Research
Axiomatic theories of parts, wholes, regions, and granularity, built for domains that no measurement carves at sharp boundaries, and an account of what it costs to impose precision on a vague configuration.
Approach
A geographic region, an infiltrating tumor, an ecosystem, and a cloud are described truly every day, although no measurement fixes the boundary of any of them. The question addressed here is what makes such descriptions correct, and the answers are given as axiomatic theories whose consequences can be checked.
In the current work, vagueness is treated as a property of the fit between a representation and its domain rather than as a defect of either, so that a theory eliminating it no longer describes the phenomenon at issue. Granularity is treated as a parameter of that fit: a judgment determinate at one resolution may be indeterminate at another, and a theory of judgment is expected to say which resolution it concerns. Precision is treated as something imposed on a vague configuration rather than found in it, and the cost of imposing it is quantified relative to a chosen dispositional specification.
- mereology
- granularity
- vagueness
- spatio-temporal reasoning
- qualitative calculi
- applied ontology
- quantum information
- interactive theorem proving
How the work developed
The work falls into six periods.
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1996 - 2002
Rough location and qualitative spatial reasoning
Doctoral work at the Technical University of Vienna, supervised by Andrew U. Frank and Thomas Eiter, on the location of an object relative to a partition of space that is coarser than the object itself. Rough sets, boundary-sensitive qualitative location, and approximate qualitative reasoning about space and time, much of it developed jointly with John G. Stell.
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2001 - 2008
Granular partitions
A general theory, with Barry Smith, of the partitions through which a domain is carved into cells: what a partition projects onto, when its cells are located, and how vagueness and approximation arise from the relation between a partition and the reality it is imposed upon. In this work granularity becomes an explicit parameter of the account.
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2004 - 2011
Mereology for bio-ontologies
Axiomatizations of parthood, componenthood, and containment for biomedical and anatomical ontologies, largely with Maureen Donnelly and Louis J. Goldberg, together with the time-dependent and qualitative spatial relations such ontologies require. Contributions to the RNA Ontology and to the normalization of medical ontologies under Basic Formal Ontology belong to the same period.
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2009 - 2017
Vagueness as a trade-off
Vague size predicates, vague distance predicates, and the demonstration that classification and delineation of geographic regions trade off against one another: a region can be classified sharply or delineated sharply, and requiring both at once generates the appearance of paradox.
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2016 - 2019
Physical entities and the move to machine checking
A formal ontology of space, time, and the entities and relations presupposed by classical mechanics, and a case study in the computational realization of such an ontology in Isabelle/HOL. Three vision papers of the same years asked whether the classificatory information of geographic regions is quantum information.
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2023 -
Quantized mereology and the stabilizer formalism
Mereological information is treated as quantum information in the 2023 article and in the article in press at the Journal of Artificial Intelligence Research: a crisp relation carries classical bits, a vague relation fewer, and vague relations are represented by qubits. A stabilizer-based algebraic framework over a real vector space was developed subsequently, in which correlated indeterminacy is representable and the cost of imposing precision is quantified.
Quantized mereology and the stabilizer formalism
A vague predicate does not merely fail to have a truth value. Which predicates are indeterminate together is itself structured, and the program represents that structure directly: correlated indeterminacy, phase information, interrogation, and a specified resolution cost are carried by one finite algebraic structure.
Consider two mutually exclusive classifications of a borderline patch in the transition zone between two ecoregions. Whether the patch falls under the one and whether it falls under the other are not independent questions: any sharpening that assigns it to the first assigns it away from the second. Such a dependence is what the program represents, and its determinate content is what the program measures.
In the algebraic framework an elementary system, a tuple of entities to which a set of predicates is jointly applied, is represented by a stabilizer state over a real vector space; only the algebra is used, and no claim is made that the entities are quantum systems. A determinate predicate is an eigenvalue commitment of that state, a gappy predicate is the absence of such a commitment, and correlated indeterminacy is a commitment to a joint observable without commitment to its factors. Determinacy, gappy indeterminacy, and correlation are then features of one algebraic object rather than three separate stipulations. Crisp mereology is recovered as the case in which all three mereological predicates receive determinate truth values, and the classical laws are recovered under the conditions stated for the strict regime of the liquid mereology paper. In the relational theory of judgment, current states are assigned to interrogator-system pairs, the elementary system itself bearing only its information capacity.
Results and work in preparation
- Three modes of determinacy. Determinacy, gappy indeterminacy, and correlated indeterminacy are distinguished. In correlated indeterminacy a determinate correlation is maintained between predicates whose individual truth values remain indeterminate. The three modes generate a finite classification of configurations; whether correlated indeterminacy is realized in a given domain is an empirical question left open.
- A cost of precision. The cost of imposing precision on a vague configuration is quantified relative to a chosen dispositional specification, and classical mereology appears as a limiting case that is stable only under the strict enforcement conditions of that specification.
- Revealing and disturbing questions. Commutation with the stabilizer sorts each question put to a configuration into those that reveal settled content and those that disturb it. The relational theory of judgment builds on this partition.
