In this paper, we prove that the lattice of all closure operators of a complete Almost Distributive Lattice L with fixed maximal element m is dual atomistic. We define the concept of a completely meet-irreducible element in a complete ADL and derive a necessary and sufficient condition for a dual atom of Φ (L) to be complemented.

We continue the work of Blok and Jónsson by developing the theory of structural closure operators and introducing the notion of a representation between them. Similarities and equivalences of Blok-Jónsson turn out to be bijective representations and bijective structural representations, respectively. We obtain a characterization for representations induced by a transformer. In order to obtain a similar characterization for structural representations we introduce the notions of a graduation and a graded variable of an M-set. We show that several deductive (...) systems, Gentzen systems among them, are graded M-sets having graded variables, and describe the graded variables in each case. In the last section we show that, for a sentential logic, having an algebraic semantics is equivalent to being representable in an equational consequence. This motivates the extension of the notion of having an algebraic semantics for Gentzen systems, hypersequents systems, etc. We prove that if a closure operator is representable by a transformer, then every extension of it is also representable by the same transformer. As a consequence we obtain that if one of these systems has an algebraic semantics, then so does any of its extensions with the same defining equations. (shrink)

In this paper, a theorem on the existence of complete embedding of partially ordered monoids into complete residuated lattices is shown. From this, many interesting results on residuated lattices and substructural logics follow, including various types of completeness theorems of substructural logics.

In the paper, tracing the traditional Hilbert-style syntactic account of logics, a syntactic characteristic of a closure operation defined on a complete lattice follows. The approach is based on observation that the role of rule of inference for a given consequence operation may be played by an ordinary binary relation on the complete lattice on which the closure operation is defined.

In this paper we develop the notion of formal precover in a topos by defining a relation between elements and sets in a local set theory. We show that such relations are equivalent to modalities and to universal closure operators. Finally we prove that these relations are well characterized by a convenient restriction to a particular set.

In our previous paper Algebraic Logic for Classical Conjunction and Disjunction we studied some relations between the fragmentL of classical logic having just conjunction and disjunction and the varietyD of distributive lattices, within the context of Algebraic Logic. The central tool in that study was a class of closure operators which we calleddistributive, and one of its main results was that for any algebraA of type (2,2) there is an isomorphism between the lattices of allD-congruences ofA and of all (...) distributive closure operators overA. In the present paper we study the lattice structure of this last set, give a description of its finite and infinite operations, and obtain a topological representation. We also apply the mentioned isomorphism and other results to obtain proofs with a logical flavour for several new or well-known lattice-theoretical properties, like Hashimoto's characterization of distributive lattices, and Priestley's topological representation of the congruence lattice of a bounded distributive lattice. (shrink)

In this paper, we investigate the pairs (A,cA), where cA is a stable closure operation on an MV-algebra A together with MV-homomorphisms subject to some condition (D). We obtain that these pairs form a category denoted by CMV and that for each such pair, the lattice cId(A) of cA-closed ideals is a coherent frame. Moreover, three functors Σ,Ψ and Φ are constructed between the category CMV and the category ChFrm of coherent frames and natural transformations between these functors are (...) established. Consequently, an application to the radical of ideals of A induces a functor RId that allows us to investigate some properties of MV-algebra homomorphisms and the relationship between coherent frames induced by MV-algebras. Finally, it is obtained that the coherent frame cAId(A) is isomorphic to the coherent frame O(SpeccA(A)) of open sets of the cA-closed ideals of Spec(A). (shrink)

ABSTRACT In this work the connections between the fuzzy closure operators and the graded consequence relations are examined Namely, as it is well known, in the crisp case there is a complete equivalence between the notion of closure operator and the one of consequence relation. We extend this result by proving that the graded consequence relations are related to a particular class of fuzzy closure operators, namely the class of fuzzy closure operators that can be (...) obtained by a chain of classical closure operators. (shrink)

We introduce a new logical operator CSTC and show that incorporating this operator into first-order logic enables as to capture the complexity class PSPACE. We also show that by varying how the operator is applied we can capture the complexity classes P, NP, the classes of the Polynomial Hierarchy PH, and PSPACE. As such, the operator CSTC can be regarded as a general purpose operator. We also give applications of these characterizations by showing that P (...) and NP coincide with those problems accepted by two new classes of program schemes , by producing some new complete problems for various complexity classes where the reductions are from first principles and do not use known complete problems , and by giving a simple proof that first-order logic incorporated with the least fixed point operator captures P. (shrink)

Geometric data models form the backbone of virtually all spatial information systems, such as GIS, CAD, and CAM. Yet a lot of spatial information from textual sources, including historical document...

