Analogy of Being: A Formal Perspective

• Analogy of Being is a framework describing how entities share structural commonalities across diverse ontological domains using formal models.
• It employs methods such as high-dimensional geometry, logical mapping, Bayesian inference, deep learning, and category theory to model analogical relations.
• The approach enables practical applications in automated analogy completion, knowledge transfer, and narrative understanding, bridging theory and computational practice.

The “Analogy of Being” denotes the notion that entities across different domains or ontological strata share commonalities in their mode of existence or structure, though these similarities are neither identical nor reducible to mere equivalence. In contemporary research, the concept is operationalized using diverse formal frameworks, including geometric models of meaning, logical-semantic mappings, Bayesian inference, deep learning architectures, and category theory. These methods provide rigorous means for capturing, analyzing, and inferring analogical relations—not only in language and cognition but also as a means for constructing metaphysical and philosophical comparisons regarding existence or ‘being.’

1. Geometric and Statistical Approaches to Analogical Structure

A geometric approach to word and concept representation models analogies as spatial relationships in high-dimensional vector spaces derived from large corpora via statistical methods such as pointwise mutual information (PMI). Words are encoded as vectors in spaces whose axes correspond to interpretable semantic properties. The classic analogy task (e.g., $$\overrightarrow{\text{king}} - \overrightarrow{\text{man}} + \overrightarrow{\text{woman}} \approx \overrightarrow{\text{queen}}$$) manifests as a parallelogram in the semantic space because the difference vectors encode consistent semantic relations.

These models address limitations of static representations by dynamically projecting contextualized subspaces optimized for specific analogy tasks. A high-dimensional PMI matrix

$$M_{w,c} = \log_2 \left(\frac{n_{w,c} \times W}{n_w \times (n_c + a)} + 1\right)$$

is constructed, and dimensions maximizing the alignment structure for analogical pairs $(A : B :: C : D)$ are selected by minimizing

$$\sum_{c} ((M_{A,c} - M_{B,c}) - (M_{C,c} - M_{D,c}))^2.$$

In practice, dynamic projections yield robust analogy completions across diverse tasks, suggesting that both simple and complex analogical relationships manifest as geometric structures—supporting a technical analogy to the “Analogy of Being.” Here, the totality of the base space corresponds to the richness of existence, while subspaces capture context-specific modes of “being” (McGregor et al., 2016).

2. Formal Logic and Preferential Structure Models

A logic of analogy formalizes the transfer of properties between domains by explicit mappings and evaluation of their success according to well-defined criteria. Domains $S$ (source) and $T$ (target) are collections of elements, relations, and functions. A one-to-one, type-preserving mapping $a: L_\alpha \rightarrow L$ supports analogical transfer; for any formula $\varphi$ in the source language, $a(\varphi)$ is its analog in the target.

Truth value preservation is analyzed by partitioning formulas into:

• $a^+ = \{\varphi : v(\varphi) = v(a(\varphi))\}$ (positive support)
• $a^- = \{\varphi : v(\varphi) \neq v(a(\varphi))\}$ (negative support)
• $a^? = \{\varphi : v(\varphi)$ known, $v(a(\varphi))$ unknown$\}$ (uncertain cases)

A set $A$ of candidate mappings is partially ordered with $<$ (preference), reminiscent of the ordering of possible worlds in counterfactual semantics. Analogical inference accepts $\varphi$ as “analogically true” iff it holds in all $<$-minimal mappings:

$$A \models \varphi \quad \text{iff} \quad \varphi \text{ holds in every } <\text{-best mapping in } A.$$

Properties such as smoothness and rankedness of $<$ guarantee robust selection of optimal analogical mappings (Schlechta, 2019). This architecture models gradable similarity and accommodates gradations in “being,” extending the philosophical Analogy of Being with formal tools for preference-based analogical inference.

3. Bayesian and Predictive Coding Perspectives

From a Bayesian perspective, analogical reasoning is mathematically equivalent to probabilistic inference. The core process is formalized as hypothesis evaluation:

$$P(\text{Mapping} \mid \text{Data}) = \frac{P(\text{Data} \mid \text{Mapping}) \cdot P(\text{Mapping})}{P(\text{Data})}$$

Here, analogies are hypotheses about the similarity of structures that explain observed features across domains. Within graded, embodied, cybernetically organized agents, the structure mapping (as in cognitive analogy) and probabilistic updating (as in Bayesian inference) are unified. Cortical architectures facilitate belief propagation through generative models, implementing both analogical and inferential reasoning via action-perception loops.

