Booleanism
One of the central theoretical commitments of this book is a thesis about the structure of propositions:
Booleanism. The set of propositions form a Boolean algebra.
We assume, moreover, that propositions form a complete, atomic Boolean algebra:
Completeness. This Boolean algebra is complete.
Atomicity. This Boolean algebra is atomic.
What follows in this section is something of a digression, but since concepts from the theory of Boolean algebras appear frequently throughout this book, this is a good opportunity to introduce them. Readers already familiar with Boolean algebras should simply skim the definitions, and skip ahead.
To understand what these principles say, we shall look at each part separately. A Boolean algebra is a set of objects B—in this case, our set of propositions—with two elements T, ⊥ ∈ B, two binary operations V, ∧ taking two elements of B to an element of B, and a unary operation — mapping B to B subject to certain constraints. Intuitively, these constraints say that two propositions are identical whenever the corresponding
Table 3.1. Axioms for Boolean algebras
biconditional is provable in the classical propositional calculus. Thus, for example, p = -'-ψ and p Λ q = q Λ p, since the corresponding biconditionals are provable in the propositional calculus. A more compact way of axiomatizing these identities is given in Table 3.1.
In addition to these axioms, it is normally stipulated that T= ⊥.
It should be emphasized that Booleanism is not the same as the assumption of classical logic: while Booleanism entails classical logic, it does not follow from the assumption of classical logic.color=black face=Cambria>[32] According to theories in which propositions are structured in a way that parallels the sentences that express them, identities like p Λ q = q Λ p fail, for the left-hand-side and right-hand-side are structurally different. Yet it is open to those who endorse structured theories to accept classical logic. Booleanism is thus a further, substantive, commitment that takes us beyond a commitment to classical logic.According to some theories, the things that play the proposition role are just linguistic constructions of some sort—perhaps sentences in some idealized language of thought. However, Booleanism is not consistent with this interpretation of propositions either, since there are distinct, but tautologically equivalent sentences. The contrast between propositional and linguistic accounts of vagueness, on this conception of propositions, is thin. The assumption of Booleanism thus registers one important sense in which propositions, as I understand them, are not derivative on language: my theory of propositional vagueness is a genuinely non-linguistic theory.[33]
The simplest example of a Boolean algebra is the algebra that has only two values— the True and the False—where Λ, V and — are interpreted by the evident truth functions. So one version of Booleanism is the view that there are only two propositions. Frege arguably held this view, but few philosophers since have defended it. Probably the most well-known Booleanist view is the one that propositions are sets of possible worlds, where Λ, V, and — are interpreted as set intersection, union, and complementation respectively. According to this view metaphysically necessarily equivalent propositions are identical.
Booleanism alone does not force us to individuate propositions this coarsely either. There are many necessary equivalences that are not provable in the propositional calculus, such as the equivalence between the proposition that water is wet and the proposition that H2O is wet: even though these propositions are necessarily equivalent, this is not something one can prove in the propositional calculus alone. Booleanism, therefore, does not force us to make these identifications. We shall see in chapter 11, that if we are to make good on the idea that the vague supervenes on the precise, we must also distinguish necessarily equivalent vague propositions.Every Boolean algebra comes with a natural ordering of entailment over its elements: say that p entails q iff p Λ q = p. In the possible worlds theory, p entails q just in case p is a subset of q, since p ∩ q = p iff p ⊆ q. A Boolean algebra is complete iff every set of elements, X, has a least upper bound under the entailment ordering: an element a such that b ≤ a for every b ∈ X, and moreover a ≤ a' whenever b ≤ a' for every b ∈ X. It is possible to show that in a complete Boolean algebra, every set X also has a greatest lower bound, namely, the least upper bound of the set of propositions that entail all members of X. In the possible worlds framework, for example, the least upper bound of a set of sets of worlds is just their union; least upper bounds are thus like disjunctions of (possibly infinite) sets of propositions. (Similarly, greatest lower bounds correspond to infinite conjunctions.) It follows that propositions in the possible worlds theory form a complete Boolean algebra.
We now come to an important definition:
Maximally strong consistent proposition: A proposition, i, is a maximally strong consistent proposition iff:
(i) It is consistent: it is not the element ⊥.
(ii) For any other consistent proposition p, if p entails i then p = i
class=31 align=left style='text-align:left;line-height:132%'>A maximally strong consistent proposition is sometimes called an atom.A Boolean algebra is called atomic iff every consistent proposition is entailed by a maximally strong consistent proposition.
According to the possible worlds account of propositions, a singleton set, {w}, containing exactly one world will be a maximally strong consistent proposition. It follows that according to the possible worlds theory, the propositions also form an atomic Boolean algebra.Indeed, any theory in which propositions are sets of things—whether they be sets of worlds, world-precisification pairs, world-time pairs, etc.—is one that accepts Booleanism, Completeness, and Atomicity. Conversely, it can be shown that every complete atomic Boolean algebra is isomorphic to a Boolean algebra of this sort: in particular, p corresponds with the set of maximally strong consistent propositions that entail p. It follows, given our assumptions, that it is always possible to think of propositions as sets of indices, where an index is just (yet) another name for a maximally strong consistent proposition.
A Boolean subalgebra of a Boolean algebra B is a non-empty subset A ⊆ B that is closed under Λ, V and —: if p, q ∈ A then p Λ q,p V q, —p ∈ A. It is easy to check that A is itself a Boolean algebra if B is. The notion of a Boolean subalgebra is useful in the context of discussions of vagueness, since it is natural to think that the precise propositions form a Boolean subalgebra of the propositions: if p is precise, so is —p, and ifp and q are precise, so is p ∧ q and p ∨ q. Indeed, throughout this book I shall assume the stronger thesis:
Boolean Precision. The precise propositions from a complete atomic Boolean algebra.
Presumably infinite disjunctions of precise propositions are precise, ensuring completeness. The atomicity of the precise propositions is less obvious, but can be motivated as follows. If I conjoin all the precise truths, the resulting proposition is precise by completeness.
Moreover, the conjunction of precise truths is surely also consistent, for it ought to be true given that its conjuncts are. Thus, the conjunction of precise truths is an atom of the algebra of precise propositions.[34] This sort of reasoning ought to be necessarily true, if true at all. So, on the assumption that every precise proposition is metaphysically possible, it follows that every precise proposition is entailed by an atomic precise proposition.[35]The preceding discussion introduced another crucial concept that will appear regularly throughout the book: the notion of an atom of the algebra of precise propositions.
Maximally strong consistent precise proposition: Apropositionp is a maximally strong consistent precise proposition iff:
(i) p is precise.
(ii) p is consistent (i.e. it's not ⊥).
(iii)style='font:7.0pt "Times New Roman"'> For any other consistent precise proposition, q, if q entails p then q = p.
A maximally strong consistent precise proposition settles all precise questions, in the sense that for any precise proposition q, it either entails q or it entails —q. However, if there are vague propositions, a maximally strong consistent precise proposition will not settle all questions: they may leave open questions about vague matters.
We can get an intuitive picture of the precise propositions by appealing to the representation of a proposition as a set of indices. Roughly, the precise propositions will partition logical space—represented by the set of all indices—into a bunch of cells: a collection of sets of indices that do not overlap each other, and are such that every index is contained in some cell. Each cell corresponds to a maximally strong consistent precise proposition. An arbitrary precise proposition consists of a union of cells. We will regularly invoke this sort of picture-thinking in later chapters.
3.3