In proof-theoretic semantics, model-theoretic validity is replaced by proof-theoretic validity. Validity of formulae is defined inductively from a base giving the validity of atoms using inductive clauses derived from proof-theoretic rules. A key aim is to show completeness of the proof rules without any requirement for formal models. Establishing this for propositional intuitionistic logic raises some technical and conceptual issues. We relate Sandqvist’s (complete) base-extension semantics of intuitionistic propositional logic to categorical proof theory in presheaves, reconstructing categorically the soundness and (...) completeness arguments, thereby demonstrating the naturality of Sandqvist’s constructions. This naturality includes Sandqvist’s treatment of disjunction that is based on its second-order or elimination-rule presentation. These constructions embody not just validity, but certain forms of objects of justifications. This analysis is taken a step further by showing that from the perspective of validity, Sandqvist’s semantics can also be viewed as the natural disjunction in a category of sheaves. (shrink)

In proof-theoretic semantics, model-theoretic validity is replaced by proof-theoretic validity. Validity of formulae is defined inductively from a base giving the validity of atoms using inductive clauses derived from proof-theoretic rules. A key aim is to show completeness of the proof rules without any requirement for formal models. Establishing this for propositional intuitionistic logic raises some technical and conceptual issues. We relate Sandqvist’s (complete) base-extension semantics of intuitionistic propositional logic to categorical proof theory in presheaves, reconstructing categorically the soundness and (...) completeness arguments, thereby demonstrating the naturality of Sandqvist’s constructions. This naturality includes Sandqvist’s treatment of disjunction that is based on its second-order or elimination-rule presentation. These constructions embody not just validity, but certain forms of objects of justifications. This analysis is taken a step further by showing that from the perspective of validity, Sandqvist’s semantics can also be viewed as the natural disjunction in a category of sheaves. (shrink)

Proof-theoretic semantics (P-tS) is the approach to meaning in logic based on proof (as opposed to truth). There are two major approaches to P-tS: proof-theoretic validity (P-tV) and base-extension semantics (B-eS). The former is a semantics of arguments, and the latter is a semantics of logical constants. This paper demonstrates that the B-eS for intuitionistic propositional logic (IPL) encapsulates the declarative content of a version of P-tV based on the elimination rules. This explicates how the B-eS for IPL works, and (...) shows the completeness of this version of P-tV. (shrink)

One might poetically muse that computers have the essence both of logic and machines. Through the case of the history of Separation Logic, we explore how this assertion is more than idle poetry. Separation Logic works because it merges the software engineer’s conceptual model of a program’s manipulation of computer memory with the logical model that interprets what sentences in the logic are true, and because it has a proof theory which aids in the crucial problem of scaling the reasoning (...) task. Scalability is a central problem, and some would even say the central problem, in appli- cations of logic in computer science. Separation Logic is an interesting case because of its widespread success in verification tools. For these two senses of model—the engineering/conceptual and the logical—to merge in a genuine sense, each must maintain their norms of use from their home disciplines. When this occurs, both the logic and engineering benefit greatly. Seeking this intersection of two different senses of model provides a strategy for how computer scientists and logicians may be successful. Furthermore, the history of Separation Logic for analysing programs provides a novel case for philosophers of science of how software engineers and computer scientists develop models and the components of such models. We provide three contributions: an exploration of the extent of models merging that is necessary for success in computer science; an introduction to the technical details of Separation Logic, which can be used for reasoning about other exhaustible resources; and an introduction to the problems, process, and results of computer scientists for those outside the field. (shrink)

This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search including proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences.