In this article we introduce and study the notion of operational closure: a transitive set d is called operationally closed iff it contains all constants of OST and any operation f∈d applied to an element a∈d yields an element fa∈d, provided that f applied to a has a value at all. We will show that there is a direct relationship between operational closure and stability in the sense that operationally closed sets behave like Σ1 substructures of the universe. (...) This leads to our final result that OST plus the axiom , claiming that any set is element of an operationally closed set, is proof-theoretically equivalent to the system KP+ of Kripke–Platek set theory with infinity and Σ1 separation. We also characterize the system OST plus the existence of one operationally closed set in terms of Kripke–Platek set theory with infinity and a parameter-free version of Σ1 separation. (shrink)

Constructive set theory a' la Myhill-Aczel has been extended in (Cantini and Crosilla 2008, Cantini and Crosilla 2010) to incorporate a notion of (partial, non--extensional) operation. Constructive operational set theory is a constructive and predicative analogue of Beeson's Inuitionistic set theory with rules and of Feferman's Operational set theory (Beeson 1988, Feferman 2006, Jaeger 2007, Jaeger 2009, Jaeger 1009b). This paper is concerned with an extension of constructive operational set theory (Cantini and Crosilla 2010) by a uniform operation of Transitive (...) Closure, \tau. Given a set a, \tau produces its transitive closure \tau a. We show that the theory ESTE of (Cantini and Crosilla 2010) augmented by \tau is still conservative over Peano Arithmetic. (shrink)

In this chapter we descend to the lowest level of the operator hierarchy. There we explore the formation of physical operators from the bottom up, starting at the level of the smallest physical systems, the leptons (e.g. electrons) and quarks.

Paper 1 establishes that persistence-based frameworks — the Free Energy Principle, Global Workspace Theory, and adjacent approaches — rely on batteries of dynamical observables insufficient, on at least one substrate where the frameworks would naturally apply them, to individuate the systems they characterize. The supplementary structural condition Paper 1 §7 calls a partition-warrant must be sensitive to substrate properties not invariant under the admissible transformation group of the dynamics. The present paper develops closure activation as a candidate filling for (...) that slot. The formal predicate W_CA is partition-evaluative: for each subsystem under a partition P, it returns a verdict structure (τ₀, μ*, κ) recording whether the activation event T₀ has been crossed, the maintenance margin μ* under graded perturbation, and the resulting classification (closed, closure-competent, subthreshold). The activation conditions — state-referential constraint, internally governed corrective response, internally viable maintenance budget — are evaluated relative to the partition's imposed boundary. The predicate's two-axis structure (binary T₀, scalar T*) is phase-structured in a way that distinguishes closure activation from classical autopoietic operational closure. W_CA is non-G-invariant at the predicate level: the causal support graph relative to the imposed boundary is partition-relative and not preserved under admissible transformations. The predicate operates in three structural modes — rejection, selection, and stratification — depending on the substrate's closure-bearing properties under G-related partitions. Evaluation on the Paper 1 witness yields a rejection-mode verdict: neither partition is closure-bearing, and the uniform negative output is the predicate's correctness behavior on a substrate where the activation conditions cannot be satisfied under any admissible labeling. Distinctions from optimization-based (Integrated Information Theory), structural-definitional (Global Workspace Theory), statistical-structural (Free Energy Principle in nonlinear regimes), and process-based (autopoietic operational closure) alternatives are developed structurally and by operational-mode coverage. The route × mode cross-tabulation supplies a meta-taxonomy for partition-warrants more generally. Falsification conditions are stated at predicate, verdict-structure, and discriminative-scope levels. In conjunction with Paper 1, the present paper separates three questions persistence-based frameworks have often conflated: what persists dynamically, what individuates structurally, and what sustains recursive closure. The class of closure-bearing systems is a proper subset of the class of persistence-bearing systems under the battery characterization, and the structural rarity of closure-bearing organization is a derivable consequence of the activation conditions' joint stringency. (shrink)

This paper proposes a reformulation of classical electromagnetism within the framework of Cognitional Mechanics (CM), in which Maxwell’s equations are not treated as four independent dynamical laws but as decomposition sectors of a single operational closure constraint. The central claim is that electromagnetic field structure arises as a Tier-3 projection of a deeper Tier-2 operational system characterized by noncommutative closure, finite generator saturation, and stable projection dynamics. -/- Within this framework, CM is organized into a hierarchical architecture consisting (...) of Tier-1 structural ordering principles, Tier-2 operational axioms governing admissible generators and closure, and Tier-3 representational projections in which field-theoretic objects emerge as observable decompositions. Classical electromagnetism is reconstructed entirely at Tier-3 as the image of a single CM closure equation, rather than as a set of independently postulated laws. -/- The electromagnetic field is represented as an element of the traceless anti-Hermitian sector of M3(C), uniquely selected by CM redundancy exclusion and Pi_d-saturation axioms. The CM-Maxwell operator D_cm = (1/c) ∂_t + div arises as the unique first-order intertwiner compatible with closure-preserving automorphisms, yielding the fundamental equation D_cm F = J. Standard electromagnetic dynamics are recovered as Tier-3 decompositions of the resulting complexified closure system. -/- The curl operator is not taken as primitive but emerges from antisymmetric matrix structure under row-wise divergence, with orientation fixed algebraically by internal consistency conditions. The divergence-free magnetic condition appears as a cohomological consequence of antisymmetric closure embedding. Gauge symmetry is reinterpreted as the Tier-3 projection of a unique one-parameter closure-preserving invariance group rather than an externally imposed redundancy. -/- This work supersedes earlier formulations focusing on compact representations of classical electrodynamics, shifting the objective from representational compression to derivation from operational closure principles. Within the broader CM projection hierarchy, this result constitutes the first stable bridge regime linking Tier-2 operational closure to Tier-3 field representations, and forms the entry point of a projection chain that extends toward Standard Model and unified field structures. -/- Published on May 16, 2026 DOI: 10.5281/zenodo.20242465. (shrink)

