In this paper we will consider the existing notions of bilateralism in the context of proof-theoretic semantics and propose, based on our understanding of bilateralism, an extension to logical multilateralism. This approach differs from what has been proposed under this name before in that we do not consider multiple speech acts as the core of such a theory but rather multiple consequence relations. We will argue that for this aim the most beneficial proof-theoretical realization is to use sequent calculi with (...) multiple sequent arrows satisfying some specific conditions, which we will lay out in this paper. We will unfold our ideas with the help of a case study in logical tetralateralism and present an extension of Almukdad and Nelson’s propositional constructive four-valued logic by unary operations of meaningfulness and nonsensicality. We will argue that in sequent calculi with multiple sequent arrows it is possible to maintain certain features that are desirable if we assume an understanding of the meaning of connectives in the spirit of proof-theoretic semantics. The use of multiple sequent arrows will be justified by the presence of congruentiality-breaking unary connectives. (shrink)

The present monograph is a slightly revised version of my Habilitations schrift Proof-theoretic Aspects of Intensional and Non-Classical Logics, successfully defended at Leipzig University, November 1997. It collects work on proof systems for modal and constructive logics I have done over the last few years. The main concern is display logic, a certain refinement of Gentzen's sequent calculus developed by Nuel D. Belnap. This book is far from offering a comprehensive presentation of generalized sequent systems for modal logics broadly conceived. (...) The proof-theory of non-classical logics is a rapidly developing field, and even the generalizations of the ordinary notion of sequent listed in Chapter 1 can hardly be presented in great detail within a single volume. In addition to further investigating the various approaches toward generalized Gentzen systems, it is important to compare them and to discuss their relative advantages and disadvantages. An initial attempt at bringing together work on different kinds of proof systems for modal logics has been made in [188]. Another step in the same direction is [196]. Since Chapter 1 contains introductory considerations and, moreover, every remaining chapter begins with some surveying or summarizing remarks, in this preface I shall only emphasize a relation to philosophy that is important to me, register the sources of papers that have entered this book in some form or another, and acknowledge advice and support. (shrink)

The topic of identity of proofs was put on the agenda of general (or structural) proof theory at an early stage. The relevant question is: When are the differences between two distinct proofs (understood as linguistic entities, proof figures) of one and the same formula so inessential that it is justified to identify the two proofs? The paper addresses another question: When are the differences between two distinct formulas so inessential that these formulas admit of identical proofs? The question appears (...) to be especially natural if the idea of working with more than one kind of derivations is taken seriously. If a distinction is drawn between proofs and disproofs (or refutations) as primitive entities, it is quite conceivable that a proof of one formula amounts to a disproof of another formula, and vice versa. A notion of inherited identity of derivations is introduced for derivations in a cut-free sequent system for Almukdad and Nelson’s constructive paraconsistent logic N4 with strong negation. The notion is obtained by identifying sequent rules the application of which has no effect on the identity of derivations. Then the notion of inherited identity is used to define a bilateralist notion of synonymy between formulas, which is a relation drawing more fine-grained distinctions between formulas than strong equivalence. (shrink)

This chapter is concerned with logical aspects of negation, i.e. with the role of negation in valid inferences and hence with the contribution negation makes to the truth and falsity conditions of declarative expressions. Negation is an important philosophical and logical concept. Often differences between logical systems can ‐ at least partially ‐ be described as differences between the notions of negation used in these logics.

In this paper, first some propositional conditional logics based on Belnap and Dunn’s useful four-valued logic of first-degree entailment are introduced semantically, which are then turned into systems of weakly and unrestrictedly connexive conditional logic. The general frame semantics for these logics makes use of a set of allowable (or admissible) extension/antiextension pairs. Next, sound and complete tableau calculi for these logics are presented. Moreover, an expansion of the basic conditional connexive logics by a constructive implication is considered, which gives (...) an opportunity to discuss recent related work, motivated by the combination of indicative and counterfactual conditionals. Tableau calculi for the basic constructive connexive conditional logics are defined and shown to be sound and complete with respect to their semantics. This semantics has to ensure a persistence property with respect to the preorder that is used to interpret the constructive implication. (shrink)

