Aboutness Theory S Yablo 1 Propositions Worlds are ways for things to be—possible ones in some sense of “possible” (Notation: w.) Logical space is the set of worlds. (Notation: W.) Propositions are subsets of logical space, or sets of worlds. (Notation: A, B, C,....). 1 A proposition A is true in world w iff w ϵ A, otherwise false. 2 Sentences are, you know. (Notation: A, B, C,.... ) Expression is the relation sentence X bears to proposition X when for all w, X is true in w iff w ϵ X. (Notation: A is the propositional content of A, B is the propositional content of B, ...) 1.1 Running Example: Propositional Calculus Worlds are classical valuations of a fixed propositional language. (Notation: ν .) Logical space is the set of all classical valuations of the language. (Notation: V) Propositions are sets of classical valuations. (Notation: A, B, C,...) Sentences are atoms p, q, r..., and truth-table combinations thereof. (Notation: A, B, C,...) Expression is the relation sentence X bears to its propositional content = the set of its classical models (the classical valuations in which X is true). (Notation: A expresses A, B expresses B , etc.) 2 Similarity and Equivalence An equivalence relation on S is a reflexive, transitive, symmetric relation on S. (Notation: ≈.) A similarity relation on S is a reflexive, symmetric relation on S.(∼.) A collection m of subsets of S is a partition of S iff its members are disjoint and jointly exhaustive of S. A partition’s members are its cells. Partitions determine equivalence relations and vice versa. x ≈ m y iff x and y cohabit a cell of m. m’s cells are maximal sets of equivalents. These are also called equivalence classes.A 1 Alternatively we could define propositions as functions from worlds to truth-values. This is the better approach when it comes to partial propositions—ones defined only on certain worlds. 2 Alternatively A is true or false in w according to whether A maps w to truth or falsity. 1