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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
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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).

  • "... 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
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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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Well the question you pose indicates that you believed that the math definition was the end all be all and then reality struck. You found out the math definition was strictly in the context of math. Well why not start off teaching that way? The topic you study is NOT logic but strictly called "Mathematical Logic". There are other types of logic. Math is not the only field that has a logic component. Philosophy has one, Psychology has one, Rhetoric has one, etc. You must not assume all logic is logic. This is where you went wrong.

In philosophy, propositions are defined as mental components. They do not have physical properties or attributes. They don't apply to your human senses. You can't see or hear a proposition. What humans do then is EXPRESS the mental component into a given language to rely the idea to other human beings. The key here is propositions are not physical. Statements on the other hand must be physical.

A statement outside of math is any physical communication method to relay an idea to another human being. This communication does not have to be verbal or written. You may tend to think of statements as verbal or written. This is an assumption because this is what you are used to and aren't directly told otherwise. A traffic sign relays a message such as to STOP or slow down. Me pointing a loaded gun at you relays a message which is a statement. If you understand the message then the communication is a statement for certain. This does not mean if you cant understand the message there is no statement. Hand gestures can relay a message. If someone can insult another person with just body gestures that is a STATEMENT. I don't literally have to say what I am thinking if I use gestures. A gorilla charges at you in a threatening manner is making a clear statement without being able to understand English. In my youth my mother would make statements with just her looks. I would see disgust on her face if I misbehaved in her presence but she was too far away from me to yelll at me or smack me for acting the fool when I knew better. Most of the latter examples clearly do not provide information on something being true or false. In math you were likely told statements are true or false (without specifically being told the domain of that discourse was IN THIS CLASSROOM). All statements do not require something to be true or false. There are also such things as meaningless statements which are neither true or false. We do know all literally meaningful statements can be translated as declarative sentences. All propositions can also be expressed as declarative sentences. But do not think they are identical because of this similarity.

In summary, a proposition is a mental concept or idea that is expressed to hold a truth value of either true or false (and specifally cannot be both or neither). Controversy will arise about what true or false means here. Scientists think only in literal sense verification whereas a philosopher can understand what objective knowledge is. That is in science if I can't show you a unicorn the scientist will say my claim "all unicorns are white" is false. Basically no sense verification then there is no truth. Objective knowledge doesn't have the sense verification requirements. I can say objectively "there is a God" which by definition must be either true or false but I the speaker may not know this to be true or false. Me not being aware of the TRUTH VALUE does not mean there is NO TRUTH VALUE. Objective expresses the idea that the truth value is independent of my awareness or my senses. Time will eventually tell the value. You are not likely to hear this in math as that is not the purpose.
I would interpret true proposition to mean the objective knowledge context unless the proposition is blatantly otherwise. Hence why "all unicorns is white" is NOT FALSE. At best it is a meaningless statement. If so it would fail to be a proposition. So again I slipped in another example there. Some statements are not propositions. All literally meaningful statements can Express propositions. Literally meaningful here means the physical connection to what is said. If I say "Al is tall as an Oak" he either is literally tall as the oak or he is not. Of course that case is not literal but just illustrating if it were then we have fulfilled the requirements of a proposition and a literal meaningful statement.

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