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  • Mathematik  (603)
  • 1
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    Reasoning about types of action and agent capabilities (2013)
    Oxford University Press
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    Publikationsdatum: 2013-09-27
    Beschreibung: We present a logical system for reasoning about types of actions (processes) and about agent capabilities to execute types of actions. The syntax of the system is based on that of Propositional Dynamic Logic ( PDL ), though the semantics we define is different (interpreting process terms as types, i.e. sets of binary relations). The standard PDL syntax is extended with capabilities statements, as in the KARO framework, atomic process types specified as precondition-effect pairs, written as , as well as backwards possibility operators. The resulting system is shown (by filtration) to have a decidable satisfiability problem and a sound and complete Gentzen-style proof system is presented.
    Print ISSN: 1367-0751
    Digitale ISSN: 1368-9894
    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 2
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    Epsilon-invariant substitutions and indefinite descriptions (2013)
    Oxford University Press
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    Publikationsdatum: 2013-09-27
    Beschreibung: It is known that an epsilon-invariant sentence has a first-order reformulation, although it is not in an explicit form, since, the proof uses the non-constructive interpolation theorem. We make an attempt to describe the explicit meaning of sentences containing epsilon-terms, adopting the strong assumption of their first-order reformulability. We will prove that, if a monadic predicate is syntactically independent from an epsilon-term and if the sentence obtained by substituting the variable of the predicate with the epsilon-term is epsilon-invariant, then the sentence has an explicit first-order reformulation. Finally, we point out that the formula gives a contextual-quantificational meaning for the indefinite descriptions, provided that one accepts Kneebone’s read of epsilon-terms.
    Print ISSN: 1367-0751
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    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 3
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    An algebraic look at filtrations in modal logic (2013)
    Oxford University Press
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    Publikationsdatum: 2013-09-27
    Beschreibung: Filtration constructions are among the oldest and best known methods for obtaining finite model properties for modal logics, and appear in the literature in both model-theoretic and algebraic versions. In this article we investigate definitions of algebraic filtrations by means of different types of binary relations on modal algebras, and the relationships between these. We generalize the notion of a model-theoretic filtration somewhat while simultaneously lifting it to the level of frames. We proceed to link algebraic filtrations with their model- or frame-theoretic counterparts by showing how our filtration notions interface neatly with the well-known duality theory of modal algebras and Kripke frames. We illustrate, by means of some examples, how this theory enables one to easily translate between algebraic and model-theoretic versions of some well-known filtrations. We obtain some order theoretic insights regarding the (usually model-theoretically specified) smallest and largest filtrations by considering their algebraic versions, thus demonstrating the utility of having ready access to both versions.
    Print ISSN: 1367-0751
    Digitale ISSN: 1368-9894
    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 4
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    Conservativity: a necessary property for the maximization of witness sets (2013)
    Oxford University Press
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    Publikationsdatum: 2013-09-27
    Beschreibung: It is generally assumed that Conservativity is a necessary ingredient for interpreting collective and cumulative predicates or, more in general, for interpreting—what I termed as—Independent Set (IS) readings (a.k.a. Scopeless readings). Under this assumption, several approaches aiming at formally providing logical representations of collective/cumulative readings implement Conservativity. This is done also in the frameworks defined in Winter (2001, Flexibility Principles in Boolean Semantics: Coordination, Plurality, and Scope in Natural Language , MIT Press) and in Robaldo (2011, J. Logic Lang. Infor. , 20, 233–271). This article investigates further the role played by Conservativity in the interpretation of IS readings, and argues that its need is not specifically due to the use of collective/cumulative predicates. Rather, it serves to maximize witness sets for quantifiers thereby guaranteeing correct truth conditions in several cases.
    Print ISSN: 1367-0751
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    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 5
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    On second-order characterizability (2013)
    Oxford University Press
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    Publikationsdatum: 2013-09-27
    Beschreibung: We investigate the extent of second-order characterizable structures by extending Shelah's Main Gap dichotomy to second-order logic. For this end we consider a countable complete first-order theory T . We show that all sufficiently large models of T have a characterization up to isomorphism in the extension of second-order logic obtained by adding a little bit of infinitary logic if and only if T is shallow superstable with NDOP and NOTOP. Our result relies on cardinal arithmetic assumptions. Under weaker assumptions we get consistency results or alternatively results about second-order logic with Henkin semantics. Mathematics Subject Classification: 03C85, 03C75.
    Print ISSN: 1367-0751
    Digitale ISSN: 1368-9894
    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 6
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    A proof procedure for adaptive logics (2013)
    Oxford University Press
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    Publikationsdatum: 2013-09-27
    Beschreibung: In this article, I present a procedure that generates proofs for finally derivable adaptive logic consequences. A proof procedure for the inconsistency adaptive logic CLuN r is already presented in [ 7 ]. In this article a procedure for CLuN m is presented and the results for both logics are generalized to all adaptive logics, on the presupposition that there exists a proof procedure for the lower limit logic. The generated proofs are so called goal-directed proofs, i.e. proofs that (i) start with the formula (the goal) of which one wants to know whether it is a consequence of a certain premise set and (ii) only consist of lines that may potentially be useful for proving or disproving the goal. The goal-directed proofs form good explications of actual problem-solving reasoning processes.
    Print ISSN: 1367-0751
    Digitale ISSN: 1368-9894
    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 7
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    Malinowski modalization, modalization through fibring and the Leibniz hierarchy (2013)
    Oxford University Press
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    Publikationsdatum: 2013-09-27
    Beschreibung: We show how various modal systems considered by Malinowski as extensions of classical propositional calculus may be obtained as fibrings of classical propositional calculus and corresponding implicative modal logics, using the fibring framework for combining logics of Fern'andez and Coniglio. Taking advantage of this construction and known results of Malinowski, we draw some useful conclusions concerning some limitations of the fibring process. Finally, Malinowski's constructions are extended to obtain some modal extensions of arbitrary equivalential logics in the context of abstract algebraic logic. These are studied with respect to their algebraic character.
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    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 8
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    Two direct proofs that LLPO implies the detachable fan theorem (2013)
    Oxford University Press
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    Publikationsdatum: 2013-09-27
    Beschreibung: It is shown, directly and in two ways, that Brouwer’s fan theorem for detachable bars can be proved within Bishop-style constructive mathematics supplemented by (countably many applications of) the omniscience principle LLPO .
    Print ISSN: 1367-0751
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    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 9
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    Editorial: Special issue on non-classical mathematics (2013)
    Oxford University Press
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    Publikationsdatum: 2013-01-16
    Print ISSN: 1367-0751
    Digitale ISSN: 1368-9894
    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 10
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    Mathematical pluralism (2013)
    Oxford University Press
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    Publikationsdatum: 2013-01-16
    Beschreibung: There is a plurality of mathematical investigations. These cannot all be reduced to proofs within the framework of Zermelo Fraenkel set theory, if only because some of them use non-classical logic (such as the various branches of intuitionist mathematics). How is one to understand this situation? In this article, I suggest that one should see this plurality as analogous to the plurality of games, any of which may be played. Various objections are considered and rejected, including the charge that the picture engenders a pernicious relativism.
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    Thema: Mathematik
    Publiziert von Oxford University Press
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