Paradoxes: What Is Settled, What Is Open, and What Is Neither

Start with the definition, because it decides everything that follows. Our research file defines a paradox as an argument that proceeds from apparently acceptable premises, by apparently valid reasoning, to a conclusion that is apparently unacceptable. The three apparentlys are the definition and they are the part that does the work. Each one names a place where something may have gone wrong, which is why the file's next move is to give three exits rather than none: reject a premise, identify a flaw in the reasoning, or revise your understanding of the conclusion. A paradox on those terms is a piece of work to be done rather than a wall to stand in front of, which is this article's compression of the file's point rather than a sentence taken from it. The file adds that paradoxes have been central to philosophy since the Greeks and that they act as engines of conceptual progress, exposing hidden assumptions and the limits of our concepts. The rest of this article asks one question of each of the seven: which exit does this one take, and has anybody actually walked through it.

Here is the whole article in one table, and every row in it is argued in its own section below. Counted off those rows: of the seven paradoxes our file carries, two have accepted technical resolutions, four are genuinely open, and one is not a contradiction at all. Zeno's arrow and his stadium are parts of the Zeno entry and are listed separately in the table rather than counted again, because the answer that settles two of his four arguments does not touch the arrow, and because the surviving text of the stadium is too compressed to grade. No sentence anywhere below contradicts a row of this table. Where a row says settled it also says in what sense, and where it says open it says what the open question is.

That is the Dichotomy, also called the Racetrack, and the halving is not where the trouble lies. To reach any point you must first cross half the distance, then half of what remains, and so on without end. The load is in what that requires: the completion of an infinite number of tasks in a finite time. A version that stops at how small the steps get has dropped the argument on the floor. The steps getting small is not the problem. Finishing infinitely many of them is.

The Achilles is the same shape with a runner in it, and it has two setup details a careless version drops. Achilles, who is faster, gives the tortoise a head start, and the tortoise keeps moving. By the time Achilles reaches the point where the tortoise started, the tortoise has moved on; by the time he reaches that point, it has moved again; and so without end. Take away the head start, or let the tortoise stop, or have Achilles simply run to a fixed line, and you have a different and far easier problem whose solution will look trivial. One note about the sources belongs here, because it is unusual. The reference entry consulted for this article does not print a block quotation from Aristotle for the Achilles at all. It quotes Simplicius instead.

The Arrow is a different animal, and this is where popular accounts most often go wrong. At any single instant an arrow in flight occupies a definite position and is therefore at rest at that instant. Time is composed of instants. So the arrow is never in motion. Notice that nothing here is a sum. There is no series to converge, so the answer that handles the Dichotomy and the Achilles has nothing to grip. What the Arrow attacks is the premise it states out loud, that time is composed of instants, and with it the question of what motion at an instant could even mean. Whether time really is built that way is Is Time Real?'s question, argued there at length, and this article leaves it there.

The fourth argument, the Stadium or the Moving Rows, is where honesty costs something. Our file gives it in a single clause: it involves the relative motion of rows passing each other, and it challenges the concept of indivisible units of space and time. The reference entry prints a fuller version, in which equal bodies move alongside equal bodies in a stadium from opposite directions at equal speeds, and Zeno takes it to follow that half the time is equal to its double. That conclusion is the part worth carrying, because it is what makes the argument recognisable as an attack on indivisible units rather than on infinite divisibility. It is also as far as this page goes. The ancient text is terse, the entry's own printing of it ends in an ellipsis, and reconstructing a worked example of three rows of bodies from memory is precisely how garbled versions get made. The Stadium's row in the table above records that rather than grading it, and that is deliberate.

Our file already contains the sentence this article needs, and it is split in two exactly as it should be. The sum of an infinite geometric series can converge to a finite value: 1/2 plus 1/4 plus 1/8 and onward without end adds to 1. That, the file says, resolves the mathematical puzzle. And then, in the same breath, it says that philosophers debate whether it addresses Zeno's deeper metaphysical point about the nature of infinity and continuous motion. Both halves or neither. Drop the first and the article claims nobody has an answer, which is false. Drop the second and it claims the matter is closed, which is also false.

