relativism and russell's paradox

Question

Relativism is the idea that there is no universal, objective truth, and instead depends on the context and perspective. There is one concerning point however, which is the following relativistic statement:

"There are no absolute truths."

One can argue that this is an absolute truth, and therefore this statement is self contradictory. This is considered a criticism of relativism. However, I argue that this statement is imprecise to begin with, because it's essentially Russell's paradox. Russel's paradox is a contradiction that arises when we have a self referential property. More precisely, let A be the set of all sets that are not members of themselves. If A is not an element of A, then by definition A is in A, and vice versa.

We can reformulate the above statement "There are no absolute truths" into Russell's paradox. (EDIT: the following argument is incorrect/flawed because it is ambiguous what kind of set theoretic object the collection of all absolute truths is.)

Let B be the set of absolute truths. If B is empty, then the statement "B is empty" is an absolute truth, so B is not empty. And if B is not empty, then the set B is an element of B because a set of absolute truths is also an absolute truth.

Another way to look at this issue to consider the following: Let's consider the determination of absolute truth to be a function f that assigns a value of 1 to a statement x if it is absolutely true, and 0 if it is not. Now, the claim "there are no absolute truths" is the claim that for all possible statements x, f(x)=0. The problem is f(x)=0 is a statement itself (let's call it y), and we can evaluate f on this statement y. This is the crux of the self referential issue and is the same one as russell's paradox.

So what is the solution to this? In mathematics, it seems like there are a couple of different solutions. One solution is to restrict the definition of "set," in line with the ZFC axioms. Another is to go into type theory, where a set is of type 1, an object that contains itself (like in russell's paradox) is type 2, an object that contains itself (or rather is self referential) "twice" is type 3, etc.

What is are the possible solutions in terms of philosophy?

Answer 1

There are several approaches here:

• We can distinguish between theory and metatheory. In this context, "there are no absolute truths" is a metatheoretical statement, which is not "talking about" itself, but merely about some reasonable domain of discourse. This is consistent but may be philosophically unsatisfying because it fails to account for the surrounding background logic.
• We can use a system of types, which is essentially a more formalized and flexible version of the previous bullet. This still ignores the surrounding background logic, but lets us have different "levels" of meta-ness, which may be more convenient. For example, we could have an infinite hierarchy of metatheories and the whole hierarchy can be incomplete with respect to the (untyped) statement "there are no absolute truths in any of the hierarchy levels."
• We can deny semantics entirely, and focus solely on syntactic manipulations (e.g. "Whenever I have '2+2' written on my paper, I can replace that sequence of symbols with the symbol '4,' but those symbols don't mean anything in particular."). In this setting, we need not use any background logic, except perhaps for the bare minimum necessary to characterize basic symbol manipulation, so it does not need to be accounted for at all. This has the disadvantage that semantics are rather convenient, and denying them makes mathematics substantially less interesting and useful.
• We can deny the truth of mathematics outright, which cuts the Gordian knot without having to deal with the background logic at all. Obviously, this approach is controversial, but it is difficult to escape for a strict physicalist (you can't pick up the number 2 and throw it across the room, for example).

Answer 2

Russel's paradox is a contradiction that arises when we have a self referential property.

Russell's Paradox is an inconsistency discovered by Russell in an early attempt to formalize set theory by G. Frege. Using the axioms of that set theory, it was possible to both prove and disprove the existence the set of all sets that are not elements of themselves. Russell discovered this inconsistency even before Frege's work was published. This "bug" was fixed by various means in subsequent versions of formal set theory (e.g. in ZFC set theory).

"There are no absolute truths."

Note that, self-reference is not a problem in logic. It is quite easily handled. In fact, it is trivial to prove that for any binary predicate R, there cannot exist an object x such that for all y, R(y,x) if and only if ~R(y,y). This would appear to be an absolute truth. It is not just the set membership relation to which it applies; it applies to all binary relations.