Sequent calculi constitute an interesting and important category of proof systems. They are much less known than axiomatic systems or natural deduction systems are, and they are much less known than they should be. Sequent calculi were designed as a theoretical framework for investigations of logical consequence, and they live up to the expectations completely as an abundant source of meta-logical results. The goal of this book is to provide a fairly comprehensive view of sequent calculi -- including a wide (...) range of variations. The focus is on sequent calculi for various non-classical logics, from intuitionistic logic to relevance logic, through linear and modal logics. A particular version of sequent calculi, the so-called consecution calculi, have seen important new developments in the last decade or so. The invention of new consecution calculi for various relevance logics allowed the last major open problem in the area of relevance logic to be solved positively: pure ticket entailment is decidable. An exposition of this result is included in chapter 9 together with further new decidability results (for less famous systems). A series of other results that were obtained by J. M. Dunn and me, or by me in the last decade or so, are also presented in various places in the book. Some of these results are slightly improved in their current presentation. Obviously, many calculi and several important theorems are not new. They are included here to ensure the completeness of the picture; their original formulations may be found in the referenced publications. This book contains very little about semantics, in general, and about the semantics of non-classical logic in particular. (shrink)

This book celebrates and expands on J. Michael Dunn’s work on informational interpretations of logic. Dunn, in his Ph.D. thesis, introduced a semantics for first-degree entailments utilizing the idea that a sentence can provide positive or negative information about a topic, possibly supplying both or neither. He later published a related interpretation of the logic R-mingle, which turned out to be one of the first relational semantics for a relevance logic. An incompatibility relation between information states lends itself to a (...) definition of negation and it has figured into Dunn's comprehensive investigations into representations of various negations. The informational view of semantics is also a prominent theme in Dunn’s research on other logics, such as quantum logic and linear logic, and led to the encompassing theory of generalized Galois logics. Dunn’s latest work addresses informational interpretations of the ternary accessibility relation and the very nature of information. The book opens with Dunn’s autobiography, followed by a list of his publications. It then presents a series of papers written by respected logicians working on different aspects of information-based logics. The topics covered include the logic R-mingle, which was introduced by Dunn, and its applications in mathematical reasoning as well as its importance in obtaining results for other relevance logics. There are also interpretations of the accessibility relation in the semantics of relevance and other non-classical logics using different notions of information. It also presents a collection of papers that develop semantics for various logics, including certain modal and many-valued logics. The publication of this book is well timed, since we are living in an "information age.” Providing new technical findings, intellectual history and careful expositions of intriguing ideas, it appeals to a wide audience of scholars and researchers. (shrink)

This book honors J. Michael Dunn, who was a preeminent relevance logician. Dunn's career spanned over 50 years and his research results had an impact on philosophy, mathematics and informatics. Dunn often used algebraic techniques in his research into logics such as relevance, orthomodular and substructural logics. He invented the logic R-mingle and the sequent calculus LR+; he proved crucial theorems about 2-valued first-order logic and non-classical higher-order logics - among many other results. The papers in this volume touch upon (...) topics that Dunn was concerned with. Some authors were students or colleagues of Dunn; some other authors had not met Dunn in person, but share his research interests. None of the articles published here have appeared in print before; indeed, most of the papers were written specifically for this collection. The diversity of the themes of the articles reflects the scope of Dunn's own research in logic. It will also ensure that anybody with an interest in logic - whether a student, a logician or a scholar in another field - will find reading this book a worthwhile endeavor. The editor, Katalin Bimbó was the 14th Ph.D. student of J. Michael Dunn at Indiana University in Bloomington, IN, U.S.A.; currently, she is a professor of philosophy at the University of Alberta in Canada. (shrink)

Four-valued semantics proved useful in many contexts from relevance logics to reasoning about computers. We extend this approach further. A sequent calculus is defined with logical connectives conjunction and disjunction that do not distribute over each other. We give a sound and complete semantics for this system and formulate the same logic as a tableaux system. Intensional conjunction and its residuals can be added to the sequent calculus straightforwardly. We extend a simplified version of the earlier semantics for this system (...) and prove soundness and completeness. Then, with some modifications to this semantics, we arrive at a mathematically elegant yet powerful semantics that we call generalized Kripke semantics. (shrink)

