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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 isIs 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 isIf 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?

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

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