Theoretical Equivalence in Science

• Theoretical Equivalence is a framework that determines when different formulations of a theory, despite varying in structure or interpretation, yield the same empirical outcomes.
• It encompasses rigorous criteria including empirical, interpretational, definitional, Morita, and categorical equivalence to assess theoretical sameness.
• Its applicability ranges from reconciling quantum mechanics and gravity formulations to advancing logical foundations and computational models through translation-based analysis.

Theoretical equivalence is a central concept in the philosophy of science and mathematical physics, delineating when two theories can be considered "the same" in an appropriate sense, despite possibly differing in formulation, interpretation, or presentation. It serves as both a formal and interpretative tool, mediating debates about scientific realism, underdetermination, and the nature of representation in physical and computational systems. The literature distinguishes multiple precise criteria—empirical equivalence, interpretational equivalence, definitional equivalence, Morita equivalence, categorical equivalence, and others—each designed to capture different aspects of what it means for theories to be "equivalent" under varying theoretical, logical, and practical constraints (Weatherall, 2018, Barrett et al., 2015, Haro, 2019, Meadows, 3 Nov 2025, Butterfield, 2018).

1. Formal Notions and Criteria

Theoretical equivalence encompasses several rigorously defined relations:

• Empirical Equivalence: Two theories $T_1$ and $T_2$ are empirically equivalent if for every experimental or observational procedure $o$ in a shared domain $\mathcal{O}$, they yield the same probability distribution over measurement outcomes: $P_{T_1}(o) = P_{T_2}(o)$. Although necessary for stronger equivalence, it is insufficient when theories differ ontologically or semantically (e.g., Bohmian vs. von Neumann–Dirac quantum mechanics) (Weatherall, 2018).
• Interpretational Equivalence: Beyond empirical data, theories are interpretationally equivalent if there exists a shared or mutually matching assignment of physical ontology, laws, and semantics such that all claims about reality coincide. This is a stringent and rarely formally codifiable requirement; significant debates persist regarding its applicability to physically distinct but empirically indistinguishable theories (Weatherall, 2018, Haro, 2019, Butterfield, 2018).
• Definitional Equivalence: In first-order logic, $T_1$ and $T_2$ are definitionally equivalent if there exist definitional extensions in a joint signature such that the theories become logically equivalent—every formula in the extended language is provably equivalent in both extensions (Barrett et al., 2015, Meadows, 3 Nov 2025). This relation ensures full back-and-forth translatability, but only within the constraints of explicit definitions.
• Morita Equivalence: As a generalization, Morita equivalence allows new sorts (domains of quantification) to be introduced or eliminated via definitional extensions, provided that inter-translatability is maintained up to logical equivalence and isomorphism of models (Barrett et al., 2015).
• Categorical Equivalence: Theories $T_1$ and $T_2$ are categorically equivalent if their categories of models ($\mathrm{Mod}(T_1)$ and $\mathrm{Mod}(T_2)$) are equivalent (i.e., there exist full, faithful, and essentially surjective functors and natural isomorphisms between them) (Barrett et al., 2015, Weatherall, 2018, Weatherall, 2019). This criterion supports "equivalence up to isomorphism" and is robust to formal changes, but may be overly coarse in some contexts.
• Bi-interpretability and Extensions: Recent work has extended these frameworks to higher-order, category-theoretic, and set-theoretic contexts, providing translation-based accounts applicable beyond first-order logic (Meadows, 3 Nov 2025).

Hierarchy of relations:

Notion	Restrictiveness	Interdefinability
Logical equivalence	Most restrictive	Only within identical signatures
Definitional equivalence	Less restrictive	Allows definitional extensions
Morita equivalence	More general	Allows new sorts
Categorical equivalence	Most general	Equivalence of categories

Each arrow in the hierarchy is, in general, not reversible; categorical equivalence strictly subsumes Morita but may admit theories lacking meaningful intertranslatability (Barrett et al., 2015).