Kleene algebras and action logic were proposed to be solutions to the finite axiomatization problem of the algebra of regular sets (of strings). They are treated here as nonclassical logics—with Hilbert-style axiomatizations and semantics. We also provide intuitive accounts in terms of information states of the semantics which provide further insights into the formalisms. The three types of "Kripke-style'' semantics which we define develop insights from gaggle theory, and from our four-valued and generalized Kripke semantics for the minimal substructural logic. (...) Soundness and completeness are proven each time. (shrink)

Nonclassical logics have played an increasing role in recent years in disciplines ranging from mathematics and computer science to linguistics and philosophy. _Generalized Galois Logics_ develops a uniform framework of relational semantics to mediate between logical calculi and their semantics through algebra. This volume addresses normal modal logics such as K and S5, and substructural logics, including relevance logics, linear logic, and Lambek calculi. The authors also treat less-familiar and new logical systems with equal deftness.

Symmetic combinatory logic with the symmetric analogue of a combinatorially complete base (in the form of symmetric λ-calculus) is known to lack the Church-Rosser property. We prove a much stronger theorem that no symmetric combinatory logic that contains at least two proper symmetric combinators has the Church-Rosser property. Although the statement of the result looks similar to an earlier one concerning dual combinatory logic, the proof is different because symmetric combinators may form redexes in both left and right associated terms. (...) Perhaps surprisingly, we are also able to show that certain symmetric combinatory logics that include just one particular constant are not confluent. This result (beyond other differences) clearly sets apart symmetric combinatory logic from dual combinatory logic, since all dual combinatory systems with a single combinator or a single dual combinator are Church-Rosser. Lastly, we prove that a symmetric combinatory logic that contains the fixed point and the one-place identity combinator has the Church-Rosser property. (shrink)

A class of ordered relational topological spaces is described, which we call orthomodular spaces. Our construction of these spaces involves adding a topology to the class of orthomodular frames introduced by Hartonas, along the lines of Bimbó's topologization of the class of orthoframes employed by Goldblatt in his representation of ortholattices. We then prove that the category of orthomodular lattices and homomorphisms is dually equivalent to the category of orthomodular spaces and certain continuous frame morphisms, which we call continuous (...) weak p‐morphisms. It is well‐known that orthomodular lattices provide an algebraic semantics for the quantum logic. Hence, as an application of our duality, we develop a topological semantics for using orthomodular spaces and prove soundness and completeness. (shrink)

Reader-friendly without compromising the precision of exposition, the book includes many new research results not found in the available literature.

The implicational fragment of the logic of relevant implication, $R_{\to}$ is one of the oldest relevance logics and in 1959 was shown by Kripke to be decidable. The proof is based on $LR_{\to}$ , a Gentzen-style calculus. In this paper, we add the truth constant $\mathbf{t}$ to $LR_{\to}$ , but more importantly we show how to reshape the sequent calculus as a consecution calculus containing a binary structural connective, in which permutation is replaced by two structural rules that involve $\mathbf{t}$ (...) . This calculus, $LT_\to^{\text{\textcircled{$\mathbf{t}$}}}$ , extends the consecution calculus $LT_{\to}^{\mathbf{t}}$ formalizing the implicational fragment of ticket entailment . We introduce two other new calculi as alternative formulations of $R_{\to}^{\mathbf{t}}$ . For each new calculus, we prove the cut theorem as well as the equivalence to the original Hilbert-style axiomatization of $R_{\to}^{\mathbf{t}}$ . These results serve as a basis for our positive solution to the long open problem of the decidability of $T_{\to}$ , which we present in another paper. (shrink)

The implicational fragment of the logic of relevant implication, $R_\to$ is known to be decidable. We show that the implicational fragment of the logic of ticket entailment, $T_\to$ is decidable. Our proof is based on the consecution calculus that we introduced specifically to solve this 50-year old open problem. We reduce the decidability problem of $T_\to$ to the decidability problem of $R_\to$. The decidability of $T_\to$ is equivalent to the decidability of the inhabitation problem of implicational types by combinators over (...) the base $\{\textsf{B},\textsf{B}',\textsf{I},\textsf{W}\}$. (shrink)

