# Do all true statements express the same proposition?

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 issue about Propositions is a complex one. Are them the "meaning" of sentences? (but what is meaning?) If so, in propositional logic the "meaning" of a propositional symbol is exactly its truth vales, but in predicate logic we can approach the issue differently. In arithmetic the two statements "2=2" and "1+1>1" express different arithmetical facts. Jan 12, 2022 at 8:59
• See Propositions: "Informally, sentences in different languages may mean “the same thing.” Formally, that “thing,” called a proposition, represents abstract, language-independent, semantic content. To bring the informal notion of proposition within the scope of formal treatment, this paper proposes a formal definition: a proposition p shall be defined as an equivalence class of sentences in some formal language L." Jan 12, 2022 at 9:12
• So says Frege, the founder of modern semantics:"Every assertoric sentence… is to be regarded as a proper name, and its Bedeutung, if it has one, is either the True or the False". "Bedeutung" is "referent" in German. It is an artifice of basic formal semantics that is convenient for technical purposes, just like the rule that anything follows from a false sentence in classical logic. More elaborate semantic theories dispense with it. Jan 12, 2022 at 12:31

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")
D==>A       D==>C
-----------------------------
D==>A&C
that's no longer generally true in substructural logics with resource awareness.

• Wait, what? If A, B, and C are true so is A AND B and also A AND C. Or am I misunderstanding you? Jan 12, 2022 at 19:42
• As the previous commenter suggested, this doesn't make sense. If A, B, and C, are true, then so are A&B and A&C. Maybe you are trying to get at a model-theoretic notion like this: if A, B, and C, are three propositions, there may be models where all three are true and other models where A&B is true and A&C is false. Jan 13, 2022 at 1:05
• @DavidGudeman Yeah, that's pretty much what I had in mind (see the example in my edit to the question). Jan 13, 2022 at 15:24
• @user107952 I don't think you're misunderstanding me, but see the example in my edit to the question which illustrates why I think what you said isn't generally true. Jan 13, 2022 at 15:30
• In linear logic the multiplicative and additive conjunction rules still hold while weakening and contraction no longer hold. Are you sure your above example is due to the failure of any conjunction rule in LL? If so, multiplicative or additive conjunction? Jan 13, 2022 at 20:09