Two independent sources agree, and they agree on both halves, which is why this is the clearest verdict on the page. The reference entry on Zeno describes the modern mathematical treatment as the received view and says it shows how modern mathematics could solve all of Zeno's paradoxes. It then states the standing condition on that answer in its own voice: it would not answer Zeno's paradoxes if the mathematical framework invoked were not a good description of actual space, time and motion. So the honest verdict has two parts. As mathematics, the Dichotomy and the Achilles are not open problems and have not been for a very long time; a reader who has met a convergent series has met the answer. What stays open is whether that mathematics describes real motion, and whether completing infinitely many tasks is a coherent notion at all.

That second question has a literature and it has a name, supertasks. Two papers mark it. J. F. Thomson published Tasks and Super-Tasks in Analysis in 1954, and Benacerraf published Tasks, Super-Tasks, and the Modern Eleatics in the Journal of Philosophy in 1962, replying to Thomson eight years later in a leading journal and naming the modern Eleatics in his title. This article read the bibliographic records for both and not the papers themselves, so it says what their existence supports and no more: the adequacy of the convergent-series answer remained a live question in analytic philosophy long after the mathematics itself was uncontested. What either paper concluded is not stated here, because it was not read here.

One thing Zeno did not do, and our file marks it incorrect in its own Tier 4 section. He did not prove that motion is impossible. The paradoxes were built to support Parmenides's metaphysics, motion obviously occurs, and what the four arguments challenge is our understanding of motion, infinity and continuity rather than the fact of motion itself. That distinction generalises to everything else on this page, and it is the definition this article opened with doing its work: an argument that ends somewhere absurd shows that something in the argument is wrong. It does not show that the absurd thing is true. Note which way that refusal runs, though. Our file refuses the overclaim on Zeno's side only, which is why the verdict below refuses it on both sides.

This sentence is false. If it is true then, by what it says, it is false; if it is false then, since that is exactly what it says, it is true. That is the liar in the form our file gives it, and it is the simple version. The formulation is the fragile part of this one. The canonical liar has to be self-referential and self-contained, and the folk versions are neither. Everything I say is a lie is not the liar: it has an easy consistent reading, namely that the speaker sometimes tells the truth. This sentence is not true is a genuinely different and stronger variant, used precisely because it survives responses that the false version does not, and it comes back a few paragraphs below.

On where it comes from, our file is careful, and the care is worth copying. It calls the liar one of the oldest paradoxes, attributes it to Eubulides of Miletus in the fourth century BCE, and says it is related to Epimenides the Cretan. Related to is not stated by, and the difference matters. The Epimenides story, in which a Cretan says that all Cretans are liars, is the ancestor of the liar and is not logically the same sentence. If any other Cretan ever told the truth, then all Cretans are liars is simply false in a Cretan's mouth, and a false statement is not a paradox. Neither Eubulides nor Epimenides has a bibliography row in our file, and this article did not independently check the attribution, so it is carried here as the standard attribution rather than as a checked fact.

A sentence on a blackboard became a problem for logic because of what it does to the most obvious principle anyone has about truth. Our file names that principle and attributes it to Tarski: the schema on which snow is white is true if and only if snow is white. Apply the schema to the liar sentence and the contradiction falls straight out. Skip this step and the liar is a party trick. Take it, and the liar is an argument that our plainest principle about truth cannot be held in full generality alongside a language able to talk about its own sentences. That is a real cost, and it is why the twentieth century spent so much on it.

Now the variant, because it shows in miniature why this one stays open. The reference entry on the liar does not work with the false version at all. It uses what it calls the simple-untruth liar, a sentence which says of itself that it is not true, and the contradiction runs the same way: if it is true then it is not true, and if it is not true then it is true. The difference from the false version is not cosmetic. This sentence is false can be answered by saying the sentence is neither true nor false, that is, by allowing a gap between the truth values. That answer does nothing at all to the other form, because a sentence with no truth value is not true, which is exactly what this one says of itself, so it comes out true after all. The repair patches one sentence and the same problem walks back in wearing the next one.