Two consecution calculi are introduced: one for the implicational fragment of the logic of entailment with truth and another one for the disjunction free logic of nondistributive relevant implication. The proof technique—attributable to Gentzen—that uses a double induction on the degree and on the rank of the cut formula is shown to be insufficient to prove admissible various forms of cut and mix in these calculi. The elimination theorem is proven, however, by augmenting the earlier double inductive proof with additional (...) inductions. We also give a new purely inductive proof of the cut theorem for the original single cut rule in Gentzen’s sequent calculus $ LK $ without any use of mix. (shrink)

Two consecution calculi are introduced: one for the implicational fragment of the logic of entailment with truth and another one for the disjunction free logic of nondistributive relevant implication. The proof technique—attributable to Gentzen—that uses a double induction on the degree and on the rank of the cut formula is shown to be insufficient to prove admissible various forms of cut and mix in these calculi. The elimination theorem is proven, however, by augmenting the earlier double inductive proof with additional (...) inductions. We also give a new purely inductive proof of the cut theorem for the original single cut rule in Gentzen’s sequent calculus $ LK $ without any use of mix. (shrink)

A sequent calculus for the positive fragment of entailment together with the Church constants is introduced here. The single cut rule is admissible in this consecution calculus. A topological dual gaggle semantics is developed for the logic. The category of the topological structures for the logic with frame morphisms is proven to be the dual category of the variety, that is defined by the equations of the algebra of the logic, with homomorphisms. The duality results are extended to the logic (...) of entailment that includes a De Morgan negation. (shrink)

Urquhart works in several areas of logic where he has proved important results. Our paper outlines his topological lattice representation and attempts to relate it to other lattice representations. We show that there are different ways to generalize Priestley’s representation of distributive lattices—Urquhart’s being one of them, which tries to keep prime filters in the representation. Along the way, we also mention how semi-lattices and lattices figured into Urquhart’s work.

Symmetric generalized Galois logics (i.e., symmetric gGl s) are distributive gGl s that include weak distributivity laws between some operations such as fusion and fission. Motivations for considering distribution between such operations include the provability of cut for binary consequence relations, abstract algebraic considerations and modeling linguistic phenomena in categorial grammars. We represent symmetric gGl s by models on topological relational structures. On the other hand, topological relational structures are realized by structures of symmetric gGl s. We generalize the weak (...) distributivity laws between fusion and fission to interactions of certain monotone operations within distributive super gGl s. We are able to prove appropriate generalizations of the previously obtained theorems—including a functorial duality result connecting classes of gGl s and classes of structures for them. (shrink)

A ternary relation is often used nowadays to interpret an implication connective of a logic, a practice that became dominant in the semantics of relevance logics. This paper examines two early manuscripts --- one by Routley, another by Meyer --- in which they were developing set-theoretic semantics for various relevance logics. A standard presentation of a ternary relational semantics for, let us say, the logic of relevant implication R is quite illuminating, yet the invention of this semantics was fraught with (...) false starts. Meyer's manuscript, in which he builds on some ideas from Routley's manuscript, essentially contains a relational semantics for which R^{ot} is sound and complete. (shrink)

The fixed point combinator (Y) is an important non-proper combinator, which is defhable from a combinatorially complete base. This combinator guarantees that recursive equations have a solution. Structurally free logics (LC) turn combinators into formulas and replace structural rules by combinatory ones. This paper introduces the fixed point and the dual fixed point combinator into structurally free logics. The admissibility of (multiple) cut in the resulting calculus is not provable by a simple adaptation of the similar proof for LC with (...) proper combinators. The novelty of our proof—beyond proving the cut for a newly extended calculus–is that we add a fourth induction to the by-and-large Gentzen-style proof. (shrink)

We define dual and symmetric combinatory calculi (inequational and equational ones), and prove their consistency. Then, we introduce algebraic and set theoretical relational and operational - semantics, and prove soundness and completeness. We analyze the relationship between these logics, and argue that inequational dual logics are the best suited to model computation.

. Relational semantics for nonclassical logics lead straightforwardly to topological representation theorems of their algebras. Ortholattices and De Morgan lattices are reducts of the algebras of various nonclassical logics. We define three new classes of topological spaces so that the lattice categories and the corresponding categories of topological spaces turn out to be dually isomorphic. A key feature of all these topological spaces is that they are ordered relational or ordered product topologies.

