Proof Theory Research Papers - Academia.edu (original) (raw)

2025

Sumilla: 1. El artículo 268 del CPP estipula que el peligrosismo procesal es un requisito indispensable, determinante de la legitimidad constitucional de la prisión preventiva. El inciso c) del citado precepto prevé que "…el imputado, en razón a sus antecedentes y otras circunstancias del caso particular, permita colegir razonablemente que tratará de eludir la acción de la justicia (peligro de fuga) u obstaculizar la averiguación de la verdad (peligro de obstaculización)". Es evidente que determinar los antecedentes y las circunstancias del caso en concreto del peligrosismo requiere de una base acreditativa razonable que permita la evaluación de los mismos. No se trata de meras elucubraciones abstractas o de meras intuiciones, sino de datos o actos de aportación de hechos (medios investigativos) que consoliden un peligro concreto de fuga o de obstaculización. 2. A su vez, los artículos 269 y 270 del CPP definen pautas que permiten calificar los peligros en cuestión. Uno de los más relevantes es el arraigo social del imputado y si tiene o no facilidades para alejarse del país o permanecer oculto, sin perjuicio de la gravedad del delito cometido y la magnitud del daño causado y su comportamiento en el procedimiento o en otras causas, así como si está en condiciones de alterar elementos de prueba o influir en órganos de prueba. 3. Lo que se exige es un peligro concreto de fuga. El imputado cuando presuntamente cometió el delito era un médico legista del Instituto de Medicina Legal y, además, como profesional de la medicina-en caso se le suspenda o inhabilite del cargo público que desempeña-tiene la posibilidad de ejercer como médico en el sector privado. Nada dice que no lo hará o que no pueda hacerlo. Además, tiene un domicilio conocido y vive con su padre de edad avanzada, a quien atiende, y su hermano. 4. No se desprende que el imputado tuvo algo que ver con una presunta desaparición de documentos, de cuya realidad aún no existe dato sólido. Nada indica que el citado encausado ocultó esos documentos o presionó al técnico para hacerlo. El que el encausado tenga un cargo directivo en la Oficina Médico Legal no dice que necesariamente realizará las conductas positivas puntualizadas en el artículo 270 del CPP. Por lo demás, es claro que existe prueba directa y una diligencia de intervención y ulterior incautación del dinero en poder del procesado ALDO RICHARD LUCANO HERRERA, unida a la prueba pericial respectiva. Las posibilidades de afectar el material probatorio obtenido son mínimas.

2025, Archive for Mathematical Logic

In this article we provide an intrinsic characterization of the famous Howard-Bachmann ordinal in terms of a natural well-partial-ordering by showing that this ordinal can be realized as a maximal order type of a class of generalized trees with respect to a homeomorphic embeddability relation. We use our calculations to draw some conclusions about some corresponding subsystems of second order arithmetic. All these subsystems deal with versions of light-face Π 1 1 -comprehension.

2025, Logic, Methodology and Philosophy of Science IX, Proceedings of the Ninth International Congress of Logic, Methodology and Philosophy of Science

2025, Proof Theory

In this paper we shall investigate fragments of Kripke-Platek set theory with Infinity which arise from the full theory by restricting Foundation to Π n Foundation, where n ≥ 2. The strength of such fragments will be characterized in terms of the smallest ordinal α such that L α is a model of every Π 2 sentence which is provable in the theory.

2025, Lecture Notes in Logic

2025, New Computational Paradigms

This paper is concerned with metamathematical properties of intuitionistic set theories with choice principles. It is proved that the disjunction property, the numerical existence property, Church's rule, and several other metamathematical properties hold true for Constructive Zermelo-Fraenkel Set Theory and full Intuitionistic Zermelo-Fraenkel augmented by any combination of the principles of Countable Choice, Dependent Choices and the Presentation Axiom. Also Markov's principle may be added. Moreover, these properties hold effectively. For instance from a proof of a statement ∀n ∈ ω ∃m ∈ ω ϕ(n, m) one can effectively construct an index e of a recursive function such that ∀n ∈ ω ϕ(n, {e}(n)) is provable. Thus we have an explicit method of witness and program extraction from proofs involving choice principles. As for the proof technique, this paper is a continuation of . [32] introduced a selfvalidating semantics for CZF that combines realizability for extensional set theory and truth.

