Proof that there are infinitely many modalities in Lewis's system S₂

Extract

In this note I show, by means of an infinite matrix M, that the number of irreducible modalities in Lewis's system S₂ is infinite. The result is of some interest in view of the fact that Parry has recently shown that there are but a finite number of modalities in the system S₂ (which is the next stronger system than S₂ discussed by Lewis).

I begin by introducing a function θ which is defined over the class of sets of signed integers, and which assumes sets of signed integers as values. If A is any set of signed integers, then θ(A) is the set of all signed integers whose immediate predecessors are in A; i.e., , so that n ϵ θ(A) is true if and only if n − 1 ϵ A is true.

Thus, for example, θ({−10, −1, 0, 3, 14}) = {−9, 0, 1, 4, 15}. In particular we notice that θ(V) = V and θ(Λ) = Λ, where V is the set of all signed integers, and Λ is the empty set of signed integers.

It is clear that, if A and B are sets of signed integers, then θ(A+B) = θ(A)+θ(B).

It is also easily proved that, for any set A of signed integers we have . For, if n is any signed integer, then