Paradoxes - types and examples

Problem Solving – Paradoxes

 1.     Introduction. Paradoxes have long been a true source of problems for professionals and the laity alike. They are often set as puzzlers. They demonstrate traps of logic, science, and life itself. Some are even linguistic, having to do with meaning. They can occur almost anywhere aside from books on them. They can be embedded within wicked problems, a home for vague language and conflicting values, uncertainty, and multiple solutions. Even logical fallacies can counted as paradoxes until they are identified. Oftentimes, paradoxes signal a major revision to the meanings of terms and even to the logical foundations of how we think.  Let’s begin with a definition.

Definition: A paradox is a seemingly absurd or self-contradictory statement or proposition that when investigated or explained may prove to be well founded or true. In many cases, it helps refine logic and understanding. They push the boundaries of logic, provoke thought, and sometimes lead to breakthroughs by exposing hidden assumptions.

2.     Characteristics. The key characteristics are simple, but deciding which a given statement may be can take some experience.  

• Veridical: Seem contradictory but are true (e.g., Birthday Paradox[1]).
• Falsidical: Appear valid but contain a flaw (e.g., flawed proofs like 1 = 2).
• Antinomic: True contradictions without resolution (e.g., Liar Paradox).

Consider a few examples.

2.1. Logical Paradoxes

These arise from inconsistencies within formal systems of logic or reasoning. They often involve self-reference or circular definitions.

• The Liar Paradox: A classic example is the statement "This sentence is false." If it’s true, then it must be false as it claims, but if it’s false, then it must be true because it accurately states it’s false. This creates a loop with no consistent resolution.
• Russell’s Paradox: In set theory, consider the set of all sets that do not contain themselves. Does this set contain itself? If it does, it shouldn’t; if it doesn’t, it should. This paradox exposed issues in early set theory and led to refinements in formal mathematics.
• Barber Paradox: A barber shaves all and only those who do not shave themselves. Does the barber shave himself? Either answer leads to a contradiction.

2.2. Semantic Paradoxes

These stem from the meanings of words or language and how they interact with truth or reference.

• Grelling-Nelson Paradox: Define "autological" as a word that describes itself (e.g., "short" is short) and "heterological" as a word that doesn’t (e.g., "long" isn’t long). Is "heterological" heterological? If it is, it isn’t; if it isn’t, it is.
• Berry Paradox: Consider "the smallest positive integer not definable in under twelve words." This phrase defines it in eleven words, contradicting the assumption that it can’t be defined so briefly.

2.3. Mathematical Paradoxes

These occur in mathematics, often revealing unexpected properties or limitations of systems.

• Zeno’s Paradoxes: For example, in the "Dichotomy Paradox," to travel a distance, one must first travel half that distance, then half of the remaining distance, and so on infinitely. Motion seems impossible, yet it happens. (Resolved mathematically via convergent series. It can also be resolved simply using set theory.)
• Banach-Tarski Paradox: In abstract geometry, a sphere can be divided into a finite number of pieces and reassembled into two spheres identical to the original. This defies physical intuition but is consistent with certain mathematical axioms (e.g., the Axiom of Choice). This means it is not a paradox.

2.4. Physical and Scientific Paradoxes

These arise in science, often highlighting gaps in understanding or counterintuitive truths.

• Grandfather Paradox: In time travel, if you go back and kill your grandfather before your parent is conceived, how could you exist to travel back? This challenges notions of causality. There are many explanations[2] for it, from multi-world ideas to locking up causality.
• Twin Paradox: In special relativity, one twin travels near light speed and returns younger than the stay-at-home twin. It seems paradoxical but is explained by the asymmetry of acceleration.
• Schrödinger’s Cat: In quantum mechanics, a cat in a box is both alive and dead until observed. This illustrates the oddity of superposition.

2.5. Philosophical Paradoxes

These explore deep questions about existence, morality, or knowledge, often without clear resolution.

