Constructing Natural Extensions of Propositional Logics

详细信息    查看全文

• 作者：Adam Přenosil
• 关键词：Abstract algebraic logic ; Consequence relations ; Propositional logic ; Natural extensions ; Transfer theorems
• 刊名：Studia Logica
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：104
• 期：6
• 页码：1179-1190
• 全文大小：495 KB
• 刊物类别：Humanities, Social Sciences and Law
• 刊物主题：Philosophy
  Logic
  Mathematical Logic and Foundations
  Computational Linguistics
  
• 出版者：Springer Netherlands
• ISSN：1572-8730
• 卷排序：104

文摘

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 Łoś 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 Łoś–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 \({\kappa}\) may have more than one conservative extension of cardinality \({\kappa}\) by a set of new variables. In this connection we then correct a mistake in the formulation of a theorem of Dellunde and Jansana.