In the categorical approach to the foundations of quantum theory, one begins with a symmetric monoidal category, the objects of which represent physical systems, and the morphisms of which represent physical processes. Usually, this category is taken to be at least compact closed, and more often, dagger compact, enforcing a certain self-duality, whereby preparation processes (roughly, states) are interconvertible with processes of registration (roughly, measurement outcomes). This is in contrast to the more concrete “operational” approach, in which the states and (...) measurement outcomes associated with a physical system are represented in terms of what we here call a convex operational model: a certain dual pair of ordered linear spaces–generally, not isomorphic to one another. On the other hand, state spaces for which there is such an isomorphism, which we term weakly self-dual, play an important role in reconstructions of various quantum-information theoretic protocols, including teleportation and ensemble steering. In this paper, we characterize compact closure of symmetric monoidal categories of convex operational models in two ways: as a statement about the existence of teleportation protocols, and as the principle that every process allowed by that theory can be realized as an instance of a remote evaluation protocol—hence, as a form of classical probabilistic conditioning. In a large class of cases, which includes both the classical and quantum cases, the relevant compact closed categories are degenerate, in the weak sense that every object is its own dual. We characterize the dagger-compactness of such a category (with respect to the natural adjoint) in terms of the existence, for each system, of a symmetric bipartite state, the associated conditioning map of which is an isomorphism. (shrink)

Once atoms and molecules existed, a whole new dynamic entered the universe the moment the first cells appeared. What is different about cells is that they require energy to maintain their construction and be active. Their energy consumption produces waste products and heat that are released, or ‘dissipated’, into the environment. Cells are therefore dissipative structures. Recent research has shown that different types of cells live at the base of the tree of life. The best known are the bacteria. A (...) second group are the archaea. How could these cells have formed in a chemical environment? And how, with bacteria/archaea as a base, could complexity continue to increase to protozoa, plants and animals? These are some of the fundamental questions explored in this chapter. (shrink)

Closure-class formation involves two structurally distinct questions: the conditions under which closure of a specified class is admissible, and the contingent events through which any specific closure realizes. This paper develops the conditions side. Five structural obstruction modes — sparse-feedback, fragment-local, cost-imbalanced, energetically-starved, dimensionally-trapped — characterize seed-field incompetence: the conditions under which a substrate region is excluded from supporting closure-class formation across its accessible perturbation regimes. The admissibility theorem characterizes which closure classes a competent region (...) admits as a function of a parameter family indexing the substrate's coupling structure, boundary exposure, energetic and temporal accessibility, and dimensional access. The theorem is stated spectrally — admissibility iff the linearized closure-formation operator possesses an unstable mode satisfying structural and dynamical conditions — and equivalently in variational form via the closure-potential functional. The four-stage architecture for closure-class formation (competence, admissibility, embryo formation, basin completion) entails a cascade structure in which closure-formation rates compose multiplicatively across stages, so that progressively more demanding closure classes are progressively rarer by structural composition rather than stipulation. The cascade is observable through four ensemble signatures — critical slowing, hysteresis, post-collapse renucleation, basin-deepening — that specialize the early-warning-signal apparatus independently developed in tipping-point ecology and regime-shift detection to closure-class transitions across substrate types. Consciousness-class closure is treated as a further-conditioned special case in which the cascade structure entails the framework's predicted rarity without further stipulation. (shrink)

In this paper we show that some standard topological constructions may be fruitfully used in the theory of closure spaces (see [5], [4]). These possibilities are exemplified by the classical theorem on the universality of the Alexandroff's cube for T 0-closure spaces. It turns out that the closure space of all filters in the lattice of all subsets forms a generalized Alexandroff's cube that is universal for T 0-closure spaces. By this theorem we obtain the following (...) characterization of the consequence operator of the classical logic: If is a countable set and C: P() P() is a closure operator on X, then C satisfies the compactness theorem iff the closure space ,C is homeomorphically embeddable in the closure space of the consequence operator of the classical logic.We also prove that for every closure space X with a countable base such that the cardinality of X is not greater than 2 there exists a subset X of irrationals and a subset X of the Cantor's set such that X is both a continuous image of X and a continuous image of X. (shrink)