2. Interpretational and Empirical Constraints

The philosophical and physical meaning of theoretical equivalence is deeply sensitive to interpretation:

• Empirical Content Is Not Sufficient: Two theories may agree on all observable predictions yet differ fundamentally in their depiction of physical ontology (e.g., structure of spacetime, existence of hidden variables) (Weatherall, 2018, Butterfield, 2018).
• Empirical Content Is Not Always Exhausted by Dynamics: Subtle distinctions, such as the treatment of boundary conditions and boundary terms, can render a pair of theories inequivalent with respect to empirical phenomena that arise solely in the presence of boundaries, despite sharing identical equations of motion (Wolf et al., 2023).
• Role of Dualities: In high-energy and gravitational physics, dualities—e.g., AdS/CFT, S/T-duality—often pair different theories via explicit isomorphisms at the level of states, observables, and correlators. Whether these dualities count as theoretical equivalence depends on whether the formal mappings extend to interpretations, especially regarding ontology and physical structure (Weatherall, 2018, Haro, 2017, Butterfield, 2018, Haro, 2019).
• Empirical-Content-Preserving Categorical Equivalence: For categorical equivalence to underwrite theoretical equivalence, the relevant functors must commute with empirical-content functors, i.e., model-to-phenomenon assignments must be preserved. Failure to meet this constraint (e.g., as in classical electromagnetic duality) demonstrates the potential gap between formal and physical equivalence (Weatherall, 2019).

3. Applications and Case Studies

The elasticity and subtlety of theoretical equivalence are illustrated across diverse domains:

• Physics:
  • Electromagnetism: The Faraday tensor and vector-potential formulations are only equivalent when boundary data are properly incorporated; otherwise, distinctions in empirical predictions and model structure persist (Wolf et al., 2023, Weatherall, 2019, Haro, 2019).
  • Gravity: The Einstein–Hilbert and teleparallel actions are theoretically equivalent only when both dynamical and boundary-possible models are matched; naive equivalence fails for spacetimes with boundaries unless careful attention is paid to boundary terms (Wolf et al., 2023).
  • Quantum Theories: Schrödinger’s wave mechanics and Heisenberg’s matrix mechanics exhibit definitional equivalence at the level of formal quantum mechanics; Bohmian and von Neumann–Dirac quantum mechanics are argued to be empirically equivalent but ontologically distinct (Weatherall, 2018).
  • Dark Energy Models: Dynamical dark energy (DDE), interacting dark energy (IDE), and running vacuum (RV) models are background-equivalent—parameterizations of the same expansion history via algebraic transformations—though only perturbative/structure-growth analyses can distinguish them (Tamayo et al., 18 Feb 2025).
• Logic Programming and Computation:
  • Strong, Uniform, and Parameterized Equivalence: In answer-set programming, strong equivalence (preservation under arbitrary extensions), uniform equivalence (preservation under addition of facts), and $H,B$-parameterized equivalences admit semantic characterizations via SE- (strong equivalence), UE- (uniform equivalence), and more general $(H,B)$-models, controlling how substitutable program fragments are under various contexts (0712.0948) [0701095].
  • Fairness in Concurrent Systems: Only certain process equivalences (failure-trace, ready-trace, and finer) guarantee preservation of strongly/weakly fair runs and thus liveness properties under fairness assumptions (Prehn et al., 2017).
  • Equivalence of Representations and Metrics: In computer vision, "equivalence" of image representations is quantifiable as invertible transformations between feature spaces; equivalence of trade-off curves (precision–recall, Lorenz, ROC, Rényi-frontier) for assessing statistical proximity reveals underlying bivariate frontiers, indicating a unification of multiple evaluation diagnostics (Lenc et al., 2014, Siry et al., 2020).
• Mathematical Foundations:
  • Univalent Foundations: Equivalence principles in univalent type theory require that well-formed properties be invariant under the appropriate notion of equivalence (isomorphism, equivalence of categories), a constraint not enforced by set-theoretic foundations without additional axioms (e.g., univalence axiom) (Ahrens et al., 2022).

4. Theoretical Equivalence and Duality

Dualities in modern physics—precise correspondences between theories with apparently different formal structure or physical interpretation—are a key testing ground for the adequacy of equivalence criteria:

• Isomorphic Cores and Ontology: Dualities often induce isomorphisms between restricted “core” model-triples (states, observables, dynamics) but may fail to align extended or "specific" structures (gauge choices, coordinate systems, or auxiliary fields) unless interpretation maps are appropriately specified (Haro, 2017, Haro, 2019).
• Limits of Formal Equivalence: It is possible for duals to be formally isomorphic while disagreeing on key physical features (e.g., spacetime dimension, physical radius, or symmetry breaking), indicating that formal structure alone is not sufficient to establish theoretical, let alone physical, equivalence (Butterfield, 2018).
• Interpretative Work and Physical Equivalence: The alignment of interpretations—whether through external or internal interpretative maps—is necessary to elevate duality to physical equivalence (i.e., "making the same claims about the same world") (Haro, 2019, Haro, 2017). De Haro's schema and subsequent variants clarify that unextendability and shared internal interpretation are essential.