• Ship of Theseus: If a ship has all its parts replaced over time, is it still the same ship? This probes identity and continuity.
• Sorites Paradox (Paradox of the Heap): If you remove one grain from a heap, it’s still a heap. Keep going—when does it stop being a heap? This questions vague boundaries. It also exhibits a scale issue where the clear meaning of heap is never given. That is there are conflicting or competing values.
• Newcomb’s Paradox: A predictor offers two boxes: one transparent with $1,000, and one opaque with either $1,000,000 or nothing, based on their prediction of your choice. Do you take both or just the opaque one? It pits free will against determinism.

2.6. Practical or Apparent Paradoxes

These seem contradictory at first but often have explanations or are illusions of reasoning.

• Birthday Paradox: In a group of just 23 people, there’s a 50% chance two share a birthday. This feels counterintuitive due to underestimating probability overlaps.
• Monty Hall Paradox: The problem is based on a game show where you choose a door between three options, with one revealing a car and the other a goat. Then the host opens one of the other doors to reveal a goat. You can then choose to switch to the remaining unopened door. What to do? The answer is to switch. Switching doors after a reveal increases your odds from 1/3 to 2/3. It defies initial intuition but is statistically sound.

3.     Stable and Unstable Paradoxes

In the context of paradoxes, "stable" and "unstable" aren’t standard formal categories but can be interpreted based on how they behave or resolve. We define

• Stable paradoxes as those that persist as thought-provoking or unresolved without breaking systems—often antinomies or philosophical conundrums that remain consistent in their contradiction.
• Unstable paradoxes as those that lead to inconsistencies, force a resolution, or destabilize a system until addressed (e.g., paradoxes that prompted changes in mathematics or logic).

In this context, Stable Paradoxes endure as consistent contradictions or puzzles that don’t necessarily demand resolution and can coexist with our understanding. Examples include

1. Liar Paradox. “This sentence is false." It is unstable because it oscillates endlessly between true and false without a definitive answer. It’s a self-contained loop that doesn’t "crash" logic but reveals limits in classical truth assignment. Philosophers and logicians (e.g., via paraconsistent logic[3]) can work with it without needing to dissolve it entirely.
2. Ship of Theseus. “A ship has all its planks replaced over time—is it still the same ship?” It is stable because this paradox about identity persists as a philosophical question with no single "correct" answer. It’s stable because it invites ongoing debate (e.g., identity as continuity vs. material composition) without breaking any system.
3. Schrödinger’s Cat[4]. “A cat in a box is both alive and dead until observed.” It is stable because in quantum mechanics, this paradox reflects superposition—a real, testable phenomenon. It’s counterintuitive but stable within the theory, resolved by observation collapsing the state, yet it remains a thought-provoking illustration.

Unstable Paradoxes create tension or inconsistency that often demands resolution, destabilizes a system, or leads to rethinking foundational assumptions. Examples include

1. Russell’s Paradox Example: Consider the set of all sets that do not contain themselves—does it contain itself? It is unstable because it destabilized naive set theory in the late 19th/early 20th century. If the set contains itself, it shouldn’t; if it doesn’t, it should. The contradiction forced mathematicians to refine set theory (e.g., with Zermelo-Fraenkel axioms), resolving the instability by restricting self-reference.
2. Zeno’s Paradox (Dichotomy). “To move a distance, you must first cross half, then half of the remaining, ad infinitum—motion seems impossible.” It was unstable because it challenged early notions of motion and infinity. It contradicted observable reality (things do move!), pushing mathematicians to develop calculus and the concept of convergent series (e.g., 1/2 + 1/4 + 1/8… = 1) to stabilize and resolve it. This took two millennia to resolve, finally, at the time of Sir Issac Newton.
3. Barber Paradox. A simple but almost identical example of Russell’s paradox, it asks, “On an isolated island, a barber shaves all and only those who do not shave themselves—does he shave himself? It is unstable because it is an unresolvable contradiction in naive logic. It’s unstable because it can’t hold in a consistent system without redefining terms or rejecting the premise (e.g., concluding no such barber exists).

Summary. Stable paradoxes are "livable". They persist as intellectual curiosities or features of a system without requiring immediate resolution. They’re often tools for reflection or inherent to certain frameworks (e.g., quantum mechanics). Unstable Paradoxes expose flaws or inconsistencies that demand action—whether a reformulation of rules (like in math) or a rejection of the setup. They’re disruptive until addressed.