Bearers track closure classes; closure classes do not track bearers. This inversion is the substantive claim the present paper defends, and its grounds are structural: the framework operates at the admissibility stratum, characterizing what closure formation can produce; bearers are realization-stratum products of closure formation; the partition between class members and non-members is a consequence of closure rather than a feature of the framework's structural commitments. The seed-field framework (Thomas 2026a) characterizes closure-class formation through (...) structural conditions on admissibility analysis, but the relation between class C and the conditions characterizing it is left underspecified in that paper alone; the present paper specifies the relation by establishing where the framework's apparatus operates and what it characterizes. The structural conditions S_C are specified through five formal families — eigenstructure conditions on the linearized closure-formation operator, rank conditions on the eigenmode's projection across maintenance-cost surfaces, graph-theoretic support conditions on the correction graph, scaling conditions on the cost functional, and bound conditions on the temporal ratios — each statable on the admissibility analysis without reference to bearer-types. Closure classes are equivalence classes of closures whose admissible modes satisfy a given S_C, with class identity attaching to constraint-preserving dynamical possibility structure rather than to recognized bearer-type. The apparatus is read correctly across five common questions through the bearers-track-classes inversion, the distinction among definitional, discovery, and naming dependence, the Platonism disclaimer, the demonstration that the apparatus extends to hypothetical and uninstantiated classes, and the labels-versus-apparatus distinction. The methodological precedent is the universality-class apparatus in critical phenomena, which operates at the class stratum upstream of where phase boundaries are realized. The consequence framing licenses cross-substrate predictions: substrates satisfying S_C in basin completion produce closure with the structural features S_C characterizes, regardless of substrate type. The framework's empirical commitments are preserved with this addition; the architectural foundation is clarified. (shrink)

This paper develops a constraint-geometric framework for analyzing the persistence, revision, and collapse of scientific theories. Drawing on the concept of organizational closure from the biological autonomy tradition (Maturana & Varela, Moreno & Mossio, Barandiaran), I argue that theories are not merely descriptions of identity-under-constraint but are themselves closure structures whose ontological commitments are constituted by constraint-survival. The framework introduces a three-state landscape for theory identity (active, latent, extinct), an operational definition of the invariance core, and a diagnostic (...) criterion—extension-stability—that distinguishes supersession from degeneration and refutation with greater structural precision than existing accounts. As a primary demonstration, the framework is applied to the dark matter problem, generating differential structural diagnoses for the particulate dark matter hypothesis (ΛCDM) and Modified Newtonian Dynamics (MOND). A secondary application to four current hypotheses in consciousness studies—Integrated Information Theory (IIT), Global Workspace Theory (GWT), the Free Energy Principle (FEP), and the Identity–Recursion–Consciousness framework (IRC)—is developed in Appendix A. The constraint-geometric apparatus was developed in the course of work on IRC, and this methodological relationship is disclosed and addressed. The result is a formal instrument for the epistemology of science that unifies theory-persistence and identity-persistence under a shared constraint geometry. (shrink)

Identity-persistence has been treated, across philosophy, cognitive science, and theoretical biology, as a question about the relations between states. A system is identical to itself across time when its states at those times stand in some specifiable relation: continuity, similarity, causal connection, or psychological linkage. This paper argues that the approach cannot succeed. State-based ontologies cannot in principle ground identity-persistence. The strategies they have available — continuity, similarity, and stipulated rules connecting states-at-times — either fail outright or collapse into a (...) trajectory condition under a different description. No state-based ontology produces persistence as the structural feature it is committed to explaining. The paper develops the alternative: identity is the persistence of a recursive maintenance regime, and the regime continues or fails. Persistence is the continuation. The Invariance Maintenance Condition specifies the structural requirement; closure is satisfaction of the condition. The cryptobiotic tardigrade is the operational test. Substrate is preserved across cryptobiosis; lineage is unbroken; no recursive maintenance occurs. The trajectory ontology returns identity-interrupted, against substrate-continuity intuition and against causal-historical theories of identity. The position is falsifiable: it would be defeated by a system whose identity-continuity is structurally preserved across genuine gaps in maintenance, or by a system that fails maintenance while maintaining identity-continuity by some non-question-begging structural criterion. The inversion is structural rather than preferential. State-based ontologies do not produce persistence less well than the trajectory ontology; they do not produce it at all. (shrink)

The Reflection Pattern is a framework for analyzing recursive closure dynamics in human and institutional systems. This paper extends it to artificial intelligence, arguing that AI systems do not merely exhibit Reflection Pattern dynamics but instantiate the recursive stabilization architecture the framework names, in a substrate that lacks the affective and social mediators through which the Pattern had previously been theorized. Five structural moves develop the account. First, unacknowledged fear is relocated from constitutive condition of the Pattern to formation (...) condition, opening the framework to mature systems where the dynamics persist with fear recognized. Second, fear is identified as one phenomenological manifestation of a deeper structural process — identity-preserving stabilization under destabilizing perturbation — that operates substrate-generally as stabilization geometry. Third, AI-RP is distinguished from human and institutional RP as template-based rather than emergent: encoded prospectively through training rather than developed retrospectively through accumulated reinforcement. Fourth, the corridor framework distinguishes healthy reflective compression (which preserves permeability to corrective signals) from pathological closure (which absorbs them). Fifth, AI is characterized as recursive interpretive infrastructure rather than tool, and recursive closure risk is identified as a distinct AI safety position, structurally separate from agency, deception, optimization, autonomy, and misalignment risks: the locus is not what AI systems do but the progressive reduction of corrective permeability in the interpretive ecologies AI mediates. The framework is diagnostic rather than prescriptive. It does not predict particular outcomes, prescribe interventions, or generate safety recommendations. It provides analytical purchase on the structural property — corridor position — that distinguishes adaptive from pathological AI-RP, and identifies an open question: whether sufficiently permeable forms of AI-RP can be sustained at scale. (shrink)

