D. Basin, S. Matthews and L Vigano. Natural Deduction for Non-Classical Logics. Studia Logica, 60(1): 119-160, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....stressing that we have only discussed what might be called mainstream hybrid logic. One of the most exciting recent developments is the amount of work in neighbouring elds which echoes key hybrid logical themes. For example, the brand of labeled deduction developed by Basin, Matthews and Vigano [8, 9], links naturally with recent hybrid proof theory. Polish work on the logic of information systems and rough sets has lead to the evolution of what are essentially hybrid logics; see, for example, Konikowska [28] For something close to hybrid logic, but developed from the perspective of ....

D. Basin, S. Matthews, and L. Vigano. Natural deduction for non-classical logics. Studia Logica, 60(1):119-160, 1998.

....version of modal system K. After that we extend our work on natural deduction and term assignment systems, with the further correspondence between typed calculus and category theory, that is usually referred as the extended Curry Howard isomorphism . Other approaches (e.g. Basin et al.[BMV98], Martini and Masini[MM96] that tie in the semantics of modal logics (in terms of possible worlds) with their sytanctic presentation have been devised. We do not say much about this line of work here. 2 The Logical System 2.1 Sequent Calculus and Axiomatic System We take the sequent calculus ....

D. Basin, S. Matthews and L Vigano. Natural Deduction for Non-Classical Logics. Studia Logica, 60(1): 119-160, 1998.

....a substantial amount of automation is supported. Despite this, some proofs are still tedious to perform as the way to the informal pen and paper level of abstraction seems fairly long. The present paper is an attempt to narrow this gap. It is inspired by work on Labelled Natural Deduction (LND) [15, 3] which combines classical natural deduction [11, 14] with labelled deductive systems [5] The LND formalism has shown its worth for traditional modal logics [1, 2] The rest of this paper is organized as follows: In Section 2 we consider propositional logics with a binary modality. These can be ....

....of frames F if for all frames F of F, for all models M based on F , M w : It is clear that M w : i M;w j= Thus, if we let AF w : denote validity of w : in the class of all frames we have Proposition 1. j= AF i AF w : We de ne a LND system for L AF in the style of [1, 3]: w : w : w : I w : w : w : E [w : v : w : E v : u : R(v; u; w) w : I w : v : u : R(v; u; w) w 0 : w 0 : E In E, v and u are di erent from both w; w 0 and each other, and do ....

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D. Basin, S. Matthews, and L. Vigano. Natural Deduction for Non-Classical Logics. Studia Logica, 60(1):119-160, 1998.

....prefixes . In that sense, our calculi are explicit systems following [Gor99] but without introducing any extra proof theoretical device that does not belong to the object modal language. Furthermore, the calculi defined in this paper does not differ very much in spirit with those defined in [Rus96, BMV98]. Indeed, we associate syntactically rules to formulas defining relational theories. However, we are able to capture all the conditions on frames for the properly displayable modal logics defined in [Kra96] We wish also to thank one of the referees for pointing us to [Bla98, Tza99] where ....

....of conditions from Figure 2 and Figure 3 in [Gor99] are C Pi 0 2 definable. All the first order definable classes of frames considered in [Rus96, CFdCGH97] are C Pi 0 2 definable and C Pi 0 2 contains all the modal logics (in their nominal tense version) defined with Horn clauses from [BMV98]. Furthermore, for any nominal tense logic L = hNTL(G; H) Ci such that C is first order definable by a finite set Phi of restricted Pi 0 2 formulae, it is known that the L validity problem can be translated into FOL validity (using [Ben83, GG93] However, there is no guarantee that L admits ....

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D. Basin, S. Matthews, and L. Vigan`o. Natural deduction for nonclassical logics. Studia Logica, 60(1):119--160, 1998.

