RDF Surfaces: Enabling Classical Negation and First-Order Expressivity on the Semantic Web

I thank the authors for the clear and detailed rebuttal, which made it very easy to understand the changes and appreciate the improvements. However, I did notice some inconsistency between the text in the rebuttal and the one in the final version. For example, the sentence “Given a concept of permitted we do not required to introduce a separate predicate for not-permitted.”, is mentioned in the rebuttal for my comment n.15, but does not appear in the revised manuscript. This is a minor issue, but it detracts from an otherwise excellent rebuttal writing style.

Overall, I believe the paper has been much improved. The formal parts of the papers are written better, and the focus of the paper is now clearer. Additional parts that were not developed to a sufficient standard for publication have been dropped, in the interest of quality and brevity. While I would have personally liked to see the theoretical study of RDF Surfaces developed more (as the authors write in the article, it was only covered briefly), I appreciate the effort put into discussing the syntax, motivating it and providing examples, which are important when proposing a new approach. Most of my previous comments have been fully addressed, and I have only a few more concerns that I have listed below. Out of those, probably the most critical one to address is the inconsistency with the claims about achieving FOL expressiveness (which I noted first in the list below). While it is essential that the authors address it before publication, I think it is a matter that can be addressed easily, and thus I do not think it requires another revision. Thus, I am happy to recommend acceptance of this article.

Comment 1:
While the authors write this “Although RDF Surfaces can be mapped into FOL, it must be noted that RDF Surfaces only supports a subset of FOL”, they also write “If a method were devised to negate triples and define variable scope, then the expressive power of FOL could be achieved” in section 4, and “We demonstrate how two additions to the RDF model under simple entailment semantics provide the full expressivity of FOL with explicit quantification”. The last two sentences arguably seem to suggest the precisely opposite fact than the first. Perhaps what the authors want to say in the second instance is something closer to the following: “If a method were devised to negate triples and define variable scope, then this would bring closer the expressive power of FOL.”?
Additionally, just before section 1.4 there is a claim that full FOL expressiveness is a requirement for RDF Surfaces. I suggest dropping this requirement, as it has not been met by the authors' own admission.

Comment 4:
While the authors agree that there are decidable fragments of FOL with negation in the rebuttal comment, the statement “A more compelling argument against logics with the expressivity of FOL is that they have been proven to have an undecidable entailment problem” remains ambiguous, as it is not clear which logics the authors are talking about, and how that relates to negation, which is the title of the subsection. I think the point the authors are trying to make is that, put simply, negation is a “problematic” feature of FOL, that once added to several decidable fragments of FOL turns them into undecidable fragments. This intuition is still missing from this section. I think that either this intuition is explained, making an example or two of fragments that require the omission of negation to be decidable, or alternatively, the section title should be changed to “Why was the full expressiveness of FOL not added to RDF?”. As it stands, the title and the contents of the subsection don’t match well.

Comment 5:
The authors have made much clearer the distinction between data and its possible interpretations, and clarified that the argument is about having a single universal notion of entailment. While this is now clear, it would be great to have a bit more discussion on how this is envisioned to work with practitioners. Currently, we have different inference systems that, although not always compatible with each other, serve different purposes. A practitioner that requires minimal reasoning capabilities might appreciate not having to delve into complex OWL fragments, while others requiring more expressiveness would appreciate not being restricted by the few entailment rules of RDFS. How would a single notion of entailment work to cover these different use cases? Is the main idea that one would carve out smaller fragments for certain simplified uses, while leaving more expressive ones for more advanced ones? I think that a few lines to explain this would go a long way to make a reader understand the whole picture.

Comment 6:
I think this sentence is still unclear. “While fragments of FOL may lead to decidability, their combination with the semantics of RDF and RDFS does not.” For the first part, I believe the authors want to say “While some fragments of FOL are decidable,…”. And about the second, it is not clear to me what a “combination” of a FOL fragment with the semantics of RDF and RDFS means, and why that necessarily leads to undecidability even for decidable fragments of FOL. If the authors are referencing a standard notion of such “combination”, then it should be cited, otherwise this sentence should be properly explained.

Comment 15-16:
I appreciate the additional notes on the journal preference example and the link to policy languages. I think it helps in highlighting the possible uses of RDF Surfaces. In its current format, I agree that the example section does not need to be moved to the Appendix.

New comments:
In page 11 line 40 variable a is used before being quantified.
In general, I found it difficult to follow the definition of interpretations the first time I read it, especially because of the overuse of the term interpretation. In definition 4.5 an interpretation is defined as a tuple, but later, when used as a function, an interpretation is defined as several other things, such as a literal, or a Boolean value, for example in sentences such as “The interpretation I[…] is considered “true””. This is not a big problem, but perhaps one could also make use of additional terms, such as “instance” to make the definitions clearer.
Page 21, line 13, typo: “*from* researcher and departmental preferences”
Page 21, line 17, typo: “by *compiling* them”
Page 24, line 33, extra space before full stop mark.

- Summary of the Paper -

The paper proposes RDF Surfaces, an extension of RDF that introduces classical (strong) negation and full first-order logic (FOL) expressivity into the RDF data model. Building on Hayes’ 2009 BLOGIC vision and Peirce’s existential graphs, the authors introduce “Hayes triples” and the notion of positive and negative surfaces to define scoped negation and explicit quantification. They provide a formal model-theoretic semantics extending RDF simple entailment and define a concrete syntax (N3S), based on a subset of Notation3, for representing such structures. The paper argues that adding classical negation and full FOL expressivity would improve portability of Web logic and unify reasoning across the Semantic Web stack. Two illustrative use cases—scholarly communication policies and medical prescription rules—demonstrate how RDF Surfaces can encode complex logical constructs.