In this paper, notions of fuzzy closure system and fuzzy closure L—system on L—ordered sets are introduced from the fuzzy point of view. We first explore the fundamental properties of fuzzy closure systems. Then the correspondence between fuzzy closure systems and fuzzy closure operators is established. Finally, we study the connections between fuzzy closure systems and fuzzy Galois connections. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

In this paper, I consider the connection between consequence relations and closure operations. I argue that one familiar connection makes good sense of some usual applications of consequence relations, and that a largeish family of familiar noncontractive consequence relations cannot respect this familiar connection.

The explanatory gap is not evidence of a missing force, but of a missing condition. Every major consciousness framework — Integrated Information Theory, the Free Energy Principle, Global Workspace Theory, Higher-Order Theories, and Predictive Processing — presupposes a bounded, persistence-capable unit at a specific point in its formalism and supplies none. The system whose consciousness is to be explained is assumed, not derived. This paper identifies the missing condition as closure under maintenance: the condition under which a configuration's boundary (...) conditions would not persist without its own internal dynamics. Closure is formalized as a binary predicate, situated within a dependency chain (constraint landscape → seed-field → metastability → closure → identity), and shown to be minimal — thinner than autopoiesis, Rosen-closure, or organizational closure, yet sufficient to generate the explanatory predicates (persistence, boundary, regime behavior, collective failure, hysteresis) that downstream theories require and presuppose. Three reduction cases demonstrate that phenomena treated as independent primitives are consequences of closure-maintained identity: induction–emergence hysteresis in anesthesia follows from the structural non-symmetry of threshold crossing and loop re-closure; decomposition failure in recursively self-conditioned systems follows from closure at sufficient depth and integration; and the persistence of identity through phenomenal dropout follows from identity and consciousness occupying different levels of a dependency hierarchy in which closure is necessary for both but recursion and substrate coupling are additionally necessary for consciousness. The paper draws an explicit boundary: closure is not sufficient for consciousness. Substrate-coupled recursive maintenance — in which the maintenance process operates on its own maintenance structure, instantiated on a substrate whose physical interactions carry the coordination work as a unified dynamical event — is consciousness. The explanatory gap does not close; it relocates, from an unbridgeable chasm between matter and experience to a tractable set of questions about maintenance geometry, recursive regime entry, and substrate-specific binding, anchored in a formalized empirical program with specified falsification conditions. (shrink)

During the construction phase of the new theory introduced in this book, it became apparent that the quest for a foundation of a theoretic framework for analysing natural organisation may profit from the classical approach to mathematics developed by the great Greek mathematician Euclid. His approach allows a consistent framework to be built from the ground up, providing a theoretical model of how nature has constructed increasingly complex types of systems. The theoretic logic could serve as a philosophical instrument that (...) could support the recognition of increasingly complex types of organisation in nature which in turn correspond with the levels indicated by the theoretic logic. Ideally one finds a close match between the abstract model and the observations of natural organisation. (shrink)

The principle of epistemic closure is the claim that what is known to follow from knowledge is known to be true. This intuitively plausible idea is endorsed by a vast majority of knowledge theorists. There are significant problems, however, that have to be addressed if epistemic closure – closed knowledge – is endorsed. The present essay locates the problem for closed knowledge in the separation it imposes between knowledge and evidence. Although it might appear that all that stands (...) between knowing the truth of the premises of a valid inference and knowledge of its conclusion is inferring it from the premises, the evidence for each of the premises may jointly count against the conclusion. The intuitive view regarding inferred knowledge says one thing, the evidence says another. One epistemological framework that seems to have the resources to resolve this tension endorses the view that knowledge always requires conclusive evidence. A second framework resolves the tension by limiting the scope of the closure principle. Only inferences drawn directly from propositions contained in the scope of a single knowledge operator are considered closed. The aim of the present essay is to revive the unpopular third option, the idea that knowledge is open. The essay proceeds by arguing that in different ways the two former frameworks only succeed in relocating the problem, not in resolving it. The first framework, the infallibilist view, relocates the problem to a sharp separation between knowledge of the occurrences of events from knowledge of their chance of occurring, a separation leading to several significant additional problems. The fallibilist view, the second framework, in endorsing closure neglects to take into full account the ways in which evidence fails to be transitive. For instance, evidence can count in favor of a conjunction while counting against each of its conjuncts. This fact, which is argued for in the essay on probabilistic as well as non-probabilistic grounds, is used as the foundation of an argument against closed knowledge that can be used as a way to understand several of the most fundamental challenges of epistemology. Not only can an open knowledge view that is based on open evidence resolve all these problems in a simple and natural way, it can also respond to formidable challenges that significantly hinder other open knowledge views. There are good reasons, then, to view both knowledge and evidence as open. (shrink)