2025, Archive for Mathematical Logic

Let T be a suitable system of classical set theory. We will show, that the Σ 1 spectrum of T, i.e. the set of ordinals having good Σ 1 definition in T is an initial segment of the ordinals.

2025, Annals of Pure and Applied Logic

The larger project broached here is to look at the generally Π 1 2 sentence " if X is well-ordered then f (X) is well-ordered", where f is a standard proof-theoretic function from ordinals to ordinals. It has turned out that a statement of this form is often equivalent to the existence of countable coded ω-models for a particular theory T f whose consistency can be proved by means of a cut elimination theorem in infinitary logic which crucially involves the function f . To illustrate this theme, we prove in this paper that the statement " if X is well-ordered then ε X is wellordered" is equivalent to ACA + 0 . This was first proved by Marcone and Montalban [7] using recursion-theoretic and combinatorial methods. The proof given here is principally proof-theoretic, the main techniques being Schütte's method of proof search (deduction chains) and cut elimination for a (small) fragment of L ω 1 ,ω .

2025, Annals of Pure and Applied Logic

We characterize the proof-theoretic strength of systems of explicit mathematics with a general principle (MID) asserting the existence of least fixed points for monotone inductive definitions, in terms of certain systems of analysis and set theory. In the case of analysis, these are systems which contain the C&axiom of choice and @-comprehension for formulas without set parameters. In the case of set theory, these are systems containing the Kripke-Platek axioms for a recursively inaccessible universe together with the existence of a stable ordinal. In all cases, the exact strength depends on what forms of induction are admitted in the respective systems.

2025, Annals of Pure and Applied Logic

The paper contains proof-theoretic investigations on extensions of Kripke-Platek set theory, KP, which accommodate first order reflection. Ordinal analyses for such theories are obtained by devising cut elimination procedures for infinitary calculi of ramified set theory with Π n reflection rules. This leads to consistency proofs for the theories KP + Π n -reflection using a small amount of arithmetic (PRA) and the well-foundedness of a certain ordinal notation system with respect to primitive recursive descending sequences. Regarding future work, we intend to avail ourselves of these new cut elimination techniques to attain an ordinal analysis of Π 1 2 comprehension by approaching Π 1 2 comprehension through transfinite levels of reflection.

2025, Annals of Pure and Applied Logic

While it is known that intuitionistic ZF set theory formulated with Replacement, IZF R , does not prove Collection it is a longstanding open problem whether IZF R and intuitionistic set theory ZF formulated with Collection, IZF, have the same proof-theoretic strength. It has been conjectured that IZF proves the consistency of IZF R . This paper addresses similar questions but in respect of constructive Zermelo-Fraenkel set theory, CZF. It is shown that in the latter context the proof-theoretic strength of Replacement is the same as that of Strong Collection and also that the functional version of the Regular Extension Axiom is as strong as its relational version. Moreover, it is proved that, contrary to IZF, the strength of CZF increases if one adds an axiom asserting that the trichotomous ordinals form a set. Unlike IZF, constructive Zermelo-Fraenkel set theory is amenable to ordinal analysis and the proofs in this paper make pivotal use thereof in the guise of well-ordering proofs for ordinal representation systems.

2025, Archive for Mathematical Logic

Universes of types were introduced into constructive type theory by . The idea of forming universes in type theory is to introduce a universe as a set closed under a certain specified ensemble of set constructors, say C. The universe then "reflects" C. This is the first part of a paper which addresses the exact logical strength of a particular such universe construction, the so-called superuniverse due to Palmgren (cf. [16,). It is proved that Martin-Löf type theory with a superuniverse, termed MLS, is a system whose proof-theoretic ordinal resides strictly above the Feferman-Schütte ordinal Γ 0 but well below the Bachmann-Howard ordinal. Not many theories of strength between Γ 0 and the Bachmann-Howard ordinal have arisen. MLS provides a natural example for such a theory.

