By Georg H. Von Wright
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From the absolutely perfect disjunctive normal form of a V E + EV-sentence can be seen with which ones of a finite number of mutually exclusive and jointly exhaustive possibilities the sentence in question expresses agreement (and with which ones it expresses disagreement). Agreement with all possibilities is a necessary and sufficient criterion of logical truth. N N N N Cf. On the Idea of Logical Truth 11, p. 21. 56 COMBINED MODALITIES Note. - There are various extensions of the Systems VE, E V , and V E + E V .
A. THE UNREDUCED MODALITIES 1. The Systems M,, etc. By an atomic M2-sentence we shall understand the sentence which we get when we prefix the operator M to a Ml-sentence. (Ml-sentences, it should be remembered, are molecular complexes of atomic Ml- and/or N,-sentences. ) By a M,-sentence we shall understand a molecular complex of atomic M,-sentences. The System M , studies M,-sentences. As to the use of brackets no new rules are needed. Consider an atomic M,-sentence. It consists of the modal operator M followed by a Ml-sentence.
A sentence of this structure has (at most) 16 VE-constituents: I. F ( E ( A& B )& E(A & - B ) & E(-A & B )& E(-A & - B ) ) . . . . . . - F(-E(A & B) & -E(A & - B ) & - E ( - A & B) & E ( N A & B)). It has further (at most) 16 EV-constituents: 16. N 1. E(F(A&B)&P(A&NB)&F(NA&B)&P(-A&NB)). . . . . . E(- P(A & B) & - - -- F(A & B) & - F(- A & B) & P(-A & B)). I n virtue of the Principle of Non-Falsification, the first E V 16. constituent expresses an always false proposition, the conjunction of all the VE-constituents expresses an always false proposition, and if all the E V-constituents express false propositions, then the last VE-constituent expresses a false proposition.