- In preparation. Distinctions among voidness, content, and identity for empty regions are investigated in manuscripts in preparation, one of which already has a public machine-checked companion.
- In preparation. A qualitative calculus of vague relations and its classical fragment are developed in manuscripts in preparation.
Papers in the series
Each public paper of the series is deposited at PhilSci-Archive under a citable eprint number. Further papers are in preparation and are described by topic.
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The algebraic framework
The anchor of the series: indeterminate predicates given an algebraic semantics, with classification, reasoning, and dispositional dynamics treated in one apparatus. A common algebraic foundation is provided for several subsequent papers, which cite it rather than restate it.
PhilSci-Archive 30569 -
Quantized mereology
Crisp mereological information is represented by bits and vague mereological information by qubits, expressed through set partitions rather than complex vector spaces. A quantized RCC5 formalism is developed, with the classical qualitative relations recovered as special cases. Forthcoming in the Journal of Artificial Intelligence Research.
PhilSci-Archive 30559 · arXiv -
Entanglement and measurement
The companion paper: entangled mereological information states, the probabilities of possible observations, and the effects of sequences of questions that cannot be asked at once are examined in a three-qubit framework.
PhilSci-Archive 30557 -
Real-valued foundations
The foundational paper: a real-valued Hilbert-space representation is proposed for relational judgments about macroscopic vagueness, on which determinate mereological facts about vague entities are established through acts of judgment, with a defense of the real-valued rather than complex structure.
PhilSci-Archive 30560 -
Mereological judgment
Mereological judgment is analyzed relationally, one interrogator-system pair at a time, an interrogation either revealing settled content or disturbing it. Endorsing and constitutive roles of judgment are distinguished and given Kantian readings, and a relational completion of the shared algebra is adopted as a postulate.
PhilSci-Archive 30864 -
Liquid mereology
Vagueness, information capacity, and the cost of precision for objects whose parts are not fixed. Strict and fluid enforcement regimes are distinguished, the cost of imposing precision is quantified, and the classical laws are recovered under stated conditions in the strict regime, with a formal realization of determinable-based ontological indeterminacy.
PhilSci-Archive 30967 -
Voidness and identity
Empty regions, content, and identity under indeterminacy, and determinate identity for objects with indeterminate extent, including the problem of the many. The machine-checked companion of the empty-region manuscript is already public.
in preparation -
Realization and composition
How vague relations compose across a network of entities: a qualitative calculus with correlated indeterminacy, and the classical fragment in which it collapses to the familiar calculus.
in preparation
Applications and planned directions
Applications and proposed extensions are considered in four domains.
Ecological classification and remote sensing
Ecosystem taxonomies, ecoregion hierarchies, and the integration of regional classification and delineation systems into national mapping products. In empirical work in preparation, whether indeterminacy between vegetation classes persists across raster resolutions is investigated, per object pair and across a ladder of scales, in the Amazon-Cerrado ecotone and three further forest-savanna transitions.
Intelligence analysis
Analytic judgments are made under ontological vagueness as well as evidential uncertainty, and the two are routinely conflated. In a framework presented at STIDS 2026, objective, relational information states are distinguished from the subsequent judgments of the analyst. The distinction is used to separate limitations of the information state from problems in the assessment.
Biomedical and upper-level ontologies
The mereogeometrical relations of the bio-ontology work were developed for anatomical and biomedical ontologies. Two papers are planned: a stabilizer encoding of the information that a realist upper-level ontology such as Basic Formal Ontology carries about the extensions of its universals, and a de re vagueness profile for such an ontology.
Learned world models
Systems that learn a model of an environment are committed, implicitly, to a claim about whether that environment has a state at all. The representation of indeterminate environments in learned world models is proposed as a future direction of the program.
Machine-checked ontology
Formal ontologies are commonly published as axiom sets whose consistency is asserted and whose consequences are derived by hand. Since the 2019 Isabelle/HOL case study, selected finite mathematical results of the papers are accompanied by Isabelle/HOL developments in which definitions, locale assumptions, and theorems are linked to the labeled environments of the manuscript by a correspondence table, so that a reader can see which claims are checked, which are not, and why. The adequacy of a formalization to the ontology it formalizes is a separate question, and it is recorded as such.
Two developments are public, each released with citation metadata:
- an Isabelle/HOL development accompanying the relational theory of mereological judgment, PhilSci-Archive 30864 (relational-judgment-isabelle);
- an Isabelle/HOL development of the empty-region lattice, the pattern characterization, partial content, consistency, content individuation, and the interaction bridge, released ahead of the manuscript it accompanies, which is in preparation (qm-empty-region-isabelle).
A further development, checking the finite algebraic claims of the liquid mereology paper, is documented in the supplement deposited with that paper.
What is checked is stated at the strength of the evidence and no higher. A development that discharges the static results of a paper is not represented as formalizing its dynamics, and results deliberately left outside the formalization are recorded as such.
Correspondence
Comments on the preprints are welcome, and corrections in particular. The intended readership is in the metaphysics of vagueness, qualitative spatial reasoning, and applied ontology.
bittner3@buffalo.edu · 109 Park Hall, University at Buffalo · ORCID 0000-0003-4204-1532