A declaration of truth or falsehood is not the reality of truth or falsehood.
Declaring something false does not make it false, the reality may be that it is true, the declaration is not effected by the reality.
So in short, the assertion is a proposition as it clearly declares the statement to be false and nothing else.Is the assertion ';This statement is false'; a proposition?
NO. It is a paradox. It is impossible to assign a consistent truth value to the statement as it contradicts itself.
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It's a self referential sentence that creates a paradox.
Yes, it's a proposition, but formal systems of axioms and theorems are generally made to forbid self reference, just so such propositions can't be derived.
That's on purpose, because it's a contradiction. If a formal system can derive a contradiction, then it's not Consistent.
That is, if a formal system isn't consistent, then it's possible to get contradictory answers from it, which is kind of useless in a formal system.
Unfortunately, if a formal system can be made completely Consistent, than it is necessarily also Incomplete...
That is, if there are True Statements which a formal system cannot derive, no matter how long you mess around with the rules and axioms, that formal system is Incomplete.
This is a limitation of such systems. And the fact that the human mind can apparently produce such propositions so easily might be thought to set it apart in some way from such systems.
I'm not the world's greatest expert on this stuff so I hope one doesn't happen to come by here and explain how I'm completely wrong.
Anyway, you should read Douglas Hofstadter's famous book, Goedel, Escher, Bach: An Eternal Golden Braid, in which he talks about this stuff a lot, and very entertainingly.
Here's something he wrote on the subject that I found with google
http://books.google.com/books?id=o8jzWF7鈥?/a>
Your concern seems to be that of philosophy of language rather than one of math, number theory or philosophical logic. Nevertheless, your definition of proposition is a bit imprecise and can pose a simplistic solution to this, a very important problem.
First, a proposition is not a sentence properly but what the sentence means, and second propositions need not to be either true or false. I give an example of both observations:
';This sentence is false';
Means the same as
';Esta oraci贸n es falsa';
and the same as
';Dies Satz ist falsch';
They are not the same sentences though.
And secondly, we can have sentences referring propositions impregnated with vagueness (vague predicates which are not true, neither false, or vague nouns or the like). For instance:
X is tall.
Where X refers to a person which is in the limit of being tall and short. (I don't know think about any other example). So, some propositions can have blurry or vague truth values, i.e., not true, nor false either, but expressing a proposition at the end. For example, that it is true that X is not tall and not short, but something in the middle of that.
We have had a simple test to know if a sentence does fix a proposition, which is to say that a sentence says something about the world, that it is a description or a verifiable saying about the world.
It is often maintained that declarative sentences oppose to normative sentences at least in the following respects through the ';is';/';ought'; classical distinction. A ';ought'; sentence, doesn't lose its validity or importance if the world doesn't supoort it. For instance: ';The world ought to live in peace';. The strength or weakeness of a sentence such as the former, doesn't change if the world changes. We can live in war or in peace and that sentence still have the same strength, validity or acceptability. But, a sentence such as ';The world is such that it lives in peace'; is sensitive to the world arrangement of facts.
If you fall into the most popular solution to the liar-like paradoxes by saying that liar sentences are simply meaningless chains of symbols, then you must explain how do you discriminate between very likely to be had as declarative chains of symbols, those that do fix a propositions, and those that doesn't.
Normally this strategy just works for sentences such as the liar sentence that you've brought. This type of solutions are had as a classical ad hoc solutions, and normally they don't fix anything given the phenomenon of the revenge of the liar. Consider, that liar like sentences make you introduce a third truth value (besides the two classical ones), indefinite or something similar. Then:
This sentence has indefinite truth value. (you make the thinking about its paradoxiness)
And even more, if you could get rid of revenge in a similar fashion for problems of ordinary language, it would not work for similar problems that raise in formal systems, results such as those of G枚del's incompleteness Theorems.
So finally, nobody truly knows a satisfactory answer to this kind of problem. Don't blame your teacher... He probably was having an honest and serious case uncertainty. Babai.
This ones got me thinking. I may be way off but Im going to take a shot in the dark.