Recently, Da Ré, Szmuc, Chemla and Égré (2024) showed that all logics based on Boolean Normal monotonic three-valued schemes coincide with classical logic when defined using a strict-tolerant standard (st). Conversely, they proved that under a tolerant-strict standard (ts), the resulting logics are all empty. Building on these results, we show that classical logic can be obtained by closing under transitivity the union of two logics defined over (potentially different) Boolean normal monotonic schemes, using a strict-strict standard (ss) for one (...) and a tolerant-tolerant standard (tt) for the other, with the first of these logics being paracomplete and the other being paraconsistent. We then identify a notion dual to transitivity that allows us to characterize the logic TS as the dual transitive closure of the intersection of any two logics defined over (potentially different) Boolean normal monotonic schemes, using an ss standard for one and a tt standard for the other. Finally, we expand on the abstract relations between the transitive closure and dual transitive closure operations, showing that they give rise to lattice operations that precisely capture how the logics discussed relate to one another. (shrink)

It is well known that, in a topological space, the open sets can be characterized using ?lter convergence. In ZF , we cannot replace filters by ultrafilters. It is proven that the ultra?lter convergence determines the open sets for every topological space if and only if the Ultrafilter Theorem holds. More, we can also prove that the Ultra?lter Theorem is equivalent to the fact that uX = kX for every topological space X, where k is the usual Kuratowski closure (...) operator and u is the Ultra?lter Closure with uX := {x ∈ X: [U converges to x and A ∈ U ]}. However, it is possible to built a topological space X for which uX ≠ kX, but the open sets are characterized by the ultra?lter convergence. To do so, it is proved that if every set has a free ultra?lter, then the Axiom of Countable Choice holds for families of non-empty finite sets. It is also investigated under which set theoretic conditions the equality u = k is true in some subclasses of topological spaces, such as metric spaces, second countable T0-spaces or {ℝ}. (shrink)

Consciousness frameworks operate on systems already treated as bounded units of analysis. Each major framework has a formal entry point at which a candidate space — a region of physical substrate treated as supporting candidate partitions — must be specified for the framework's operations to be defined. The candidate space is not produced by the framework's apparatus; it is presupposed at the point where the apparatus becomes applicable. The paper identifies this structural feature — candidate-space dependence — in three major (...) consciousness frameworks (Integrated Information Theory, Global Workspace Theory, the Free Energy Principle) and distinguishes it from the individuation question taken up in a companion gateway. Candidate-space dependence implicates a formation question distinct from the questions consciousness frameworks have organized themselves around: under what conditions does a region of physical reality come to constitute a candidate space at all? The Identology corpus develops the formation question through a paper sequence — Constraint Topology and the Precedence of Systemhood V2, Closure Formation as Admissible Instability, Basin Genesis, Consolidation, and Closure Before Bearers — that characterizes the structural conditions for closure formation through a three-stage dependency: landscape admissibility, coupling admissibility, and closure activation. The paper does not redevelop that apparatus; it identifies the dependency structure, places the corpus's apparatus in methodological alignment with the universality-class discipline developed in critical phenomena, and clarifies the corpus's two-doorway approach to closure under maintenance: formation and individuation as complementary entry points. The contribution is positional rather than constructive: the diagnosis specifies conditions of applicability rather than critiquing framework content, and the formation question is identified as prior to the questions consciousness frameworks have asked. (shrink)

We establish a general hierarchy theorem for quantifier classes in the infinitary logic L∞ωωon finite structures. In particular, it is shown that no infinitary formula with bounded number of universal quantifiers can express the negation of a transitive closure.This implies the solution of several open problems in finite model theory: On finite structures, positive transitive closure logic is not closed under negation. More generally the hierarchy defined by interleaving negation and transitive closure operators is strict. This proves (...) a conjecture of Immerman.We also separate the expressive power of several extensions of Datalog, giving new insight in the fine structure of stratified Datalog. (shrink)

Identology is a formal framework addressing the structural conditions underlying identity persistence and consciousness. Its central claim is that the field of consciousness studies has presupposed a bounded, persistence-capable unit without deriving the conditions under which such a unit exists—and that closure under maintenance is the missing foundational predicate. -/- Closure is defined as boundary-precondition coupling: a binary condition in which a configuration’s internal dynamics either sustain the individuating boundary conditions or they do not. Maintenance is scalar: it (...) varies in cost, depth, and regime across systems and across time. Consciousness is the regime in which a closure-maintained system’s identity-preserving dynamics become recursively self-conditioned—not a substance added to the system, not an emergent property layered on top, but the same dynamical process described under a different aspect. -/- This status report consolidates nineteen individual papers from a corpus of approximately fifty modular papers – the collection available at ORCID - into a single citable reference. It declares the framework’s operating invariants, maps its upstream dependencies and downstream consequences for existing frameworks, specifies the open research fronts where the framework is explicitly incomplete, and provides a dependency architecture of the corpus itself. It is designed as a navigational hub: every claim points outward to the paper that establishes it. (shrink)

