Predicate logics without the structure rules

In our previous paper [5], we have studied Kripke.type semantics, for propositional logics without the contraction rule. In this paper, we will extend our argument to predicate logics without the structure rules. Similarly to the propositional case, we can not carry out Henkin's construction in the predicate case. Besides, there exists a difficulty that the rules of inference (~V) and (=1~) are not always vali4 in our semantics. So, we have to introduce a notion of normal models. In this paper , we will in t roduce five predicate logics wi thout the some s t ruc tu re rules (except ZJ), Z.BCA, .LBCB, ZBCC,, .LBC.K, and JLJ. 1%r each of them, the cut el imination theorem holds. Then, we will introduce: a Kr ipke type semantics wi th vary ing domain (a semantics with cons tan t domain has been given b y I t . Ono [4]). In the proofs of the completeness theorems for those logics (except .SJ), we can no t cons t ruc t t t en ldn ' s theory . I t is closely re la ted with the fac t t ha t nei ther the sequent A ^ (B v C) -->(An B ) v (AA C) nor the sequent A ^ 3xB(x)--->3x(AA B(x)) can be p roved in those logics wi thout the contract ion rule. So, we mus t p rove the completeness theorems for those logics wi thout using Henkin ' s construct ion b y changing the in terpre ta t ion of v and 3 . We assume the familiarity" wi th [3] and [5]. 1. Syntact ica l analys i s I n the following, we fix a language ~ for predicate logics. The language .Sf contains __1, T ( truth) , ~ , v , ^ , &, V and 3 as logical connect ives. W e suppose t ha t 5e contains nei ther funct ion symbols nor individual cons tants . W e r e m a r k here tha t it does no t necessarily hold in L.BCA t h a t A-->-V is p rovab le even if -~A is provable . On the other hand, in ZBCA, T-->A is also provable if -+A is provable . I n any sys tem with the weakening rule, bo th sequents A-->T and m -+A are provable if -~A is provable . :~ow we will in t roduce a basic logical sys tem called ZBCA. l~oughly speaking, ZBC~A is a formal sys tem obta ined f rom Gentzen 's sys tem Z J for intuit ionist ic predicate logic, b y eliminating all the s t ruc ture rules. In i t i a l sequents of .~.BCA are either of the form F~ J_~ A->A