This book is a specialized monograph on the development of the mathematical and computational metatheory of reductive logic and proof-search, areas of logic that are becoming important in computer science. A systematic foundational text on these emerging topics, it includes proof-theoretic, semantic/model-theoretic and algorithmic aspects. The scope ranges from the conceptual background to reductive logic, through its mathematical metatheory, to its modern applications in the computational sciences. Suitable for researchers and graduate students in mathematical, computational and philosophical logic, and in (...) theoretical computer science and artificial intelligence, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism (2nd Edition), Dov M. Gabbay, Mark A. Reynolds, and Marcelo Finger's Temporal Logic Mathematical Foundations and Computational Aspects, J. M. Dunn and G. Hardegree's Algebraic Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning, and P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2. (shrink)

The lambdaPi-calculus, a theory of first-order dependent function types in Curry-Howard-de Bruijn correspondence with a fragment of minimal first-order logic, is defined as a system of (linearized) natural deduction. In this paper, we present a Gentzen-style sequent calculus for the lambdaPi-calculus and prove the cut-elimination theorem. The cut-elimination result builds upon the existence of normal forms for the natural deduction system and can be considered to be analogous to a proof provided by Prawitz for first-order logic. The type-theoretic setting considered (...) here elegantly illustrates the distinction between the processes of normalization in a natural deduction system and cut-elimination in a Gentzen-style sequent calculus. (shrink)

"We argue that the computation of a logic program can be usefully divided into two distinct phases: the first being a proof- valued computation or proof-search; the second a residual computation, or answer extraction. Extension of extraction techniques to various theories then permits more extensive languages and proof procedures to be employed for the computational solution of problems. We illustrate these ideas with a simple propositional logic and show that SLD-resolution computes presentations of proofs in which the residual computation may (...) be interleaved with the proof-search, whereas a more general proof procedure yields shorter presentations of (the same) proofs, but which require more extensive residual computations.". (shrink)

In proof-theoretic semantics, meaning is based on inference. It may seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a ‘base’ of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems |$K$|⁠, |$KT$|⁠, |$K4$| and (...) |$S4$|⁠, with |$\square $| as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between |$\square $| and a natural presentation of |$\lozenge $|⁠. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases. (shrink)

Proof-theoretic semantics (P-tS) is the paradigm of semantics in which meaning in logic is based on proof (as opposed to truth). A particular instance of P-tS for intuitionistic propositional logic (IPL) is its base-extension semantics (B-eS). This semantics is given by a relation called support, explaining the meaning of the logical constants, which is parameterized by systems of rules called bases that provide the semantics of atomic propositions. In this paper, we interpret bases as collections of definite formulae and use (...) the operational view of them as provided by uniform proof-search—the proof-theoretic foundation of logic programming (LP)—to establish the completeness of IPL for the B-eS. This perspective allows negation, a subtle issue in P-tS, to be understood in terms of the negation-as-failure protocol in LP. Specifically, while the denial of a proposition is traditionally understood as the assertion of its negation, in B-eS we may understand the denial of a proposition as the failure to find a proof of it. In this way, assertion and denial are both prime concepts in P-tS. (shrink)

We introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises from an analysis of the proof-theoretic relationship between conjunction and implication; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality of BI can be seen categorically: models of propositional BI's proofs are given by bicartesian doubly closed categories, i.e., categories which freely combine the semantics of propositional intuitionistic (...) logic and propositional multiplicative intuitionistic linear logic. The predicate version of BI includes, in addition to standard additive quantifiers, multiplicative (or intensional) quantifiers ∀ new and ∃ new which arise from observing restrictions on structural rules on the level of terms as well as propositions. We discuss computational interpretations, based on sharing, at both the propositional and predicate levels. (shrink)

