1.

2 The Objects of Belief

In the following we will see several proposed models for the structure of belief. Most of these
proposals take the objects of belief to be either propositions, or sentences in a formalized language.
This section reviews the basic notions required to work with propositions and sentences in a formal
language. If the reader feels overwhelmed with the technicalities in this section, they should feel free to
postpone them, and refer back to it on-the-fly. Readers who are accustomed to working with these
objects may freely skip this section.
The received view is that the objects of belief are proposition and propositions are sets of possible
worlds. But what are these supposed to be? This is a rather difficult question (see the entry on possible
worlds). On one picturesque view, a possible world is a complete description of an alternative reality.
To pick out a possible world is to specify–in a way careful to avoid contradiction–every fact that holds
in some possible reality that is not necessarily our own. On this view, the set of all possible worlds W
is like a giant library that contains the complete history of every possible reality. The actual world
picks out the volume that corresponds to our own reality.
It is not necessary–and perhaps unhelpful–to think of possible worlds as total metaphysical
possibilities. At this extremely fine level of granularity, each possibility specifies an infinity of obscure
and uninteresting details. But context usually determines which features of the world we can take for
granted; which we are uncertain about but would prefer not to be; and which are of no interest. For
example, Sophia may be interested in the identity of the next mayor of Vienna, but whether they are left
or right-handed is of no importance. For our purposes, a possible world is a complete specification of
all and only those features of the world that are relevant given the context. The set W
, therefore, is the set of all contextually relevant epistemic possibilities. Narrowing down the set of
possibilities to an individual w∈W would completely settle some interesting question under discussion.
A proposition P⊆W is a set of possible worlds, i.e. it is a partial specification of the way the world is.
To be certain that P is true is to be certain that the actual world is among the set of worlds {w:w∈P}
since P is true in a possible world w iff w∈P
.
Propositions enjoy a set-theoretic structure. The relative complement of P
, ¬P=W∖P, is the set of all worlds in which P is false. If P,Q are arbitrary propositions, then their
intersection P∩Q is the set of all worlds in which P and Q are both true. The disjunction P∪Q is the set
of worlds in which at least one of P,Q is true. The material conditional P→Q is the set of worlds
¬P∪Q, in which either P is false or Q is true. If P⊆Q we say that P entails Q and also that P is
logically stronger than Q. If P⊆Q and Q⊆P we write P≡Q and say that P and Q are logically
equivalent. The tautological proposition W is true in all worlds and the contradictory proposition, the
empty set ∅, is not true in any world. A set of propositions A is consistent iff there is a world in which
all the elements of A are true, i.e. if ∩A≠∅. Otherwise, we say that A is inconsistent. A set of
propositions A is mutually exclusive iff the truth of any one element implies the falsehood of all other
elements. The set of logical consequences of A, written Cn(A), is the set {B⊆W:∩A entails B}. Note
that if A is inconsistent, then Cn(A) is P(W), the set of all propositions over W
.
A set of propositions F
is a field (sometimes algebra) iff F contains W and it is closed under intersection, union and
complementation. That is to say that if A,B are both elements of F then W,A∪B,A∩B and ¬A are also
elements of F. A set of propositions F is a σ-field (sometimes σ-algebra) iff it is a field that is closed
under countable intersections, i.e. if S⊆F is a countable collection of propositions, then the intersection
of all its elements ∩S is also an element of F. That definition implies that a σ-field is also closed under
countable unions. It is not difficult to prove that the intersection of σ-fields is also a σ-field. That
implies that every collection of propositions F generates σ(F), the least σ-field containing F, by
intersecting the set of all σ-fields containing F
.
Propositions, although usually expressed by sentences in a language, are not themselves sentences.
That distinction is commonly drawn by saying that propositions are semantic objects, whereas
sentences are syntactic objects. Semantic objects (like propositions) are meaningful, since they
represent meaningful possibilities, whereas bits of syntax must be “interpreted” before they become
meaningful. In a slogan: sentences are potentially meaningful, whereas propositions already are.
For our purposes, a language L
is identified with the set of all grammatical sentences it contains. Sentences will be denoted by
lowercase letters Greek α,β,…. The language L is assumed to contain a set of atomic sentences α,β,…
which are not built out of any other sentences, as well as all the sentences generated by combining the
atomic sentences with truth-functional connectives from propositional logic. In other words: if α,β are
sentences in L then ¬α, α∨β, α∧β, α→β, and α↔β are also sentences in L. These are meant to be read
respectively as “not α”, “α or β”, “α and β”, “if α, then β” and “α if and only if β”. The symbol ⊥
(pronounced “falsum”) denotes an arbitrarily chosen contradiction (e.g. α∧¬α) and the symbol ⊤
(pronounced “top”) denotes an arbitrary tautology.
Some of the sentences in L
follow “logically” from others. For example, under the intended interpretation of the truth-functional
connectives, α follows from the sentence α∧β and also from the set of sentences {β,β→α}. To capture
the essentials of deductive consequence, we introduce a consequence relation, ⊢, which holds between
any two sentences α⊢β, whenever β is a deductive consequnce of α . The consequence operator is
assumed to satisfy the following properties, which abstract the characteristic features of deductive
logic: