Languages involving modalities and languages involving vagueness have each been thoroughly studied. On the other hand, virtually nothing has been said about the interaction of modality and vagueness. This paper aims to start filling that gap. Section 1 is a discussion of various possible sources of vague modality. Section 2 puts forward a model theory for a quantified language with operators for modality and vagueness. The model theory is followed by a discussion of the resulting logic. In Section 3, the (...) framework will permit us to address a puzzle raised by Elizabeth Barnes and Robert Williams. (shrink)

Karl Popper developed a theory of deductive logic in the late 1940s. In his approach, logic is a metalinguistic theory of deducibility relations that are based on certain purely structural rules. Logical constants are then characterized in terms of deducibility relations. Characterizations of this kind are also called inferential definitions by Popper. In this paper, we expound his theory and elaborate some of his ideas and results that in some cases were only sketched by him. Our focus is on Popper's (...) notion of duality, his theory of modalities, and his treatment of different kinds of negation. This allows us to show how his works on logic anticipate some later developments and discussions in philosophical logic, pertaining to trivializing connectives, the duality of logical constants, dual-intuitionistic logic, the conservativeness of language extensions, the existence of a bi-intuitionistic logic, the non-logicality of minimal negation, and to the problem of logicality in general. (shrink)

In the 1920s, Paul Hertz (1881-1940) developed certain calculi based on structural rules only and established normal form results for proofs. It is shown that he anticipated important techniques and results of general proof theory as well as of resolution theory, if the latter is regarded as a part of structural proof theory. Furthermore, it is shown that Gentzen, in his first paper of 1933, which heavily draws on Hertz, proves a normal form result which corresponds to the completeness of (...) prepositional SLD-resolution in logic programming. (shrink)

In this paper we consider the class of truth-functional modal many-valued logics with the complete lattice of truth-values. The conjunction and disjunction logic operators correspond to the meet and join operators of the lattices, while the negation is independently introduced as a hierarchy of antitonic operators which invert bottom and top elements. The non-constructive logic implication will be defined for a subclass of modular lattices, while the constructive implication for distributive lattices (Heyting algebras) is based on relative pseudo-complements as in (...) intuitionistic logic. We show that the complete lattices are intrinsically modal, with banal identity modal operator. We define the autoreferential set-based representation for the class of modal algebras, and show that the autoreferential Kripke-style semantics for this class of modal algebras is based on the set of possible worlds equal to the complete lattice of algebraic truth-values. The philosophical assumption is based on the consideration that each possible world represents a level of credibility, so that only propositions with the right logic value (i.e., level of credibility) can be accepted by this world, then we connect it with paraconsistent properties and LFI logics. The bottom truth value in this complete lattice corresponds to the trivial world in which each formula is satisfied, that is, to the world with explosive inconsistency. The top truth value corresponds to the world with classical logics, while all intermediate possible worlds represent the different levels of paraconsistent logics. (shrink)

Hypotheticals, conditionals, and their connecting relation, implication, dramatically changed their meanings during the nineteenth and early part of the twentieth century. Modern logicians ordinarily do not distinguish between the terms hypothetical and conditional. Yet in the late nineteenth century their meanings were quite different, their ties to the implication relation either were unclear, or the implication relation was used exclusively as a logical operator. I will trace the development of implication as an inference operator from these earlier notions into the (...) first third of the twentieth century using as the starting point the ideas primarily of R. Whately and W. Hamilton before discussing the work of the transitional logicians, A. De Morgan and G. Boole on these topics. Then we discuss the relevant views of four prominent but relatively unknown nineteenth century British logicians, W.E. Johnson, J.N. Keynes, E.E.C. Jones, and H. MacColl, as well as those of the more influential logicians, W.S. Jevons and J. Venn, closing with a section on ‘implication as inference’ where we explore some key ideas of B. Russell and sketch the work of D. Hilbert, P. Hertz, and G. Gentzen who together are responsible for the development of the modern ideas related to the subjects of this paper. (shrink)

In this paper we elaborate a conception of entailment based on what we call the Ackermann principle, which explicates valid entailment through a logical connection between sentences depending on their informational content. We reconstruct Dunn’s informational semantics for entailment on the basis of Restall’s approach, with assertion and denial as two independent speech acts, by introducing the notion of a ‘position description’. We show how the machinery of position descriptions can effectively be used to define the positive and the negative (...) information carried by sentences of a given language and to present a formalized version of the Ackermann principle as an inclusion relationship between the informational contents of the conclusions and the premises of a valid entailment. Being so interpreted, the entailment relation exhibits certain properties, including the property of transitivity. Whereas properties such as Anderson and Belnap’s variable sharing property or Parry’s proscriptive principle are normally presented as imposing a relevance requirement on valid entailment, the suggested formalization of the Ackermann principle supports all of Gentzen’s structural rules, including weakening, a rule that is normally given up in sequent-style proof systems for relevance logics. In this way we propose an Ackermann-inspired explication of the nature of entailment as a relation between the informational contents of sentences. (shrink)

The status of the equality predicate as a logical constant is problematic. In the paper we look at the problem from the proof-theoretic standpoint and survey several ways of treating equality in formal systems of different sorts. In particular, we focus on the framework of sequent calculus and examine equality in the light of criteria of logicality proposed by Hacking and Došen. Both attempts were formulated in terms of sequent calculus rules, although in the case of Došen it has a (...) nonstandard character. It will be shown that equality can be characterised in a way which satisfies Došen’s criteria of logicality. In the case of Hacking’s approach the fully satisfying result can be obtained only for languages with a nonempty, finite set of predicate constants other than equality. Otherwise, cut elimination theorem fails to hold. (shrink)

Structural reasoning is simply reasoning that is governed exclusively by structural rules. In this context a proof system can be said to be structural if all of its inference rules are structural. A logic is considered to be structuralizable if it can be equipped with a sound and complete structural proof system. This paper provides a general formulation of the problem of structuralizability of a given logic, giving specific consideration to a family of logics that are based on the Dunn–Belnap (...) four-valued semantics. It is shown how sound and complete structural proof systems can be constructed for a spectrum of logics within different logical frameworks. (shrink)

Attempts to articulate the real meaning or ultimate significance of a famous theorem comprise a major vein of philosophical writing about mathematics. The subfield of mathematical logic has supplie...

Entailment relations, introduced by Scott in the early 1970s, provide an abstract generalisation of Gentzen’s multi-conclusion logical inference. Originally applied to the study of multi-valued logics, this notion has then found plenty of applications, ranging from computer science to abstract algebra. In particular, an entailment relation can be regarded as a constructive presentation of a distributive lattice and in this guise it has proven to be a useful tool for the constructive reformulation of several classical theorems in commutative algebra. In (...) this paper, motivated by these concrete applications, we state and prove a cut-elimination result for inductively generated entailment relations. We analyse some of its consequences and describe the existing connections with analogous results in the literature. (shrink)

This open access book is a superb collection of some fifteen chapters inspired by Schroeder-Heister's groundbreaking work, written by leading experts in the field, plus an extensive autobiography and comments on the various contributions by Schroeder-Heister himself. For several decades, Peter Schroeder-Heister has been a central figure in proof-theoretic semantics, a field of study situated at the interface of logic, theoretical computer science, natural-language semantics, and the philosophy of language. -/- The chapters of which this book is composed discuss the (...) subject from a rich variety of angles, including the history of logic, the proper interpretation of logical validity, natural deduction rules, the notions of harmony and of synonymy, the structure of proofs, the logical status of equality, intentional phenomena, and the proof theory of second-order arithmetic. All chapters relate directly to questions that have driven Schroeder-Heister's own research agenda and to which he has made seminal contributions. The extensive autobiographical chapter not only provides a fascinating overview of Schroeder-Heister's career and the evolution of his academic interests but also constitutes a contribution to the recent history of logic in its own right, painting an intriguing picture of the philosophical, logical, and mathematical institutional landscape in Germany and elsewhere since the early 1970s. The papers collected in this book are illuminatingly put into a unified perspective by Schroeder-Heister's comments at the end of the book. Both graduate students and established researchers in the field will find this book an excellent resource for future work in proof-theoretic semantics and related areas. (shrink)

This collection of papers, celebrating the contributions of Swedish logician Dag Prawitz to Proof Theory, has been assembled from those presented at the Natural Deduction conference organized in Rio de Janeiro to honour his seminal research. Dag Prawitz’s work forms the basis of intuitionistic type theory and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics in Logic, Linguistics and Theoretical Computer Science. The range of contributions includes material on the extension of natural deduction with higher-order (...) rules, as opposed to higher-order connectives, and a paper discussing the application of natural deduction rules to dealing with equality in predicate calculus. The volume continues with a key chapter summarizing work on the extension of the Curry-Howard isomorphism, via methods of category theory that have been successfully applied to linear logic, as well as many other contributions from highly regarded authorities. With an illustrious group of contributors addressing a wealth of topics and applications, this volume is a valuable addition to the libraries of academics in the multiple disciplines whose development has been given added scope by the methodologies supplied by natural deduction. The volume is representative of the rich and varied directions that Prawitz work has inspired in the area of natural deduction. (shrink)

This book explains the first published consistency proof of PA. It contains the original Gentzen's proof, but it uses modern terminology and examples to illustrate the essential notions. The author comments on Gentzen's steps which are supplemented with exact calculations and parts of formal derivations. A notable aspect of the proof is the representation of ordinal numbers that was developed by Gentzen. This representation is analysed and connection to set-theoretical representation is found, namely an algorithm for translating Gentzen's notation into (...) Cantor normal form. The topic should interest researchers and students who work on proof theory, history of proof theory or Hilbert's program and who do not mind reading mathematical texts.​. (shrink)

The seminal book on graph theory by Dénes Kőnig, published in the year 1936, collected notions and results from precursory works from the mid to late nineteenth century by Hamilton, Cayley, Sylvester and others. More importantly, Kőnig himself contributed many of his own results that he had obtained in the more than twenty years that he had been working on this subject matter. What is noteworthy is the fact that the fundamentals of what he calls directed graphs are taken almost (...) exhaustively from Paul Hertz' 1922 article on structural reasoning about sentences of the form a→b. This is not a fact that is well known in logical circles, even though Kőnig fully acknowledges this in his book. In view of the numerous trends in the recent decades to describe and explicate logical matters by means of graphs, the fact that it was Hertz' foundation of structural reasoning that informed basic notions of graph theory in the first place is highly significant. The main goal of this paper is to summarize Hertz' article and demonstrate how Kőnig integrates the notions and results presented therein in his book. This is followed by an exposition of how and when Hertz' results were reinvented in terms of graph theory. A critical discussion of the opinion expressed by both Hertz and Kőnig that the more general sentences of the form (a1,…,an)→b, introduced by Hertz in a companion article in 1923, cannot be interpreted by graphs concludes this paper. (shrink)