2025, ResearchGate

Our remark first discusses the Birch - Swinnerton - Dyer conjecture and its possible proof in the context of mathematical emergence

2025, Archive for Mathematical Logic

The main goal of the present note is not to give a new proof of McAloon's result, but to attempt to mirror this result in arithmetic. By "arithmetic" I shall initially mean primitive recursive arithmetic, PRA, formulated in the language of ordinary arithmetic with Elinduction. Eventually, I shall mean Peano arithmetic, PA. In place of PRA and PA, one could take any pair T S T' of ne. extensions of PRA of sufficient difference in strength. For the sake of definiteness, however, I shall stick to PRA and PA. The "arithmetisation" of McAloon's construction is immediately suggested by rewriting Prov °°((x,rcp1) as PrZF F(p1). Formula (4) becomes ZFF-(p H `da[PrZF rcp') -3 R <_ a PrZF, (r-,cps) ]. (5) To obtain arithmetical McAloon-Rosser sentences, I simply replace the hierarchy of admissible set theories,

2025, Annals of Pure and Applied Logic

S AUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, TCo, of Transitive Containment, we shall refer as MAC. His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasizes, one that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke-Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory Z, and obtain an apparently new proof that Z is not finitely axiomatisable; we study Friedman's strengthening KP P + AC of KP + MAC, and the Forster-Kaye subsystem KF of MAC, and use forcing over ill-founded models and forcing to establish independence results concerning MAC and KP P ; we show, again using ill-founded models, that KP P + V = L proves the consistency of KP P ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret and Boffa

2025, Incompleteness for Higher-Order Arithmetic

Gödel's true-but-unprovable sentence from the first incompleteness theorem is purely logical in nature, i.e. not mathematically natural or interesting. An interesting problem is to find mathematically natural and interesting statements... more

2025, Publications of the Research Institute for Mathematical Sciences

2025, Proceedings of the Design Society

The literature on design distinguishes between exploration-based experimentation and validationbased experimentation. This typology relies on an assumption that exploration and validation cannot and should not be performed simultaneously in the same experimentation. By contrast, some practitioners, such as les Sismo, propose that proof of concept might combine these two logics. This raises the question of what design logic might enable this type of combination of exploration and validation. We first use design theory to build an experimentation design framework. This framework highlights a typology of proof logics in experimentation related to proof of the known and proof of the unknown. Second, we show that these proof models are supported by les Sismo's cases and describe a diversity of arrangements of exploration and validation mechanisms: sequential, parallel, and combinational. Through the formalisation of proof of concept as a double proof (proof of the known and proof of the unknown), we show that proof of concept can be more than a tool for the go/no-go decision by gradually validating propositions, questioning the relevance of propositions, and discovering new propositions to be investigated and tested.

2025, Logique & Analyse, Volume 52, pp. 3-20

Locke, Berkeley, Gentzen gave different justifications of universal generalization. In particular, Gentzen's justification is the one currently used in most logic textbooks. In this paper I argue that all such justifications are problematic, and propose an alternative justification which is related to the approach to generality of Greek mathematics.

2025, arXiv (Cornell University)

This paper defines the (first-order) conflict resolution calculus: an extension of the resolution calculus inspired by techniques used in modern SAT-solvers. The resolution inference is restricted to (first-order) unit-propagation and the calculus is extended with a mechanism for assuming decision literals and a new inference rule for clause learning, which is a first-order generalization of the propositional conflict-driven clause learning (CDCL) procedure. The calculus is sound (because it can be simulated by natural deduction) and refutationally complete (because it can simulate resolution), and these facts are proven in detail here.