Relevance logics are known to be sound and complete for relational semantics with a ternary accessibility relation. This paper investigates the problem of adequacy with respect to special kinds of dynamic semantics (i.e., proper relation algebras and relevant families of relations). We prove several soundness results here. We also prove the completeness of a certain positive fragment of R as well as of the first-degree fragment of relevance logics. These results show that some core ideas are shared between relevance logics (...) and relation algebras. Some details of certain incompleteness results, however, pinpoint where relevance logics and relation algebras diverge. To carry out these semantic investigations, we define a new tableaux formalization and new sequent calculi (with the single cut rule admissible) for various relevance logics. (shrink)

Combinatory logic is known to be related to substructural logics. Algebraic considerations of the latter, in particular, algebraic considerations of two distinct implications, led to the introduction of dual combinators in Dunn & Meyer 1997. Dual combinators are "mirror images" of the usual combinators and as such do not constitute an interesting subject of investigation by themselves. However, when combined with the usual combinators, the whole system exhibits new features. A dual combinatory system with weak equality typically lacks the Church-Rosser (...) property, and in general it is inconsistent. In many subsystems terms "unexpectedly" turn out to be weakly equivalent. The paper is a preliminary attempt to investigate some of these issues, as well as, briefly compare function application in symmetric -calculus and dual combinatory logic. (shrink)

We briefly overview some of the historical landmarks on the path leading to the reduction of the number of logical connectives in classical logic. Relying on the duality inherent in Boolean algebras, we introduce a new operator ( Nallor ) that is the dual of Schönfinkel’s operator. We outline the proof that this operator by itself is sufficient to define all the connectives and operators of classical first-order logic ( Fol ). Having scrutinized the proof, we pinpoint the theorems of (...) Fol that are needed in the proof. Using the insights gained from the proof, we show that there are four binary operators that each can serve as the only undefined logical constant for Fol . Finally, we show that from every n -ary connective that yields a functionally complete singleton set of connectives two Schönfinkel-type operators are definable, and all the latter are so definable. (shrink)

The implicational fragment of the relevance logic "ticket entailment" is closely related to the so-called hereditary right maximal terms. I prove that the terms that need to be considered as inhabitants of the types which are theorems of $T_\rightarrow$ are in normal form and built in all but one casefrom B, B' and W only. As a tool in the proof ordered term rewriting systems are introduced. Based on the main theorem I define $FIT_\rightarrow$ - a Fitch-style calculus (related to (...) $FT_\rightarrow$ ) for the implicational fragment of ticket entailment. (shrink)

The implicational fragment of the relevance logic "ticket entailment" is closely related to the so-called hereditary right maximal terms. I prove that the terms that need to be considered as inhabitants of the types which are theorems of $T_\rightarrow$ are in normal form and built in all but one casefrom B, B' and W only. As a tool in the proof ordered term rewriting systems are introduced. Based on the main theorem I define $FIT_\rightarrow$ - a Fitch-style calculus for the (...) implicational fragment of ticket entailment. (shrink)

The challenge of giving a semantics for relevance logic in terms of worlds or situations intrigued several logicians. As a solution, Fine gave a two-sorted semantics. We overview the semantics as well as some further work of Fine in the area of relevance logic. Then we show that beyond supplying technical results such as soundness, completeness and the finite model property (fmp) for many logics, the operational–relational semantics provides footing for an informal interpretation and it naturally leads to an interpretation (...) in terms of relevance. (shrink)

This paper briefly overviews some of the results and research directions. In the area of substructural logics from the last couple of decades. Substructural logics are understood here to include relevance logics, linear logic, variants of Lambek calculi and some other logics that are motivated by the idea of omitting some structural rules or making other structural changes in LK, the original sequent calculus for classical logic.

Dual combinators emerge from the aim of assigning formulas containing ← as types to combinators. This paper investigates formally some of the properties of combinatory systems that include both combinators and dual combinators. Although the addition of dual combinators to a combinatory system does not affect the unique decomposition of terms, it turns out that some terms might be redexes in two ways (with a combinator as its head, and with a dual combinator as its head). We prove a general (...) theorem stating that no dual combinatory system possesses the Church-Rosser property. Although the lack of confluence might be problematic in some cases, it is not a problem per se. In particular, we show that no damage is inflicted upon the structurally free logics, the system in which dual combinators first appeared. (shrink)

