Projective implication

One of the very simplest implication relations can be obtained by regarding an element B of a set S as implied by finitely many members listed as A1, …, An, just in case B is Aj for some j in {1, …, n}. The verification of the conditions for implication relations is straightforward.

Millean implication

This is a kind of implication relation for which A1, …, An implies B if and only if some Aj implies B for some j in {1, …, n} (see Chapter 4, note 2).

Exercise 8.1. Let “#” any reflexive and transitive binary relation on a set S of at least two members. Let “⇒#” hold between A1, …, An and B if and only if Aj # B for some j in {1, …, n}. Show that “⇒#” is a Millean implication relation.

Bisection implication

A much more theoretically interesting example whose properties we shall now study in some detail arises from a consideration of the various ways in which a nonempty set S can be partitioned into two nonempty, mutually exclusive subsets.