2025, Automated Deduction – CADE 26

This paper introduces Scavenger, the first theorem prover for pure first-order logic without equality based on the new conflict resolution calculus. Conflict resolution has a restricted resolution inference rule that resembles (a first-order generalization of) unit propagation as well as a rule for assuming decision literals and a rule for deriving new clauses by (a first-order generalization of) conflict-driven clause learning. Author order is alphabetical by surname. 1 Not to be confused with the homonymous calculus for linear rational inequalities .

2025, Journal of Automated Reasoning

2025, Lecture Notes in Computer Science

This is an account of the semantics of a family of logics whose paradigm member is the relevant logic R of Anderson and Belnap. The formal semantic theory is well worn, having been discussed in the literature of such logics for over a quarter of a century. What is new here is the explication of that formal machinery in a way intended to make sense of it for those who have claimed it to be esoteric, `merely formal' or downright impenetrable. Our further goal is to put these logics in the service of practical reasoning systems, since the basic concept of our treatment is that of an agent a reasoning to conclusions using as assumptions the theory of agent b, where a and b may or may not be the same. This concept is fundamental to multi-agent reasoning.

2025, Zenodo (Preprint)

This paper presents a direct proof of the Beal Conjecture, which concerns the Diophantine equation A x + B y = C z , where A, B, and C are positive integers and x, y, z > 2. The conjecture asserts that these integers must share at least one common prime factor. The proof analyzes the prime factorization of each term and shows that, in the absence of a common prime factor, the combination of distinct prime powers necessarily induces intrinsic structural instability. By applying logarithmic comparisons and Diophantine stability arguments, the assumption of coprimality leads to a contradiction. Therefore, any solution must involve a common prime factor, thereby confirming the validity of the Beal Conjecture.

2025, Principles of Knowledge Representation and Reasoning

A Logic of Arbitrary and Indefinite Objects, LA, has been developed as the logic for knowledge representation and reasoning systems designed to support natural language understanding and generation, and commonsense reasoning. The motivations for the design of LA are given, along with an informal introduction to the theory of arbitrary and indefinite objects, and to LA itself. LA is then formally defined by presenting its syntax, proof theory, and semantics, which are given via a translation scheme between LA and the standard classical First-Order Predicate Logic. Soundness is proved. The completeness theorem for LA is stated, and its proof is sketched. LA is being implemented as the logic of SNePS 3, the latest member of the SNePS family of Knowledge Representation and Reasoning systems.

2025, Notre Dame Journal of Formal Logic

Let J and K be sets of (interpreted) logical primitives and let LJ and LK be languages based on J and if respectively, but having a common set of variables and non-logical constants. Let £Jbe a logic on LJ. Suppose t is a function which carries formulas of LJ into logically equivalent formulas of LK. It has been known since at least 1958 [6] that the completeness of the logic on LK (<£K), resulting from the translation (by t) of JQJ is not assured by the completeness of <£J. This result may not be widely known; in 1972 Crossley [2] made a mistake by overlooking it. Crossley constructed a logic, here called J£[Ί, &, Ξ)], by translating a logic known to be complete, 1 here called -C[" > -> v ί Crossley thought that ^[Ί, &, 3] is complete, but it is not. 2 Similar examples may have motivated William Frank's recent article in this Journal concerning the reasons why some translations do not preserve completeness. Unfortunately, there are two errors in the latter; it is the purpose of this article to set them straight. Frank's main theorem reads as follows: If Ί(A) is the closure of a formal system in a language £, with axioms Al, . . ., AN; and rules Rl, . . ., RM and t a rule of translation from -C to -C r , then T f , the closure of t(Al), . . .,t(AJV),

2025, Foundations of Science , Volume 18, pp. 93–106

The philosophy of mathematics of the last few decades is commonly distinguished into mainstream and maverick, to which a 'third way' has been recently added, the philosophy of mathematical practice. In this paper the limitations of these trends in the philosophy of mathematics are pointed out, and it is argued that they are due to the fact that all of them are based on a top-down approach, that is, an approach which explains the nature of mathematics in terms of some general unproven assumption. As an alternative, a bottom-up approach is proposed, which explains the nature of mathematics in terms of the activity of real individuals and interactions between them. This involves distinguishing between mathematics as a discipline and the mathematics embodied in organisms as a result of biological evolution, which however, while being distinguished, are not opposed. Moreover, it requires a view of mathematical proof, mathematical definition and mathematical objects which is alternative to the top-down approach.

