After introductory reminder of and comments on Gödel’s ontological proof, we discuss the collapse of modalities, which is provable in Gödel’s ontological system GO. We argue that Gödel’s texts confirm modal collapse as intended consequence of his ontological system. Further, we aim to show that modal collapse properly fits into Gödel’s philosophical views, especially into his ontology of separation and union of force and fact, as well as into his cosmological theory of the nonobjectivity of the lapse of time. (...) As a result, modal collapse should not be conceived in Gödel so much as a deficit, but rather as a kind of the rise of modalities to the “perfect” being. We further show that, in accordance with Gödel’s ontology, the concepts of modality and time should be derived in terms of the “fundamental philosophical concept” of cause. To give an example of how a formalization of such causative Gödelian ontology and ontological proof might look, we propose the transformation of GO into a kind of causally re-interpreted justification logic. (shrink)
The concept of burden of proof is used in a wide range of discourses, from philosophy to law, science, skepticism, and even in everyday reasoning. This paper provides an analysis of the proper deployment of burden of proof, focusing in particular on skeptical discussions of pseudoscience and the paranormal, where burden of proof assignments are most poignant and relatively clear-cut. We argue that burden of proof is often misapplied or used as a mere rhetorical gambit, with (...) little appreciation of the underlying principles. The paper elaborates on an important distinction between evidential and prudential varieties of burdens of proof, which is cashed out in terms of Bayesian probabilities and error management theory. Finally, we explore the relationship between burden of proof and several (alleged) informal logical fallacies. This allows us to get a firmer grip on the concept and its applications in different domains, and also to clear up some confusions with regard to when exactly some fallacies (ad hominem, ad ignorantiam, and petitio principii) may or may not occur. (shrink)
This book is a collection of materials concerned not only with the law of evidence, but also with the logical and rhetorical aspects of proof; the epistemology of evidence as a basis for the proof of disputed facts; and scientific aspects of the subject. The editor also raises issues such as the philosophical basis for the use of evidence.
In this paper we give some formal examples of ideas developed by Penco in two papers on the tension inside Frege's notion of sense (see Penco 2003). The paper attempts to compose the tension between semantic and cognitive aspects of sense, through the idea of sense as proof or procedure – not as an alternative to the idea of sense as truth condition, but as complementary to it (as it happens sometimes in the old tradition of procedural semantics).
Proof, Logic and Formalization addresses the various problems associated with finding a philosophically satisfying account of mathematical proof. It brings together many of the most notable figures currently writing on this issue in an attempt to explain why it is that mathematical proof is given prominence over other forms of mathematical justification. The difficulties that arise in accounts of proof range from the rightful role of logical inference and formalization to questions concerning the place of experience (...) in proof and the possibility of eliminating impredictive reasoning from proof. Students and lecturers of philosophy, philosophy of logic, and philosophy of mathematics will find this to be essential reading. A companion volume entitled Proof and Logic in Mathematics is also available from Routledge. (shrink)
How is the burden of proof to be distributed among individuals who are involved in resolving a particular issue? Under what conditions should the burden of proof be distributed unevenly? We distinguish attitudinal from dialectical burdens and argue that these questions should be answered differently, depending on which is in play. One has an attitudinal burden with respect to some proposition when one is required to possess sufficient evidence for it. One has a dialectical burden with respect to (...) some proposition when one is required to provide supporting arguments for it as part of a deliberative process. We show that the attitudinal burden with respect to certain propositions is unevenly distributed in some deliberative contexts, but in all of these contexts, establishing the degree of support for the proposition is merely a means to some other deliberative end, such as action guidance, or persuasion. By contrast, uneven distributions of the dialectical burden regularly further the aims of deliberation, even in contexts where the quest for truth is the sole deliberative aim, rather than merely a means to some different deliberative end. We argue that our distinction between these two burdens resolves puzzles about unevenness that have been raised in the literature. (shrink)
In his "Ontological proof", Kurt Gödel introduces the notion of a second-order value property, the positive property P. The second axiom of the proof states that for any property φ: If φ is positive, its negation is not positive, and vice versa. I put forward that this concept of positiveness leads into a paradox when we apply it to the following self-reflexive sentences: (A) The truth value of A is not positive; (B) The truth value of B is (...) positive. Given axiom 2, sentences A and B paradoxically cannot be both true or both false, and it is also impossible that one of the sentences is true whereas the other is false. (shrink)
