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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$\cdots$ then$\cdots$”-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.

##### MSC:
 03B60 Other nonclassical logic 68T27 Logic in artificial intelligence 03F05 Cut-elimination; normal-form theorems