| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Concerned with formal languages, structures, signatures, and interpretations; includes domains, functions, relations, and symbol assignments. Excludes semantic content not expressible in first-order definability. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates entirely at the formal/logical scale: symbolic expressions, abstract structures (sets with relations/functions), and homomorphic mappings between them. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Structures (đ), domains ( |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Arity, interpretation of symbols, definability, satisfaction, isomorphism type, elementary equivalence, and preservation properties under embeddings. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Languages, signatures, term algebras, formulas, structures, morphisms (homomorphisms/embeddings), definable sets/functions, elementary diagrams, substructures. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Assignments, valuations, tuples from the domain, truth values of formulas under interpretations, definable element-sets, definable functions. |
| | Parameterization | How variables encode and represent the systemâs state. | Parameterization through substitution of variables with domain elements, interpretation of nonlogical symbols, and the satisfaction relation đ âš Ï(aÌ). |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Treat languages as fixed and finite; treat structures as set-theoretic; assume closure under substitution; ignore computational or cardinality constraints; assume exact symbolic interpretation. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Break down with higher-order semantics, infinitary languages, ultraproduct anomalies, category-theoretic semantics, or failures of definability or compactness assumptions. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Assumes classical first-order logic, Tarskian semantics, bivalent truth conditions, standard set-theoretic foundations, and stable interpretation of language symbols. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the fieldâs conceptual structure. | Assumes satisfaction is absolute under isomorphism, definability behaves predictably, diagrams encode structures faithfully, and embeddings preserve the essential logical structure. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Requires consistent signatures, coherent interpretation of symbols, and non-contradictory satisfaction assignments across structures. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires formulas, structures, embeddings, and interpretations to align; satisfaction invariant under isomorphism; substructure and diagram relations consistent with signature constraints. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Truth values of formulas in structures, definable sets/functions, satisfaction patterns, homomorphism behavior, embedding properties, isomorphism invariants. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Limits set by expressiveness of the language: first-order inability to distinguish elementarily equivalent models, undefinability of certain sets, compactness constraints, LöwenheimâSkolem bounds. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Logical units: arity, quantifier rank, formula complexity, cardinality of language, size of domains, degree of definability. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Logical tools: satisfaction relation (âš), diagrams, syntactic calculi, homomorphism tests, embedding tests, back-and-forth systems, EhrenfeuchtâFraĂŻssĂ© games, type spaces. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Definability via formulas; interpretation of terms, relations, and functions; elementarily definable sets; schema-based definitions; diagrammatic encodings. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Evaluating satisfaction đ âš Ï(aÌ); constructing diagrams; checking embeddings; performing EF-game rounds; computing types; verifying preservation under maps. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Constructing structures, substructures, and elementary extensions; forming reducts/expansions; evaluating definability; applying TarskiâVaught criteria. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Choosing representative tuples, definable subsets, types, or finite partial isomorphisms; selecting fragments of structures (finite subdiagrams). |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Formulas, types, satisfaction data, diagrams, elementary embeddings, EF-game outcomes, definable sets/functions, isomorphism invariants. |
| | Resolution | The granularity or precision with which data is captured. | Fineness of logical discrimination: quantifier rank, alternation depth, expressive power of language, complexity of formulas, precision of type classification. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Ensuring mappings preserve structures: verifying homomorphisms, embeddings, and elementary embeddings; calibrating definability by equivalence of formulas. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Logical error sources: misinterpreted signatures, incorrect substitution, failure of preservation, non-elementary embeddings, ambiguity in definability, compactness/pathology effects. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Satisfaction relation đ âš Ï(aÌ); preservation theorems; compactness; completeness; LöwenheimâSkolem effects; definability patterns. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Isomorphism type, automorphism groups, definable closure, elementary equivalence, quantifier-rank invariants, back-and-forth invariants, type spectra. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Logical consequence, definability mechanisms, preservation under embeddings, ultraproduct construction, type realization, diagram expansion. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Chains of embeddings, elementary chains, back-and-forth sequences, saturation pathways, construction of models via diagrams or Fraïssé limits. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domainâs structure (force, gene, equilibrium, field). | Structure, signature, language, formula, term, model, theory, type, elementary substructure, quantifier rank, definability, interpretation. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Model classes, theory classes (stable, unstable, simple, o-minimal), isomorphism classes, definable sets, saturation levels, algebraic vs. elementary closure. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Logical equivalences, satisfaction relations, definitions of embeddings/isomorphisms, compactness statements, ultraproduct constructions. |
| | Models | Structured representationsâmathematical, computational, or conceptualâused to predict and explain phenomena. | Structures interpreting a language; elementary extensions; reducts/expansions; saturated models; prime models; limit models; ultraproduct models. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Pure relational structures; pure sets with simple signatures; finite substructures; atomic diagrams; toy models for definability. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | First-order expressiveness boundaries; compactness-induced phenomena; non-definability regions; breakdown under higher-order or infinitary languages. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | First-order logic, Tarskian semantics, model-theoretic stability theory, classification theory, interpretability theory. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Links to algebra (groups, fields, modules), topology (Stone spaces), combinatorics (Fraïssé limits), computer science (automata, formal languages), and category theory. |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Varying languages, signatures, or axioms to test definability, preservation, elementary equivalence, or expressiveness. Constructing alternative structures to probe logical distinctions. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Studying models without manipulating them: examining satisfaction patterns, definable sets, type spaces, EF-games, or automorphism behavior. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Checking whether structures satisfy specific sentences, whether embeddings preserve formulas, whether two models are elementarily equivalent, or whether definability claims hold. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Reproducing satisfaction results across isomorphic structures; repeating definability tests across extensions, reducts, or ultraproducts; verifying EF-game outcomes. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Logical analogs: extracting invariants from EF-games, analyzing type frequencies, counting realizations of formulas, studying definability spectra. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing structures or theories by expressive power, quantifier complexity, definability strength, type behavior, saturation, or classification-theoretic profile (stable vs. unstable, simple vs. complex). |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Identifying failures of embeddings, misinterpreted signatures, incorrect substitutions, definability errors, compactness misapplications, or non-elementary embeddings. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Ensuring neutrality in language choice, avoiding overfitting via overly rich signatures, preventing selection bias in chosen substructures or types, controlling assumptions in diagrams. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Critical examination of definability claims; reviewing preservation theorems; evaluating construction methods (ultraproducts, diagrams); refining classification results. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Adjusting axioms, languages, or interpretations; revising diagrams; modifying theories to reflect newly discovered preservation failures or definability boundaries. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Full specification of signatures, languages, structures, diagrams, embeddings, EF-game parameters, and construction methods. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Intellectual honesty in definability claims; accurate reporting of preservation limits; clarity in assumptions; proper attribution of theorems and constructions. |