Statements that are not propositions

[AcidRainLiTE]

I just started reading a book ("The Number Systems of Analysis" - by Little, Teo, and Brunt) which attempts to resolve the paradox resulting from the statement "this statement is false" by claiming that it is neither true nor false. However, I am a little confused because I don't think that solves the problem. Here's why:

(1) Let p = this statement is false.

(2) According to the book, p is not true and p is not false. Let's suppose the book is correct (I will show that we will get a contradiction).

(3) Let q = p is not false.

So, q is true (since, as stated in (2), we are assuming the book is correct).

(4) Let r = q is false.

(5) But, since we know that q is true, r is false.

(6) Now, it seems to me that r is equivalent to p. If this is true (which I will show in a minute), then we have a problem, because we said r is false and p is not false.



Here is why I think r is equivalent to p:

r = q is false (By definition from (4))
= "p is not false" is false (By substituting for q)

So r says that it is false that "p is not false". In other words, r says that it would be a lie to say "p is not false." Thus, r says that p is false. We thus have,

(7) r = p is false.
= "this statement is false" is false (by substituting for p)

But it turns out that p is equivalent to that statement. You can see this by substituting as follows:

p = this statement is false
= "this statement is false" is false." (by substituting p for 'this statement')
This is exactly what we got for r in (7).

So, we have that r = p.


But from (5), r is false and from (2), p is not false. Since r and p are equivalent, we have a contradiction.



I hope that made sense. I tried to state it clearly, but it is inevitably convoluted.

[AcidRainLiTE]

I think what I wrote above is unnecessarily complicated. I think I can come up with a similar thing that gets at the same underlying issue.

Let's say we are trying to figure out how to deal with the statement p = "this statement is false." We do so by saying p is neither true nor false (i.e. we say that some things just don't have truth value. For instance, the command "go do your laundry" is not true and it is not false. It just doesn't have truth value).

Now, if p is not false, then it would be incorrect to claim that it is false. However, p is claiming that it is false. Hence, p's claim is incorrect and p is thus false.

How do you deal with this?

[JSuarez]

The book's assertion is correct; I don't know the authors's explanation, so I cannot comment on it, but it seems that your reasoning ends with a contradiction, which is precisely what happens when we try to ascribe a truth-value to that particular statement, so I don't understand what is exactly your problem.

But I can say a little more about your question's title: statements and propositions are indeed two different things.

A statement is linguistic object (a phrase) that refers to a proposition.

A proposition is something that describes (rightly or wrongly) a state of affairs (roughly speaking), but it doesn't have to be codified in any language, that is, it's not necessary to have a statement that refers to the given proposition which, nevertheless, may be true or false.

Consider the following example: somewhere in a desert there are three red rocks; this is a proposition, referred by the statement "somewhere in a desert there are three red rocks"; it is true if the three rocks are indeed red and false if, for example, there are only two rocks or they have distinct colours.

But if there were a proposition referred by the statement "this statement is false" (or any of its variants), then it would be simultaneously true and false; this is called a dialetheia, which means an actual, existing, contradiction and the present consensus is that they don't exist in possible worlds.

[AcidRainLiTE]

Okay, those were actually some very helpful comments. It has helped to clarify what I am wrestling with. Now, let me rephrase what I am saying in the terms you just mentioned.

There does exist a proposition which is referred to by the statement " 'this statement is false' is not false." Correct? Call that proposition q.

But, propositions come in pairs. For every proposition, there exists another proposition--namely, its negation. What, then, is the negation of q?

I guess you could deny that propositions come in pairs. But is that really the solution?

[JSuarez]

There does exist a proposition which is referred to by the statement " 'this statement is false' is not false." Correct?

Why is it so obvious that that statement denotes a proposition? Note that if it doesn't, then the problem disappears.

[AcidRainLiTE]

I guess I would say that " 'this statement is false' is not false" denotes a proposition because I think that, if there isn't any proposition that referrs to the statement 'this statement is false', then it is true that 'this statement is false' is not false since the only statements that can be considered to be true or false are those that represent propositions. To me it seems correct to say that if any statement doesn't represent a proposition, then it is neither true nor false.