Modal systems in which necessity is ``factorable''.
J. Jay Zeman · Notre Dame Journal of Formal Logic · 1969
We will say that necessity is "factorable" in a modal system S if there are modal functions Xip, . . ., X n p-L itself being none of the Xi -such that in S the conjunction KX 1 pKX 2 p . . .X n p is equivalent to Lp.For the systems discussed in this paper, n in the above formulas will be 2 and X 1 p will be simply p.An obvious example of a system in which necessity is factorable is the system S4.4,which contains as a thesis (1) EKpMLpLp.We shall redirect our attention to S4.4 later on in this paper.1. S images in the S ° systems.We shall now show that by considering the operator usually read as "necessity" in the systems Sl°-S4° to be a factor of necessity rather than necessity itself, we may find in each of these systems an image of its respective (without the ίO ') ordinary Lewis-modal system.As bases for Sl°-S4°, we may use the C-N-L formulations of [l]; for our present purposes, however, let us employ for these systems the letter Q in place of L, and reserve L for the necessity operator in the "images" we will discover in Sl°-S4°.In all of these systems, then, we will define L and Mas follows:Df. L: Lφ for Kφ Qφ Df. M: Mφ for ANQNφ φAxioms and rules for the systems will be drawn from the following stock, as in [l], with Q read for L: Jla.CQCpCqrQCQpCQqQr Jib.CQCpqCQpQq J2.CKQCpqQCqrQCpr Jα.If hφ, then hQφ.Jb.If φ is an axiom or PC theorem, \-Qφ.Jc. // hQCφψ, then \-QCQφ Qψ.Jd.