Abstract
A formal model of abductive inference is provided in which abduction is conceived as expansive and contractive movements through a topological space of theoretical and practical commitments. A pair of presheaves over the (Heyting algebra) space of commitments corresponds to communities sharing commitments on the one hand and possible obstructions to commitments on the other. In this framework, abductive inference is modeled by the dynamics of redistributed communities of commitment made in response to obstructive encounters. This semantic-pragmatic model shows how elementary category theory tools can be used to formalize abductive inference while hewing close to ordinary intuitions about collective agency and reasoning.
Space is the place.
Sun Ra, cosmic jazz maestro
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Notes
- 1.
We omit the corresponding formal characterizations for simplicity of presentation.
- 2.
From a logical point of view, this assumption amounts to conceiving of the commitment-space as a Lindenbaum algebra.
- 3.
Formally, a Heyting algebra is defined as follows [6]: First, we define a bounded lattice as a partially ordered set \(\mathcal {A}\) in which for every pair \(a, b \in A = \text{ Ob }(\mathcal {A})\) there exists a supremum or least upper bound \(a \vee b\) and an infimum or greatest lower bound \(a \wedge b\) and there exists a top and a bottom, 1 and 0, respectively, such that for every \(a \in A\), \(a \preceq 1\) and \(0 \preceq a\). In categorical terms, the supremum of a and b is their coproduct \(a + b\), while the infimum is \(a \times b\), and the bottom is an initial object such that for every object a, there exists a unique morphism \(0 \rightarrow a\). Analogously, the top is a terminal object such that there exists a unique \(a \rightarrow 1\) for every object a. A Heyting algebra is then a bounded lattice \(\mathcal {A}\) with two functors \(\wedge : A \times A \rightarrow A\) and \(\Rightarrow : A \times A \rightarrow A\), left and right adjoints, respectively. The interpretation is that given \(c, a, b \in A\), \(x \wedge a \preceq b\) is equivalent to \(x \preceq a \Rightarrow b\). In logical terms, \(x\wedge a\) is a conjunction between x and a, while \(a \Rightarrow b\) is an implication with antecedent a and consequent b. In categorical terms \(a \Rightarrow b\) is the exponential \(b^a\).
- 4.
It should be noted that this particular interpretation serves a merely heuristic purpose. The structures and dynamics discussed in the following sections do not depend upon any particular interpretation of the commitments in the Heyting algebra presented, only upon the structure of the algebra (category) itself.
- 5.
\(\mathbf {Sets_{\subseteq }}\) notates the category whose objects are sets and whose arrows or morphisms are all inclusion functions between sets. This category is a faithful subcategory of the category Set of sets and functions.
- 6.
Here, as is standard, \(\lnot c\) is shorthand for \(c \Rightarrow \bot \).
- 7.
Of significant interest, but beyond the reach of the present exposition, is the system of controlled variation among such concrete timeslices. This system may be formalized in terms of the category of natural transformations among the relevant pairs of \(\mathbf {OB}\) and \(\mathbf {COM}\) functors.
- 8.
The ancient Greek concept of \(\kappa \iota \nu \eta \sigma \iota \sigma \) would hold resources here for understanding and expressing the direct connection between movement and change, even to the point of identifying them.
- 9.
We omit the technical details for reasons of space.
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Gangle, R., Caterina, G., Tohmé, F. (2021). Abductive Spaces: Modeling Concept Framework Revision with Category Theory. In: Shook, J.R., Paavola, S. (eds) Abduction in Cognition and Action. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 59. Springer, Cham. https://doi.org/10.1007/978-3-030-61773-8_3
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