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From the entry "Propositions" in the Stanford Encyclopedia of Philosophy:

The term ‘proposition’ has a broad use in contemporary philosophy. It is used to refer to some or all of the following: the primary bearers of truth-value, the objects of belief and other “propositional attitudes” (i.e., what is believed, doubted, etc.), the referents of that-clauses, and the meanings of sentences. (McGrath and Frank 2024)

From a high school mathematics textbook in South Korea:

문장 ‘2는 소수이다.’는 참이고, 식 ‘√2 + √3 = √5’는 거짓이다. 이와 같이 참, 거짓을 명확하게 판별할 수 있는 문장이나 식을 명제라고 한다. (전인태 n.d., 83)

The sentence ‘2 is prime’ is true and the expression ‘√2 + √3 = √5’ is false. We refer to a sentence or expression whose truth or falsehood can be clearly determined, like these examples, as a proposition.

Does the usage of the term 'proposition' in Korean school mathematics coincide with that in contemporary philosophy? I don't think so for two reasons.

First, high school mathematics teachers in South Korea would say that 'Bulhwi Cha loves himself' isn't a proposition. On the contrary, it seems the authors of the entry "Propositions" would regard it as a proposition, judging from the following quotations:

E.g., if the proposition that a loves b is the ordered triple <loving, a, b>, it is distinct from the proposition that b loves a, which would be the ordered triple <loving, b, a>. (McGrath and Frank 2024)

Is the proposition that John loves Mary different from the proposition that Mary is loved by John? (ibid.)

Second, Goldbach's conjecture states that every even natural number greater than 2 is the sum of two prime numbers. We don't know whether it's true or false as of 2024, so it's not clear whether it's a proposition in Korean high school mathematics. However, I think it is since it has a truth value. Therefore, what Goldbach's conjecture states is a proposition in the sense that it's a primary bearer of truth-value.

Edit: I also want to mention that the teachers and the textbooks they use always present an example like '99는 큰 수이다' (99 is a large number) as a non-proposition. See Question 1 in 전인태 n.d., 98.

Recently, a high school student from Korea asked two math teachers which of the following is a proposition: (a) a subjective value judgment and (b) a mathematical statement whose truth value we don't know yet. Both responded that (a) isn't a proposition.

However, they were divided on whether (b) is a proposition. One of them said that (b) is also a proposition since we can determine its truth or falsehood. The other argued that linguistically speaking ("국어적으로 해석한다면"), we can determine the truth or falsehood of (b), but mathematically speaking, it can't be a proposition.

References

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    No, but it coincides with use of "proposition" in English language mathematics (statement which is either true or false, i.e. has determinate truth value). This also happens to be one of the standard dictionary meanings, see Merriam-Webster. The philosophical meaning (truth-value-bearer detached from sentences expressing it) is as esoteric in English as it is in Korean. The Goldbach conjecture has that special form that if it is undecidable in Peano Arithmetic then it is true. So it is a mathematical proposition, in English and Korean.
    – Conifold
    Commented Oct 19 at 6:21
  • @Conifold Hmm, the meaning 1c of the word in the dictionary is "a theorem or problem to be demonstrated or performed." Therefore, √2 + √3 = √5 isn't a proposition in that sense. However, it is in Korean school mathematics.
    – Bulhwi Cha
    Commented Oct 19 at 8:07
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    Mirriam-Webster also has 2a and is not controlling for mathematical uses anyway. √2 + √3 = √5 and the like are propositions in English mathematics, and not just of school kind, see e.g. Dolciani's text: proposition is a statement that is either true or false, but not both true and false. Mathematical terminology is one of the most internationalized.
    – Conifold
    Commented Oct 19 at 8:29
  • True, but in my experience 12 years ago, Korean high school mathematics teachers and the textbook they used never discussed whether we should regard a sentence or expression whose truth value we don't know yet as a proposition. They only showed examples of true or false propositions. I'm pretty sure it's the same nowadays, but I admit I need more evidence to support my claim.
    – Bulhwi Cha
    Commented Oct 19 at 10:38
  • See also Professor Kevin Buzzard's message: leanprover.zulipchat.com/#narrow/channel/….
    – Bulhwi Cha
    Commented Oct 19 at 10:41

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The SEP article you cited also quotes David Lewis as saying that the concept of a proposition is "something of a jumble of conflicting desiderata". The term means a bunch of closely related things, so it is helpful to be specific when using it. In the way that philosophers tend to use the term, minimally a proposition is a statement that is truth-apt, meaning that it is capable of having a truth value. It doesn't matter whether anyone knows its truth value, whether its truth value is beyond being knowable, or whether the proposition is vague.

