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

In this thesis, we will study the * embedding of classical first-order logic in first-order S4, which is based on the * translation, originally introduced in Fitting (1970). The initial, main part is dedicated to a detailed modeltheoretic proof of the soundness of the embedding. This will follow the proof sketch in Fitting (1970). We will then outline a proof procedure for a proof-theoretic replication of the soundness result. Afterwards, a potential proof of faithfulness of the * embedding, read in terms of soundness and completeness, will be discussed. We will particularly highlight the many difficulties coming with it. In the final section, we will relate this discussion to the debate on notational variance in French (2019). We will do this by showing how a weaker version of French's notion of 'expressive equivalence' conforms to the model-theoretic soundness result. We will then conclude that the soundness result without completeness might contain rather little overall insight by relating it to the extensibility of classical logic to S4.

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- Modal Logic, Logic, Proof Theory, Mathematical Logic

L’errore giudiziario è una manchevolezza che può derivare, e non di rado deriva, da un uso inadeguato degli strumenti lato sensu logici che il giudice ha a disposizione.
La presente indagine si propone di approfondire proprio il rapporto tra logica del giudice ed errore giudiziario, seguendo un percorso tanto lineare quanto impegnativo.
Innanzitutto, illustrerò le caratteristiche generali dell’errore umano, le peculiarità dell’errore giudiziario e le varie accezioni di “errore giudiziario” che queste consentono di definire (in senso latissimo, lato, stretto e strettissimo) (§ 2).
Inoltre, ponendomi nella prospettiva delle due grandi concezioni della conoscenza giudiziale (intellettualismo e razionalismo), tratterò le cause salienti delle due componenti dell’errore giudiziario (errore qualificatorio ed errore probatorio), in particolare le cause razionali (suddistinte in cause ineluttabili e cause contrastabili dal giudice) (§ 3).

- by Gaetano Carlizzi
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- Criminal Law, Criminal Procedure, Jurisprudence, Proof Theory
- by hugo segundo
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- Proof Theory, Teoria Geral do Processo, Onus Da Prova, Direito Processual

Abstract. Nel dibattito sulla prova specialistica, in particolare scientifica, si evoca spesso una formula, “Iudex peritus peritorum”, per rendere l’idea dell’autonomia valutativa riservata al giudice anche in questa sfera. Tuttavia, non solo l’idea si particolarizza in varianti molteplici e difficili da discernere, ma neppure è chiaro quali di esse siano compatibili con l’epistemologia giudiziaria accolta nel nostro sistema penale. Il presente saggio tenta di affrontare entrambi i problemi. Dopo aver illustrato le grandi articolazioni del ragionamento probatorio specialistico, nonché i principali modi in cui il giudice può contribuire al suo svolgimento, esso sottopone a critica questi ultimi, considerati come altrettante immagini del iudex peritus peritorum.
Abstract. In the debate on the specialized evidence, especially on the scientific one, the formula “Iudex peritus peritorum” is often evoked to express the idea of the evaluation autonomy reserved to the judge also in this sphere. However, not only does the idea differentiate itself into multiple and difficult to discern variants, but it is not even clear which of them are compatible with the judicial epistemology accepted in our criminal system. This essay attempts to address both problems. Having illustrated the great articulations of the specialized evidential reasoning and the main ways in which the judge can contribute to its development, it critically scrutinizes the latter, considered as many images of the iudex peritus peritorum.

- by Gaetano Carlizzi
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- Philosophy of Science, Proof Theory, Law of evidence, Evidence

The systematic implications of irrational numbers, for those interested.

- by Nathan Coppedge
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- Mathematics, Number Theory, Set Theory, Proof Theory

STRONG INDUCTION and its underlying definitions are presented in this white paper (knowledge base).

- by Matheus Lobo
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- Mathematics, Proof Theory, Philosophy Of Mathematics, Mathematics Education

Indicates that everything standing on top of naive set theory is also proven.

