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In propositional logic, a proposition is a statement that is either true or false, but not both. In a text I am reading and in many others, "this statement is false" is not considered a proposition. But is this because it is both true and false or because it is neither true or false, i.e., doesn't have a truth value?
The text I am reading says that the truth or falsity of a proposition may be clearly understood or arbitrarily assigned, which I interpret as meaning that what is important is that a proposition must be able to hold a single "stable" truth value. When we attempt to assign a truth value to "this statement is false" what is the problem?
EDIT 2: I understand that in order for an assertion to be considered a proposition, we must be able to associate a truth value to it. I have seen the following terse reasoning about why the assertion "this sentence is false" is not a proposition: If it is true, then it is false, and if it is false, then it is true. Does this mean that assigning it a truth value leads to it being both true and false, or that assigning it a truth value leads to a contradiction? Please explain.