brittleness and incompleteness of traditional knowledge bases. We present a method for performing deductive reasoning directly in such a vector space, combining analogy, association, and deduction in a straightforward way at each step in a chain of reasoning, drawing on knowledge from diverse sources and ontologies.

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The Logic of Partitions: Introduction to the Dual of the Logic of Subsets

Cited by 30 publications

References 22 publications

An introduction to partition logic

An introduction to partition logic

The quantum logic of direct-sumdecompositions: the dual to the quantum logic of subspaces

Deductive and Analogical Reasoning on a Semantically Embedded Knowledge Graph

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