Our file names four responses to the liar and crowns none of them, which is correct. Tarski's hierarchy of languages: no language can contain its own truth predicate. Kripke's fixed-point theory of truth, from 1975, which the file says allows truth-value gaps. Dialetheism, attributed to Graham Priest: some contradictions are true. And revision theory, attributed to Gupta and Belnap. These are not four versions of one idea. They are four different things to give up. Tarski gives up a single universal language. Kripke gives up bivalence. Priest gives up non-contradiction. Gupta and Belnap give up truth as a fixed property in favour of a process that revises it. Saying that plainly is the most useful thing this section can do, because it shows what kind of problem the liar is: not a puzzle with a missing piece, but a bill that has to be paid out of one account or another.

The reference entry on the liar goes further and says why the question stays open. It states that the past few thousand years have yielded a great number of proposals and that it will not be able to examine all of them, and it lays out at least seven distinct families: paracomplete approaches, paraconsistent approaches and dialetheism, substructural logics, classical approaches including the Tarskian hierarchy and the Kripke fixed-point construction and compositional theories, contextualist approaches, revision theory, and inconsistency views. Then it names the thing that keeps happening, and this is the strongest single fact this article can carry about the liar. It calls it revenge: a proposed solution is rejected on the basis of what might be taken to be a slightly modified form of the paradox. The move above, where the truth-value gap defeats one sentence and fails against the next, is that phenomenon in miniature. The liar is open, and revenge is why it stays that way.

One of the seven is not this article's to tell. Our file reports the Ship of Theseus from Plutarch's Life of Theseus: if the ship is maintained by replacing old planks with new ones, is it the same ship once every plank has been replaced. It adds Hobbes's extension, which this page names and leaves to the article that owns it. That is the whole of it here. Personal Identity, in this same wing, owns the ship outright: it is that article's title concept and its opening move, it carries a Plutarch passage fetched from a published translation, and it runs the rival criteria of identity properly. Go there for it.

There is one thing worth adding that the sibling article does not need to say, and it is a characterization rather than a finding, which is why it is marked Tier 2. Apply the definition this page opened with. A paradox derives an apparently unacceptable conclusion from apparently acceptable premises. The Ship of Theseus derives no conclusion at all. It poses a question, is this the same ship, and rival criteria of identity answer it differently; our own file presents it in exactly that shape, as competing accounts rather than as a derived absurdity. So it is the odd one out in this cabinet, and solved against open is the wrong axis to hang it on. That is why the table above grades it as neither.

It is not a puzzle about sand. Our file states what the sorites actually exposes, which is vagueness, and it adds that many, perhaps most, natural language predicates lack sharp boundaries. Keep the perhaps; it is doing real work. The file's own examples are tall, bald, red and rich. If the argument runs, it runs for every vague predicate in the language, and that is why this one belongs in a band about the limits of knowing. It is not a fact about heaps. It is a fact about how our words meet the world.

Our file names four responses. Epistemicism, attributed to Williamson in 1994: there is a sharp boundary, and we cannot know where it is. Supervaluationism, attributed to Fine in 1975: a sentence involving a vague predicate can be true on every admissible way of setting the boundary even when no particular setting is the correct one. Degree theory: truth comes in degrees. And contextualism: the extension of a vague predicate shifts with conversational context. The first two carry names and dates in our file and the last two do not, and this article does not invent attributions for them. Epistemicism is the startling one and it is worth saying plainly, because readers do not expect it. On that view there is an exact grain at which a non-heap becomes a heap, and our failure to find it is ignorance rather than indeterminacy.

The reference entry on the sorites says something precise that is easy to misread. Most theorists of vagueness suppose the paradox is solvable, that is, that the paradoxical argument is defective and that we can discover the defect. Solvable is not solved, and running the two together would be a real error rather than a loose one. The same entry then lists the rival families with their named defenders: ideal-language approaches, the epistemic theory, supervaluationism, three-valued logic, contextualism, multiple range theory, and embracing the paradox, which is to say denying that it has a solution at all. Seven families and no consensus. And it is one of the questions the discipline was formally polled on, which is not what happens to a settled matter.