If the statement ';This statement is false'; is false does not mean the proposition is necessarily true. As u state a propostion declares that a statement is true or false but not both. If ';This statement is false'; is false and therefore it is true - then the propostion is declaring ';either this proposition is false or the statement is true.'; So you see its declaring the statement is true or false at the same time but in another way.
Jones is 5 feet tall
Proposition - This statement is false
Conclusion - Jones is 5 feet tall
Is this true? Can we deduce that Jones is 5 feet tall because the propostion was false? No! Maybe he has stilts on or he`s wearing thick soled boots. Now a person might say well then the propostion was not really false. But therein lies the rub. Whenever we make a statements about statements theres no way to validity deduce its veracity. Statements about statements sets no parameters and gives no limitations as to what might be true. Theres no way to objectively verify a propostions validity by simply stating its true or false.
Edit: I`ve thought of another way to explain this. What if one part of the statement is false but the other is true.
Jones is 5ft tall
This statement is false
Jones is 5ft tall
Problem - What if the person in the statment is not Jones, but he`s 5ft tall? The statement ';This statement is false'; would be false but it would not mean that Jones is 5ft tall.
Another way to look at this is in terms of falsifiability. Karl Popper articulated that in order for a statement to be true it must be in principle falsifiable.
I.e. All polar bears are white
Falsifability - I found a brown polar bear.
Can we falsify ';This statement is false?'; No, saying the statement is true does not falsify this statement is false. Because ';This is point'; in order for a statement to be falsifiable it must have refrence to the real world. Whenever we make a statements about statements theres no way to test its veracity.
A proposition is something that you declare that can be either true or false, but not something you declare that can be both.
To know whether or not something is true or false, it must be proven. To prove it, it must be observed in someway for verification.
What we observe are things from the outside and real world, such as the way things are, where things are, what things look like, etc.
Therefore, I would say a proposition must not only be a statement but also be a statement of something we have a reason to believe or observed in some way for provability or disprovability.
The statement ';The sun is yellow'; is either right or wrong, and we can try to prove it by observing the sun and seeing if it is yellow or not. Therefore it is a proposition, and it is true, not false, because it is yellow.
The statement, ';An apple is blue'; is either right or wrong, and we can try to prove it by observing the apple and seeing if it is blue or not. Therefore, it is a proposition, and it is false, not true, because the apple is red.
';The sun'; and ';An apple'; are not propositions because they aren't stating their relationship to anything that needs to be proven and observed as either true or false.
Now if I say something like ';The sky is blue'; and then I say, ';That previous statement is true';. Both of those are propositions, because something can be done to (at least try) prove whether or not both of these are true. If the sky is blue, then the second statement is true. But if the sky is not blue , then it is false. Therefore, the second statement is a proposition because it is either true or false. And it is either true or false *because it has provability and vericationability*
If I left out the first statement and just said, ';That statement is true'; What would I be talking about? There is nothing to prove or observe whether or not it's true because we're not really talking about anything, so it is lacking. We're just stating something without any connection to the real world or ability to prove or disprove it, therefore it can't be true or false because there's no reason for it to. Therefore, that statement alone would not be a proposition.
So likewise with the sentence, ';This statement is false'; How can we prove whether or not the assertion is true or false? It's just a statement and nothing more and has no provablity. We can't go anywhere outside and look and see if the statement is true or false. We can't look in our history book or encyclopedia to see whether or not the statement is true or false.
If you first said, ';A horse has five legs'; and then said, ';That sentence is false';, then both of those statements would be propositions. But the second one is only a proposition because it is connected to the one that has provability. If you said, ';That sentence is false'; alone it wouldn't have no meaning because there would be no specific sentence to know whether or not to prove or disprove.
Therefore, the statement alone ';This statement is false'; is meaningless because to claim something as intended truth (even if you are saying something is false) you have to have a way for another person to verify it. It needs to be longer and have more meaning in order for it to be a proposition and have the ability to be either true or false.
Even ';This statement is true'; is not a proposition. Because in order to state something as a proposition it must be provable. ';This statement is true'; is not provable or disprovable. It's only a statement. ';This statement is false'; is also just a statement and has no basis in provablity therefore its not a proposition. Those types of sentences go more along in the category (if there is any) of statements without provability.
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