2025, In Bharath Sriraman (ed.), Humanizing mathematics and its philosophy: Essays celebrating the 90th birthday of Reuben Hersh, pp. pp. 223–252

Reuben Hersh is a champion of maverick philosophy of mathematics. He maintains that mathematics is a human activity, intelligible only in a social context; it is the subject where statements are capable in principle of being proved or disproved, and where proof or disproof bring unanimous agreement by all qualified experts; mathematicians' proof is deduction from established mathematics; mathematical objects exist only in the shared consciousness of human beings. In this paper I describe my several points of agreement and few points of disagreement with Hersh's views. Keywords Maverick philosophy of mathematics • Front and back of mathematics • Deductive proof • Analytic proof • Inexhaustibility of Mathematics • Nature of mathematical objects

2025, Boringhieri, pp. 315

This book provides the elements of proof theory, that is, the part of mathematical logic in which proofs are taken as the object of mathematical study. The study of proofs can be conducted with two different purposes: to provide a general analysis of the notion of proof and its properties, or to develop a tool for Hilbert's Program. Corresponding to these two forms are two different forms of proof theory, general proof theory and reductive proof theory. The volume presents the fundamental concepts and methods of general proof theory, but does not neglect applications to reductive proof theory, although these are only a collateral aspect of the work. The treatment is developed around two types of results, the normalization theorem and the univocity property, and is essentially self-sufficient. Each chapter is accompanied by a number of exercises, the solutions to which are provided to enable the reader to proceed without outside help.

2025, Logic Journal of the IGPL, Volume 33, Issue 3

This work is motivated by the problem of finding the limit of the applicability of the first incompleteness theorem (G1). A natural question is, can we find a minimal theory for which G1 holds? We examine the Turing degree structure of recursively enumerable (RE) theories for which G1 holds and the interpretation degree structure of RE theories weaker than the theory R with respect to interpretation for which G1 holds. We answer all questions that we posed in [2], and prove more results about them. It is known that there are no minimal essentially undecidable theories with respect to interpretation. We generalize this result and give some general characterizations, which tell us under what conditions there are no minimal RE theories having some property with respect to interpretation.

2025, Journal of Logic and Computation

Rosser theories play an important role in the study of the incompleteness phenomenon and meta-mathematics of arithmetic. In this paper, we first define the notions of n-Rosser theories, exact n-Rosser theories, effectively n-Rosser theories and effectively exact n-Rosser theories (see Definition 1.6). Our definitions are not restricted to arithmetic languages. Then we systematically examine properties of n-Rosser theories and relationships among them. Especially, we generalize some important theorems about Rosser theories for recursively enumerable sets in the literature to n-Rosser theories in a general setting.

2025, Logic Journal of the IGPL, Volume 32, Issue 5, Pages 880-908

In this work, we aim at understanding incompleteness in an abstract way via metamathematical properties of formal theories. We systematically examine the relationships between the following twelve important metamathematical properties of arithmetical theories: Rosser, EI (effectively inseparable), RI (recursively inseparable), TP (Turing persistent), EHU (essentially hereditarily undecidable), EU (essentially undecidable), Creative, 0 (theories with Turing degree 0), REW (all RE sets are weakly representable), RFD (all recursive functions are definable), RSS (all recursive sets are strongly representable), RSW (all recursive sets are weakly representable). Given any two properties P and Q in the above list, we examine whether P implies Q.