This paper offers an analysis of the effect of the identity combinators in dual systems. The result is based on an easy technical trick, namely, that the identity combinators collapse all the combinators which are dual with respect to them. After reviewing dual combinators I consider the possible combinatory systems and l-calculi in which the functions and/or the application operation are bidirectional. The last section of the paper shows the devastating effect the identity combinators have for a dual system: they (...) half trivialize simple combinatory bases, although they are not sufficient to cause real triviality for what cancellative combinators are needed. (shrink)

The decidability of the logic of pure ticket entailment means that the problem of inhabitation of simple types by combinators over the base { B, B′, I, W } is decidable too. Type-assignment systems are often formulated as natural deduction systems. However, our decision procedure for this logic, which we presented in earlier papers, relies on two sequent calculi and it does not yield directly a combinator for a theorem of ${T_\to}$. Here we describe an algorithm to extract an inhabitant (...) from a sequent calculus proof—without translating the proof into another proof system. (shrink)

This paper investigates set-theoretical semantics for logics that contain unary connectives, which can be viewed as modalities. Indeed, some of the logics we consider are closely related to linear logic. We use insights from the relational semantics of relevance logics together with a new version of the squeeze lemma in our semantics for logics with disjunction. The ideal-based semantics, which takes co-theories to be situations, dualizes the theory-based semantics for logics with conjunction.

The dissertation deals with problems in "logic", more precisely, it deals with particular formal systems aiming at capturing patterns of valid reasoning. Sequent calculi were proposed to characterize logical connectives via introduction rules. These systems customarily also have structural rules which allow one to rearrange the set of premises and conclusions. In the "structurally free logic" of Dunn and Meyer the structural rules are replaced by combinatory rules which allow the same reshuffling of formulae, and additionally introduce an explicit marker (...) for the application of the rule---the combinator. The dissertation gives syntactic extensions of such systems introducing conjunction and disjunction . Cut elimination is proved for these systems. ;Syntactic systems as a rule are accompanied by a semantics which interprets them. The extended systems are supplied with semantics using algebraic methods. First, the Lindenbaum algebras are formed, and then these algebras are represented in an algebra of sets. This gives a so called Kripke-style semantics for the systems. The representation for the distributive extension utilizes well-known techniques; it uses sets of prime filters and set theoretical operations on them to deal with conjunction and disjunction, plus a ternary accessibility relation to accommodate the intensional operators as fusion and implication. The representation theorem is proven generally, i.e., for arbitrary sets of combinators using the "squeeze lemma" of Routley and Meyer. ;The non-distributive calculi require a different representation since union and intersection are distributive. Two previous representations for lattices are overviewed, and a modification of the Hartonas-Dunn representation is successfully augmented with the intensional operations. Instead of sets of filters, sets of pairs of minimally inconsistent filters and ideals are used and the representation theorem is proven. The representation results provide soundness and completeness for the systems. The dissertation also investigates properties of dual combinators , proving a series of lemmas about dual combinatory systems not being Church-Rosser and being inconsistent. Combinatorially definable classes of lambda-terms are considered as well. (shrink)

Symmetic combinatory logic with the symmetric analogue of a combinatorially complete base (in the form of symmetric λ-calculus) is known to lack the Church-Rosser property. We prove a muchstrongertheorem that no symmetric combinatory logic that containsat least two proper symmetric combinatoryhas the Church-Rosser property. Although the statement of the result looks similar to an earlier one concerning dual combinatory logic,the proof is differentbecause symmetric combinators may form redexes in both left and right associated terms. Perhaps surprisingly, we are also able (...) to show that certain symmetric combinatory logics that include justone particular constantare not confluent. This result (beyond other differences) clearly sets apart symmetric combinatory logic from dual combinatory logic, since all dual combinatory systems with a single combinator or a single dual combinator are Church-Rosser. Lastly, we prove that a symmetric combinatory logic that contains the fixed point and the one-place identity combinator has the Church-Rosser property. (shrink)

Two consecution calculi are introduced: one for the implicational fragment of the logic of entailment with truth and another one for the disjunction free logic of nondistributive relevant implication. The proof technique—attributable to Gentzen—that uses a double induction on the degree and on the rank of the cut formula is shown to be insufficient to prove admissible various forms of cut and mix in these calculi. The elimination theorem is proven, however, by augmenting the earlier double inductive proof with additional (...) inductions. We also give a new purely inductive proof of the cut theorem for the original single cut rule in Gentzen’s sequent calculus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ LK $$\end{document} without any use of mix. (shrink)

This is a book review, which I promised to write for the PIR last month.