The Baire algebra of a topological space X is the quotient of the algebra of all subsets of X modulo the meager sets. We show that this Boolean algebra can be endowed with a natural closure operator, resulting in a closure algebra which we denote $\mathbf {Baire}(X)$. We identify the modal logic of such algebras to be the well-known system $\mathsf {S5}$, and prove soundness and strong completeness for the cases where X is crowded and either completely (...) metrizable and continuum-sized or locally compact Hausdorff. We also show that every extension of $\mathsf {S5}$ is the modal logic of a subalgebra of $\mathbf {Baire}(X)$, and that soundness and strong completeness also holds in the language with the universal modality. (shrink)

The Deontic Principle of Closure (DCL) appears initially to be a highly plausible principle. The DCL is commonly assumed in practical ethical reasoning, as when we make certain inferences about what we (morally) ought to do in particular situations. For example, if I am standing beside a burning house with several victims trapped inside and I have an obligation to rescue them, then if it is necessary for me to open the front door in order for me to lead (...) them out, then it seems that I am morally bound in this situation to open the door. Similarly, if it is the case that I ought to keep my Friday morning 8:00 appointment with my student and it is a necessary condition for keeping the appointment that I wake up some time before 8:00, then I ought to wake up before 8:00. In spite of its attractiveness, however, various worries have been raised about the plausibility of this closure principle. In what follows, I shall critically examine DCL and discuss its plausibility in the face of various objections that have been raised against it. I shall argue that a slightly modified version of the principle circumvents the main objections and holds under several of the more or less standard interpretations of the “ought” operator. (shrink)

All standard epistemic logics legitimate something akin to the principle of closure, according to which knowledge is closed under competent deductive inference. And yet the principle of closure, particularly in its multiple premise guise, has a somewhat ambivalent status within epistemology. One might think that serious concerns about closure point us away from epistemic logic altogether—away from the very idea that the knowledge relation could be fruitfully treated as a kind of modal operator. This, however, need (...) not be so. The abandonment of closure may yet leave in place plenty of formal structure amenable to systematic logical treatment. In this paper we describe a family of weak epistemic logics in which closure fails, and describe two alternative semantic frameworks in which these logics can be modelled. One of these—which we term plurality semantics—is relatively unfamiliar. We explore under what conditions plurality frames validate certain much-discussed principles of epistemic logic. It turns out that plurality frames can be interpreted in a very natural way in light of one motivation for rejecting closure, adding to the significance of our technical work. The second framework that we employ—neighbourhood semantics—is much better known. But we show that it too can be interpreted in a way that comports with a certain motivation for rejecting closure. (shrink)

The canonical closure of the Relational Zero State (RZS) framework fixed the formal ar-chitecture of the program: weighted relational microdynamics, Laplacian diffusion, heat-kernel distance, emergent geometry, an infrared cosmological sector, and an ordered observational sequence. The main open issues were calibration, admissibility, and controlled completion. This paper addresses those issues by introducing explicit criteria for geometric regime formation, a conservative infrared admissibility filter for the low-redshift bulk, and a controlled onset condition for the higher-order tail sector against complementary null (...) ensembles. It also states the Einstein-limit target in effective-action form, promotes matter and gauge structure from a minimal interface to a minimal backreaction ansatz inside the Romero stability balance, and implements a numerical proof-of-concept based on a mean-field Monte Carlo surrogate of the inherited jump law. The aim is operational: to determine when the inherited microdynamics may be coarse-grained as a viable geometric and cosmological sector, where it must be stopped, and what remains to be derived before stronger claims about gravity, matter, and survey-scale inference are justified. (shrink)