2025, Notre Dame J. Formal Logic, 64(4): 425-439

This paper belongs to the research on the limit of the first incompleteness theorem. Effectively inseparable (EI) theories can be viewed as an effective version of essentially undecidable (EU) theories, and EI is stronger than EU. We examine this question: Are there minimal effectively inseparable theories with respect to interpretability? We propose tEI, the theory version of EI. We first prove that there are no minimal tEI theories with respect to interpretability (i.e., for any tEI theory T , we can effectively find a theory which is tEI and strictly weaker than T with respect to interpretability). By a theorem due to Pour-EI, we have that tEI is equivalent with EI. Thus, there are no minimal EI theories with respect to interpretability. Also, we prove that there are no minimal finitely axiomatizable EI theories with respect to interpretability.

2025, Journal of Logic and Computation, Volume 34, Issue 6, Pages 1010–1031

Effectively inseparable pairs and their properties play an important role in the meta-mathematics of arithmetic and incompleteness. Different notions are introduced and shown in the literature to be equivalent to effective inseparability. We give a much simpler proof of these equivalences using the strong double recursion theorem. Then we prove some results about the application of effective inseparability in meta-mathematics.

2025, Review of Analytic Philosophy, Vol. 2, No. 1

Gödelʼs incompleteness theorems, published in 1931, are important and profound results in the foundations and philosophy of mathematics. On the basis of new advances in research on incompleteness in the literature, we discuss the correct interpretations of Gödelʼs incompleteness theorems, their influence on various fields, and the limit of their applicability. The motivation of this paper is threefold: to explore the foundational and philosophical significance of new advances in research on incompleteness since Gödel, to introduce new advances in research on incompleteness to the general philosophy community, and to commemorate the 90th anniversary of the publication of Gödelʼs incompleteness theorems.

2025, Philosophia Mathematica, Volume 30, Issue 2, pp. 173-199

We use Gödel's incompleteness theorems as a case study for investigating mathematical depth. We examine the philosophical question of what the depth of Gödel's incompleteness theorems consists in. We focus on the methodological study of the depth of Gödel's incompleteness theorems, and propose three criteria to account for the depth of the incompleteness theorems: influence, fruitfulness, and unity. Finally, we give some explanations for our account of the depth of Gödel's incompleteness theorems.

2025, Bulletin of Symbolic Logic, Volume 27, Issue 2, pp. 113-167

We give a survey of current research on G\"{o}del's incompleteness theorems from the following three aspects: classifications of different proofs of G\"{o}del's incompleteness theorems, the limit of the applicability of G\"{o}del's first... more

2025, Bulletin of Symbolic Logic, Volume 26, Issue 3-4, pp. 268-286

In this paper, we examine the limit of applicability of G ödel's first incompleteness theorem (G1 for short). We first define the notion "G1 holds for the theory T ". This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which G1 holds. To approach this question, we first examine the following question: is there a theory T such that Robinson's R interprets T but T does not interpret R (i.e., T is weaker than R w.r.t. interpretation) and G1 holds for T? In this paper, we show that there are many such theories based on Jeřábek's work using some model theory. We prove that for each recursively inseparable pair A,B , we can construct a r.e. theory U A,B such that U A,B is weaker than R w.r.t. interpretation and G1 holds for U A,B . As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree 0 < d < 0 ′ , there is a theory T with Turing degree d such that G1 holds for T and T is weaker than R w.r.t. Turing reducibility. As a corollary, based on Shoenfield's work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which G1 holds.

