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Constructing Natural Extensions of Propositional Logics

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    0459809 - ÚI 2017 RIV NL eng J - Journal Article
    Přenosil, Adam
    Constructing Natural Extensions of Propositional Logics.
    Studia Logica. Roč. 104, č. 6 (2016), s. 1179-1190. ISSN 0039-3215. E-ISSN 1572-8730
    R&D Projects: GA ČR GA13-14654S
    Institutional support: RVO:67985807
    Keywords : abstract algebraic logic * consequence relations * propositional logic * natural extensions * transfer theorems
    Subject RIV: BA - General Mathematics
    Impact factor: 0.589, year: 2016

    The proofs of some results of abstract algebraic logic, in particular of the transfer principle of Czelakowski, assume the existence of so-called natural extensions of a logic by a set of new variables. Various constructions of natural extensions, claimed to be equivalent, may be found in the literature. In particular, these include a syntactic construction due to Shoesmith and Smiley and a related construction due to Lo´s and Suszko. However, it was recently observed by Cintula and Noguera that both of these constructions fail in the sense that they do not necessarily yield a logic. Here we show that whenever the Lo´s–Suszko construction yields a logic, so does the Shoesmith–Smiley construction, but not vice versa. We also describe the smallest and the largest conservative extension of a logic by a set of new variables and show that contrary to some previous claims in the literature, a logic of cardinality K may have more than one conservative extension of cardinality K by a set of new variables. In this connection we then correct a mistake in the formulation of a theorem of Dellunde and Jansana.
    Permanent Link: http://hdl.handle.net/11104/0259968

     
     
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