I present an account of truth values for classical logic, intuitionistic logic, and the modal logic S5, in which truth values are not a fundamental category from which the logic is defined, but rather, an idealisation of more fundamental logical features in the proof theory for each system. The result is not a new set of semantic structures, but a new understanding of how the existing semantic structures may be understood in terms of a more fundamental notion of logical (...) consequence. (shrink)
This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic (intuitionistic as well as classical); the theory of logic programming; category theory; modal logic; linear logic; first-order arithmetic and second-order logic. In each case the aim is to illustrate the methods in relatively simple situations and then apply them elsewhere in (...) much more complex settings. There are numerous exercises throughout the text. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence. For the new edition, many sections have been rewritten to improve clarity, new sections have been added on cut elimination, and solutions to selected exercises have been included. (shrink)
The paper presents a proof-theoretic semantics (PTS) for a fragment of natural language, providing an alternative to the traditional model-theoretic (Montagovian) semantics (MTS), whereby meanings are truth-condition (in arbitrary models). Instead, meanings are taken as derivability-conditions in a dedicated natural-deduction (ND) proof-system. This semantics is effective (algorithmically decidable), adhering to the meaning as use paradigm, not suffering from several of the criticisms formulated by philosophers of language against MTS as a theory of meaning. In particular, Dummett’s manifestation argument (...) does not obtain, and assertions are always warranted, having grounds of assertion. The proof system is shown to satisfy Dummett’s harmony property, justifying the ND rules as meaning conferring. The semantics is suitable for incorporation into computational linguistics grammars, formulated in type-logical grammar. (shrink)
Using Carnapâs concept explication, we propose a theory of concept formation in mathematics. This theory is then applied to the problem of how to understand the relation between the concepts formal proof (deduction) and informal, mathematical proof.
The requirement of proof-theoretic harmony has played a pivotal role in a number of debates in the philosophy of logic. Different authors have attempted to precisify the notion in different ways. Among these, three proposals have been prominent in the literature: harmony–as–conservative extension, harmony–as–leveling procedure, and Tennant’s harmony–as–deductive equilibrium. In this paper I propose to clarify the logical relationships between these accounts. In particular, I demonstrate that what I call the equivalence conjecture —that these three notions essentially come to (...) the same thing—is erroneous. (shrink)
In this paper, we discuss the prevailing view amongst philosophers and many mathematicians concerning mathematical proof. Following Cellucci, we call the prevailing view the “axiomatic conception” of proof. The conception includes the ideas that: a proof is finite, it proceeds from axioms and it is the final word on the matter of the conclusion. This received view can be traced back to Frege, Hilbert and Gentzen, amongst others, and is prevalent in both mathematical text books and logic (...) text books. (shrink)
Validity, the central concept of the so-called ‘proof-theoretic semantics’ is described as correctly applying to the arguments that denote proofs. In terms of validity, I propose an anti-realist characterization of the notions of truth and correct assertion, at the core of which is the idea that valid arguments may fail to be recognized as such. The proposed account is compared with Dummett’s and Prawitz’s views on the matter.
When Archimedes, while bathing, suddenly hit upon the principle of buoyancy, he ran wildly through the streets of Syracuse, stark naked, crying "eureka!" In The Moment of Proof, Donald Benson attempts to convey to general readers the feeling of eureka--the joy of discovery--that mathematicians feel when they first encounter an elegant proof. This is not an introduction to mathematics so much as an introduction to the pleasures of mathematical thinking. And indeed the delights of this book are many (...) and varied. The book is packed with intriguing conundrums--Loyd's Fifteen Puzzle, the Petersburg Paradox, the Chaos Game, the Monty Hall Problem, the Prisoners' Dilemma--as well as many mathematical curiosities. We learn how to perform the arithmetical proof called "casting out nines" and are introduced to Russian peasant multiplication, a bizarre way to multiply numbers that actually works. The book shows us how to calculate the number of ways a chef can combine ten or fewer spices to flavor his soup (1,024) and how many people we would have to gather in a room to have a 50-50 chance of two having the same birthday (23 people). But most important, Benson takes us step by step through these many mathematical wonders, so that we arrive at the solution much the way a working scientist would--and with much the same feeling of surprise. Every fan of mathematical puzzles will be enthralled by The Moment of Proof. Indeed, anyone interested in mathematics or in scientific discovery in general will want to own this book. (shrink)
Proof Theory of Modal Logic is devoted to a thorough study of proof systems for modal logics, that is, logics of necessity, possibility, knowledge, belief, time, computations etc. It contains many new technical results and presentations of novel proof procedures. The volume is of immense importance for the interdisciplinary fields of logic, knowledge representation, and automated deduction.