Predictive coding operationalizes this theory biologically: the mind minimizes prediction error by matching current patterns with stored analogies and updating internal models to align with sensory experience. Empirical priors, acquired through embodied action, reduce the hypothesis space for analogical inference, enriching the space of possible mappings that support robust analogies of “being” (Safron, 2019).

4. Neural, Connectionist, and Deep Learning Models

Neural architectures for analogy operationalize structural alignment as networked processes. For instance, recurrent networks (Analogator) learn to perform analogy by example, uniting low-level perceptual segmentation (figure-ground separation) with high-level relational mapping through distributed hidden-layer activations. Such networks demonstrate generalization across symbolic, spatial, and perceptual domains, handling novel analogy problems despite the absence of explicit, hard-coded rules.

More abstract deep learning architectures (e.g., Analogical Matching Networks) employ dual embeddings (label graphs for abstract structure, signature graphs for identity preservation), structure-aware LSTMs, and Transformer-based selection to discover and enforce systematic structural correspondences. The process involves iterative attention-based selection, maximization of systematicity, and projection of candidate inferences to complete analogies in line with cognitive constraints. These neural models quantitatively capture the degree to which entities “share being” by mapping deep structure and enabling automatic inference about ontological commonality (Blank, 2020, Crouse et al., 2020).

5. Category-Theoretic Formalization of Analogical Relations

Category theory provides a universal language for expressing analogical structure across scientific and philosophical domains. In this framework, knowledge domains are represented as categories ($\mathcal{S}$, $\mathcal{H}$), whose objects and morphisms encode entities and relations. Analogies are structure-preserving functors $F: \mathcal{S} \rightarrow \mathcal{H}$ satisfying:

$$F(1_A) = 1_{F(A)}, \qquad F(f \circ g) = F(f) \circ F(g)$$

For example, mapping between the solar system and hydrogen atom involves functors that align “sun” to “nucleus,” “planet” to “electron,” and their relational structures accordingly.

Pullbacks identify the core (shared) substructure:

$\begin{tikzcd} P \arrow[r, "p_2"] \arrow[d, "p_1"'] & Y \arrow[d, "g"] \ X \arrow[r, "f"'] & Z \end{tikzcd}$

Pushouts blend domain structures into unified frameworks:

$\begin{tikzcd} Z \arrow[r, "i"] \arrow[d, "j"'] & Y \arrow[d, "k"] \ X \arrow[r, "l"'] & P \end{tikzcd}$

The approach extends to metaphysical analogies, where various categories model different conceptions of Being, functors express systematic correspondences, pullbacks abstract shared ontological cores, and pushouts generate blended frameworks of existence. These constructions preserve relational structure and provide a rigorous method for comparing, blending, and analyzing both scientific and philosophical analogies of being (Ott et al., 26 May 2025).

6. Taxonomies of Analogical Dimensions in Narrative and Cognitive Science

Taxonomic frameworks delineate the diversity of analogical similarity:

• Shallow Attribute Analogy (SAA): observable, physical properties
• Deep Attribute Analogy (DAA): abstract, inferred qualities
• Relational Analogy (RA): correspondence between relationships
• Event Analogy (EA): mapping of similar events
• Structural Analogy (SA): alignment of causally related events
• Moral/Purpose Analogy (MP): alignment of underlying lessons or purposes

Collections of narrative units are annotated along these dimensions, enabling empirical paper of how surface features, relations, event structure, and purpose collectively comprise the “being” of stories and their analogical connections. Benchmarking AI systems (e.g., LLMs, neuro-symbolic reasoners) on these dimensions exposes the challenges in scaling analogical reasoning to higher-order, structural, and moral analogies—key aspects of the analogy of being in narrative contexts (Nagarajah et al., 2022).

7. Applications, Implications, and Future Directions

The formal and computational methods described are applicable to automated analogy completion, knowledge base construction, metaphor modeling, transfer learning between scientific models (e.g., planetary and atomic systems), and narrative understanding. Dynamic projections, category-theoretic blends, probabilistic mappings, and deep neural alignments each provide mechanisms for the discovery and operationalization of analogical “being” in both symbolic and numeric representations.

Challenges remain in scaling analogical reasoning to domains requiring abstraction, hierarchy, or cross-domain transfer, especially where the analogy of being requires synthesizing structural commonality and preserving distinctness. Open research areas include the unification of cognitive constraints with deep learning, the development of rigorous annotation and evaluation protocols for analogical reasoning, and the extension of blending operations to philosophical and metaphysical analogies.

In sum, contemporary research grounds the “Analogy of Being” in explicit mathematical, logical, probabilistic, and architectural frameworks, revealing that structured relationships, gradations of similarity, and context-sensitive mappings are central to formalizing and understanding the analogical ties that underpin existence, cognition, and meaning across domains.