In natural language. If I speak about a small cat in a small house,
then the definition of small is different for a cat and for a house. So, a
mathematician, or a logician would propose to make more rigorous this
text by putting an index at the predicate ”small”: smallcat having a distinct
definition that smallhouse. In the extreme it could seem extremely rigourous,
indexing ANY object, predicate constant or function, in a mathematical
text, and even more in a computing program, by types. However, the texts,
in vernacular languages, are not so ambiguous. We actually succeed to
communicate, with a lot of nuances sometimes, when we perfectly dominate
our language, why? We shall only remark two thinks.
1- ”small” is not so distinct in its definition when we speak about a
cat, or about a house : The definition in either case is of the form ”less than
...”
2- Actually the predicate ”small” has a hidden index, which consist in the
whole text around.
In mathematical language. It seems that the concept of context does
not appear in the formulations of traditional mathematics ... but it turn out
that when we practice relative methods, the introduction of the context is
natural, and gives a formal speech which resembles the natural language. I
will illustrate this through one example. I

- by Yves Peraire
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- Proof Theory

O Supremo Tribunal Federal teria aparentemente alterado seu anterior entendimento de que qualquer constatação de um crime em estado de flagrância dentro do domicílio era capaz de validar uma busca domiciliar realizada sem mandado judicial. Desde final de 2015 as buscas domiciliares são inválidas se não era possível divisar a ocorrência de um crime em estado de flagrância a partir do exterior do domicílio. No entanto, se o estado de flagrância for percebi-do apenas de dentro do domicílio e a entrada tenha se dado por consentimento do morador, é válida a prisão em flagrante dela decorrente? O direito fundamental da inviolabilidade domi-ciliar pode ser renunciado? Como deve ocorrer a renúncia à inviolabilidade do domicílio para que se entenda o consentimento como válido? Palavras-chave: BUSCA DOMICILIAR SEM MANDADO. LEGALIDADE. CONSENTI-MENTO. RENÚNCIA A DIREITOS FUNDAMENTAIS. RENÚNCIA À INVIOLABILI-DADE DOMICILIAR. REQUISITOS. ABSTRACT The Brazilian Federal Supreme Court seems to have altered its previous holding that any crime in flagrante state found within a house was able to validate a search and seizure conducted in a house without a court order. Since the end of 2015 warrantless house search and seizure are invalid if the flagrante state of a crime was not perceivable from the outside of the house. However, if the flagrante state is perceived only within the house and the entrance occurred by resident's consent, is it valid the arrest made that arises from the consented en-trance? Can the fundamental right of house inviolability be waived? How should a waiver of the house inviolability occur in order to be valid? Keywords: WARRANTLESS SEARCH AND SEIZURE. LEGALITY. CONSENT. WAIVER OF FUNDAMENTAL RIGHTS. WAIVER OF THE HOUSE INVIOLABILITY. REQUIREMENTS. Introdução Este estudo de caso tenta responder se e como o consentimento pode ser utilizado para autorizar a realização de busca e apreensão em domicílio sem mandado judicial e, por conseguinte, validando a prisão em flagrante dela decorrente e se há razão na nova ratio decidendi aparentemente adotado pelo STF.

- by Henrique de S . V . Lopes
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- Proof Theory, Proof and Reasoning, Miranda Rights, Waiver

First days of a logic course
This short paper sketches one logician’s opinion of some basic ideas that should be presented on the first days of any logic course. It treats the nature and goals of logic. It discusses what a student can hope to achieve through study of logic. And it warns of problems and obstacles a student will have to overcome or learn to live with. It introduces several key terms that a student will encounter in logic.

- by Andy Galloway
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- Distributed Computing, Proof Theory, Software Development, Domain Theory
- by Bob Coecke
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- Quantum Physics, Quantum Teleportation, Proof Theory, Quantum Information
- by Tetsuo Ida
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- Mathematics, Proof Theory, Computational Geometry, Relational Algebra

Nel volume viene esposta analiticamente la dimostrazione del teorema di eliminazione del Cut (Hauptsatz) dimostrato da G. Gentzen nelle sue “Untersuchungen über das logische Schliessen”. La dimostrazione originale di Gentzen è una dimostrazione fondata su una doppia induzione, sul rango delle derivazioni e sul grado della formula soggetta al taglio. In questo testo la dimostrazione è una dimostrazione a doppia induzione, ma sulla lunghezza della derivazione e sul grado della formula soggetta al taglio. Il metodo seguito dall’autore, è ripreso con miglioramenti nel rigore formale e nella completezza analitica dei passaggi da L. Heindorf, Elementare Beweistheorie, Wissenschaftsverlag, Mannheim 1994, pp. 103-140. Inoltre, ci sono due differenze. In primo luogo, mentre Heindorf dimostra il teorema per il calcolo intuizionistico LJ, qui è dimostrato per il il calcolo classico LK. In secondo luogo, il calcolo usato nel testo contiene solo la regola strutturale di Weakening e questo consente di semplificare la dimostrazione del teorema di eliminazione del taglio, evitando di dover far ricorso alla regola di fusione equivalente a quella del taglio.
Il volume si articola in tre parti: la prima comprende una esposizione preliminare del calcolo dei sequenti LK. Nella seconda si enuncia e si dimostra l’Hauptsatz. Nella terza sono illustrate alcune importanti conseguenze dell’Hauptsatz, alcune di carattere formale, altre di significato più filosofico. La seconda parte è arricchita di una appendice tecnica, nella quale si studiano le ragioni dell’uso della regola del Mix al posto di quella del Cut.