§ 1. Prolegomena 1. Die Logik, die in diesem Lehrbuch entwickelt wird, ist bestimmt durch die folgenden Kennzeichen: (1) Sie fuBt auf derselben Ontologie wie die von erkennbaren Wider­ sprlichen befreite und in diesem Sinne vertretbare klassische Mathe­ matik. Flir diese Ontologie ist charakteristisch die Grundvoraussetzung, daB die Objekte der Mathematik und mit ihnen die mathematischen Bereiche an sich existieren, wie die platonischen Ideen. Mit Bezug auf diesen An-sich-Charakter sprechen wir von einer platonischen Ontologie. Flir diese Ontologie existieren die unendlichen (...) Bereiche beliebig hoher Machtigkeit als fertig vorliegende Objekte in derselben Art wie die durch Aufzahlung ihrer Elemente erfaBbaren endlichen Mengen und in gleichem Range mit ihnen. Die weittragenden Folgen dieser Auffassung sind in zwei Haupt­ punkten konzentriert. Erster Hauptpunkt: die Beurteilung der Potenz­ mengen. Mit den abzahlbaren Mengen existieren, genauso wie im end­ lichen Falle, auch ihre Potenzmengen, mit dem ihnen zukommenden Charakter der Uberabziihlbarkeit. Das Uberabzahlbare steht also gleich­ berechtigt neben dem Abzahlbaren. Zweiter Hauptpunkt: der flir den Platonismus charakteristische Gehalt des ausgeschlossenen Dritten. Es genligt hier, die mengentheoretische Formulierung dieses Prinzips ins Auge zu fassen. Sie besagt, daB eine flir die Elemente einer beliebigen Menge, also mit EinschlieBung der unendlichen Mengen von einer be­ liebigen Machtigkeit erklarte Eigenschaft allen Elementen der Menge zukommt oder es gibt (wenigstens) ein Mengenelement, dem sie nicht zukommt, unabhangig davon, ob ein solches Element angegeben werden kann oder nicht. Ein Anwendungsfall ist das der elementaren Zahlen­ theorie angeh6rige Prinzip der kleinsten Zahl. (shrink)

A theory of substructural negations as impossibility and as unnecessity based on bi-intuitionistic logic, also known as Heyting-Brouwer logic, has been developed by Takuro Onishi. He notes two problems for that theory and offers the identification of the two negations as a solution to both problems. The first problem is the lack of a structural rule corresponding with double negation elimination for negation as impossibility, DNE, and the second problem is a lack of correspondence between certain sequents and a characterizing (...) frame property. While the identification of negation as impossibility and negation as unnecessity does solve Onishi’s problems, in general it nevertheless seems desirable to keep the two notions separate. The present paper addresses the first problem by introducing a reformulation of Onishi’s display sequent calculus in a language of sequents that incorporates Boolean negation. Instead of identifying negation as impossibility and negation as unnecessity, a notion of weak frame correspondence is defined. It is observed that DNE weakly corresponds with a certain frame property, and two structural sequent rules are presented that together also weakly correspond with that frame property and allow one to derive DNE. Moreover, the reformulated display calculus has an independent motivation by considerations on proof-theoretic semantics. (shrink)

Einführende Bemerkungen zu einem umstrittenen Begriff.

En el artículo se hace la presentación de las ideas que san Agustín destaca sobre la Virgen María en sus sermones de Navidad. Se hace en primer lugar una presentación de dichos sermones, para posteriormente abordar el tema de los dos nacimientos de Cristo, que es usado para explicar las dos naturalezas en Cristo (divina y humana). Se destacan asimismo los sermones navideños en los que san Agustín señala el paralelismo entre la Virgen María y la Iglesia, ya ambas son (...) vírgenes y madres. Como un tema central de los sermones de Navidad de san Agustín, el artículo destaca la virginidad perpetua de María. Finalmente se ofrecen unas conclusiones en las que se pone de manifiesto que los sermones de Navidad de san Agustín que tratan sobre la Virgen María son parte de su “teología predicada”. Así como sus grandes tratados desarrollan la Mariología, sus sermones son la popularización de dicho pensamiento mariano. (shrink)

Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.