number-theory

106 7 The converse principal-type algorithmFirst note that a-+a E A' by assumption, so by (Sub),§(z) - s(a) E V-.But we have assumed AF = A°- and ADS-BD-, sos(2)->s(r) E BD-.Now it is easy to see from Definition 7D1 thatD(s(i)->s(r), r) = S(T).Hence by rule (D) applied to the D-theorems S(T)--+S(T) and T we gets(2) E I°F.7D10 Theoremis D-complete.The axiom-set {(B), (C), (K), (W)} for Intuitionist logic given in 6D3Proof By 7D7 and 7D9, since this logic is a D-extension of R..7D11 Note (Other D-complete logics) (i) Classical logic. The implicational fragmentof classical logic is a Hilbert-style system defined by the axioms (B), (C), (K), (W)and(PL)((PL) is called Peirce's law, see .6A1.2 and 6B7.4.) It can be shown that a formulais provable in this system if it is a tautology in the usual truth-table sense. (Prior1955 Part I, Ch. III.) By 7D9-10 this system is D-complete.(ii) Ticket entailment. A Hilbert-style logic defined by the axioms (B), (B'), (I) and(W) listed in 6B2.1 was introduced in Anderson and Belnap 1975 (see especiallyCh. 1 §§6, 8.3.2), where it was called T_, or the logic of ticket entailment. The detailsof T_, and its motivation are not the concern of this book, but in a sense (B) and(B') are "right-handed" and "left-handed" replacement properties: if a-+b holds, (B)says that a can be replaced by b in the formula c-.a and (B') says that b can bereplaced by a in a-*c. (Meanings for (I) and (W) were suggested in 6D6.1(ii).) Thislogic can be shown to be weaker than R_, but by 7D6 and 7C8 it is neverthelessD-complete. 1(iii) A set of axioms whose logic is strictly weaker than T. was proved D-completeby N. Megill in unpublished notes in 1993, and an infinite series of ever weakeningD-complete axiom-sets has since been constructed (Megill and Bunder 1996).But not all axiom-sets are D-complete, as the next theorem will show.7D12 TheoremD-incomplete.The axiom-sets {(B), (C), (K)} and {(B), (C), (I)} given in 6D5-6 are1 By the way, it is not yet known whether there is a decision-procedure for provability in T...,.