Explicit Paradigms Overview

• Explicit paradigms are research frameworks with transparent, formalized principles and operational steps that distinctly contrast with implicit methods.
• They enable rigorous auditability, reproducibility, and systematic evaluation in domains such as symbolic AI, mathematical analysis, and software design.
• Their well-defined workflows facilitate algorithm extraction and integration with hybrid approaches, such as neuro-symbolic systems, to enhance practical applications.

An explicit paradigm is a research or design framework whose principles, assumptions, and operational steps are made transparent and formally articulated—often in contrast to implicit, hidden, or background conventions that organize scientific, technical, or methodological practice. The term “explicit” highlights the presentation of foundational commitments, process steps, or mappings, rendering theoretical structure and workflow visible to scrutiny and application. This article surveys the role of explicit paradigms in the sciences, engineering, and formal methods, including their mathematical structure, epistemic basis, operationalizations across domains, representative examples, and current debates over their necessity and limitations.

1. Foundational Definition and Contrast with Implicit Paradigms

An explicit paradigm can be characterized as a system whose axioms, representational forms, and inference rules are accessible in symbolic or algorithmic terms. In logic and mathematics, explicitness entails full formulation of quantifier alternations, explicit rule-systems, or constructive procedures. In empirical sciences and engineering, it refers to methodical protocols, articulated decision or design flows, and unambiguous mappings from hypothesis or requirement to result or artifact. In contrast, implicit paradigms are those in which structure, process, or commitment are left as background assumptions, heuristics, or are operationalized only informally.

For instance, in AI, symbolic (“explicit”) paradigms encode knowledge as sets of formulas in a language $\mathcal{L}$, e.g., $KB = \{\varphi_1, \ldots, \varphi_n\}$, with defined inference rules such as modus ponens or resolution; the inferential process is transparent and auditable step by step (Xiong et al., 11 Jul 2024). In contrast, connectionist systems operate through distributed activation patterns and weights, with inference and representation remaining largely opaque—an “implicit” knowledge paradigm.

The distinction recurs in domains such as wireless communication, where explicit NeurJSCC maps signal $x$ to a latent $z$ with learned priors for channel allocation and explicit decoders (Wang et al., 2023), as opposed to implicit approaches parameterizing the entire data manifold or function space.

2. Theoretical Foundations: Logic, Mathematics, and Epistemology

In foundational mathematics, explicit paradigms are often linked to formalism and constructive mathematics, where all deductive processes and objects must be generated by effective procedures or written rules. A prototypical case is the contrast between two paradigms in analysis:

• Archimedean (ε–δ) paradigm (“A-track”): All convergence, continuity, and function-theoretic relations are given by explicit quantifier alternations, e.g.,

$$\forall\varepsilon>0\,\exists N\in\mathbb{N}\,\forall n>N\,\forall x\in I:\;\left|f_n(x)-f(x)\right|<\varepsilon$$

This renders the logical structure “explicit” in terms of the dependencies and stringency required for uniformity (Bascelli et al., 2017).

• Infinitesimal-enriched (Bernoullian, “B-track”) paradigm: Quantification is replaced by a direct assertion involving infinitesimals and hyperreal extensions:

$$\forall x\in{}^*I,\;\forall H\in{}^*\mathbb{N}\setminus\mathbb{N}: f_H(x)\approx f(x)$$

Here, the extension of the number system and the use of direct manipulations with infinitesimals are explicit features, matching the procedural steps found in Cauchy's historical work.

In software engineering, the explicit encoding of the design process is central to the rational design paradigm (Ralph, 2013), where each phase and decision criterion (requirements $\rightarrow$ analysis $\rightarrow$ design $\rightarrow$ implementation) is documented and traceable.

3. Methodologies and Operational Workflows

Explicit paradigms are implemented through formal specifications, methodical workflows, and standard-form representations. Key operationalizations include:

• Symbolic AI and Neuro-Symbolic Systems: Knowledge representation as sets of explicit formulas, e.g., $P(t_1,\ldots,t_k)$, with computation mediated by inference rules directly mapped to code or rule-engines. Automated reasoning proceeds through formal derivations, e.g., $(\Gamma, P \implies Q)$. Autonomous agents leverage explicit plan, memory, and verification modules; hybrid neuro-symbolic systems additionally encapsulate symbolic representations within neural architectures (Xiong et al., 11 Jul 2024).
• Physics and Education Research: The recurrence-oriented paradigm is an explicit framework seeking reproducible, generalizable patterns; it relies on statistical designs and formal hypothesis testing (e.g., normalized gains via Hake's formula), whereas the case-oriented paradigm explicitly articulates meaning-making in local contexts by linking theory to concrete observational episodes (Robertson et al., 2013).
• Software Design: Explicit process-theories such as SDLC and plan-driven models document each design stage with specified roles, deliverables, and evaluation criteria (e.g., Waterfall, UP, TP-S), contrasting with amethodical or agile approaches where process emerges ad hoc (Ralph, 2013).
• Communication Systems: Explicit NeurJSCC architectures delineate separate encoder, latent prior, and decoder blocks, each trained and allocated resources according to defined objectives and constraints (see Table below) (Wang et al., 2023).

Domain	Explicit Paradigm Features	Implicit Paradigm
AI/Logic	Symbolic rules; transparent inference	Subsymbolic, weight-based, opaque
Math/Analysis	ε–δ quantifiers; formal alternations	Infinitesimal/heuristic reasoning
Software Design	Plan-driven, documented stages	Improvisational, emergent process
Communications (NeurJSCC)	Learned latent, explicit prior, rate allocation	Parameterization of function spaces

Explicit paradigms permit fine-grained auditing, reproducibility, and transferability, supporting both empirical validation and theoretical extension.

