# What is the difference between a statement and a proposition?

I'm doing a MOOC on mathematical philosophy and the lecturer drew a distinction between a proposition and a statement. This is very puzzling to me. My background is in math and I regard those two words as synonymous. I looked on Wikipedia and it says:

Often propositions are related to closed sentences to distinguish them from what is expressed by an open sentence. In this sense, propositions are "statements" that are truth bearers. This conception of a proposition was supported by the philosophical school of logical positivism.

http://en.wikipedia.org/wiki/Proposition

This also went right over my head. I (naively) regard both a proposition and a statement to be well-formed formulas that, once a suitable interpretation is chosen, have the ability to be either true or false. For example 2 + 2 = 4 is a proposition or statement because once I assume the Peano axioms along with the usual interpretations of the symbols '2', '4', '+', and '=', this statement is capable of being determined to be true or false.

Can anyone shed some light?

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Did your lecturer provide any examples? – Mozibur Ullah Apr 22 '14 at 1:50

## 3 Answers

Leitgeb distinguishes between statements, which are declarative sentences (he calls them 'descriptive sentences'), from propositions, which, unlike statements, are not linguistic objects. Propositions are the sort of objects that can have truth-values. E.g., [that snow is white] is a true proposition (Lecture 2-1).

Once the distinction is made, the key idea is this: statements express propositions, which are then said to be true or false. E.g. "snow is white" is a statement that itself doesn't have a truth-value, but instead expresses the proposition that snow is white, which happens to be true. That's pretty much it.

As regards your "2 + 2 = 4" example, Leitgeb could say this: "2 + 2 = 4" and "two plus two equals four" are two different sentences that express the same proposition. If you call them both 'proposition', then since the two sentences are syntactically distinct, you'll be committed to the claim that "2 + 2 = 4" and "two plus two equals four" are different propositions (this might be okay with you, but I think something is wrong with that). You might find the following analogy between algorithms and programs useful: given a single algorithm (~proposition), there are often multiple programs (~sentences) that implement it.

Leitgeb, Hartmann (2014 Spring) Introduction to Mathematical Philosophy (Coursera).

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"... statements express propositions, which are then said to be true or false. E.g. "snow is white" is a statement that itself doesn't have a truth-value, but instead expresses the proposition that snow is white, which happens to be true." -- Is this a syntax/semantics kind of thing? As in, "Snow is white" isn't true or false until I choose a model in which to interpret it: a meaning for the words "snow" and "white," and a particular planet whose weather I'm interested in. Am I close? – user4894 Apr 22 '14 at 3:47
Lecture 2 is developing toward Tarski's indefinability of truth theorem, so it's essential that Leitgeb develop the object/meta language distinction. I think the statement/proposition distinction is one of the bricks he uses to build a foundation for that task. So yes, it's a syntax/semantics kind of thing. But "snow is white" isn't a proposition even if predicates 'is snow', 'is white' and the logical constant '∀' are interpreted; ∀x(Snow(x) → White(x)) is a proposition, which becomes true/false depending on the interpretation, but "snow is white" is just a string of symbols. – Hunan Rostomyan Apr 22 '14 at 3:57

In philosophy of language (and metaphysics), statements are linguistic objects, like sentences of a natural language. Propositions are (traditionally understood as) the meanings of sentences (of a language) (in a context of utterance).

To illustrate:

The German statement "Schnee ist Weiss." expresses the same proposition as the English statement "Snow is white."

The distinction is arguably not immediately relevant for model-theoretic semantics of formal languages. Very few (if any) take the well-formed formulas of a formal language of mathematics to express propositions, although the connection between the semantics of formal languages and the semantics of natural languages is a hotbed of linguistic and philosophical issues of active research since (at least) Montague.

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I hope the following statements help you see the difference.

A proposition is a type of (logical) statement.
A statement does not have to be a proposition (logical).

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