The reference [4] proved the consistency of S1 and S2 among Lewis' five strict implication systems in the modal logic by using the method of the Boolean-valued model. But, in this method, the consistency of S3, S4...The reference [4] proved the consistency of S1 and S2 among Lewis' five strict implication systems in the modal logic by using the method of the Boolean-valued model. But, in this method, the consistency of S3, S4 and S5 in Lewis five strict implication systems is not decided. This paper makes use of the properties: (1) the equivalence of the modal systems S3 and P3, S4 and P4; (2) the modal systems P3 and P4 all contained the modal axiom T(□p→p); (3) the modal axiom T is correspondence to the reflexive property in VB. Hence, the paper proves: (a) ‖As31‖=1; (b) ‖As41‖=1; (c) ‖As51‖=1 in the model (V^B,R,‖ ‖)(where B is a complete Boolean algebra, R is reflexive property in V^B).Therefore, the paper finally proves that the Boolean-valued model V^B of the ZFC axiom system in set theory is also a Boolean-valued model V^B of the ZFC axiom system in set theory is also a Boolean-valued model of Lewis the strict implication system S3, S4 and S5.展开更多
文摘The reference [4] proved the consistency of S1 and S2 among Lewis' five strict implication systems in the modal logic by using the method of the Boolean-valued model. But, in this method, the consistency of S3, S4 and S5 in Lewis five strict implication systems is not decided. This paper makes use of the properties: (1) the equivalence of the modal systems S3 and P3, S4 and P4; (2) the modal systems P3 and P4 all contained the modal axiom T(□p→p); (3) the modal axiom T is correspondence to the reflexive property in VB. Hence, the paper proves: (a) ‖As31‖=1; (b) ‖As41‖=1; (c) ‖As51‖=1 in the model (V^B,R,‖ ‖)(where B is a complete Boolean algebra, R is reflexive property in V^B).Therefore, the paper finally proves that the Boolean-valued model V^B of the ZFC axiom system in set theory is also a Boolean-valued model V^B of the ZFC axiom system in set theory is also a Boolean-valued model of Lewis the strict implication system S3, S4 and S5.
