Abstract

1. Anderson and Belnap's most explicit characterization of the fallacy of modality is as follows: "Modal fallacies arise when it is claimed that entailments follow from, or are entailed by, contingent propositions." The view which Nelson attributes to Anderson and Belnap, on the other hand, is "that necessary propositions are entailed only by necessary ones, never by contingent ones." Anderson and Belnap speak of "entailments," whereas Nelson generalizes to "necessary propostitions." The move is far from innocent, as we shall see. For though Anderson and Belnap hold that every true entailment is a necessary proposition, not every necessary proposition is an entailment. In order to grasp the implications of this distinction more clearly, let us see how Anderson and Belnap's statement of the modal fallacy might legitimately be restated in terms of necessity. To that end we must draw a related distinction between necessary propositions and necessity propositions. A expresses a necessary proposition iff A → A → A holds. A expresses a necessity proposition iff A is of the form B → C. The particular form of these definitions is dependent on the fact that entailment rather than necessity is primitive in Anderson and Belnap's system; the forms in parentheses are the equivalents in terms of necessity. Now it follows from these definitions that every entailment is a necessity proposition and that every necessity proposition is an entailment. The modal fallacy may thus be equivalently characterized as the claim that a necessity proposition is entailed by a contingent proposition. That this is different from Nelson's formulation is clear from the following considerations. First, not every necessary proposition is a necessity proposition; example: "Either it is raining or it is not raining." Second, not every necessity proposition is a necessary proposition; example: any false necessity proposition. Thus the possibility is left open that some necessary proposition might be entailed by a contingent one. This is exactly the case with Nelson's example 2, "That every polychromatic surface is red entails that every polychromatic surface is colored." This might well be a theorem in a suitable extension of E. Such cases do not, however, arise among the theorems of EI. In that limited system, all that a contingent proposition can entail is itself.