5. Extensions Beyond First-Order Logic

Not all theories relevant in mathematical physics or logic are naturally presented as first-order systems. Recent work has:

• Translation-Based Notions: Extended the theory of definitional equivalence and bi-interpretability to non-first-order, category-theoretic, or set-theoretic settings, providing flexible frameworks for “found in translation” criteria for equivalence (Meadows, 3 Nov 2025).
• Automorphism-Based Distinctions: Shown that structural inequivalence can often be detected through invariants such as automorphism groups, even when categorical equivalence or bi-interpretability appears to hold under more liberal definitions (Meadows, 3 Nov 2025).

6. Contemporary Challenges and Pragmatic Considerations

No fully general notion of theoretical equivalence has settled all foundational questions; rather, context and domain inform which criteria are appropriate:

• Pluralist and Pragmatic Perspectives: Dynamic practice in physics and computer science often leads to situations where multiple, non-unique action principles, models, or formulations are used interchangeably. Pluralist approaches advocate equivalence only where translations or interpretations are fully spelled out for the relevant empirical, boundary, or computational structures (Wolf et al., 2023).
• Interpretational Relativity: Even the most precise formal equivalences may be relativized to interpretative choices, boundary structures, or other extra-formal data, highlighting that theoretical equivalence is as much a semantic as a syntactic or structural matter (Weatherall, 2019, Butterfield, 2018).
• Future Directions: Ongoing research investigates the role of generalized definitions (e.g., in higher categories, homotopy type theory), as well as computational and constructive foundations that operationalize inter-definability beyond classical model theory (Ahrens et al., 2022, Meadows, 3 Nov 2025).

7. Summary Table: Principal Equivalence Relations

Notion	Description	Key References
Empirical	Same predictions for all observables/experiments	(Weatherall, 2018)
Interpretational	Coinciding ontologies, laws, semantics	(Weatherall, 2018, Haro, 2019, Butterfield, 2018)
Definitional	Logically equivalent after recursive definitions in extensions	(Barrett et al., 2015, Meadows, 3 Nov 2025)
Morita	As above, but with new sorts allowed	(Barrett et al., 2015)
Categorical	Equivalence of categories of models (full, faithful, essentially surjective functors)	(Barrett et al., 2015, Weatherall, 2018, Weatherall, 2019)
Bi-interpretability	Mutual interpretations with definable isomorphism back-and-forth	(Meadows, 3 Nov 2025)
Internal/Core Isomorphism	Dualities between isomorphic “bare” model-triples plus shared internal interpretation	(Haro, 2019, Haro, 2017)

• (Weatherall, 2018) "Theoretical Equivalence in Physics"
• (Barrett et al., 2015) "Morita Equivalence"
• (Weatherall, 2019) "Equivalence and Duality in Electromagnetism"
• (Wolf et al., 2023) "Respecting Boundaries: Theoretical Equivalence and Structure Beyond Dynamics"
• (Haro, 2019) "Theoretical Equivalence and Duality"
• (Haro, 2017) "Spacetime and Physical Equivalence"
• (Meadows, 3 Nov 2025) "Found in Translation: at the limits of the Hudetz program"
• (Butterfield, 2018) "On Dualities and Equivalences Between Physical Theories"
• (Ahrens et al., 2022) "Univalent foundations and the equivalence principle"
• (0712.0948) "A Common View on Strong, Uniform, and Other Notions of Equivalence in Answer-Set Programming"
• (Lenc et al., 2014) "Understanding image representations by measuring their equivariance and equivalence"
• (Siry et al., 2020) "On the Theoretical Equivalence of Several Trade-Off Curves Assessing Statistical Proximity"
• (Prehn et al., 2017) "Keep it Fair: Equivalences"
• (Tamayo et al., 18 Feb 2025) "Equivalence of Dark Energy Models: A Theoretical and Bayesian Perspective"

The ongoing refinement of equivalence notions continues to drive both foundational clarity and rigorous practice in logic, physics, and computation.