 4. Computers and Paradoxes.

Can computers save us by resolving paradoxes we cannot? Computers can resolve some paradoxes but not all—it depends on the type of paradox and whether it’s rooted in formal systems, empirical reality, or human interpretation. It also depends on how the computer has been trained to analyze complex problems. Keep in mind that for some now-resolved paradoxes, millennia of thought were required to resolve them. In more mathematical logical areas, new mathematics was needed. In semantical areas, it required the recognition of the notion of what self-referential means. And then computers needed to be trained on these. To date, computers have not innovated new methods to resolve paradoxes, only to not similarities. They have, however, completed calculations too complex for humans to demonstrate the solution of problems previously unknown. Thus, these major achievements were not the resolution of paradoxes as such, but the solution of open (math) problems. And they required extensive human interaction.  

5. Can Computers Resolve Paradoxes?

Computers excel at processing formal systems like logic, mathematics, or simulations, so they can often address paradoxes that have clear rules or reducible components. By this time, if you ask an AI to resolve a paradox, it has probably been trained to solve it. However, behind the training is intense human interaction to help. Included in this lot are the Zeno paradoxes and the proof that 1=2, which involves somewhere a illegal operation, division by zero. On the other hand, its resolution of the Monte Hall paradox can be resolved by computer simulation, but AI would not write the code to do this unless this is programmed into the system. However, the Monte Hall paradox can be derived mathematically using conditional probabilities.

Some paradoxes resist resolution because they involve self-reference, vagueness, or philosophical concepts beyond computational scope. For example, consider the Liars paradox, “This sentence is false.” The reason is that the computer winds up with an infinite loop of “true”, the “false.” Computers rely on decidable algorithms, but this is inherently undecidable in classical systems. In truth, this paradox is self-referential, and these are the bane of all logical-type paradoxes, and they required a major adjustment in mathematical logic. A similar example is Russell’s paradox. Another is the “Ship of Theseus” paradox. Here the problem is with the definition of identity. Computers process data, not meaning. This paradox is similar to the quote from Herodotus, who said you can never step into the same river twice. Again, the problem is meaning; the river flows and is therefore always changing. Hence, we have the logical problem of what a “river” means.

Some paradoxes resist resolution because they involve self-reference, vagueness, or philosophical concepts beyond computational scope. What computers can do instead is to model the situation by simulating it, detect it by identifying contradictions in the code, and assist humans by providing data, test hypotheses, and simply massive computations. Finally, we mention the the Halting Problem Connection. The limits of computation itself tie into paradoxes. Alan Turing’s Halting Problem shows that no computer can universally determine if any program will halt or loop forever. This undecidability mirrors paradoxes like the Liar Paradox—some questions are inherently unresolvable within a system, a result computers can’t escape either.

Computers can resolve paradoxes that boil down to misapplied logic, probability, or calculable systems (stable or falsidical ones). But they falter with self-referential, philosophical, or truly antinomic paradoxes, where resolution isn’t about computation but about redefining the problem or accepting ambiguity. Humans and computers together are stronger than either separately. Indeed, many unsolvable problems of only a few decades back are tractable today because of computers.

 

 

 

©2025

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[1] The birthday paradox is a probability theory that states that in a group of 23 people, there is more than a 50% chance that two people will share a birthday. Very counter intuitive but correct, this is a classic problem/paradox taught in every probability class.

[2] Here is a cute explanation. Imagine you are living on the river of time and you go upstream (back in time) and block the river so it changes course. The river you were on keeps flowing (with you not in it), but there is now a new river of time with you not in it beyond your life there.

[3] Paraconsistent logic is a type of logic that allows contradictory statements to coexist without leading to absurdity.

[4] Imagine a cat in a box with a device that could kill it. The subatomic device has a 50% chance of killing the cat in an hour. Until you look in the box, the cat is both alive and dead. This is because the cat's fate is linked to a random subatomic event. Welcome to quantum theory!