| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Concerned with the satisfaction relation (⊨), truth in structures, definability of sets/functions, and expressibility of formulas; excludes semantics not reducible to first-order or the governing logic. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates at the formal/logical scale: domain elements, variable assignments, formulas, quantifier structure, definable subsets, and interpretations inside mathematical structures. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Structures (𝔐), domains ( |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Definability status, truth values under assignments, closure under definable operations, arity, quantifier complexity, stability under elementary equivalence. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Formulas, terms, definable sets, definable relations, definable functions, satisfaction instances, types, definability classes (first-order, quantifier-free, etc.). |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Variable assignments, tuples from the domain, truth conditions for formulas, definability predicates, type realizations. |
| | Parameterization | How variables encode and represent the system’s state. | Encoding system state through assignments, interpretations of symbols, substitution of tuples, and definable-characterization of sets or relations. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Treat languages as fixed; assume perfect definability checks; idealize variable assignments; treat satisfaction as exact; assume closure under substitution and definability. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Break down under higher-order logic, infinitary languages, ambiguous semantics, non-standard models, or definability gaps caused by compactness/expressiveness limits. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Assumes first-order logic (or chosen base logic), Tarskian semantics, bivalence, stable truth conditions, and well-formed formulas with clear interpretation. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes absoluteness of satisfaction across isomorphism, predictable definability hierarchy, coherent substitution behavior, and stable semantic grounding for formulas. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Requires internally coherent interpretations, non-contradictory definability claims, and compatibility of satisfaction across substructures and expansions. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires formulas, assignments, structures, and definability predicates to align; satisfaction must be invariant under isomorphism; definability behavior must integrate with model-theoretic assumptions. |
| 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 under assignments, definable sets/functions, failure or success of definability, quantifier-elimination behavior, type realizations, preservation under embeddings. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Expressive boundaries of the logic: inability to distinguish elementarily equivalent structures, undefinability of sets, quantifier-rank thresholds, compactness constraints, Skolem limitations. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Logical units: quantifier rank, arity, formula length, domain cardinality, complexity of definability, size of signature. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Tools such as satisfaction relation (⊨), syntactic evaluation, definability tests, Diagrams, EF-games, type spaces, Skolem functions, quantifier-elimination procedures. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Explicit definability via formulas; interpretation-based definitions; definable closure; elementary diagram definitions; quantifier-free vs. first-order definability tests. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Evaluating satisfaction 𝔐 ⊨ φ(ā); constructing definable sets; computing closures; running EF-game rounds; checking equivalence of formulas; performing quantifier elimination. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Constructing models, reducts/expansions, elementary substructures; extracting definability spectra; forming diagrams; probing definability via tuples and assignments. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Selecting representative tuples, types, definable subsets, finite fragments of structures, partial isomorphisms, or definability witnesses. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Formulas, variable assignments, definable sets/functions, type distributions, EF-game outcomes, satisfaction tables, diagram fragments. |
| | Resolution | The granularity or precision with which data is captured. | Logical discrimination power: quantifier rank, alternation depth, expressive strength, type granularity, fineness of definability hierarchy. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Ensuring correctness of definability claims; verifying satisfaction consistency; checking embeddings; calibrating interpretations across isomorphic structures. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Sources of logical error: misinterpreted signatures, incorrect substitutions, non-elementary embeddings, definability illusions, compactness-induced anomalies, Skolem paradox phenomena. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Tarskian satisfaction; definability criteria; preservation theorems; compactness effects; monotonicity of definability; quantifier-elimination behavior; type realization properties. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Truth under isomorphism, definable-closure invariants, type invariants, quantifier-rank invariants, elementary equivalence, EF-game invariants, stability of definability across expansions/reducts. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | How formulas determine truth in structures; how definability arises from syntactic form; mechanisms of preservation under embeddings; effects of quantifiers on definability; ultraproduct mechanisms. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Sequences of embeddings, elementary chains, quantifier-elimination sequences, definability refinement steps, EF back-and-forth strategies, type-construction pathways. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Satisfaction, definability, interpretation, type, quantifier rank, elementary substructure, definable closure, Skolem function, reducibility, reduct/expansion, expressiveness. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Definability classes (quantifier-free, existential, first-order), type classes, ranks of formulas, definability hierarchies, theory classes (stable, simple, o-minimal), preservation categories. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Logical equivalences, satisfaction conditions 𝔐 ⊨ φ(ā), formal definability conditions, diagrammatic constraints, quantifier-elimination identities, Skolemization transformations. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Structures interpreting a language, definable-set structures, reducts/expansions, Skolemized structures, saturated models relative to definability, models witnessing definability failures. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Pure relational structures, simple signatures, quantifier-free frameworks, finite variable fragments, toy models for definability boundaries. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Expressiveness limits of first-order logic, compactness-driven phenomena, undefinability of well-ordering, quantifier-rank thresholds, behavior under infinitary or higher-order logics. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | First-order semantics, model-theoretic definability theory, classification theory, type theory, quantifier-elimination frameworks, semantics–syntax correspondence. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Connections to algebra (definable groups/fields), analysis (definable sets in o-minimal structures), computer science (descriptive complexity), topology (Stone spaces), logic foundations. |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Manipulating formulas, quantifier complexity, signature richness, or parameter sets to test definability boundaries, preservation behavior, or satisfaction under varying assignments. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing satisfaction patterns, definability behavior, type distributions, EF-game outcomes, and invariance across isomorphic or elementarily equivalent structures without direct manipulation. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing whether structures satisfy certain formulas; checking definability claims; verifying quantifier-elimination success; probing equivalence of formulas or types. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Reproducing satisfaction results across different embeddings, isomorphic models, reducts, expansions, ultraproducts, or varying presentations of the same theory. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Logical analogues: identifying definability frequencies, analyzing type multiplicities, counting realizations of formulas, estimating definability complexity across models. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing definability power, expressive strength, quantifier-elimination performance, type spectra, and preservation behavior across different theories or structures. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Identifying incorrect satisfaction evaluations, misinterpreted signatures, faulty substitutions, non-elementary embeddings, definability illusions, and compactness-induced artifacts. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding biased choice of signatures or parameters; ensuring neutral selection of structures; preventing overfitting of definability claims by artificially enriched languages. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Examination of definability proofs, satisfaction claims, quantifier-elimination steps, type analyses, and preservation theorems by other logicians or model-theorists. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Refining languages, modifying axioms, adjusting definability frameworks, recalibrating Skolem functions, or changing diagrammatic encodings in response to new counterexamples. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Full disclosure of signatures, languages, structures, definability criteria, satisfaction procedures, EF-game parameters, and all assumptions. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Honest reporting of definability limits; avoidance of hidden assumptions; accurate attribution of theorems; clear delimitation between semantic and syntactic claims. |