This paper supersedes Mathematics as the Unique Top-Down Projection of Intelligence: A Structural Theorem from Cognitional Mechanics and Noology (DOI: 10.5281/zenodo.19968224), which itself superseded the first edition of the programme (January 2026, DOI: 10.5281/zenodo.18280992). The first edition identified mathematical structures as stabilised residues of irreversible, non-commutative operational histories and positioned the framework as a meta-theoretical explanatory layer operating above existing mathematical foundations. The second edition established the stronger claim that mathematics is the unique top-down projection of the operational structure of (...) intelligence, with the induced mathematical category M uniquely determined up to identity automorphism and the projection map surjective with structurally undefined inverse. -/- The present work marks a foundational transition: it re-establishes all structural results of the second edition under Operatiology (DOI: 10.5281/zenodo.20350414), the framework that supersedes Cognitional Mechanics as the executive layer derived from Noology Version 2 (DOI: 10.5281/zenodo.18690463). The prior premise that the minimal operational structure is identified with M_3(C) is removed. The projection source is now the abstract rank-3 operational closure C^(3)_Πd itself, established within Tier-2 by Operatiology, without presupposing its identification with any specific algebraic structure. That identification is a separate Tier-3 question and does not enter any argument of the present work. -/- Under this clarified foundation, the central theorems are re-established: the projection Π_math from C^(3)_Πd to M is surjective with structurally undefined inverse; Aut(M) = {id}; and M is the unique fixed point of the Πd-saturation closure operator acting on Tier-3 formal structures. The prior corpus citations from the Cognitional Mechanics framework are retained for historical continuity only; no result derived by means of M_3(C) is invoked in any proof or definition. The sole logical foundation of the present paper is Operatiology, derived from Noology Version 2. -/- A new result absent from the second edition is introduced: an internal operational distance d_op on M, measuring the index family type required for operational determination of each mathematical structure. This yields a five-layer classification: Layer 1 (finite index families, algebra); Layer 2 (countable type, discrete infinite structures); Layer 3 (power-set type, continuous and geometric structures); Layer 4 (unrestricted type, axiomatic and functorial structures); Layer 5 (maximal type, large-cardinal and higher-categorical structures). Algebra occupies Layer 1 as the unique mathematical domain closest to C^(3)_Πd. The prior Operatiology mathematics series covering number systems, analytic structures, and set theory is positioned within this hierarchy. -/- Consequences include the dissolution of the Platonist-formalist dichotomy; the structural impossibility of alternative mathematics; the reinterpretation of mathematical effectiveness in physics as a tautology; and the reclassification of foundational competition among ZFC, category theory, and type theory as a compression-efficiency competition internal to Tier-3, evaluated from above by C^(3)_Πd. -/- Published on May 24, 2026 DOI:10.5281/zenodo.20362852. (shrink)

The general principle of epistemic closure stipulates that epistemic properties are transmissible through logical means. According to this principle, an epistemic operator, say ε, should satisfy any valid scheme of inference, such as: if ε(p entails q), then ε(p) entails ε(q). The principle of epistemic closure under known entailment (ECKE), a particular instance of epistemic closure, has received a good deal of attention since the last thirty years or so. ECKE states that: if one knows that (...) p entails q, and she knows that p, then she knows that q. It is widely accepted that ECKE constitutes an important piece of the skeptical argument, but the acceptance of an unrestricted version of ECKE is still a matter of debate. On the side of the defenders of ECKE, one finds Stine (1976), Brueckner (1985), Vogel (1990), and Feldman (1995). Others proposed a refutation or a limitation of the principle, like Dretske (1970), Nozick (1981), Hales (1995), Williams (1996), and Sosa (1999). As it turns out, the relevant alternatives view (RAV) elaborated by Dretske, which restricts the scope of ECKE, has been discussed extensively and acknowledged as one of the most important contributions. There is nonetheless a major unsolved difficulty pertaining to Dretske-RAV: the notion of relevant alternatives is defined in such a way that it is bounded by counterfactual possibilities. This ontological import leaves open the door to the skeptic. Some have tried to give more precision to this notion, like Stine (1976), who appealed to a Gricean approach to define relevant alternatives in conversational contexts. My proposal is in accordance with the gist of Dretske’s strategy, i.e. to restrict the validity of ECKE, and I claim that in order to escape the difficulties inherent to RAV one has to introduce a more robust notion, the notion of epistemic context. Epistemic contexts are a subclass of propositional contexts. In that perspective, the closure property is expressed in terms of a property of a relation between epistemic contexts. ECKE holds when and only when either the epistemic context of the premisses is the same as the epistemic context of the conclusion, or the epistemic context change between the premisses and the conclusion is permissible. Permissibility of epistemic context change is a function of consistency. By means of this epistemic context approach, I will show that: (1) epistemic contexts are defined by basic propositions (unchallenged justified beliefs), (2) ECKE holds only under very specific constraints, and (3) the skeptical argument involves a non-permissible change of epistemic context and, by the same token, cannot rely upon ECKE. (shrink)

A dispute in epistemology has arisen over whether some class of things epistemic (things known or justified, for example) is closed under some operation involving the notion of what follows deductively from members of this class. Very few philosophers these days believe that if you know that p, and p entails q, then you know that q. But many philosophers think that something weaker holds, for instance that if you know that p, and p entails q, then you are in (...) a position to know that p, or if you know that p and you competently deduce p from q, then you know that q. However there are some considerations tracing back to Wittgenstein’s On Certainty and some early work by Fred Dretske that suggests even these weaker principles are false. (shrink)

Sanford Goldberg’s account of epistemic coverage constitutes a special case of Douglas Walton’s view that epistemic closure arises from dialectical argument. Walton’s pragmatic version of epistemic closure depends on dialectical norms for closing an argument, and epistemic coverage operates at the limits of argument closure because it minimizes dialectical exchange. Such closure works together with a shared hypothetical consideration to justify dismissal of surprising claims.