- by Sergio Galvan
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- Proof Theory

Le seguenti riflessioni hanno un fine teorico-pratico: imparare a trattare adeguatamente i due giudizi in cui si articola la decisione del giudice. Esse nascono da un’esperienza personale, che forse può essere utile comunicare. Nel corso della mia attività di giudice penale, ho notato che, sebbene la differenza concettuale tra giudizio probatorio e giudizio sussuntivo sia alquanto chiara, grazie alla netta differenza di senso delle quaestiones che essi mirano a risolvere, nondimeno tali giudizi interferiscono variamente tra di loro, in modi non sempre afferrabili. In particolare, in diverse occasioni mi sono trovato disorientato, in quanto ciò che credevo di star facendo o di aver fatto in funzione sussuntiva (in fase di istruzione, di deliberazione o di motivazione1) mi sembrava servire, al contempo, a fini probatori, o viceversa. Donde il desiderio di comprendere se queste interferenze siano di vari tipi, quali sono le loro cause, se siano tutte patologiche, se siano in parte evitabili e così via. D’altro canto, nello svolgimento dell’indagine, mi propongo di resistere alle opposte, diffuse tentazioni di far valere visioni astratte dell’attività del giudice o semplici intuizioni professionali: tentazioni parimenti incompatibili con un’aspirazione davvero teorica. Così, il contributo che vorrei offrire costituirà l’esito di un percorso graduale, articolato nei seguenti passaggi. Nel § 1, indicherò i principali fattori di complicazione del rapporto tra prova e sussunzione e cercherò di mostrare il carattere in buona misura apparente della problematicità che ne deriva. Nel § 2, sulla scorta di tale analisi preliminare, suddistinguerò in due tipi le principali tesi formulate sul nostro rapporto, tesi sovrappositive e tesi discretive. Nel § 3, col corredo di vari esempi tratti dal processo penale, proverò a sviluppare una delle tesi discretive, quella dell’Ermeneutica Giuridica Continentale, che tratta prova e sussunzione come attività di giudizio e si interessa al loro svolgimento naturale. Nel § 4, infine, ricapitolerò in forma poco più che stenografica i risultati dell’intera indagine.

- by Gaetano Carlizzi
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- Proof Theory, Hermeneutics (Research Methodology), Law of evidence, Hermeneutics

An introductory text to the analysis of Prover9’s proofs.

- by Edgar Vigário
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- Algebra, Computer Science, Logic, Proof Theory

The most scientific evidence-yet of simple perpetual motion.

- by Sam Buss
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- Proof Theory, First-Order Logic
- by Dale Griffin
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- Entrepreneurship, Decision Making, Proof Theory, Overconfidence

