A Solution To A Problem Of Popper
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چکیده
plus a principle of indirect proof for atomic sentences"(p. 244), and \one may doubt whether this is the proper way of analysing classical inferences"(p. 244{245). Prawitz suggested that a good candidate for this role could be the classical sequent calculus, since its rules are \closer to the classical meaning of the logical constants"(p. 245). More recently this criticism of natural deduction has been taken up and extended by Girard, Lafont and Taylor 1989 and Cellucci 1992. 7 This does not apply, however, to the multi-conclusion system proposed by Cellucci 1992 which is, in fact, a mixture of natural deduction and sequent calculus. 10 A proof system S 0 p-simulates another proof system S if there exists a polynomial p such that for every tautology T, if there is an S-proof of T, then there is also an S 0-proof 0 of T such that j 0 j p(jj), where jj indicates the length of proof (i.e. the number of symbols in it). This approach to classifying the complexity of proof-systems dates back to Cook and Rechow 1974 and 1979. For a recent textbook based on this approach see Eder 1992. 11 By this we mean that their search can be easily mechanized by means of standard techniques. 12 There is a three-valued semantics which validates all the rules of the classical sequent calculus, but not the cut rule. Since the cut-free sequent calculus and the calculus with cut prove exactly the same sequents, this three-valued semantics characterizes the same set of theorems as the standard two-valued one. For a discussion see Given the close correspondence between the cut-free sequent calculus and the tableau method (see Smullyan 1968), this discussion applies also to the tableau method. F P The last of these rules is the classical reductio ad absurdum referred to in the text. Another variant of the classical calculus is described in Tennant 1978 where the author uses the following classical rule of dilemma: P ] Q :P ] Q Q TABLEAU RULES 5 We assume the reader is familiar with both the tableau method and natural deduction (in its several variants). For the reader's convenience the rules of these systems are listed in the Appendix. For the tableau method see Smullyan 1968, for natural deduction and its variants see Prawitz 1965 and Tennant 1978. All currently used introductory textbooks are based on one or the other of these two …
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تاریخ انتشار 1995