Constructing Natural Extensions of Propositional Logics

0459809 - ÚI 2017 RIV NL eng J - Článek v odborném periodiku
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
Grant CEP: GA ČR GA13-14654S
Institucionální podpora: RVO:67985807
Klíčová slova: abstract algebraic logic * consequence relations * propositional logic * natural extensions * transfer theorems
Kód oboru RIV: BA - Obecná matematika
Impakt faktor: 0.589, rok: 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.
Trvalý link: http://hdl.handle.net/11104/0259968