zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Structure, consequence relation and logic. (English) Zbl 0823.03013
Gabbay, D. M. (ed.), What is a logical system? Oxford: Oxford University Press. Stud. Log. Comput. 4, 239-259 (1994).

This is a paper on sequent systems for logics containing a conditional, that is a binary operator which is meant to formalize subjunctive “if then”-sentences. The presentation is couched in considerations on the combination of data structures, its impact on logical operations, and on other by now rather familiar ideas from the field of substructural logics. The emphasis on operations on structured data and their correspondence with logical connectives is very much in the spirit of Gabbay’s theory of structured consequence relations [D. M. Gabbay, “A general theory of structured consequence relations”, in: K. Došen and P. Schroeder-Heister (eds.), Substructural logics, Stud. Log. Comput. 2, 109-151 (1994; Zbl 0811.68056)].

It is argued that the conditional connective represents in the object language deductions from implicit hypotheses. Since such deductions fail to be transitive, a structural account of conditionals may be based on restricting the composition of proofs. Therefore, second-degree sequents are introduced in which a principal deduction relation P may relate sets of auxiliary sequents X a Y. The authors point out that various conditional logics which are related to well-known systems of conditional logic can be classified by restrictions on the composition of proofs. In almost all cases these restrictions can be expressed by purely structural rules in the higher-level sequent calculus.

As the authors admit, they “have not answered the main question of this book” (namely “What is a logical system?”). They seem, however, to have some sympathy with the notion of a structured consequence relation.

03B60Other nonclassical logic
68T27Logic in artificial intelligence
03F05Cut-elimination; normal-form theorems