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47
Fresh Logic
 Journal of Applied Logic
, 2007
"... Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with resp ..."
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Cited by 186 (22 self)
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Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with respect to metavariables. We present oneandahalfthorder logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of oneandahalfthorder logic derivability, show them equivalent, show that the derivations satisfy cutelimination, and prove correctness of an interpretation of firstorder logic within it. We discuss the technicalities in a wider context as a casestudy for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation
Logics of Formal Inconsistency
 Handbook of Philosophical Logic
"... 1.1 Contradictoriness and inconsistency, consistency and noncontradictoriness In traditional logic, contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory ..."
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Cited by 45 (19 self)
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1.1 Contradictoriness and inconsistency, consistency and noncontradictoriness In traditional logic, contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory
The complexity of reasoning with boolean modal logics
 LTCSReport 0002, LuFG Theoretical Computer Science, RWTH
, 2001
"... 1 Motivation Since Modal Logics are an extension of Propositional Logic, they provide Boolean operators for constructing complex formulae. However, most Modal Logics do not admit Boolean operators for constructing complex modal parameters to be used in the box and diamond operators. This asymmetry i ..."
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Cited by 35 (5 self)
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1 Motivation Since Modal Logics are an extension of Propositional Logic, they provide Boolean operators for constructing complex formulae. However, most Modal Logics do not admit Boolean operators for constructing complex modal parameters to be used in the box and diamond operators. This asymmetry is not present in Boolean Modal Logics, in which box and diamond quantify over arbitrary Boolean combinations of atomic modal parameters [9]. Boolean Modal Logics have been considered in various forms and contexts: 1. &quot;Pure &quot; Boolean Modal Logic has been studied in [9]. Negation and intersection of modal parameters occur in some variants of Propositional Dynamic Logic, see, e.g., [7, 16, 22].
Relation algebras in qualitative spatial reasoning
 Fundamenta Informaticae
, 1999
"... The formalization of the “part – of ” relationship goes back to the mereology of S. Le´sniewski, subsequently taken up by Leonard & Goodman (1940), and Clarke (1981). In this paper we investigate relation algebras obtained from different notions of “part–of”, respectively, “connectedness” in var ..."
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Cited by 34 (13 self)
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The formalization of the “part – of ” relationship goes back to the mereology of S. Le´sniewski, subsequently taken up by Leonard & Goodman (1940), and Clarke (1981). In this paper we investigate relation algebras obtained from different notions of “part–of”, respectively, “connectedness” in various domains. We obtain minimal models for the relational part of mereology in a general setting, and when the underlying set is an atomless Boolean algebra. 1
Step by Step  Building Representations in Algebraic Logic
 Journal of Symbolic Logic
, 1995
"... We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterised according to the outcome of certain games. The Lyndon conditions defini ..."
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Cited by 28 (15 self)
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We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterised according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Countable relation algebras with homogeneous representations are characterised by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is !categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another twoplayer game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this ap...
Complete Representations in Algebraic Logic
 JOURNAL OF SYMBOLIC LOGIC
"... A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary. ..."
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Cited by 19 (8 self)
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A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary.
A RelationAlgebraic Approach to the Region Connection Calculus
 Fundamenta Informaticae
, 2001
"... We explore the relationalgebraic aspects of the region connection calculus (RCC) of Randell et al. (1992a). In particular, we present a refinement of the RCC8 table which shows that the axioms provide for more relations than are listed in the present table. We also show that each RCC model leads ..."
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Cited by 19 (0 self)
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We explore the relationalgebraic aspects of the region connection calculus (RCC) of Randell et al. (1992a). In particular, we present a refinement of the RCC8 table which shows that the axioms provide for more relations than are listed in the present table. We also show that each RCC model leads to a Boolean algebra. Finally, we prove that a refined version of the RCC5 table has as models all atomless Boolean algebras B with the natural ordering as the "part  of" relation, and that the table is closed under first order definable relations iff B is homogeneous. 1 Introduction Qualitative reasoning (QR) has its origins in the exploration of properties of physical systems when numerical information is not sufficient  or not present  to explain the situation at hand (Weld and Kleer, 1990). Furthermore, it is a tool to represent the abstractions of researchers who are constructing numerical systems which model the physical world. Thus, it fills a gap in data modeling which often l...
Logic and Relativity (in the light of definability theory)
, 2002
"... Introduction The combined investigation of mathematical logic and relativity theory is not at all new, as follows. Direction (1): Already at the beginnings, i.e. around 1920, Einstein's friend Reichenbach set to himself the goal of building up relativity as a logical theory, i.e. a theory pure ..."
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Cited by 14 (8 self)
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Introduction The combined investigation of mathematical logic and relativity theory is not at all new, as follows. Direction (1): Already at the beginnings, i.e. around 1920, Einstein's friend Reichenbach set to himself the goal of building up relativity as a logical theory, i.e. a theory purely in rstorder logic, cf. [30]. Similarly, Carnap [8, 9] pursued and advocated the same goal using a more sophisticated version of mathematical logic. De nability theory, one of the most important branches of modern logic, was brought to existence in the relativity book [30] because of the special needs of relativity theory. (This eld reached maturity via extensive work by Tarski on de nability as indicated in the dissertation.) Direction (2): The foundations of logic presuppose a kind of worldview (Weltanschaung) which has an eect on the \structure" of the theory of logic. In this sense the latest developments of relativity and related areas (e.g. Godel spacetime, Kerr spacetime) provid
Beyond modalities: Sufficiency and mixed algebras
 In E. Orłowska & A. Szałas (Eds.), Relational Methods in Computer Science Applications, 277– 299
, 2000
"... this paper for a discussion on the merits or otherwise of Kripke semantics and its "sufficiency" extension. Just as Kripke frames are dual to a class of Boolean algebras with modal operators [18, 24], one can build a duality for frames and Boolean algebras with sufficiency operators. Mixed ..."
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Cited by 13 (11 self)
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this paper for a discussion on the merits or otherwise of Kripke semantics and its "sufficiency" extension. Just as Kripke frames are dual to a class of Boolean algebras with modal operators [18, 24], one can build a duality for frames and Boolean algebras with sufficiency operators. Mixed structures occur when modal and sufficiency operators arise from the same accessibility relation. In this paper we introduce the classes of sufficiency algebras and that of mixed algebras which include both a modal and a sufficiency operator, and study representation and duality theory for these classes of algebras. We also give examples for classes of firstorder definable frames, where such operators are required for a "modalstyle" axiomatisation. 2 Why sufficiency and mixed algebras?