By Georg H. Von Wright
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The System M E covers an important part of the ground of inquiry in traditional modal logic, viz. the study of the so-called modal syllogism. We shall deal with the modal syllogism in an appendix. The reason, why we prefer here to deal with the System V E rather than with the traditionally more important System M E lies in the fact that the System V E can be supplemented with the Systems E V and V E E V , whereas a corresponding supplement- + 43 THE SYSTEM VE + ation of the System M E with Systems E M and M E E M appears dubious.
In the derivation of constituents care should be taken that they axe all “in terms of” the same set of atomic predicates. Thus a V E + EV-sentence, in which there are n atomic predicates, has (at most) 2” E-, 2n V-, 2(2”) VE-, and 2(2n) EV-constituents. Between the VE- and EV-constituents there are logical interdependencies which restrict the distribution of truth-values. The decision problem of the System V E + E V, therefore, is not trivial relative to the decision problem of the Systems V E and E V which was trivial relative to the decision problem of the “underlying” or “basic” Systems V and E themselves.
For instance: UV(A & B). By a E V-sentence, finally, we understand an atomic E V-sentence or an atomic UV-sentence or a molecular complex of atomic E V and/or UV-sentences. For instance : EVA & U (FA v VB). The same rules for brackets apply as in the System VE. The System E V studies E V-sentences. The governing principles of the System E V are the same as the governing principles of the System VE. Every E V-sentence has V- and E V-constituents. The procedure by means of which they are derived is mutatis mutundis the same as the one by means of which we derive the E- and VE-constituents of VE-sentences.