Journal of Logic, Language and Information 10 (2):339-352 (2001)
|Abstract||In Zielonka (1981a, 1989), I found an axiomatics for the product-free calculus L of Lambek whose only rule is the cut rule. Following Buszkowski (1987), we shall call such an axiomatics linear. It was proved that there is no finite axiomatics of that kind. In Lambek's original version of the calculus (cf. Lambek, 1958), sequent antecedents are non empty. By dropping this restriction, we obtain the variant L 0 of L. This modification, introduced in the early 1980s (see, e.g., Buszkowski, 1985; Zielonka, 1981b), did not gain much popularity initially; a more common use of L 0 has only occurred within the last few years (cf. Roorda, 1991: 29). In Zielonka (1988), I established analogous results for the restriction of L 0 to sequents without left (or, equivalently, right) division. Here, I present a similar (cut-rule) axiomatics for the whole of L 0. This paper is an extended, corrected, and completed version of Zielonka (1997). Unlike in Zielonka (1997), the notion of rank of an axiom is introduced which, although inessential for the results given below, may be useful for the expected non-finite-axiomatizability proof.|
|Keywords||axiomatizability cut rule Lambek calculus|
|Categories||categorize this paper)|
|Through your library||Configure|
Similar books and articles
Maria Bulińska (2009). On the Complexity of Nonassociative Lambek Calculus with Unit. Studia Logica 93 (1):1 - 14.
Wojciech Zielonka (2002). On Reduction Systems Equivalent to the Lambek Calculus with the Empty String. Studia Logica 71 (1):31-46.
Wojciech Zielonka (1990). Linear Axiomatics of Commutative Product-Free Lambek Calculus. Studia Logica 49 (4):515 - 522.
Wojciech Zielonka (1989). A Simple and General Method of Solving the Finite Axiomatizability Problems for Lambek's Syntactic Calculi. Studia Logica 48 (1):35 - 39.
Maria Bulińska (2005). The Pentus Theorem for Lambek Calculus with Simple Nonlogical Axioms. Studia Logica 81 (1):43 - 59.
Brian Hill & Francesca Poggiolesi (2010). A Contraction-Free and Cut-Free Sequent Calculus for Propositional Dynamic Logic. Studia Logica 94 (1):47 - 72.
Wojciech Buszkowski (1996). The Finite Model Property for BCI and Related Systems. Studia Logica 57 (2-3):303 - 323.
Rajeev Gore, Linda Postniece & Alwen Tiu, Cut-Elimination and Proof-Search for Bi-Intuitionistic Logic Using Nested Sequents.
Heinrich Wansing (2002). A Rule-Extension of the Non-Associative Lambek Calculus. Studia Logica 71 (3):443-451.
Wojciech Zielonka (2000). Cut-Rule Axiomatization of the Syntactic Calculus NL. Journal of Logic, Language and Information 9 (3):339-352.
Added to index2009-01-28
Total downloads2 ( #245,513 of 722,698 )
Recent downloads (6 months)0
How can I increase my downloads?