2025, Dowodzenie w procesach kościelnych

"Proving Innocence of Clergymen Accused of Crimes Reserved for the Congregation for the Doctrine of the Faith"
Document of the Holly See and Conference of Polish Bishops regarding the law for clerics who committed an offense against the sixth commandment does not mention much about proofs. Considering the experience of the Catholic Church in other countries in which such problems had already intensified in the eighties the author referred to the experience of canon law specialists from these regions to present the proofs which can be used to defend the clergymen accused of pedophilia. T he introduction presents actions the accused should avoid not to hinder or ruin the defense plot. Then the actions that should be taken by the defense before the formal beginning of the trial to prove that the accusation is unjustified are pictured. As the evidence opinions of experts in sexology, moral theology, anthropology, psychology and lie detecting is presented. In the part of the article focused on witness testimony legal analogy to the guidelines of Congregation for the Doctrine of the Faith for cases involving solicitation in the confessional is presented. In the end the role of the attorney who should accompany the accused from the moment the accusation enters bishop’s office is highlighted. His advice should help to avoid the situations when the accused who does not know the canon law will deliver evidence against himself instead of proving his own innocence.

2025, HAL (Le Centre pour la Communication Scientifique Directe)

This paper studies how dependent types can be employed for a refined treatment of event types, offering a nice improvement to Davidson's event semantics. We consider dependent event types indexed by thematic roles and illustrate how, in the presence of refined event types, subtyping plays an essential role in semantic interpretations. We consider two extensions with dependent event types: first, the extension of Church's simple type theory as employed in Montague semantics that is familiar with many linguistic semanticists and, secondly, the extension of a modern type theory as employed in MTT-semantics. The former uses subsumptive subtyping, while the latter uses coercive subtyping, to capture the subtyping relationships between dependent event types. Both of these extensions have nice meta-theoretic properties such as normalisation and logical consistency; in particular, we shall show that the former can be faithfully embedded into the latter and hence has expected meta-theoretic properties. As an example of applications, it is shown that dependent event types give a natural solution to the incompatibility problem (sometimes called the event quantification problem) in combining event semantics with the traditional compositional semantics, both in the Montagovian setting with the simple type theory and in the setting of MTT-semantics.

2025, arXiv: General Topology

We prove a general categorical theorem for the extension of dualities. Applying it, we present new proofs of the de Vries Duality Theorem for the category CHaus of compact Hausdorff spaces and continuous maps, and of the recent Bezhanishvili-Morandi-Olberding Duality Theorem which extends the de Vries duality to the category Tych of Tychonoff spaces and continuous maps. In the process of doing so we obtain new duality theorems for the categories CHaus and Tych.

2025

En este trabajo exploramos el artículo original de Gödel sobre la prueba de suficiencia de los axiomas, también llamada completud (débil), para la lógica de primer orden. Antes haremos una introducción histórica sobre qué entendemos por completud y cómo fue evolucionando esta noción a lo largo de los últimos siglos, pues creemos pertinente dar una mínima aclaración de las distintas nociones empleadas y frecuentemente confundidas de dicho término. Para la elaboración del presente artículo hemos consultado las traducciones española e inglesa de las obras completas de Kurt Gödel, en las que aparece el artículo original de Gödel de 1930, traducidas y con los comentarios de Jesús Mosterín y S.C. Kleene, respectivamente.

2025

A common practice of ML systems development concerns the training of the same model under different data sets, and the use of the same (training and test) sets for different learning models. The first case is a desirable practice for identifying high quality and unbiased training conditions. The latter case coincides with the search for optimal models under a common dataset for training. These differently obtained systems have been considered akin to copies. In the quest for responsible AI, a legitimate but hardly investigated question is how to verify that trustworthiness is preserved by copies. In this paper we introduce a calculus to model and verify probabilistic complex queries over data and define four distinct notions: Justifiably, Equally, Weakly and Almost Trustworthy which can be checked analysing the (partial) behaviour of the copy with respect to its original. We provide a study of the relations between these notions of trustworthiness, and how they compose with each other and under logical operations. The aim is to offer a computational tool to check the trustworthiness of possibly complex systems copied from an original whose behavour is known.

2025, Does Adding Tennant's Principle of Uniform Primitive Recursive Reflection to PA Yield A Decidable Axiom Set?at follows explains why it is important to me and perhaps others. In 1931 Godel announced his two incompleteness results for formal systems of arithmetic equivalent to PA-- Peano Axioms in ...

