The distinction between u-propositions and m-propositions is not motivated by a blind desire for analogy with the distinction between sets and collections. On the one hand, the distinction is forced on us by the propositional paradox, the moral of which is that not all m-propositions are u-propositions. On the other hand, even if, per impossibile, all m-propositions were u-propositions, the distinction between the two notions would still be intelligible. Setting aside the paradoxes that he completely ignored, Husserl is to my knowledge the sole logician-philosopher to have expressed this intuition. In terms borrowed from the inventor of phenomenology (except that of “u- or m-proposition”), one can say that the consciousness of an m-proposition is a polythetic, plural consciousness that directs itself on its objective (Gegenstandlichkeit) in a many-rayed way. By contrast, the consciousness of a u-proposition is a monothetic, singular consciousness, that reaches its objective in a one-rayed way, and in which this objective constitutes itself as object (Gegenstand) in the strict sense of the term. (Cf. HUSSERL, 1913. § 119.)

The distinction between u-propositions and m-propositions is not motivated by a blind desire for analogy with the distinction between sets and collections. On the one hand, the distinction is forced on us by the propositional paradox, the moral of which is that not all m-propositions are u-propositions. On the other hand, even if, per impossibile, all m-propositions were u-propositions, the distinction between the two notions would still be intelligible. Setting aside the paradoxes that he completely ignored, Husserl is to my knowledge the sole logician-philosopher to have expressed this intuition. In terms borrowed from the inventor of phenomenology (except that of “u- or m-proposition”), one can say that the consciousness of an m-proposition is a polythetic, plural consciousness that directs itself on its objective (Gegenstandlichkeit) in a many-rayed way. By contrast, the consciousness of a u-proposition is a monothetic, singular consciousness, that reaches its objective in a one-rayed way, and in which this objective constitutes itself as object (Gegenstand) in the strict sense of the term. (Cf. HUSSERL, 1913. § 119.)

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Again, there should be two ways of calling the principle into question, a “moderate” way and a “radical” way. The moderate way would consist in weakening the principle to shelter the notion of proposition (u-proposition) from the contradiction. The problem thus posed would be “the basic problem of the theory of propositions.” The radical way would consist in purely and simply abandoning this principle, along with the notion of proposition (u-proposition) and all that accompanies it, notably the notion of propositional (u-propositional) truth. One would then elaborate a theory of types for m-propositions.

 

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