The Expressive Truth Conditions of Two-Valued Logic

Abstract

In a finitary closure space, irreducible sets behave like two-valued models, with membership playing the role of satisfaction. If f is a function on such a space and the membership of $fx_1 ,\ldots, x_n$ in an irreducible set is determined by the presence or absence of the inputs $x_1 ,\ldots, x_n$ in that set, then f is a kind of truth function. The existence of some of these truth functions is enough to guarantee that every irreducible set is maximally consistent. The closure space is then said to be expressive. This paper identifies the two-valued truth functional conditions that guarantee expressiveness.