We give a novel approach to proving soundness and completeness for a logic (henceforth: the object-logic) that bypasses truth-in-a-model to work directly with validity. Instead of working with specific worlds in specific models, we reason with eigenworlds (i.e., generic representatives of worlds) in an arbitrary model. This reasoning is captured by a sequent calculus for a _meta_-logic (in this case, first-order classical logic) expressive enough to capture the semantics of the object-logic. Essentially, one has a calculus of validity for the (...) object-logic. The method proceeds through the perspective of reductive logic (as opposed to the more traditional paradigm of deductive logic), using the space of reductions as a medium for showing the behavioural equivalence of reduction in the sequent calculus for the object-logic and in the validity calculus. Rather than study the technique in general, we illustrate it for the logic of Bunched Implications (BI), thus IPL and MILL (without negation) are also treated. Intuitively, BI is the free combination of intuitionistic propositional logic and multiplicative intuitionistic linear logic, which renders its meta-theory is quite complex. The literature on BI contains many similar, but ultimately different, algebraic structures and satisfaction relations that either capture only fragments of the logic (albeit large ones) or have complex clauses for certain connectives (e.g., Beth’s clause for disjunction instead of Kripke’s). It is this complexity that motivates us to use BI as a case-study for this approach to semantics. (shrink)

The development of logic has largely been through the deductive paradigm: conclusions are inferred from established premisses. However, the use of logic in the context of both human and machine reasoning is typically through the dual reductive perspective: collections of sufficient premisses are generated from putative conclusions. We call this paradigm, reductive logic. This expression of logic encompass as diverse reasoning activities as proving a formula in a formal system to seeking to meet a friend before noon on Saturday. This (...) paper is a semantical analysis of reductive logic. In particular, we provide mathematical foundations for representing and reasoning about reduction operators. Heuristically, reduction operators may be thought of as ‘backwards’ inference rules. In this paper, we address their mathematical representation, how they are used in the context of reductive reasoning, and, crucially, what makes them valid. (shrink)

Further difficulties are encountered when we consider the addition of the modality! (of course) to our fragment of linear logic. We provide an elementary quantale semantics for our logic programs and give an appropriate completeness theorem. We consider a translation -- resembling those of Girard -- of the intuitionistic hereditary Harrop formulae and intuitionistic uniform proofs into our framework, and show that certain properties are preserved under this translation. We sketch the design of an interpreter for linear logic programs.

"We present a proof-theoretic foundation for logic programming in Girard's linear logic. We exploit the permutability properties of two-sided linear sequent calculus to identify appropriate notions of uniform proof, definite formula, goal formula, clause and resolution proof for fragments of linear logic. The analysis of this paper extends earlier work by the present authors to include negative occurrences of [cross] (par) and positive occurences of! (of course!) and? (why not?). These connectives introduce considerable difficulty. We consider briefly some of the (...) issues related to the mechanical implementation of our resolution proofs.". (shrink)

Proof-theoretic semantics (P-tS) is an innovative approach to grounding logical meaning in terms of proofs rather than traditional truth-conditional semantics. The point is not that one provides a proof system, but rather that one articulates meaning in terms of proofs and provability. To elucidate this paradigm shift, we commence with an introduction that contrasts the fundamental tenets of P-tS with the more prevalent model-theoretic approach to semantics. The contribution of this paper is a P-tS for a substructural logic, intuitionistic multiplicative (...) linear logic (IMLL). Specifically, we meticulously examine and refine the established P-tS for intuitionistic propositional logic. Subsequently, we present two novel and comprehensive forms of P-tS for IMLL. Notably, the semantics for IMLL in this paper embodies its resource interpretation through its number-of-uses reading (restricted to atoms). This stands in contrast to the conventional model-theoretic semantics of the logic, underscoring the value that P-tS brings to substructural logics. (shrink)