4. Representative Examples Across Disciplines

Symbolic Paradigms in AI: Autonomous agents such as LLM-empowered LAAs combine explicit symbolic workflows—planning, reasoning, step parsing—with neural controllers. Knowledge graphs exemplify explicit symbolic structuring of information for query and reasoning, distinct in their deterministic behavior and schematic rigidity (Xiong et al., 11 Jul 2024).

Physics Education Research: Recurrence-oriented studies (e.g., Pollock & Finkelstein’s measurement of normalized learning gains) exemplify explicit hypothesis-driven methods with formal statistical modeling. Case-oriented analyses (e.g., Berland & Hammer’s paper of classroom “framing”) explicitly connect analytic constructs to observed interaction patterns, making their interpretative moves transparent and systematic (Robertson et al., 2013).

Software Design: The rational design paradigm operationalizes explicitness via phase-based life cycles, technical problem-solving approaches, and specifications. For example, SDLC and the Function-Behavior-Structure (FBS) frameworks specify requirements, constraints, and evaluation metrics at each design step (Ralph, 2013).

Quantifier Elimination and Algorithm Extraction: Proof-mining methods translate nonconstructive model-theoretic embedding tests into explicit, terminating procedures—algorithms that convert first-order formulas into quantifier-free equivalents for specific logical theories. This extraction of explicit algorithms (e.g., QE_ACF_Small for algebraically closed fields with a small subgroup) is central to formal methods and computer algebra (Towsner, 2023).

Domain-Specific Languages: DSLs in Haskell support explicit sharing via let_ or share combinators, yielding expression DAGs that represent sharing at the AST level. Pure hash-consing provides implicit common subexpression elimination, but explicit sharing enables efficient, transparent management of complex computational graphs (Kiselyov, 2011).

5. Epistemic Implications, Advantages, and Trade-Offs

Explicit paradigms confer significant benefits:

• Auditability and Replicability: Formal rules and workflows can be reviewed, re-executed, and analyzed for error or improvement.
• Theoretical Clarity: Assumptions and dependencies are visible, exposing hidden premises and making the architecture of explanations or predictions clear.
• Transferability and Extension: Explicit parameterizations facilitate adaptation and modular replacement, e.g., in scientific modeling, software engineering, and communications.
• Integration and Comparison: Well-articulated paradigms can be juxtaposed, hybridized, or dialectically contrasted—as in mixed-methods research in PER or neuro-symbolic integration in AI (Robertson et al., 2013, Xiong et al., 11 Jul 2024).

However, explicitness may impose rigidity and over-specification, potentially hindering adaptability in dynamic, ambiguous, or emergent contexts. A plausible implication is that explicit paradigms are most effective when underlying phenomena are stable, well-understood, or can be adequately captured by the paradigm's formalism. In settings marked by ambiguity, as in case-oriented research, purely explicit models may fail to capture the richness or contingency of local interpretation. In software design, reliance on explicit process (rationalism) can conflict with the realities of tightly-coupled, coevolutionary projects, necessitating amethodical or empirical responses (Ralph, 2013).

6. Hybridization, Contemporary Developments, and Future Directions

The contemporary landscape features both the refinement of explicit paradigms and their integration with data-driven or heuristic (implicit) approaches:

• Hybrid AI Systems: LLM-empowered agents fuse explicit chain- or tree-of-thought symbolic workflows with connectionist (neural) subsystems, and advances in neuro-vector-symbolic architectures promise algebraic manipulation of explicitly bound vectors for enhanced reasoning (Xiong et al., 11 Jul 2024).
• PARADIGMS OF EVALUATION IN AI: AI evaluation research now delineates explicit paradigms aligned with distinct goals—benchmarking (systematic, fixed-test), red-team “evals,” construct-oriented psychometrics, exploratory hypothesis-driven probing, real-world impact (RCTs, field studies), and formal TEVV (Test, Evaluation, Verification, Validation). Each adopts explicit methodological and conceptual commitments, supporting transparency, comparison, and rigorous culture (Burden et al., 21 Feb 2025).
• Pragmatic and Dialectic Mixes: In fields such as PER, explicit paradigms may be combined in mixed-methods research strategies (pragmatic/dialectic), balancing predictive generalization and interpretive depth by making plain the commitments and limitations of each (Robertson et al., 2013).
• Formalization of Nonconstructive Proofs: Proof mining techniques are increasingly used to render classical theorems (formerly proved nonconstructively or via embedding tests) into explicit algorithmic form—enabling practical computation and automated reasoning in logic, algebra, and computer science (Towsner, 2023).
• Semantic Communications: Explicit NeurJSCC replaces isolated, hand-designed modules with trained, end-to-end explicit pipelines governing representation, resource allocation, and error-minimization by optimizing a transparent, joint rate-distortion loss (Wang et al., 2023).

7. Ongoing Challenges and Controversies

Debate persists regarding the necessity, sufficiency, and generality of explicit paradigms:

• Overfitting to Formalization: Strict adherence to explicit frameworks may lead to overfitting or neglect of complex contextual features that are better handled by flexible, local, or empirical methods.
• Loss of Practitioner Intuition: Overly explicit processes may marginalize skilled, context-sensitive judgment, leading to “box-checking” or mechanical application of rules.
• Integration with Implicit Knowledge: Balancing explicit transparency with the capacity to leverage rich, implicit knowledge—particularly in hybrid and neuro-symbolic systems—remains a central research frontier, as does the development of architectures enabling explicit reasoning over implicit, distributed representations.

Explicit paradigms remain foundational in formal, empirical, and engineering disciplines. Their ongoing evolution is marked by the requirement to render not only the form but the operationalization of assumptions, mappings, and workflow components accessible, auditable, and adaptable across a spectrum of research and development contexts.