“[It was an] amazing fact…that our logical intuitions (i.e., intuitions concerning such notions as: truth, concept, being, class, etc.) are self-contradictory.” –Kurt Godel
The sorites paradox presents a challenge to the notion of clearly definable and graspable concepts such as gender and identity. However, many people may still think that the problem doesn’t extend that far. Even brilliant minds such as that of Ludwig Wittgenstein believed that we are capable of complete understanding through the mutual use of clear language. Thus, there must be a solution to this apparent problem of vagueness that conflicts with our fundamental intuitions regarding semantics (meaning) and logic.
In order to resolve the issue, philosophers have done various things including denying that logic applies to soritical expressions like the one described, denying some premise(s), denying the validity of the arguments demonstrating the problem (i.e., denying that that the conclusion follows from the premises), or by accepting the arguments as sound (i.e., the conclusion does follow from the premises and the premises are true). However, each strategy comes with its own set of problems.
Let’s start with the first. In order to argue that logic doesn’t apply to these expressions, some have argued that there is no actual problem once we are able to eliminate vagueness in our language, thus saving classical logic, which relies on binary truth values: True and False (hence classical logic being referred to as bivalent). Unfortunately, the pursuit to eliminate vagueness hasn’t produced satisfactory results and has largely been abandoned. As it turns out, vagueness is inherent in our thought as well as in our language.
Okay, well the second strategy might work. What if instead the problem wasn’t a problem inherent to meaning or logic, but was instead inherent to our ability to identify false premises? In other words, a sharp line between heapness and non-heapness exists. The problem is we just don’t know where the line is. That may be a route we could take, but it should be noted that this doesn’t seem to work out very will with other intuitions we might have. Indeed, intuitions are not enough to rule this possibility out, but there are also other reasons (which will be covered soon) to doubt that this is a simple problem of not knowing where the line is.
What about the other two strategies? In an attempt to successfully use them, philosophers have appealed to multi-valent (or many-valued) logics. For example, instead of a binary system only using the values True and False, we can use a three-valued system using True, Indeterminate and False. However, a trivalent logic leads to the same problems. What about a “fuzzy logic” that has infinite values where instead of a binary of truth values we have a spectrum of values? As it turns out, that doesn’t work either because it’s not exactly clear why we need to accept degrees of truth outside of solving these particular problems. Also, whatever degree of truth we assign is going to involve an element of choice. By this point we should be perfectly aware that we cannot objectively determine whether or not a statement is 1/2 true, 1/3 true, 1/4 true, etc. The final reason why is that even if we accepting degrees of truth solves this set of problems, it leads to its own set of problems. Binary logic: can’t live with it, can’t live without it, amirite?!
By now my dear readers are probably either in denial or in the depths of despair because we naturally desire certainty. It provides us comfort because it reassures us that the world is understandable and predictable. I mean, that’s how our ancient ancestors survived. They needed to come up with ways of navigating a very harsh and cruel world. In order to do that, they evolved certain cognitive skills that not only helped their chances of survival and successful reproduction, but also allowed them to think logically. This adaptation gives us an intuitive sense that we can really know our world, so I really apologize for what I am about to do, dear readers.
Enter the liar paradox (a paradox of self-reference): “this statement is not true”. What truth value should we assign to such a statement? It cannot be true because if it is then it’s not true and it cannot be false because if it is then it is true. Well, shit! What does this mean? In the philosophy of mathematics, a famous version of a self-reference paradox known as Russell’s paradox presents a challenge to the rational foundations of mathematics, the ultimate paradigm of rationality and truth. I mean, if 1+1 isn’t 2, then what is it? Fish?!
The paradox is as follows:
“According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves.”
Bertrand Russell, the paradox’s namesake, is credited with coming up with this particular paradox of self-reference. With the help of Alfred North Whitehead, he sought to dissolve this paradox in their work Principia Mathematica by attempting to ground arithmetic in logic. As impressive as it was, it was eclipsed by its massive failure at the hands of mathematician Kurt Godell (I know that’s not how your technically spell his name, but I don’t know how to configure the text on my keyboard to fix it–so shut up, nerds) and his incompleteness theorems which showed that arithmetic is necessarily incomplete. This means that arithmetic cannot be grounded in logic because it must rely on assumptions that cannot be proven to be true and it cannot show itself to be consistent. Well, what the hell does that all mean and what does it have to do with anything?
The thing about paradoxes is they seem to be indications of flaws in the concepts that we employ in our thought and in our language. For example, the sorites paradox points out that language is inherently vague, and that can be tough to deal with if we expect to have categories with sharp, definable boundaries. Take male and female as examples. The problem of vagueness suggests that any attempt to come up with clear, definable sets of criteria for belonging in either category is futile. It will necessarily involve a degree of choice and arbitrariness.
With self-reference paradoxes like the liar paradox and Russell’s paradox we are confronted with even more fundamental challenges:
The liar paradox is a significant barrier to the construction of formal theories of truth as it produces inconsistencies in these potential theories…the central question then becomes: How may the formal setting or the requirements for an adequate theory of truth be modified to regain consistency–that is, to prevent the liar paradox from trivializing the system?…Set-theoretic paradoxes [such as Russell’s Paradox] constitute a significant challenge to the foundations of mathematics.
If our notions of truth and the rational basis for mathematics are suspect, what else is at stake? Consider another paradox of self-reference known as the paradox of the knower: “No one knows this statement”. This paradox relies on two assumptions: (1) proof is sufficient for knowledge and (2) knowledge is sufficient for truth. Philosophers use a working definition of knowledge which is justified true belief. In order for a statement to be known, it must be a true statement that a knower has good reasons to believe is true. After all, how could someone know a statement to be true if it is actually false? Also, proof appears to be a high enough of a standard for being justified in believing that a statement is true. Okay, now that we got that all out of the way, let’s get back to the paradox.
If we were to take “No one knows this statement” to be true, then we would be claiming to know a statement that is not known by anyone. If we were to take it as not true, then we would be claiming to know that a statement that is not known by anyone is not true. Regardless, we would be making claims to knowledge that we would at the same time claim we do not have since we are members of the set that is excluded from knowing. The knower paradox is a part of a class of paradoxes known as epistemic paradoxes that present challenges to our understanding of knowledge. If truth and the knowability of statements are suspect, then can we know anything? Is 1+1 actually fish?!
Attempts to solve or get around the paradoxes of self-reference have been made by weakening some of the basic assumptions that precede them. Remember our working definition of knowledge? If “knowledge is justified true belief” is suspect, then what can we know? What does this all mean for us? Am I even real? Are there only two genders? Yes; it depends on what you mean by “know”, “true”, and “justified”; you’ll find out; it depends on what you mean by “I”; Lol hell no – haven’t you been paying attention?