- Motivation and Practical Relevance -

While the paper presents a technically careful proposal, its motivation raises important questions about timeliness and practical necessity.

The authors build heavily on Hayes’ 2009 BLOGIC vision and argue that RDF still lacks classical negation and full first-order logic (FOL) expressivity. From a strictly logical perspective, this claim is correct: RDF under simple entailment supports only conjunction and existential quantification (via blank nodes), and OWL 2 DL intentionally restricts expressivity to preserve decidability. In this sense, RDF Surfaces addresses a genuine theoretical gap in the Semantic Web stack.

However, it is less clear that this gap corresponds to a pressing need in current practice. As of 2026, the Semantic Web ecosystem is mature and widely deployed in industrial settings. OWL 2 DL provides decidable reasoning with controlled forms of negation; SHACL supports constraint validation; Datalog-style rule systems (often with negation-as-failure) are routinely used in knowledge graph platforms; and many real-world reasoning tasks are handled in hybrid architectures combining symbolic components with probabilistic or LLM-based reasoning layers.

In this landscape, the community has largely converged on a pragmatic design philosophy: favour decidable fragments, layered architectures, and tractable reasoning over maximal logical expressivity. The deliberate restriction of classical negation and unrestricted quantification in RDF and OWL was not an oversight but a design choice motivated by computability, interoperability, and engineering feasibility.

The paper argues that portability of Web logic requires full FOL expressivity and classical negation within RDF itself. This is a coherent position, and the formalisation presented is elegant. Nonetheless, the authors do not provide convincing evidence that current deployments are hindered by the absence of such expressivity. The use cases demonstrate that RDF Surfaces can encode complex logical patterns, but they do not clearly show that these scenarios cannot be adequately addressed using existing combinations of OWL, SHACL, rule engines, or application-level logic.

Moreover, the reintroduction of undecidability at the core RDF layer reopens a foundational debate. It is unclear whether the community still seeks a fully expressive classical logic at the data layer, or whether the implicit consensus has shifted toward controlled fragments and layered reasoning approaches.

That said, the proposal has theoretical merit. It provides:

• A clean, model-theoretic extension of RDF simple entailment.
• A principled integration of classical negation.
• A unified framework capable of expressing arbitrary FOL within RDF syntax.
• A concrete realization of the long-discussed BLOGIC vision.

As such, RDF Surfaces may be best understood as an exploration of the logical limits of RDF rather than as an immediately deployable extension of the Semantic Web stack.

Contribution and Methodology

The proposal effectively shifts RDF from a tractable data model to an undecidable logical framework without providing corresponding reasoning infrastructure. This raises concerns about whether the proposal is actionable beyond its theoretical formulation.

While the paper provides a formal semantic extension of RDF simple entailment and introduces a concrete syntax (N3S), it does not propose:
• A proof calculus,
• A semi-decision procedure for entailment,
• A translation to existing automated theorem provers,
• A prototype reasoning implementation.

Given that the central claim is the enablement of full FOL expressivity within RDF, the absence of a proof-theoretic or operational account significantly weakens the technical contribution.

Undecidability does not preclude useful reasoning procedures: classical FOL has well-established calculi (resolution, tableau, sequent systems) and mature automated provers. If RDF Surfaces is indeed a faithful embedding of FOL, one would expect at least a formal reduction to a standard calculus or an outline of how entailment would be computed in practice.

Instead, the paper remains at the model-theoretic level. It demonstrates what can be expressed, but not how it would be reasoned over, at what computational cost, or with what engineering implications.

For a full research article in this journal, this gap is substantial. Vision-level arguments may be appropriate for workshops or position papers, but a journal contribution typically requires a more complete formal, algorithmic, or empirical development.

In its current form, the contribution remains primarily definitional rather than algorithmic or experimentally validated.

Strengths and Weaknesses

Strengths
• Formally precise and carefully defined model-theoretic semantics.
• Conceptually elegant integration of classical negation into RDF.
• Clear and systematic encoding of FOL constructs using surfaces and graffiti nodes.
• Thoughtful engagement with foundational debates about portability and expressivity.
• Transparent discussion of undecidability implications.
Weaknesses
• Limited evidence of practical demand in the current Semantic Web ecosystem.
• Use cases demonstrate expressivity but not necessity.
• No proof calculus, reasoning procedure, or implementation strategy.
• No complexity analysis or discussion of computational feasibility.
• Largely conceptual, with no empirical validation.

Overall Assessment and Recommendation

RDF Surfaces is a well-articulated and intellectually serious attempt to revisit a foundational question: what would a fully expressive classical Web logic look like if embedded directly in RDF? As a theoretical exploration, it is coherent.

However, as a full research paper submitted to this journal, the absence of a proof-theoretic framework, reasoning procedure, complexity analysis, or implementation evidence significantly limits its contribution. The work currently reads more as a foundational research programme or position piece than as a completed technical framework.

For these reasons, I cannot recommend acceptance in its current form. I would encourage the authors to substantially extend the work with a proof calculus, operational reasoning framework, complexity characterisation, or implementation-based validation and consider resubmission as a more complete contribution.