Logic nonsense/paradox

I'm not sure if this is a paradox or a nonsense or neither of both. Anyway this is the "problem" if we can call it like that:

A: B is True
B: A is False

How can you solve it?

It is not much different from the plain liar paradox:

A: A is false

There are various more or less contrived "philosophical" attempts to resolve it, but by far the most common resolution is to deny that the statement means anything in the first place; therefore it is also meaningless to ask for its truth value.

The formalist mathematical approach to the paradox is to note that a sentence that speaks directly and explicitly about its own truth value (or, as in your original example, a set of sentences that speak mutually about each other's truth value) cannot be formalized in the systems that are commonly used as foundations for mathematics.

I fact, it is a classical result (by Tarski) that it is not possible for a sentence in formal mathematics to express "A is true" (at the same metalevel) of any (symbolically given) sentence $A$.

(See also here for additional discussion about the unwillingness of mathematics to deal with such self-reference).

This does indeed belong to the Liar-paradox family. But I'd dissent from Henning Makholm's rather quick remark that "by far the most common resolution is to deny that [Liar statements] mean anything in the first place". I don't think this is in fact at all common these days either among logically minded philosophers or among philosophically minded logicians. For some of the options for responding to Liar paradoxes, see e.g. http://plato.stanford.edu/entries/liar-paradox/

Here's one reason why the "meaningless" strategy looks difficult to consistently pull off. Consider the statements

1. Most things Nixon said about Watergate are false.
2. Most things Dean said about Watergate are true.

These are surely, by any normal standards, perfectly ordinary statements, which are perfectly meaningful in English. And in benign circumstances (Nixon's and Dean's statements are clear, unambiguous, etc.) are straightforwardly true or false. Yet they can give rise to paradox. What if, most unfortunately, Dean says fifty one things about Watergate, namely he says (1) plus twenty five plain truths plus twenty five plain falsehoods. Meanwhile (you can see where this is going!) Nixon also says fifty one things about Watergate, he says (2) plus twenty five plain truths and twenty five plain falsehoods. In these circumstances, we cannot give stable truth-values to either (1) or (2) -- think about it!

Trouble! And the trouble can't plausibly be resolved by saying that (1) and (2) are meaningless, ill-formed claims. We'd understand them perfectly well in the newspaper. It is precisely because they are meaningful that we see that -- in these very special circumstances -- we can't assign them stable truth-values. (And it seems unattractive to suggest we should deal with the "classic" Liar paradox by the "it's meaningless" gambit, while giving a different resolution for this kind of more "accidental" paradox.)

The Nixon/Dean example is, of course, Saul Kripke's. Kripke's own response is described in §3.2.1 of http://plato.stanford.edu/entries/self-reference/

The fact that may be surprising at first is that natural-language methods for reasoning about truth values are just not consistent with the assumption that every sentence is either true or not true, but not both. For example, if we analyze the statement

This sentence is not true

we find that if it is true then, as it says, it is not true, and if it is not true then, because it says it is not true, it is in fact true. This is a genuine "semantic paradox": it shows that the usual way we would reason about truth values in English is problematic.

As Henning Makholm has explained, the way that we usually avoid this in mathematics is to use formal systems, rather than natural-language reasoning. These formal systems are more limited than natural languages in what they can express, but this limitation is sufficient to make them consistent.

There are two other paradoxes that are less well known but worth knowing:

• Yablo's paradox obtains a semantic paradox without any self reference. The paradox consists of an infinite sequence of sentences each of which only refers to the later ones. But there is still no consistent way to assign truth values to all the sentences.

• Curry's paradox uses a sentence of the form "If this sentence is true then P" where P can be an arbitrary (fixed) proposition. This sentence is actually provable using normal natural-language methods regardless of what proposition is used for P (which another sign that natural-language methods are inconsistent).

It is also worth knowing that the use of the word "this" can be eliminated. For example,

The result of appending the following phrase after itself in quotes is false: "The result of appending the following phrase after itself in quotes is false:"

This phrase does not directly refer to itself, but if you perform the construction that is described you will arrive at the sentence you started with. This general method was explored by Carnap and was used by Gödel to prove his incompleteness theorems.

There are several ways to "avoid" the paradoxes, but they all rely on somehow reducing the ability of a system to perform the paradoxical deductions that can be performed in usual natural-language reasoning. There are axiomatic theories of truth that keep classical logic but reduce the way the truth predicate ("X is true") can be used, and there are paraconsistent logics that, though they are classically inconsistent, prevent "local" inconsistencies from spreading elsewhere, by restricting the deduction rules that can be used. But there is no resolution of the paradoxes that maintains all the features of normal natural-language reasoning - this is why they are genuinely paradoxes.

As a programmer I would concluse this problem as following:

Thesis:

A: B is True
B: A is False

Definition:

A = (B = TRUE)
B = (A = FALSE)

Solution:

Let's try explaining it mathematically with solving A.

A = (B = TRUE)                  | INSERT B
A = ((A = FALSE) = TRUE)
A = A = FALSE = TRUE
A = FALSE = TRUE

... but A cannot be FALSE and TRUE at the same time. I.e.

A = FALSE ≠ TRUE
⇒ A CANNOT BE SOLVED.

If A cannot be solved, then B should also be unsolvable. Let's check:

B = (A = FALSE)                 | INSERT A
B = ((B = TRUE) = FALSE)
B = B = TRUE = FALSE
B = TRUE ≠ FALSE
⇒ B CANNOT BE SOLVED.

Conclusion:

1. If A is TRUE, B has to be TRUE.
2. But if B is TRUE, A has to be FALSE.
3. If A is FALSE now, then B is FALSE.
4. And if B is FALSE, A must be TRUE.

⇒ It's recursive. You have to go to the begin (1) again.

A refers to B, to its own referent. So A is referring to itself. This semantic paradox is also known as liar paradox.

A or B has to lie. Then the approach above is solvable. Or you have to spend the rest of your life searching for a mathematical solution. You won't find. ;-)

The combined statements do indeed form a paradox. To check that, note that if $A$ is true, then $A$ is false, and vice versa.