Paradoxes tend to be sold as mysteries, things that the best minds cannot unravel, and so most people tend to pay about as much attention to them as anything else put outside of their reach or interests. They’re actually just points in an intellectual framework where results disagree with one another. A chain of reasoning that says 1 = 0, for example, can be considered a paradox.
Much like the chain of reasoning that can be used to bamboozle non-mathematicians, however, most paradoxes can be solved by identifying a flaw in the chain of reasoning, or the assumptions in which it is embedded. The classical paradoxes attributed to the philosopher Zeno, for example, aren’t problematic when considered in the framework of modern calculus - infinite divisibility won’t prevent Achilles from winning a race, or an arrow from moving, since movement considered according to modern calculus is distance over time rather than merely distance.
Of course, the particular disagreements brought to light by particular paradoxes will vary, and understanding of the paradox will depend on a person’s understanding of the subject within which the paradox is identified. One particular paradox, however, shows that you have to be very smart indeed not to solve it. It’s called the Liar Paradox.
A gentleman named Alfred Tarski identified the Liar Paradox as a semantic paradox, as one in which a particular sentence named S says: “Sentence S is false.” So sentence 1 is false if it was true, or true if it was false. And since, where we assume something must be either true or false but both (so no clever appeal to superposition), those are both inconsistent and hence consistent a paradox. Tarski identified the problem as self-reference and prescribed avoiding self-reference in order to avoid similar problems. Self-reference isn’t a problem, however, and Tarski’s prescription was essentially the ignore the problem so that it would go away. This let to the development of well-founded set theory.
Since then all sorts of clever people like Peter Aczel have pointed out that there’s plenty of logical frameworks, non-well-founded set theories, that handle self-reference without returning paradoxical results. But people still find paradoxes like the Liar paradox troubling, because as they’re typically laid out they’re like a magician’s magic trick: a mental sleight-of-hand. The Liar’s Paradox, laid out in a table of rows concerning what the sentence actually says, and columns of what it potentially means, reveals that the paradox is the result of considering the facts of being false if it is true, and true if it is false, as paradoxical.
Let’s consider otherwise, that S is not paradoxical, and that its potential truth value does not contradict its stated truth value, and that we can know the actual truth value if we take the aggregate of its stated and potential values. Then we can figure it out:
(1) If S says it is false, and that might be true, and we suppose such a combination or ‘conjunct’ of truth and/or false to have an aggregate value of ‘false,’ then we can deduce that where S might be true, and it says it is false, it is false.
(2) Likewise when S says it is false and that such a statement might also be false, then that combination of potentially false and stated false would also have an aggregate value of ‘false.’
(3) And where these two combinations are exclusively disjunct, and cannot both the case, we can combine them a third time under the logical rule of exclusive disjunction, where either one or the other and not both, can be true, or false. But both alternatives, the potential for the sentence to be either true or to be false, are false in aggregate.
Considering the Liar Paradox as stated truth over potential truth, rather than just as stated truth, we avoid the classical paradox of infinite regression (if truth then false, if false then true, etc…) and turn a paradox from something baffling into something akin to arithmetic.
Now here's something new and interesting that I haven't noted before, but had my attention and interest jogged by a question on Quora: Liar sentences are the logical equivalent to 0, in that the conjunction of any arithmetical value with zero is 0, and the bivalent conjunction of any semantic value with S is false. In a complementary fashion, so-called "Truth-teller" sentences that say something like "Sentence R is true" default to an actual value of true when we consider their stated truth over their potential truth. Because if R says it is true and it might be true, then it's true. And if R says it is true and it might be false, then it's false. Where the actual value of such a sentence R must be either true or false, which is to say bivalent and exclusively disjunct, then such sentence are true following the truth conditions of such disjunct propositions.
To say such sentences as R are vacuously true missed something of a point about how zero works in arithmetic as a place-holder and null-value. Zero is likewise vacuous, but in that vacuity it does some serious work in making sure that modern mathematics treats place-values efficiently. I'll discuss this place-holder value further in another post.