In my dissertation I address some questions concerning the proof theory of second order logic and its constructive counterpart, System F (Girard 1971). These investigations follow two distinct (though historically related) viewpoints in proof theory, which are compared throughout the text: on the one side, the proof theoretic semantics tradition inaugurated by Dummett and Prawitz (Prawitz 1971, Dummett 1991), focusing on the analysis of the inferential content of proofs; on the other side, the interactionist tradition arising from Kleene's realizability (Kleene 1945) and the Tait/Girard reducibility technique (Tait 1967, Girard 1971), which interprets proofs as untyped programs and focuses, rather, on the behavioral content of proofs, i.e. the way in which they interact through the cut-elimination algorithm.
A distinction is made between the issues of justifying and understanding ("explaining why" and "explaining how", as in Girard 2000) impredicative reasoning, i.e. between non elementary results like the Hauptsatz and the combinatorial analysis of proofs, seen as programs, i.e. recursive objects.
As for justifi cation, an epistemological analysis of the circularity involved in the second order Hauptsatz is developed; it is shown that the usual normalization arguments for second order logic do not run into the vicious circularity claimed by Poincaré and Russell, but involve a
diff erent, epistemic, form of circularity. Still, this weaker circularity makes justifi cation, in a sense, pointless; in particular, some examples of inconsistent higher order theories admitting epistemically circular normalization arguments are discussed.
As for the explanation issue, a constructive and combinatorial (i.e. independent from normalization) analysis of higher order order quantifi cation is developed along two directions, with some related technical results. The fi rst direction arises from the parametric and dinatural interpretations of polymorphism (Reynolds 1983, Girard-Scott-Scedrov 1992), which provide a clear mathematical meaning to Carnap's defense of impredicative quanti fication (Carnap 1983). In particular, the violation of the parametric condition leads to paradoxes which are often ignored in the philosophical literature (with the exception of Longo-Fruchart 1997). The analysis of the combinatorial content of these interpretations leads to a 1-completeness theorem (every normal closed -term in the universal closure of a simple type is typable in simple type theory), which connects the interactionist and the inferential conceptions of proof.
The second direction follows the analysis of the typing conditions of the -terms associated with intuitionistic second order proofs. To the \vicious circles" in the proofs there correspond recursive (i.e. circular) speci fications for the types of the -terms. The geometrical structure of these vicious circles is investigated (following Lechenadec 1989, Malecki 1990, Giannini - Ronchi Della Rocca 1991), leading to a combinatorial characterization of typability in some inconsistent extension of System F: since, as Girard's paradox shows, a typable term need not be normalizing, one is indeed naturally led to consider not normalizing theories.
Such investigations go in the direction both of a mathematical understanding of the structure generated by the vicious circles of impredicative theories and of the development of a proof-theoretic analysis of potentially incorrect or uncertain proofs.

- by Paolo Pistone
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- Type Theory, Proof Theory, Philosophy of Logic, Cut-Elimination
- by Paolo Pistone
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- Proof Theory, Linear Logic, Philosophy of Logic, Transcendentalism

El objetivo del presente trabajo es mostrar la correspondencia entre el cálculo de deducción natural intuicionista y el cálculo lambda con tipos, conocida como isomorfismo Curry-Howard. Se expone cómo se constituye un sistema lógico, cuáles son las propiedades que debe satisfacer y en qué marco teórico se realiza el estudio de estas propiedades. Sobre esto presentamos los cálculos lógicos de Gerhard Gentzen: el cálculo de deducción natural y el cálculo de secuentes, tanto para lógica clásica y la lógica intuicionista. Hecho esto, presentamos el cálculo lambda de en dos versiones, sin tipos y con tipos simples. Este cálculo fue presentado y usado por A. Church como un formalismo para definir funciones computables, siendo un planteamiento más abstracto de la noción de computabilidad que el de las máquinas de Turing. Por último mostramos cómo corresponden las fórmulas que representan los tipos en el cálculo lambda con tipos con las fórmulas del calculo de deducción natural intuicionista para proposiciones, mostrando ciertas limitaciones de este resultado. Quedamos entonces con un sistema que podría interpretarse como la definición formal de una semántica para el cálculo de decucción natural intuicionista donde la noción de construcción queda interpretada como cómputo expresable via cálculo lambda. El resultado podría ser extendido a una semántica intensional donde los procesos de deducción son entendidos como el uso de patrones inferenciales específicos.

- by Numa Tortolero
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- Logic, Proof Theory, Mathematical Logic, Lambda Calculus
- by Gaetano Carlizzi
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- Criminal Law, Jurisprudence, Comparative Law, Civil Law

A variety of projects in proof theory of relevance to the philosophy of mathematics are surveyed, including Gödel's incompleteness theorems, conservation results, independence results, ordinal analysis, predicativity, reverse mathematics,... more

- by Andrew Arana
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- Proof Theory, Philosophy Of Mathematics, Philosophy of Logic

In this paper a first-order version of hybrid logic is presented. The language is obtained by adding nominals, satisfaction operators and the down-arrow binder to classical first-order modal logic (including constants and function symbols). The satisfaction operators are applied to both formulas and terms. Moreover adding the universal modality is discussed. This first-order hybrid language is interpreted over varying domains and a sound and complete, fully internalized tableau system for this logic is given.