Consider the set of all sets that are not members of themselves. Does it contain itself? If it does, then by its own definition it should not; if it does not, then by its own definition it should. That is Russell's paradox as our file states it, dated 1901, and the file records what it did: it led to the reconstruction of set theory, naming the Zermelo and Fraenkel axiomatisation and Russell's own theory of types. One tier note, carried exactly as the file assigns it. Our file files this under Tier 2 rather than Tier 1, even though it is arguably the most securely established item in the whole document, and this page does not silently promote it. Where the article says below that the mathematics is settled, that verdict rests on the external sources named there and not on this mark.

The reference entry states the same argument in symbols, and it corrects one date that our file gives bare. It places Russell's discovery in the late spring of 1901 and notes explicitly that the reports conflict, listing June 1901, the spring of 1901, and May of that year. So the year is safe and a confident month is not. The dated, documented event is the letter. Russell wrote to Frege on 16 June 1902, while the second volume of Frege's work was at the printer, which is also the better scene of the two.

Here is the part the popular framing gets wrong, and it gets it wrong in both directions at once. Nobody is waiting for a solution to Russell's paradox. Naive unrestricted comprehension was abandoned, and the standard replacements do not permit the argument to be run. On Zermelo's approach, as the entry puts it, all the contradiction shows is that V is not a set, that V is an empty name. On von Neumann's method the Russell class cannot be a member of any class and is therefore a proper class rather than a set. Type theory blocks it a third way, by making the sentence x is a member of x ill-formed rather than false. Two of those three were published in the same year by the men who made them, Russell's type theory and Zermelo's axiomatisation, and both are cited below. Von Neumann's method is named here from the entry and is not separately cited. What is not settled is which repair is philosophically right. The entry presents them as various ingenious and mathematically and philosophically significant ways of dealing with the paradox, and this article read no sentence in it declaring the matter finally resolved. Mathematics moved on. The philosophy of why it moved that way is still discussed.

A teacher announces that there will be a surprise test next week. The students reason backwards from the last day. If the test has not happened by Thursday evening it must fall on Friday, and then it would not be a surprise, so it cannot be Friday. With Friday removed the same reasoning removes Thursday, and then Wednesday, and so on back to the start of the week. They conclude that a surprise test is impossible. Then the test happens on Wednesday and they are surprised. Our file's ending is the point and it must not be cut, because it is a different shape of failure from everything else in this cabinet: the argument looks airtight and the world simply ignores it. The direction matters too. The elimination has to start at the last possible day; a retelling that starts on Monday is a different and much weaker thing.

One reply announces itself in its own title. In 1953 Quine published a short deflation of it in Mind under the heading On a So-Called Paradox. This article read the bibliographic record and not the paper, so the only thing it attributes to Quine is what the title announces: that on his view this is a so-called paradox rather than a paradox.

Whether Quine settled it is a different question, and the honest answer is that this article does not know and says so. What it can show is that the argument did not stop there. A search of the bibliographic records returns replies to Quine's deflation published in a philosophy journal twelve years later, and again fifty-five years later under a title announcing a rejection of it. Neither was read beyond its record and neither is cited here, so what those replies concluded is not stated. What their existence supports is narrow and real: this is not a closed question. That is why the table above grades it as no agreed analysis rather than as open or settled, and why this row sits at Tier 2 rather than Tier 1.

A predictor has already placed money in two boxes. Box A contains 1,000 dollars, always. Box B contains 1,000,000 dollars if the predictor predicted that you would take only Box B, and nothing if the predictor predicted that you would take both. The predictor has been highly accurate. Do you take both boxes, or only Box B? Three details hold the problem up and a careless retelling drops them. The prediction is already made and the boxes are already filled before you choose. The contents of Box A are not in doubt. And the predictor's accuracy is stipulated as high, not as perfect.

Two decision theories give opposite answers and both have a case. Evidential decision theory says take only Box B: taking one box is evidence that you were predicted to take one box, and the expected value of doing so is 1,000,000 dollars. Causal decision theory says take both: the prediction has already been made and nothing you do now can change it, so taking both dominates, because whatever is in Box B you get 1,000 dollars more by also taking Box A. Our file's closing line for that section, in its own voice, is this: no consensus resolution. That sentence matters beyond Newcomb's problem, because it shows that our own file does distinguish the open cases from the closed ones even though it never says so systematically.

The field was asked. David Chalmers's own discussion document for the 2020 PhilPapers Survey carries the question in his wording: "Newcomb's problem: one box or two boxes?" A discipline does not put a settled question to its own members as a two-option choice. Two more of this article's paradoxes are on the same list. "Vagueness: epistemic, metaphysical, or semantic?" is the sorites. "Liar sentence: neither true nor false, but true and false, not truth-apt ...?" is the liar, and the trailing dots are printed in the source, where the option list runs on past what was read here. One hard limit belongs on the page rather than in a footnote. This article did not retrieve the survey's results and prints no percentage from it. The published paper is cited below. The numbers are not here, because they were not read here.

There is one more position, and our file files it at Tier 3, which is where it stays here. Some philosophers, and the file names Priest's 2006 book, argue that certain paradoxes reveal genuine limits of classical logic, and that reality itself may be inconsistent in ways that require a paraconsistent or dialethic logic. If that is right then the liar is not a sentence to be repaired but a true contradiction to be accepted, and the revenge phenomenon is what you would expect from trying to repair it. Our file adds, in its own voice, that this remains highly controversial, and that hedge belongs inside the sentence rather than in a note at the bottom. It is a tempting closing line, and it is exactly the wrong one, because a Tier 3 claim used as a peroration reads as a conclusion.

Aristotle reaches this page at second hand and the page says so rather than implying otherwise. Both primary-text routes tried for this article refused the connection, so the four arguments arrive through the Stanford Encyclopedia's own block quotations of him. The one sentence quoted above was fetched twice and came back character for character identical both times, which is why this page carries it as a quotation at all rather than as a summary. The passage on the Arrow was fetched twice and came back at two different lengths, one of them silently abridged, so its content is stated above and none of it is quoted. The passage on the Stadium contains dashes this house does not print even inside quotations, so it is paraphrased and never quoted. No translator is named anywhere on this page, because the page these were read from names none. And one image was audited the same way. The liar-paradox diagram was uploaded in 2025, and a recent diagram is the hardest class in which to spot a generated picture, so it was checked twice before it was kept: the raster was inventoried element by element, and the vector source was fetched and read. It is hand-drawn in Inkscape. Two of its three paths carry that editor's own node-editing marks, the third path is an arrowhead marker inside the definitions block, the panel behind the sentence is a rectangle rather than a path, and its three pieces of text are still live text rather than converted to outlines. That is the signature of a person moving points, not of a model producing a picture.

Two identifiers in our own file's bibliography point at the wrong things, and neither is reproduced here. The row for Sainsbury's Paradoxes carries an ISBN that resolves at Open Library to Time Paradox by Eoin Colfer, a children's novel published in the same year. Its check digit computes correctly, its year matches, and the word paradox appears in both titles, which is almost certainly how it was matched, so no structural test catches it and only resolving the number and reading what comes back does. The Cambridge monograph is cited below in its place. The row for Priest's In Contradiction carries a DOI that resolves to a four-page chapter of that book rather than to the book itself, and no book-level identifier exists at all, so the book is cited below by name without one and Priest's 1979 paper is cited with one. A third row, Salmon's collection on Zeno, has no record of its own, and the record that comes back under its title is a 1972 review of it; that review is not attached to the book here. The contrast worth naming is inside the same bibliography: one of its two ISBNs was checked and repaired at some earlier point, and that repair is confirmed correct by two independent routes. The other was never checked.