In any particular structure, each proposition is simply true or false. But one proposition can be true in some structure and false in another structure. The universe could instantiate many structures, with non-indexical terms being interpreted the same way in each of them, but indexical terms being interpreted differently. Then sentences not containing indexical terms would have the same truth value in each of these structures, and sentences containing indexical terms would not. None of this contradicts using classical logic to reason about each of these structures.
I’m sympathetic to the notion that indexical language might not be meaningful, but it does not conflict with classical logic.
The point is that the meaning of a classical proposition must not change throughout the scope of the problem being considered. When we write A1, …, An |= P, i.e. “A1 through An together logically imply P”, we do not apply different structures to each of A1, …, An, and P.
The trouble with using “today” in the Sleeping Beauty problem is that the situation under consideration is not limited to a single day; it spans, at a minimum, both Monday and Tuesday, and arguably Sunday and/or Wednesday also. Any properly constructed proposition used in discussing this problem should make sense and be unambiguous regardless of whether Beauty or the experimenters are uttering the proposition, and whether they are uttering it on Sunday, Monday, Tuesday, or Wednesday.
In any particular structure, each proposition is simply true or false. But one proposition can be true in some structure and false in another structure. The universe could instantiate many structures, with non-indexical terms being interpreted the same way in each of them, but indexical terms being interpreted differently. Then sentences not containing indexical terms would have the same truth value in each of these structures, and sentences containing indexical terms would not. None of this contradicts using classical logic to reason about each of these structures.
I’m sympathetic to the notion that indexical language might not be meaningful, but it does not conflict with classical logic.
The point is that the meaning of a classical proposition must not change throughout the scope of the problem being considered. When we write A1, …, An |= P, i.e. “A1 through An together logically imply P”, we do not apply different structures to each of A1, …, An, and P.
The trouble with using “today” in the Sleeping Beauty problem is that the situation under consideration is not limited to a single day; it spans, at a minimum, both Monday and Tuesday, and arguably Sunday and/or Wednesday also. Any properly constructed proposition used in discussing this problem should make sense and be unambiguous regardless of whether Beauty or the experimenters are uttering the proposition, and whether they are uttering it on Sunday, Monday, Tuesday, or Wednesday.