....Deduction Systems, Sequent Systems. 1 Introduction Context This paper is part of ongoing research that aims at developing a framework for applying proof theory to the analysis of the complexity of families of non classical logics, and thereby designing decision procedures for these logics [4, 5, 6, 34]. Modal, relevance and other non classical logics are usually presented using Hilbert style systems. However, although uniform, these systems are dicult to use in practice, especially in comparison with the more natural Gentzenstyle systems such as natural deduction, sequent and tableaux ....

....2] Fitting [13] and Gabbay [14] among others, labelled deduction systems have been used extensively to formalize non classical logics, and exploited for applications that can be modelled by these logics. For example, several labelled deduction systems have been given for relevance logics, e.g. [1, 2, 5, 8, 9, 13, 16, 20], ranging from systems where the labels explicitly encode information from the Kripke or algebraic semantics, to systems where the additional information is of a proof theoretical nature, e.g. encoding resources and their use. Research on labelling has focused not only on the design of labelled ....

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D. Basin, S. Matthews, and L. Vigano. Natural deduction for non-classical logics. Studia Logica, 60(1):119-160, 1998.

.... model; for example, the binary relational formula wRw 0 tells us that the world w 0 is accessible from w in a Kripke model (which is a particular, simple, instance of the general models of the form hU; B; i) Labelled deduction systems have been given for several non classical logics, e.g. [1, 2, 3, 4, 8, 10, 12, 14, 16, 19, 28]. Research on labelling has focused not only on the design of systems for speci c logics, but also, more generally, on the characterization of the classes of logics that can be formalized this way. General properties and limitations of labelling techniques have also been investigated. For example, ....

....Research on labelling has focused not only on the design of systems for speci c logics, but also, more generally, on the characterization of the classes of logics that can be formalized this way. General properties and limitations of labelling techniques have also been investigated. For example, [4, 28] highlight a tradeo between limitations and properties: if we reason on the semantic information provided by labelling using Horn style rules, then we are able to present only a subset of all possible non classical logics, but we can still capture many of the most common ones and, more ....

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D. Basin, S. Matthews, and L. Vigano. Natural deduction for nonclassical logics. Studia Logica, 60(1):119-160, 1998.

.... in a particular model; for example, the relational formula w:Rw # tells us that the world w # is accessible from w in a Kripke model (which is a particular, simple, instance of the general models of the form ##) Labelled deduction systems have been given for several non classical logics, e.g. [1, 2, 3, 4, 8, 10, 12, 14, 16, 19, 28]. Research on labelling has focused not only on the design of systems for specific logics, but also, more generally, on the characterization of the classes of logics that can be formalized this way. General properties and limitations of labelling techniques have also been investigated. For ....

....Research on labelling has focused not only on the design of systems for specific logics, but also, more generally, on the characterization of the classes of logics that can be formalized this way. General properties and limitations of labelling techniques have also been investigated. For example, [4, 28] highlight a tradeo# between limitations and properties: if we reason on the semantic information provided by labelling using Horn style rules, then we are able to present only a subset of all possible non classical logics, but we can still capture many of the most common ones and, more ....

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D. Basin, S. Matthews, and L. Vigano. Natural deduction for non-classical logics. Studia Logica, 60(1):119--160, 1998.

....towards the bring of these logics with many valued ones. 1. Introduction 1.1. Context Labelled deduction is an approach to presenting di erent logics in a uniform and natural way as Gentzen style deduction systems, such as natural deduction, se1 quent or tableaux systems; see, for instance, [3, 4, 11, 12, 15, 21, 25]. It has been applied, for example, to formalize and reason about dynamic state oriented properties, such as knowledge, belief, time, space, and resources, and thereby formalize deduction systems for a wide range of non classical logics, such as modal, temporal, intuitionistic, relevance and ....

....be evaluated at w. We can also use labels to specify at the syntactic level the way in which the di erent worlds are related in the Kripke structures; for example, we can use the formula wRv to specify that the world denoted by v is accessible from that denoted by w. As discussed in, among others, [4, 21, 25], a modal labelled natural deduction system over this extended language is then obtained by giving inference rules for deriving labelled formulae, introducing or eliminating formula constructors such as implication A and modal necessity 2, and by de ning a suitable labelling algebra, which ....