The title is my question. What follows explains why it is important to me and perhaps others. In 1931 Godel announced his two incompleteness results for formal systems of arithmetic equivalent to PA-Peano Axioms in first order logic. The first result famously created a sentence G that was materially equivalent to a universal statement "saying" nothing was a proof of G in PA; however, while PA could prove of each number that it did not number a proof of G, it couldn't yield the full generalization nothing at all does. The problem was that while PA handled each number of possible relevant proofs of G, PA famously has nonstandard models (in 1934 Skolem showed how to create one) and some nonstandard models (sets including integers but with other items "floating above") make G true and some false. Thus, no theorem status for G in PA. It seemed clear to most that we knew the full generalization of G is true, but how? And how if at all can this gap between truth and derivability be closed? Many have argued that this gap from Godel shows that there must be thick notion of arithmetic truth, more than just derivability. Incompleteness supposedly thus rules out deflationist accounts of arithmetic truth.

2025, Working Draft

This is very brief exploratory work looking for others' input: In several papers, starting in Deflationism and the Godel Phenomenon, Neil Tennant had suggested that to close the apparent gap between proof and truth created by Kurt Gödel's first incompleteness theorem, a system like PA should be supplemented with a Principle of Uniform Primitive Recursive Reflection: If PA itself can prove every numerical instance n of the primitive recursive one-place “has a proof,” which it can, then the full universal generalization (over everything m) may be asserted in the the enriched system PAupr. I ask if adding such an axiom to PA yields a new system with decidable axioms and proof system, and consider possible results.

2025, Bézout's identity

This here is a Proof of "Bézout's identity" in a diffrent way as i am using "Proof by induction"

2025

We investigate the power of first-order logic with only two variables over ω-words and finite words, a logic denoted by FO 2 . We prove that FO 2 can express precisely the same properties as linear temporal logic with only the unary temporal operators: "next", "previously", "sometime in the future", and "sometime in the past", a logic we denote by unary-TL. Moreover, our translation from FO 2 to unary-TL converts every FO 2 formula to an equivalent unary-TL formula that is at most exponentially larger, and whose operator depth is at most twice the quantifier depth of the first-order formula. We show that this translation is optimal. While satisfiability for full linear temporal logic, as well as for unary-TL, is known to be PSPACE-complete, we prove that satisfiability for FO 2 is NEXP-complete, in sharp contrast to the fact that satisfiability for FO 3 has non-elementary computational complexity. Our NEXP time upper bound for FO 2 satisfiability has the advantage of being in terms of the quantifier depth of the input formula. It is obtained using a small model property for FO 2 of independent interest, namely: a satisfiable FO 2 formula has a model whose "size" is at most exponential in the quantifier depth of the formula. Using our translation from FO 2 to unary-TL we derive this small model property from a corresponding small model property for unary-TL. Our proof of the small model property for unary-TL is based on an analysis of unary-TL types. * Part of the research reported here was conducted while the authors were visiting DIMACS as part of the Special Year on Logic and Algorithms.

2025, Underreview

This paper undertakes a foundational inquiry into logical inferentialism with particular emphasis on the normative standards it establishes and the implications these pose for classical logic. The central question addressed herein is: 'What is Logical Inferentialism & How do its Standards challenge Classical Logic?' In response, the study begins with a survey of the three principal proof systems that is, David Hilbert's axiomatic systems and Gerhard Gentzen's natural deduction and his sequent calculus, thus situating logical inferentialism within a broader proof-theoretic landscape. The investigation then turns to the core tenets of logical inferentialism by focusing on the role of introduction and elimination rules in determining the meaning of logical constants. Through this framework, natural deduction is evaluated as a system that satisfies key inferentialist virtues including harmony, conservativeness and the subformula property. Ultimately, the paper presents challenges to classical logic from intuitionist and revisionist perspectives by arguing that certain classical principles fail to uphold inferentialist standards, consequently undermining their legitimacy within a meaning-theoretic framework.