Do all true statements express the same proposition? I know that, for example, the statements "2=2" and "1+1>1" are distinct sequences of symbols. However, I want to know, do they express the same proposition? In my view, a proposition is, or can be defined as, an equivalence class of statements. So, then, are there just two propositions, the first being the set of all true statements, and the second being the set of all false statements? And has any philosopher made this argument?
The answer to this turns on your views on propositional granularity. You say
a proposition is, or can be defined as, an equivalence class of propositions.
This sounds right but is underinformative, since to know when two propositions are identical we need to know what equivalence relation over propositions you have in mind. Some kind of translatability might be an equivalence relation over propositions (i.e. transitive, reflexive, and symmetric), but it is not the case that "1+1=2" is a good translation of "Schnee ist weiss" even though both are true.
Here are three views on propositional granularity. If you want you could think of them as arising from different views about what the relevant equivalence relation ought to be:
- Extensionalism is the view that there are only two propositions, T and F. On this view two propositions P and Q are identical just in case they share a truth value. This was Frege's view. Almost no one believes it today.
- Intensionalism treats propositions as sets of possible worlds. In this view, there are very many different propositions. "Grass is green" is true and "1+1=2" are both true, but there are possible worlds in which they come apart (e.g. worlds where grass is blue). On this view, propositions P and Q are identical just in case they necessarily share a truth value. I would guess that this is quite a widely held view, although less popular than 30/40 years ago. A disadvantage of intentionalism is that necessary equivalents express the same proposition. This is implausible in some contexts; knowing that "1+1=2" doesn't seem to have any bearing on the truth of Fermat's Last Theorem, but these things express the same proposition according to intentionalists.
- Hyperintentionalists allow that even propositions that necessarily share a truth value might be distinct. The most common way of motivating a view like this is by adopting what is called a "structured view" of propositions, according to which two propositions are different if they are made up of distinct entities. So "Socrates=Socrates" is distinct from "Plato=Plato" because Socrates and Plato are distinct. The problem for this view is that it seems to give rise to a version of the Russell-Myhill paradox.
Which account you prefer might turn on your purposes. If you are interested in propositions as the objects of propositional attitudes, you probably want a fine-grained i.e. hyperintensional view, since necessary equivalents are not intersubstitutable in many attitude-contexts. If you are interested in a language for mathematics, as Frege was, perhaps extensionalism will do fine. If, though, you are a metaphysical realist about propositions and want to give the true facts about how many propositions there are, then you face a tricky and much debated question.
For a short introduction to these topics you might try this piece by Peter Fritz: https://philpapers.org/archive/FRIHFI.pdf
No, I don't think so. Suppose A, B, C are three true propositions. Nevertheless, we might have a situation where A&B is true, whereas A&C is false. Therefore, B and C can't mean the same thing.
___ E d i t / E x a m p l e R e l a t i n g t o C o m m e n t s ___
Are you guys aware of "resource awareness" (e.g., as in linear logic)? The situation that I specifically had in mind, though you can conjure up many others, is the following. Suppose you're standing in front of a vending machine with two dollars in your hand. Apples cost one dollar, Bananas cost one dollar, and Coca Cola costs two dollars. And now, proposition A is "I can buy an Apple", proposition B is "I can buy a Banana", proposition C is "I can buy a Coke". So there you go, right?... A, B, C are all true. And A&B is true, whereas A&C is false.
It's the familiar rule of inference (where suppose that D is the proposition "I have two Dollars")
that's no longer generally true in substructural logics with resource awareness.