- by Jens Ulrik Hansen
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- Modal Logic, Proof Theory, Hybrid Logic, First-Order Logic
- by Dimiter Vakarelov
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- Semantics, Proof Theory, Modeling, Paraconsistent logic

Historical analysis and new concepts enable understanding of Riemann's Hypothesis (RH), his zeta function, mathematics, the principles enabling them, and ontological proofs. They enable new, post-modern metamathematics. New terms and definitions support comprehensive proof of new number theory, set theory, and proof theory. They enable deep understanding of the full scope of the historic context and causes of RH. The "Results" section provides basics and the technical context of work on RH and related problems. The "History" section summarizes the background of useful work and attempted solutions. Section 2 enables understanding of the work, enabling metalogic, and the domain of discourse. Section 3 provides comprehensively definitive, deeply explanatory proof of RH, and an integral critique of prior metatheory. It includes proofs of closely related problems, explanations, and how primal numbers may be rapidly, economically located. The new metatheorems of ontology, mathematics, numbers, sets, and proof also support new conjectures and possibilities. Disproof of the P/NP problem is included. Section 4 provides summary comments and predictions, for inspiration and verification.

- by Michael Lucas Monterey
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- Semiotics, Sociology, Psychology, Number Theory
- by Sam Buss
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- Proof Theory

My Oxford 1983doctoral dissertation

- by Göran Sundholm
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- Proof Theory, Metamathematics, Arithmetic, Ordinals
- by Mihai Ganea
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- Logic And Foundations Of Mathematics, Proof Theory, Arithmetic

La sentenza Cozzini, pronunciata dalla Corte Suprema di Cassazione nel 2010, ha stabilito i principali requisiti di
affidabilità delle informazioni rese dagli esperti (periti e consulenti) nel processo penale. Il presente articolo esamina
contenuti e funzioni di tale regime, anche in confronto all’analoga disciplina vigente nel processo americano, e
affronta alcuni interrogativi che impegneranno nei prossimi anni i giudici chiamati ad applicarlo.
Deciding the Cozzini case in 2010, the Italian Supreme Court of Cassation has ruled the main requirements which
expert witnesses must satisfy to be considered reliable in a criminal trial. The paper investigates contents and
functions of this regulation, on the one hand comparing it with the similar standard in force in the USA, on the
other hand dealing with some questions about it which criminal judges must address in the next years.

- by Gaetano Carlizzi
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- Epistemology, Proof Theory, Law of evidence, Evidence
- by Ludovico Fusco
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- Algebra, Universal Algebra, Model Theory, Logic

We offer a critical overview of two sorts of proof systems that may be said to characterize classical propositional logic indirectly (and non-standardly): refutation systems, which prove sound and complete with respect to classical contradictions, and rejection systems, which prove sound and complete with respect to the larger set of all classical non-tautologies. Systems of the latter sort are especially interesting, as they show that classical propositional logic can be given a paraconsistent characterization. In both cases, we consider Hilbert-style systems as well as Gentzen-style sequent calculi and natural-deduction formalisms.

- by Achille C . Varzi
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- Logic, Proof Theory, Philosophical Logic, Philosophy of Logic

ABSTRACT: This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich contemporary discussion. Much of Stoic logic appears surprisingly modern: a recursively formulated syntax with some truth-functional propositional operators; analogues to cut rules, axiom schemata and Gentzen's negation-introduction rules; an implicit variable-sharing principle and deliberate rejection of Thinning and avoidance of paradoxes of implication. These latter features mark the system out as a relevance logic, where the absence of duals for its left and right introduction rules puts it in the vicinity of McCall's connexive logic. Methodologically, the choice of meticulously formulated meta-logical rules in lieu of axiom and inference schemata absorbs some structural rules and results in an economical, precise and elegant system that values decidability over completeness.

- by Susanne Bobzien
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- Logic, Proof Theory, History of Logic, Structural Proof Theory

The Spanish translation of Haack's very successful book, PHILOSOPHY OF LOGICS (1978). This book covers virtually every important topic in philosophy of logic.

The version here is a scan of a second-hand copy, but all seems to be legible.

