Does a proposition have to have a true conclusion?

“...understand that a belief is propositional, which means it can be expressed in a declarative sentence—a sentence that is either true or false...Recall that in logic, a proposition has been demonstrated when it has been shown to be the conclusion of a sound argument—an argument in which (1) all premises are true and (2) it is impossible for the premises to be true and for the conclusion to be false” (Critical Thinking Moore and Parker). Aren’t these two quotes contradicting because the first one says a proposition can be true or false while the second one states that it is impossible for the proposition to be false? Am I misunderstanding this and can someone clarify this for me?

• The first one is about propositions in general, the second is about the propositions in a "sound argument". All the propositions in a sound argument must be true, but you can have logically "valid" arguments which are not "sound" because at least one proposition in the premises is false, and you can also have arguments that are neither valid nor sound. Commented May 10, 2021 at 5:08
• The second one only says that it is impossible for a proposition that "has been demonstrated" to be false. Not even every true proposition has been demonstrated, and false propositions cannot be (soundly) demonstrated at all. Commented May 10, 2021 at 10:46
• The passage is stating that you can't derive a false proposition from an argument that has all true premises & follow all the rules of argumentation. You must violate at least one rule to purposely come up with all true premises & your conclusion is false. In Mathematical logic one can make premises any kind of way but this is not true on other systems of logic. There are other types of logic. Mathematical is the main one used these days. The definition of proposition is not accurate in math. Propositions are not literally verifiable. You don't see or hear propositions. Other logic show this. Commented May 10, 2021 at 12:20
• An argument has a conclusion. Commented May 11, 2021 at 7:35

Your two quotes are consistent in classic propositional logic. Regarding your "the first one says a proposition can be true or false", according to IEP here:

The term proposition is sometimes used synonymously with statement... A statement can be defined as a declarative sentence, or part of a sentence, that is capable of having a truth-value, such as being true or false.

propositional logic is classical truth-functional propositional logic, which studies logical operators and connectives that are used to produce complex statements whose truth-value depends entirely on the truth-values of the simpler statements making them up, and in which it is assumed that every statement is either true or false and not both.

So a proposition generally have a truth value, of course it may be true or false depends on further available information.

Regarding your "second one states that it is impossible for the proposition to be false", according to IEP here:

A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. Otherwise, a deductive argument is said to be invalid.

A deductive argument is sound if and only if it is both valid, and all of its premises are actually true. Otherwise, a deductive argument is unsound... Consider, then an argument such as the following:

All toasters are items made of gold.

All items made of gold are time-travel devices.

Therefore, all toasters are time-travel devices.

Obviously, the premises in this argument are not true. It may be hard to imagine these premises being true, but it is not hard to see that if they were true, their truth would logically guarantee the conclusion’s truth.

So an argument can be logically valid but not sound and a sound argument can demonstrate the conclusion proposition as true.

• Your answer applies strictly to mathematical logic. Propositions existed for thousands of years prior to Mathematical logic. Propositions in Aristotle's time line we not declarative sentences nor were they statements. Sentences of any kind are distinct from statements, statements do not have to be sentences and propositions are totally distinct from sentences or statements. Different sentences can Express the same proposition as in different languages can translate the same idea. This sentences are NOT propositions. What you would end up doing is counting each sentence as a new instance. Commented May 10, 2021 at 12:13