So in this sense, the Goldbach conjecture is a proposition, even though nobody knows whether it is true or not. The statement, "One day in the distant future there will be no sentient life," is a proposition, even though ex hypothesi there will be nobody around to witness it. "John loves Mary," is a proposition, even though it is vague as to exactly what love is and exactly how much of it would justify the assertion.

It would be weird to try to exclude vague statements from counting as propositions, since when we examine things closely, lots of things turn out to be vague, at least outside of mathematics. "John is tall," is vague, but we can think of it as a proposition. If you say, "John is tall," and I say "No, he's not," then we are disagreeing about the truth of the proposition. We can also say that Mary believes John is tall. This means that Mary believes that the proposition, "John is tall" is true.

So, if your South Korean textbooks are excluding unknown or vague statements from counting as propositions, then there is definitely a difference in usage.

As to the link you provided to the Lean forum, I am surprised at Kevin Buzzard's comment. In my experience, mathematicians usually call an "easy, true statement" a lemma, not a proposition. Propositions have truth values and can be false.

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    I imagine that Buzzard is referring to how the word "Proposition" is used in mathematical writing, as an element of style, e.g. when you write "Proposition 4. The fundamental group of a loop space is abelian." You only write "Proposition" in this way if you're about to give a proof or cite somewhere where a proof has appeared. So it's only used for true statements. This usage of the word "Proposition" as an element of the mathematical style of writing is quite different from the use of the word "proposition" in philosophy and in propositional logic, where propositions can of course be false.
    – user509184
    Commented Oct 23 at 17:56
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The Korean text you quoted makes no sense as translated, as my response to this question shows. A sentence or an expression is neither true nor false. The definition of proposition in the Stanford Encyclopedia of Philosophy makes it very clear that a proposition must have the property of truth value. A statement is formulated in some language, a proposition is not. Propositions are mental neurological entities so they exist in brains, and have the properties of truth or falsity. A sentence/expression/statement cannot have those properties.

Edit- space is three dimensional, so a proposition cannot exist outside the universe, hence propositions aren't abstract objects. The only thing that is in space is matter, hence if propositions exist they must be material. That's why I referred to them as mental neurological entities, although neurological entities would be sufficient.

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  • "Entities that exist in brains"? You mean brain cells? Neurons? Electric potentials? I guess you don't really mean that. But if not, what do you mean and why would you introduce some kind of invisible extra entities? Because meanings are what correspond to the same statement translated into two different languages? How do you know? Isn't this a case of obscurum per obscurius?
    – mudskipper
    Commented Oct 19 at 13:54
  • @mudskipper, for something to have a property it must exist. Propositions have the property of truth value, so they exist. Only matter exists, so propositions are matter. Therefore propositions exist somewhere. The answer as to where they are located is inside brains, so they are neurological entities. I'm a physicist not a neuroscientist, I would say a proposition is smaller than a brain cell, NOT a voltage difference, and possibly a neuron, or part of a neuron.
    – lee pappas
    Commented Oct 19 at 14:06
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    In usual mathematical textbooks, proposition means simply matematical statement. No philosophical overtones. See propositional calculus.
    – Mauro ALLEGRANZA
    Commented Oct 19 at 15:33
  • What is a mental neurological entity? Do you have a reference or are you simply making up terms? I'm glad to see you've abandoned the claim that propositions are material and have mass! Abstract objects are defined as being outside of the physical universe. I'll withdraw my downvote for a rather marked improvement if you edit again.
    – J D
    Commented Oct 22 at 15:08

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