模态系统与蕴涵系统

<正> 在本文中我们采用(?)ukasiewicz 符号,以(?)分別表示“非α或β”(实质蕴涵)“α或β”(析取)“α与β“(舍取)“C_(αβ)且C_(αβ)”(实质等价)“非α”(否定).此外,我们更引入下列符号“□α”——α是必然的,必然α;“◇α”——α是可能的,可能α;“Fαβ”表示口 C_(αβ),G_(αβ)”表示 KF_(αβ)F_(αβ)(注意,它与□E_(αβ)未必相同).但须注意,在最后一节内,F_(αβ)另有意义.我们采用公理模式而不采用合有命题变元的公理,因此在证明过程中可以用不到代入规则.

THE MODAL SYSTEMS AND IMPLICATION SYSTEMS

We make use of Eukasjewicz symbols with the following addition:□a(it is necessarythat a) F_(αβ)(=□C_(αβ)and G_(αβ)(=KF_(αβ)F_(βα)).We start with the weakest(with quite few exceptions) basical modal system A,axiomatized as follows:1.Whenever a is an axiom of the traditional two-valued system,then □a is anaxiom of the system A.2.F□αα3.F_(αβ)→β.4.F_(αβ),□α→□β5.a,β→K_(αβ).6.G_(αβ)→G□α□β. We strengthen thesystem A by addition of the respective axioms or rules:B:F_(αβ)→C□α□β.C:CF_(αβ)□α□β D:FFαβC□α□β.E:F_(αβ)→F□α□βF:C + E.G: D + E.H:CF_(αβ)F□α□βI:D + H.J:FF_(αβ)□α□β.The relation of them may be schematized as follows(from weak to strong):(?)When the system X is added the axiom/rule CG_(αβ)G□α□β, FG_(αβ)G□α□βα→FβK_α(equivalent to α→□α),α→ FF_(αββ),C_αFF_(αββ),F_αFF_(αββ),then the resulting systems willdenoted X_1,X_2,X,X_a,X_b,X_c respectively.By combining some of these additions,we would have,apparently,240 systems.Yetmany of them are identacal.In fact,we have(1)H=H_1,I=I_1,J=J_1=J_2.(2)B=E,C=D=F=G,H=I=J, X_1= X_2.(3)Bi=C_1,B_2=C_2,E_1=H_1(= H)=F_1,G_1=I_1(=I).(4)E_2= F_2=G_2=H_2=I_2=J_2.(5)X_b=X_c,X_a+□F_(aa)=X(Hence X_a=X)(6)B_b+□F_(aa)=E_c,both are identical With two-valued system Hence we have atmost 65 distinct systems (two-valued system excluded).We show that G=S2,J=S3,and the asserted propositions in J are the same asthat in S4,yet the rules of procedure of them are different.We would meet paradoxes of implication even in the weakest system A,e.g.,wehave □α→Fβα and □N_α→F_(αβ).Hence the writer of the present paper would propose-two new implication systems instead.In these two new systems,the concepts F_(αβ)and □αare both primitive,neither of them can be defined by the other.The first implication system may be axiomatized as follows:1.Let the axioms of two-valued system be given in the form C_(αβ),then F_(αβ)are theaxloms of this system.2.FF_(αβ)C_(αβ).3.F_(αβ),α→β.4.FC_(αβ)C_(γδ),F_(αβ).5.α,β→K_(αβ).6.GC_(αβ)C(γδ)→GF_(αβ)F(γδ).7.FK□α□βK_(αβ).8.FF_(αβ)F□α□β.9.F□□αα.10.FF_(αβ)□αβ.11.F□Kαβ□α.The second implication system reads as follows:The rute of detachment and the rule of conjunctin.1.FF_pF_(pq)F(pq).2.F_pFF_(pqq).3.,FF_(pq)FF_(qr)F_(pr),.4.F_(pp).5.FF_pN_qF_qN_p.6.FNN_(pp).7.FK_(pq)K_(qp).8.FK_pK_(qr)KK_(pqr).9.FK_(ppp).10.F_pK_pp.11.FF_()pqFK_(pr),K_(qr),.12—16(the same as 7—11,with“K” replaced by“A”)17.FK_p,A_(qr)AK_(pqr).18.FF_(pq)AN_(pq).19.FK_pNK_(pq)N_q.20.AN_pA_(pq).21.FKN_pN_qNA_(pq).22.FNA_(pq)KN_p,N_q.23.FK□α□βK_(αβ).24.F□K_(αβ)□α.25.FF_(αβ)F□α□β.26.F□aa.27.FF_(αβ)□_(αβ).