A textbook on proof in mathematics, inspired by an Aristotelian point of view on mathematics and proof. It explains how to prove statements in mathematics, from evident premises. It deals with the proof of "all" statements, "some" statements, multiple quantifiers and mathematical induction.
We consider two versions of truth as grounded in verification procedures: Dummett's notion of proof as an effective way to establish the truth of a statement and Hintikka's GTS notion of truth as given by the existence of a winning strategy for the game associated with a statement. Hintikka has argued that the two notions should be effective and that one should thus restrict one's attention to recursive winning strategies. In the context of arithmetic, we show that the two (...) notions do not coincide: on the one hand, proofs in PA do not yield recursive winning strategies for the associated game; on the other hand, there is no sound and effective proof procedure that captures recursive GTS truths. We then consider a generalized version of Game Theoretical Semantics by introducing games with backward moves. In this setting, a connection is made between proofs and recursive winning strategies. We then apply this distinction between two kinds of verificationist procedures to a recent debate about how we recognize the truth of Gödelian sentences. (shrink)
From the point of view of proof-theoretic semantics, it is argued that the sequent calculus with introduction rules on the assertion and on the assumption side represents deductive reasoning more appropriately than natural deduction. In taking consequence to be conceptually prior to truth, it can cope with non-well-founded phenomena such as contradictory reasoning. The fact that, in its typed variant, the sequent calculus has an explicit and separable substitution schema in form of the cut rule, is seen as a (...) crucial advantage over natural deduction, where substitution is built into the general framework. (shrink)
Proof-theoretical notions and techniques, developed on the basis of sentential/symbolic representations of formal proofs, are applied to Euler diagrams. A translation of an Euler diagrammatic system into a natural deduction system is given, and the soundness and faithfulness of the translation are proved. Some consequences of the translation are discussed in view of the notion of free ride, which is mainly discussed in the literature of cognitive science as an account of inferential efficacy of diagrams. The translation enables us (...) to formalize and analyze free ride in terms of proof theory. The notion of normal form of Euler diagrammatic proofs is investigated, and a normalization theorem is proved. Some consequences of the theorem are further discussed: in particular, an analysis of the structure of normal diagrammatic proofs; a diagrammatic counterpart of the usual subformula property; and a characterization of diagrammatic proofs compared with natural deduction proofs. (shrink)
The aim of this paper is to reconsider several proposals that have been put forward in order to develop a Proof-Theoretical Semantics, from the by now classical neo-verificationist approach provided by D. Prawitz and M. Dummett in the Seventies, to an alternative, more recent approach mainly due to the work of P. Schroeder-Heister and L. Hallnäs, based on clausal definitions. Some other intermediate proposals are very briefly sketched. Particular attention will be given to the role played by the so-called (...) Fundamental Assumption. We claim that whereas, in the neo-verificationist proposal, the condition expressed by that Assumption is necessary to ensure the completeness of the justification procedure ( from the outside , so to speak), within the definitional framework it is a built-in feature of the proposal. The latter approach, therefore, appears as an alternative solution to the problem which prompted the neo-verificationists to introduce the Fundamental Assumption. (shrink)
In this paper, we provide a general setting under which results of normalization of proof trees such as, for instance, the logicality result in equational reasoning and the cut-elimination property in sequent or natural deduction calculi, can be unified and generalized. This is achieved by giving simple conditions which are sufficient to ensure that such normalization results hold, and which can be automatically checked since they are syntactical. These conditions are based on basic properties of elementary combinations of inference (...) rules which ensure that the induced global proof tree transformation processes do terminate. (shrink)
This volume contains articles covering a broad spectrum of proof theory, with an emphasis on its mathematical aspects. The articles should not only be interesting to specialists of proof theory, but should also be accessible to a diverse audience, including logicians, mathematicians, computer scientists and philosophers. Many of the central topics of proof theory have been included in a self-contained expository of articles, covered in great detail and depth. The chapters are arranged so that the two introductory (...) articles come first; these are then followed by articles from core classical areas of proof theory; the handbook concludes with articles that deal with topics closely related to computer science. (shrink)
This work is derived from the SERC "Logic for IT" Summer School Conference on Proof Theory held at Leeds University. The contributions come from acknowledged experts and comprise expository and research articles which form an invaluable introduction to proof theory aimed at both mathematicians and computer scientists.
The article discusses burden of proof rules in social criticism. By social criticism I mean an argumentative situation in which an opponent publicly argues against certain social practices; the examples I consider are discrimination on the basis of species and discrimination on the basis of one's nationality. I argue that burden of proof rules assumed by those who defend discrimination are somewhat dubious. In social criticism, there are no shared values which would uncontroversially determine what is the reasonable (...) presumption and who has the burden of proof, nor are there formal rules which would end the debate and determine the winner at a specific point. (shrink)
We define proof nets for cyclic multiplicative linear logic as edge bi-coloured graphs. Our characterization is purely graph theoretical and works without further complication for proof nets with cuts, which are usually harder to handle in the non-commutative case. This also provides a new characterization of the proof nets for the Lambek calculus (with the empty sequence) which simply are a restriction on the formulas to be considered (which are asked to be intuitionistic).
This paper discusses proof-theoretic semantics, the project of specifying the meanings of the logical constants in terms of rules of inference governing them. I concentrate on Michael Dummett’s and Dag Prawitz’ philosophical motivations and give precise characterisations of the crucial notions of harmony and stability, placed in the context of proving normalisation results in systems of natural deduction. I point out a problem for defining the meaning of negation in this framework and prospects for an account of the meanings (...) of modal operators in terms of rules of inference. (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.
Opponents of biotechnology ofteninvoke the Precautionary Principle to advancetheir cause, whereas biotech enthusiasts preferto appeal to ``sound science.'' Publicauthorities are still groping for a usefuldefinition. A crucial issue in this debate isthe distribution of the burden of proof amongthe parties favoring and opposing certaintechnological developments. Indeed, the debateon the significance and scope of thePrecautionary Principle can be fruitfullyre-framed as a debate on the proper division ofburdens of proof. In this article, we attemptto arrive at a more refined way (...) of thinkingabout this problem in order to escape from theexisting polarization of views between ``guiltyuntil proven innocent'' and ``innocent untilproven guilty.'' This way of thinking alsoenables a critical review of currentdemarcations between risk assessment and riskmanagement, or science and politics, and of themorally laden controversy on the relativeimportance of type-I and type-II errors instatistical testing. (shrink)
A proof by Allan Gibbard (Ifs: Conditionals, beliefs, decision, chance, time. Reidel, Dordrecht, 1981) seems to demonstrate that if indicative conditionals have truth conditions, they cannot be stronger than material implication. Angelika Kratzer's theory that conditionals do not denote two-place operators purports to escape this result [see Kratzer (Chic Linguist Soc 22(2):1–15, 1986, 2012)]. In this note, I raise some trouble for Kratzer’s proposed method of escape and then show that her semantics avoids this consequence of Gibbard’s proof (...) by denying modus ponens. I also show that the same holds for Anthony Gillies’ semantics (Philos Rev 118(3):325–349, 2009) and argue that this consequence of these theories is not obviously prohibitive—hence, both remain viable theories of indicative conditionals. (shrink)
Machine generated contents note: Prologue: Hilbert's Last Problem; 1. Introduction; Part I. Proof Systems Based on Natural Deduction: 2. Rules of proof: natural deduction; 3. Axiomatic systems; 4. Order and lattice theory; 5. Theories with existence axioms; Part II. Proof Systems Based on Sequent Calculus: 6. Rules of proof: sequent calculus; 7. Linear order; Part III. Proof Systems for Geometric Theories: 8. Geometric theories; 9. Classical and intuitionistic axiomatics; 10. Proof analysis in elementary geometry; (...) Part IV. Proof Systems for Nonclassical Logics: 11. Modal logic; 12. Quantified modal logic, provability logic, and so on; Bibliography; Index of names; Index of subjects. (shrink)
Jean van Heijenoort was best known for his editorial work in the history of mathematical logic. I survey his contributions to model-theoretic proof theory, and in particular to the falsifiability tree method. This work of van Heijenoort’s is not widely known, and much of it remains unpublished. A complete list of van Heijenoort’s unpublished writings on tableaux methods and related work in proof theory is appended.