- by Susan Haack
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- Modal Logic, Intuitionistic Logic, Metaphysics, Ontology
- by JY B and +8
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- Jainism, Artificial Intelligence, Music Theory, Political Philosophy

While the LGB literature (Chomsky 1981) contains extensive discussion of empty categories in languages typologically like English on the one hand (non-pro-drop) and Italian on the other (pro-drop), relatively little has been said about languages like Russian, Polish or Czech, which fit neither of these two types of languages. The main purpose of this paper is to correct this typological imbalance by attempting to demonstrate whether current approaches need to be substantially revised in the light of data from Slavic and some other languages. The paper proceeds as follows: after making explicit the basic assumptions underlying the several versions of the theory of empty categories (EC), I will argue on the basis of Russian, Polish and Czech null-subjects and null-objects and their distribution that a number of revisions in theory's rules and basic principles is indeed necessary. Whereas the Minimalist program seems to correctly predict - on the basis of the principle of FULL INTERPRETATION (FI) and some additional assumptions (economy, greed, last ressort) - the existence of EC's, there is a problem with the notion of impoverished AGR as the only licensing formal condition, since languages without morphology, being at the same time pro-drop (e.g. Chinese, Japanese), cannot be explained this way due to lack of inflection. Although the notion of Morphological Uniformity of inflectional paradigm (Jaeggli/Safir 1989) seems to correctly predict the existence of Null-Subjects, there is no way how to include Null-Objects into this kind of explanation (p. 34). In both cases, pro must meet the licensing conditions of locality, agreement, case assignment, and, in case of thematic (referential) null subjects and objects, assignment of a referential theta-role. For languages without rich morphology, we therefore propose that both, null-subjects and null-objects, are bound - similarly to variables, by an Operator. This hypothesis is explained by the following facts that are observable in both pro and variables: 1) both variables and null-subjects/-objects stand in A-position; 2) Variables must be bound from A-bar positions, and null pronouns can be bound (and so they must be bound) from A-bar-positions (long scrambling); 3) null-subjects/-objects must have case, this is also a property of variables; 4) null-subjects/-objects must receive a theta-role, so do variables; 5) both, pronouns and variables are /- anaphors/.

Connections between Algebraic Logic and (ordinary) Logic. Algebraic co- unterpart of model theoretic semantics, algebraic counterpart of proof theory, and their connections. The class Alg(L) of algebras associated to any logic L. Equivalence theorems stating that L has a certain logical property iff Alg(L) has a certain algebraic property. (E.g. L admits a strongly complete Hilbert- style inference system iff

- by István Németi
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- Proof Theory, Algebraic Logic
- by Nathan Coppedge
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- Engineering, Mechanical Engineering, Physics, Proof Theory
- by Richard Zach
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- Logic, Proof Theory, Many-Valued Logic
- by Martin Salzmann
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- Proof Theory, Optimality Theory, LFG, English Syntax

In this thesis we discuss interpretations of Markov's Principle in constructive and computational systems. First an historical perspective is given, with the interpretations of Markov's Principle provided by various intuitionistic realizabilities. Then we present systems based on the Curry-Howard correspondence, and we develop a new system that is constructive, has a computational interpretation and is equivalent to Markov's principle. Finally, we present a way to transform proofs in classical logic into proofs in the new system.

- by Matteo Manighetti
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- Type Theory, Proof Theory, Proof-Theoretic Semantics, Constructive Mathematics

In 1933 G ¨ odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G ¨ odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G ¨ odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and-calculus.

According to the logical inferentialist, the meaning of a logical connective is determined by the inference rules that govern its use. Proof theoretic semantics attempts to make this idea precise in a proof theoretic framework, using for example natural deduction or sequent calculus rules. Since Prior's infamous connective tonk much of proof theoretic semantics have been occupied with formal anti-tonk conditions which rule out ill-behaved connectives (e.g. conservativeness, harmony). Common between them is that inference rules only succeed in determining the meaning of a connective if the proof theoretic conditions are fulfilled. On the traditional account, however, such conditions are insensitive to substructural dimensions of proof theory, e.g. the distinction between additive and multiplicative connectives. We argue that proof theoretic semantics ought to have the resources to attribute different meanings to substructurally distinct connectives. Subsequently we show how to develop a notion of proof theoretic harmony that preserves substructural distinctions from introduction to elimination rules. The substructural account of harmony can rule out cases of nonconservativeness that previous accounts have not dealt with.

- by Ole T Hjortland
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- Proof Theory, Inferentialism, Substructural Logics, Natural Deduction

This paper shows why the non-trivial zeros of the Riemann zeta function ζ will always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the... more

- by Bertrand Wong
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- Mathematics, Number Theory, Analytic Number Theory, Algebraic Number Theory