The _logic of bunched implications_ (BI) can be seen as the free combination of _intuitionistic propositional logic_ (IPL) and _intuitionistic multiplicative linear logic_ (IMLL). We present here a base-extension semantics (B-eS) for BI in the spirit of Sandqvist’s B-eS for IPL, deferring an analysis of proof-theoretic validity (in the sense of Dummett and Prawitz) to another occasion. Essential to BI’s formulation in proof-theoretic terms is the concept of a ‘bunch’ of hypotheses, a notion familiar from relevance logic. Bunches amount to (...) trees whose internal vertices are labelled with either the IMLL or the IPL context-former and whose leaves are labelled with propositions or units for the context-formers. This structure presents significant technical challenges in setting up a base-extension semantics for BI. Our approach starts from the B-eS for IPL and the B-eS for IMLL and provides a systematic combination. Such a combination requires that base rules carry bunched structure, and so requires a more complex notion of derivability in a base and a correspondingly richer notion of support in a base. One reason why BI is a substructural logic of interest is that the ‘resource interpretation’ of its semantics, given in terms of sharing and separation and which gives rise to Separation Logic in the field of program verification, is quite distinct from the ‘number-of-uses’ interpretation of propositions in linear logic. This resource reading of BI provides useful intuitions in the formulation of its proof-theoretic semantics. We discuss a simple example of the use of the given B-eS in security modelling. (shrink)

The reductive, as opposed to deductive, view of logic is the form of logic that is, perhaps, most widely employed in practical reasoning. In particular, it is the basis of logic programming. Here, building on the idea of uniform proof in reductive logic, we give a treatment of logic programming for BI, the logic of bunched implications, giving both operational and denotational semantics, together with soundness and completeness theorems, all couched in terms of the resource interpretation of BI’s semantics. We (...) use this set-up as a basis for exploring how coalgebraic semantics can, in contrast to the basic denotational semantics, be used to describe the concrete operational choices that are an essential part of proof-search. The overall aim, toward which this paper can be seen as an initial step, is to develop a uniform, generic, mathematical framework for understanding the relationship between the deductive structure of logics and the control structures of the corresponding reductive paradigm. (shrink)

In proof-theoretic semantics, meaning is based on inference. It may be seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a ‘base’ of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems K, KT , (...) K4, and S4, with □ as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between □ and a natural presentation of ♢. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Constructive Postmodernism: Toward Renewal in Cultural and Literary StudiesDavid CarrierConstructive Postmodernism: Toward Renewal in Cultural and Literary Studies, by Martin Schiralli. Westport, CT, and London: Bergin and Garvey, 1999, 165pp., $55 cloth.Concerned with the consequences of Jacques Derrida's philosophy of deconstruction for cultural and literary studies, Martin Schiralli's elegantly written book offers, first, a critique of these claims and, then, a constructive alternative analysis. He admires Derrida's brilliantly (...) innovative writing, which as he rightly notes is "serious and philosophical" even if "the most patient reader can easily become disoriented and confused" (24). But even if, as Derrida argues, Ferdinard de Saussure's theory of meaning is flawed, we need not—he urges—become deconstructionists. Looking to the work of Stephen Toulmin, John Dewey, and late Wittgenstein, Schiralli finds suggestive ways of identifying concerns with cognitive value in literary texts. It then is possible, he argues, to offer a more reasonable perspective on recent debates about culture and politics than the recent academic arguments between "postmodern revisionists... [and] often hostile and defensive traditionalists" (73).In the second part of his book Schiralli illustrates the value of this perspective by offering a great deal of productive positive discussion. He discusses how "theoretically pertinent knowledge" (95) may be found in the novels of Jane Austen, Barbara Pym, and Anita Brookner. Appealing to Michael Polanyi's conception of tacit knowledge, he argues that Austen's ability to intuit "the rules governing social meaning... represents the tacit emergence of insight, just as her ability to use those insights explicitly in her fiction marks the emergence of more focal knowledge" (123). In analyzing an emotional response, he claims, we can ask whether it is "warranted or unwarranted in a given circumstance" (31). Using E. M. Forster's Howards End as his model, he develops this claim in an analysis that is brief, clear, and convincing. Finally, in his concluding pages he looks at the studies of character provided by Henry James and Marcel Proust. A proper theory of literary intelligence, he argues, should "treat those forms of cognitive acuity involved imaginatively in miming and representing verbally the intricacies and particularities of all aspects of human sensibility" (150).This bold, far-reaching book avoids the fanaticism that, as Schiralli notes, has marred all too much recent debate in the humanities. At this time, when the influence of Derrida has faded, I think this argumentation about his philosophical position is of less value than the constructive parts of the analysis. Treating Derrida as a theorist whose arguments tell us how to analyze literature is an unpromising project, as Schiralli well shows. Once that point is acknowledged, then we are ready to look seriously at literature as a form of social knowledge. Humanists teaching literature will find much of value in this graceful volume—and along with their colleagues in philosophy, these scholars will benefit from study of this well-argued analysis. [End Page 122]David CarrierCase Western Reserve University and Cleveland Institute of ArtCopyright © 2008 Board of Trustees of the University of Illinois... (shrink)