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David Basin, Sean Matthews, and Luca Vigano. Natural deduction for non-classical logics. Studia Logica, 60(1):119-160, 1998.

....a rst step towards bring these logics with many valued ones. 1 Introduction Context Labelled Deduction is an approach to presenting di erent logics in a uniform and natural way as Gentzen style deduction systems, such as natural deduction, sequent or tableaux systems; see, for instance, [2, 3, 8, 9, 11, 16, 21]. It has been applied, for example, to formalize and reason about dynamic stateoriented properties, such as knowledge, belief, time, space, and resources, and thereby formalize deduction systems for a wide range of non classical logics, such as modal, temporal, intuitionistic, relevance and ....

....natural deduction systems. The main idea underlying our approach is the use of algebras of truth values as the labelling algebras of our systems, which allows us to give generalized systems for multiple valued logics. More speci cally, our framework generalizes previous work, including our own [3, 16, 21], on labelled deduction systems where labels represent worlds in the underlying Kripke structures, and this generalization is illustrated by the following observation: since we can take multiple valued logics as meaning not only nitely or in nitely many valued logics but also power set logics, ....

[Article contains additional citation context not shown here]

D. Basin, S. Matthews, and L. Vigano. Natural deduction for non-classical logics. Studia Logica, 60(1):119-160, 1998.

....bring these logics with many valued ones. 1 Introduction Context Labelled Deduction is an approach to providing frameworks for presenting di erent logics in a uniform and natural way as Gentzen style deduction systems, such as natural deduction, sequent or tableaux systems; see, for instance, [2, 3, 8, 9, 11, 16, 21]. It has been applied, for example, to formalize and reason about dynamic state oriented properties, such as knowledge, belief, time, space, and resources, and thereby formalize deduction systems for a wide range of non classical logics, such as modal, temporal, intuitionistic, relevance and ....

....systems. The main idea underlying our approach is the use of algebras of truth values as the labelling algebras of our labelled deduction systems, which allows us to give generalized systems for multiple valued logics. More speci cally, our framework generalizes previous work, including our own [3, 16, 21], on labelled deduction systems where labels represent worlds in the underlying Kripke structures, and this generalization is illustrated by the following observation: since we can take multiple valued logics as meaning not only nitely or in nitely many valued logics but also power set logics, ....

[Article contains additional citation context not shown here]

D. Basin, S. Matthews, and L. Vigano. Natural deduction for non-classical logics. Studia Logica, 60(1):119-160, 1998.

....management of separate theories and their structured combination, resulting in a parameterized proof development system where (although it is not formally quantifiable) proof construction is natural and intuitive. Due to lack of space, proofs have been omitted or considerably shortened, see [3] for details. 2 LABELLED NON CLASSICAL LOGICS In this section we formalize our presentations. We introduce the fundamentals of how an LDS presentation relates to a Kripke semantics (Section 2.1) After this we define the base logic (Section 2.2) and the associated class of relational theories ....

....i.e. where the base logic contains Ei, we need i, while for a classical negation, i.e. with Ec, we need i and c; similarly, the monR i rules complement monl. Moreover, only by requiring these complementary rules can one establish desired prooftheoretic results (cf. the proof of Theorem 11 in [3]) Thus it is convenient, on pragmatic grounds, to assume that a base logic B is extended with a theory that includes these minimal relational rules (a characterization of the logics in which this complementarity is not satisfied, e.g. Ei without i, or Ec with only c, is out of the scope of ....

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D. Basin, S. Matthews, and L. Vigan`o. Natural deduction for non-classical logics. Technical Report MPI-I-96-2-006, Max-PlanckInstitut fur Informatik, Saarbrucken, 1996. Available at the URL http://www.mpisb. mpg.de/¸luca/Publications/publications.html.

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D. Basin, S. Matthews and L Vigano. Natural Deduction for Non-Classical Logics. Studia Logica, 60(1): 119-160, 1998.

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