# Some questions regarding Propositions and the Liar's Paradox.

1. Oct 9, 2008

### TheDarkElf

Hi Folks,
I am stuck at a certain logic problem which is probably very common and easy for undergrad compsci students so forgive me if my question seems rather trivial. The question is this:

1. I believe the statement "This statement is false" is not a proposition and is paradoxical in nature because if it were true it implies that it is false and it goes against the law of excluded middles.
2. Now based on the same logic, Consider the problem,
For a sequence of 100 statements, nth statement is that
Exactly n of these statements is false.
The book does state that the answer is that 99th statement is true. I kinda do follow the reasoning because every statement does contradict every other statement by a process of elimination you can very well suggest that the 99th statement can be valid.

But look at it closer:
Assume there was only two statements:
Exactly 1 of these statements are false
Exactly 2 of these statements are false
1 Will be true only if the second statement is false.
Let us look at the second statement: Obviously if true its paradoxical. But if false, what does it imply? It implies that Either more statements are false (which is impossible because the domain is just of two statements) or it implies that only one of the statements if false which should obviously be this statement only.
My question is Is this reasoning right? Can a statement which if true is paradoxical but if false is valid be considered to be a valid proposition?

Next up, (Clearly assuming that such a statement is valid), I looked at another problem in the book. Which was
At least n of the statements is false (Applied to the previous question with the same number of statements).
The book said that around 50 statements were true and the rest were false. (Not difficult to reason out).
A sub division was to solve the previous question assuming there were only 99 statements.
The book stated here that this was a paradoxical statement and cannot be solved.

My question is If one does accept that the statement in the first question if false is a valid proposition, the third question is also valid.
Reasoning
Assume there are three statements only:

At least one statement is false
At least two statements are false
At least three statements are false.

-> 1. being true implies that either 2. is false or both 2 and 3 are false or 3 is false. So it can be a valid proposition. Although if 1. were true, let us look at 2.
2. cannot be TRUE because if it were true it would negate one being true, so 2 is false, 3 being true is paradoxical so assume it to be false. Hence we have a valid solution.

-> 2 alone being true is also valid because then statement 1 and statement 3 would be false.

Do clarify what on earth is the flaw in my reasoning? or if the book is just wrong?

2. Oct 13, 2008

### gel

I've never really looked at such self-referential statements, but the questions don't sound so hard...
What is the definition of a paradoxical statement? It sounds like it should be that a statement is paradoxical if it cannot be either true or false. So, I don't know what "if true is paradoxical" means. Does it mean anything?

if statement 2 is true and statements 1,3 are false then at least one of them is false, so 1 is true and this is not a valid solution.

3. Oct 13, 2008

### Hurkyl

Staff Emeritus
In formal logic, whether or not something is a proposition is purely a matter of syntax. The semantics of the statement are irrelevant.

When we make statements in natural language (intending them to be convertible into formal statements), I imagine some semantics is necessary... but only so far as to understand the grammar of the statement. The 'truth' of the statement is irrelevant.

The construction of the usual formal logic explicitly forbids self-reference; each expression has an 'order', and the variables of an n-th order expression must have order less than n.

(And the reason formal logic is built this way is precisely because of problems like the Liar's paradox. It's the same reason set theory is built 'constructively' via axioms like ZFC, rather than using Cantor's 'naive' set theory)

Other logics avoid the problem in different ways. e.g. some computer languages have no problem offering a predicate that acts on itself. But if you try to implement the liar's paradox, such as in the following C++ program fragment:
Code (Text):

typedef bool (*proposition)();
bool is_this_false(proposition p)
{ return !p(); }
bool liar()
{ return is_this_false(&liar); }

when you evaluate liar you get one of the implicit 'truth' values that is neither true nor false. Most likely, the one you'll get is Segmentation fault.

So, going back to gel's question... how is the logic that you are using in this problem defined?

Last edited: Oct 13, 2008
4. Oct 14, 2008

### TheDarkElf

@ gel & Herkyl
Thanks for the reply.

I realize after rereading my post that I have not been quite clear in what I wanted to ask.Let me clarify:
I am working on Propositional Calculus. The book does state that a Proposition is a declarative sentence that is either true or false. Nothing in the middle.
Now coming back to what I didn't clearly define previously:
Exactly 1 of these statements is false
Exactly 2 of these statements are false
* 1 Will be true only if the second statement is false.
* The second statement if true indicates that the second statement is also false. If false, it indicates that Either one of the statements is true or None of the statements is false. This at least acccording to me seems like a valid assertion because then it would fit in with the logic that the first statement is true.
As per wikipedia, the definition of a paradoxical statement depends on three rules:

1. A statement that is self referential. ( The second statement is self referential)
2. A statement that is contradictory (Its truth does contradict the statement)
3. A statement that is viciously circular ( The circular logic doesn't progress ad infinitum: The truth of the statement is clearly absurd because it implies that the statement is false. If false, we don't enter a problem).

Well, I am not sure if the statement "Exactly 2 of these statements are false" fulfills all the conditions of becoming a paradoxical statement, it does fulfill two of the conditions.

Yes, I missed that there. Thanks for pointing it out :)
That of course implies that there is no valid solution because the only thing left is the following combination.
At least one of these statements is false - TRUE
At least two of these statements is false - FALSE
At least three of these statements is false - FALSE

The second statement" At least two of these statements is false" being false implies. There is one false statement or none. But because the statement 2 and statement 3 being false renders the second statement true, which is impossible. So because we are going into a viciously circular logic there is no solution.
Is that right?

Thanks again folks.
Regards
TDE