This book is a specialized monograph on interpolation and definability, a notion central in pure logic and with significant meaning and applicability in all areas where logic is applied, especially computer science, artificial intelligence, logic programming, philosophy of science and natural language. Suitable for researchers and graduate students in mathematics, computer science and philosophy, this is the latest in the prestigous world-renowned Oxford Logic Guides, which contains Michael Dummet's Elements of intuitionism (second edition), J. M. Dunn and G. Hardegree's Algebraic (...) Methods in Philosophical Logic, H. Rott's Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning, P. T. Johnstone's Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2, and David J. Pym and Eike Ritter's Reductive Logic and Proof Search: Proof theory, semantics and control. (shrink)

I teach a workshop on mending and repair at a local haberdashery shop called RayStitch in North London, and I usually begin each session by describing how my interest started. It was because of a sweater that was all worn out and full of holes, with thin and threadbare knitting in its forearms. It had already been repaired in several places, and that repair was visible. The thick white sweater had belonged to my great-uncle Roland and had been mended over (...) the years by his sister, my great-aunt Elizabeth. My great-uncle was a painter, and in old age he used to sit with a board across the arms of his chair and lean forward drawing on paper taped to the board.Roly and Elizabeth.The sleeves on Roly's sweater... (shrink)

For Gadamer, translation operates as an illustrative “extreme case” of interpretation, of interest to the extent that it can push the logics of less-extreme interpretative practices. Yet the main thing Gadamer consistently says about what is extreme in translation seems to be that it is a strangely intellectual process, bereft of lived experience. One can nevertheless trace threads of lived experience within translation knowledge, both through what translators say and from what translation process research reveals. Further, the nature of that (...) experience, in exceeding its interpretations, can justify an empirical attitude to its study. Hence hermeneutics could do worse than incorporate empirical attitudes into its work on translation, rather than endlessly repeat inherited insights. (shrink)

The central question in this article concerns the implications of developments in the rearrangement of tasks in health care, particularly for nurses in terms of their duties and responsibilities. Attention is focused on the transfer of medical tasks from physicians to nurses. An investigation was carried out on the impact of the rearrangement of tasks from the perspective of health care law on the legal framework of the nursing profession. From case law it can be concluded that a number of (...) specific conditions have to be met, such as mentoring, supervision and audit, orders and instructions, clear agreement about task allocation, comparing information, and making certain that the person who is given the instruction is sufficiently competent and experienced to carry it out. Although the rearrangement of tasks fits into the existing legal framework, changes in the health professions have to be met more flexibly. (shrink)

The Theory and Philosophy of Organisations assesses and analyzes the assumptions upon which our understanding of organizations is based and in doing so aims to redirect the ways in which organizational research is conceived and executed. Contributions to the volume emphasize how all approaches to the study of organizations are influenced by deep metatheoretical assumptions about the nature of science and society. It is argued that these differences create a spectrum of valid perspectives and methods, and the book outlines how (...) conceptual problems in the field have given rise to innovative forms of empirical enquiry such as symbolism, postmodernism, ethnomethodology and structuralism. Criticizing the conservatism which has characterized much of the work in this field, the authors offers this range of methods as a starting point for obtaining richer and more grounded explanations of organizations. (shrink)

The Description for this book, David Hume: Common-Sense Moralist, Sceptical Metaphysician, will be forthcoming.