This paper is a contribution to the study of the rôle of disjunction inAlgebraic Logic. Several kinds of (generalized) disjunctions, usually defined using a suitable variant of the proof by cases property, were introduced and extensively studied in the literature mainly in the context of finitary logics. The goals of this paper are to extend these results to all logics, to systematize the multitude of notions of disjunction (both those already considered in the literature and those introduced in this (...) paper), and to show several interesting applications allowed by the presence of a suitable disjunction in a given logic. (shrink)
The paper suggests a revision of the notion of harmony, a major necessary condition in proof-theoretic semantics for a natural-deduction proof-system to qualify as meaning conferring, when moving to a bilateral proof-system. The latter considers both forces of assertion and denial as primitive, and is applied here to positive logics, lacking negation altogether. It is suggested that in addition to the balance between (positive) introduction and elimination rules traditionally imposed by harmony, a balance should be imposed also (...) on: (i) negative introduction and elimination rules, and (ii) positive and negative introduction rules. The paper suggests a proof-theoretical definition of duality (not referring to truthtables), using which double harmony is defined. The paper proves that in a doubly-harmonious system, the coordination rule, typical to bilateral systems, is admissible. (shrink)
In this paper, we show the equivalence between the provability of a proof system of basic hybrid logic and that of translated formulas of the classical predicate logic with equality and explicit substitution by a purely proof–theoretic method. Then we show the equivalence of two groups of proof systems of hybrid logic: the group of labelled deduction systems and the group of modal logic-based systems.
In this paper we give relational semantics and an accompanying relational proof theory for full Lambek calculus (a sequent calculus which we denote by FL). We start with the Kripke semantics for FL as discussed in  and develop a second Kripke-style semantics, RelKripke semantics, as a bridge to relational semantics. The RelKripke semantics consists of a set with two distinguished elements, two ternary relations and a list of conditions on the relations. It is accompanied by a Kripke-style valuation (...) system analogous to that in . Soundness and completeness theorems with respect to FL hold for RelKripke models. Then, in the spirit of the work of Orlowska , , and Buszkowski and Orlowska , we develop relational logic RFL. The adjective relational is used to emphasize the fact that RFL has a semantics wherein formulas are interpreted as relations. We prove that a sequent Γ → α in FL is provable if and only if a translation, t( $\gamma_1 \bullet \cdots \bullet \gamma_n \supset \alpha)\varepsilon \upsilon u$ , has a cut-complete fundamental proof tree. This result is constructive: that is, if a cut-complete proof tree for $t(\gamma_1 \bullet \cdots \bullet \gamma_n \supset \alpha)\varepsilon \upsilon u$ is not fundamental, we can use the failed proof search to build a relational countermodel for $t(\gamma_1 \bullet \cdots \bullet \gamma_n \supset \alpha)\varepsilon \upsilon u$ and from this, build a RelKripke countermodel for $\gamma_1 \bullet \cdots \bullet \gamma_n \supset \alpha$ . These results allow us to add FL, the basic substructural logic, to the list of those logies of importance in computer science with a relational proof theory. (shrink)
In this paper, we continue our research on a hybrid narrative-argumentative approach to evidential reasoning in the law by showing the interaction between factual reasoning (providing a proof for ‘what happened’ in a case) and legal reasoning (making a decision based on the proof). First we extend the hybrid theory by making the connection with reasoning towards legal consequences. We then emphasise the role of legal stories (as opposed to the factual stories of the hybrid theory). Legal stories (...) provide a coherent, holistic legal perspective on a case. They steer what needs to be proven but are also selected on the basis of what can be proven. We show how these legal stories can be used to model a shift of the legal perspective on a case, and we discuss how gaps in a legal story can be filled using a factual story (i.e. the process of reasoning with circumstantial evidence). Our model is illustrated by a discussion of the Dutch Wamel murder case. (shrink)
While direct proof is widely considered the paradigm of the acquisition of knowledge by deductive means, indirect proof has traditionally been criticized as showing merely ‘that’ its conclusion is true and not ‘why’ it is true. This paper accounts for the traditional objection by emphasizing the argumentative role in indirect proof of logical principles such as excluded middle and non-contradiction.
