By Georg H. Von Wright

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After doing this, we distribute the operator E in front of every one of the normal forms. The atomic E-sentences which occur in the VE-sentence after the distribution we shall call the E-constituents of the (original) VE-sentence. N” 44 COMBINED MODALITIES We now make a complete list of all the E-constituents. We thereupon replace every one of the E-sentences which occur in the VE-sentence by its perfect disjunctive normal form in terms of all E-constituents in our list. After doing this, we replace the operator F, wherever it stands without a negation-sign in front of itself, by F and distribute the N F in front of every one of the last mentioned normal forms.

E F ( A & B v A & 12. E P ( A & B v A & B v - A & B ) --f F - E ( - A & B). 13. E P ( A & B v - A & B v - A & - B ) + P - E ( A & - B ) . 14. E F ( A & BV &Bv -A & B +P - E ( A & B). The combinations of truth-values which are excluded by these fourteen cases of entailment can be immediately “read off” from the perfect disjunctive normal forms of the 14 implication-sentences in terms of the VE- and EV-constituents. Generally speaking, a V E + EV-sentence which contains n atomic predicates has (at most) 2(2n) VE- and (at most) 2(p) EVconstituents.

The study can conveniently be pursued in three stages. We shall successively develop a System V E , a System E V , and a System V E + EV. The System V E deals with expressions, in which an epistemic modal phrase is prefixed to a quantification statement. For example : “it is known that something is red”. I n these expressions the modalities are used exclusively de dicto. The System E V deals with expressions, in which a quantifier is prefixed to the name of an “epistemically modalized” property.

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