This paper is concerned with the structure of texts in which aproof is presented. Some parts of such a text are assumptions, otherparts are conclusions. We show how the structural organisation of thetext into assumptions and conclusions helps to check the validity of theproof. Then we go on to use the structural information for theformulation of proof rules, i.e., rules for the (re-)construction ofproof texts. The running example is intuitionistic propositional logicwith connectives , and. We give new proofs of (...) some familiar results aboutthe proof theory of this logic to indicate how the new techniques workout. (shrink)
Since the analysis by John Bell in 1965, the consensus in the literature is that von Neumann’s ‘no hidden variables’ proof fails to exclude any significant class of hidden variables. Bell raised the question whether it could be shown that any hidden variable theory would have to be nonlocal, and in this sense ‘like Bohm’s theory.’ His seminal result provides a positive answer to the question. I argue that Bell’s analysis misconstrues von Neumann’s argument. What von Neumann proved was (...) the impossibility of recovering the quantum probabilities from a hidden variable theory of dispersion free (deterministic) states in which the quantum observables are represented as the ‘beables’ of the theory, to use Bell’s term. That is, the quantum probabilities could not reflect the distribution of pre-measurement values of beables, but would have to be derived in some other way, e.g., as in Bohm’s theory, where the probabilities are an artefact of a dynamical process that is not in fact a measurement of any beable of the system. (shrink)
The formulas-as-types isomorphism tells us that every proof and theorem, in the intuitionistic implicational logic $H_\rightarrow$, corresponds to a lambda term or combinator and its type. The algorithms of Bunder very efficiently find a lambda term inhabitant, if any, of any given type of $H_\rightarrow$ and of many of its subsystems. In most cases the search procedure has a simple bound based roughly on the length of the formula involved. Computer implementations of some of these procedures were done in (...) Dekker. In this paper we extend these methods to full classical propositional logic as well as to its various subsystems. This extension has partly been implemented by Oostdijk. (shrink)
Dynamic and proof-conditional approaches to discourse (exemplified by Discourse Representation Theory and Type-Theoretical Grammar, respectively) are related through translations and transitions labeled by first-order formulas with anaphoric twists. Type-theoretic contexts are defined relative to a signature and instantiated modeltheoretically, subject to change.
This paper shows how proof nets can be used to formalize the notion of incomplete dependency used in psycholinguistic theories of the unacceptability of center embedded constructions. Such theories of human language processing can usually be restated in terms of geometrical constraints on proof nets. The paper ends with a discussion of the relationship between these constraints and incremental semantic interpretation.
A combination of epistemic logic and dynamic logic of programs is presented. Although rich enough to formalize some simple game-theoretic scenarios, its axiomatization is problematic as it leads to the paradoxical conclusion that agents are omniscient. A cut-free labelled Gentzen-style proof system is then introduced where knowledge and action, as well as their combinations, are formulated as rules of inference, rather than axioms. This provides a logical framework for reasoning about games in a modular and systematic way, and to (...) give a step-by-step reconstruction of agents omniscience. In particular, its semantic assumptions are made explicit and a possible solution can be found in weakening the properties of the knowledge operator. (shrink)
With respect to the confirmation of mathematical propositions, proof possesses an epistemological authority unmatched by other means of confirmation. This paper is an investigation into why this is the case. I make use of an analysis drawn from an early reliability perspective on knowledge to help make sense of mathematical proofs singular epistemological status.
In the last decade the concept of context has been extensivelyexploited in many research areas, e.g., distributed artificialintelligence, multi agent systems, distributed databases, informationintegration, cognitive science, and epistemology. Three alternative approaches to the formalization of the notion ofcontext have been proposed: Giunchiglia and Serafini's Multi LanguageSystems (ML systems), McCarthy's modal logics of contexts, andGabbay's Labelled Deductive Systems.Previous papers have argued in favor of ML systems with respect to theother approaches. Our aim in this paper is to support these arguments froma (...) theoretical perspective. We provide a very general definition of ML systems, which covers allthe ML systems used in the literature, and we develop a proof theoryfor an important subclass of them: the MR systems. We prove variousimportant results; among other things, we prove a normal form theorem,the sub-formula property, and the decidability of an importantinstance of the class of the MR systems. The paper concludes with a detailed comparison among the alternativeapproaches. (shrink)
Aim of this work is to investigate from a proof-theoretic viewpoint a propositional and a predicate sequent calculus with an ω–type schema of inference that naturally interpret the propositional and the predicate until–free fragments of Linear Time Logic LTL respectively. The two calculi are based on a natural extension of ordinary sequents and of standard modal rules. We examine the pure propositional case (no extralogical axioms), the propositional and the first order predicate cases (both with a possibly infinite set (...) of extralogical axioms). For each system we provide a syntactic proof of cut elimination and a proof of completeness. (shrink)