The Unicorn, the Wall, and the Probability of Nonsense (The Unicorn Standard)
Why “possible” is the cheapest word in argument, probability is not likelihood, and feasibility is where attractive nonsense is finally asked to produce identification.
There is a wonderfully inexpensive way to make an argument sound profound: say that something is possible. The word is nearly impossible to defeat because it asks so little of the speaker. Unicorns may exist. Elves may inhabit some valley nobody has searched properly. Fairies may have perfected camouflage. A person may, in some quantum-mechanical description, pass through a wall. An unknown civilisation may be arranging the stars for aesthetic reasons. None of these sentences has yet told us whether the proposition is probable, well supported, statistically likely in the technical sense, practically feasible, or worth spending a single pound to investigate. Yet public argument routinely promotes “possible” through all of those ranks without so much as an interview. A proposition begins life as something not obviously contradictory and, after several repetitions by people wearing conference badges, retires as a scenario demanding a budget.
The difficulty is not that ordinary language is imprecise; ordinary language is allowed to enjoy itself. The difficulty begins when the looseness of ordinary language is imported into philosophy, science, risk analysis, forecasting, statistics, law, engineering, and policy, where the distinctions are the entire point of the exercise. Modal logic asks what could or must be true under specified conditions. Probability theory assigns numerical weights within a formal structure. Statistical likelihood asks how well observed data accord with one model or parameter relative to another. Epistemology asks what degree of belief the evidence warrants. Decision theory asks what action is rational under uncertainty. Feasibility asks whether an action can actually be carried out given physics, technology, money, time, coordination, and competence. These questions converse with one another, certainly, but they are not aliases. Calling them all “possibility” is rather like calling every liquid champagne because the glasses happen to be similar.
The useful discipline is therefore to separate five questions before allowing anyone to proceed: Is the proposition possible in the relevant sense? Is it plausible given what is already known? What probability, if any, can defensibly be assigned? How does the observed evidence bear on competing hypotheses or models—that is, what is the likelihood structure? And is the proposed event or action feasible under real constraints? This order matters because each question adds information that the previous question did not contain. A logically coherent unicorn is not thereby an evidentially credible unicorn; an unlikely event may be perfectly feasible to attempt, as every lottery ticket demonstrates; a physically allowed microscopic process may become functionally unavailable when scaled to a human body; and a dataset may be highly likely under a hypothesis without making that hypothesis itself highly probable. The English language may forgive these substitutions. Mathematics, philosophy, and engineering are rather less sentimental.
1. “Possible” is not one thing
The first insult to simplicity is that philosophers do not recognise one single undifferentiated species of possibility. The Stanford Encyclopedia of Philosophy’s survey of modality distinguishes several families of modal notions, and the familiar possible-worlds tradition treats a proposition as possible when it is true in at least one relevant possible world and necessary when it is true in all relevant possible worlds. But the word “relevant” is doing serious work. Logical possibility concerns consistency with the rules of logic. Metaphysical possibility asks, more controversially, whether reality genuinely could have been that way. Nomological or physical possibility asks whether a state of affairs is compatible with the laws of nature. Epistemic possibility concerns what is not excluded by what an agent knows. Practical possibility concerns what an agent can actually do. Deontic possibility concerns what is permitted. A thing may be possible in one of these senses and impossible in another without contradiction; indeed, much of adult reasoning consists of discovering which modality a speaker quietly changed halfway through the sentence.
Consider a winning lottery ticket. It is logically possible that the ticket wins because no contradiction follows from its winning. It is physically possible because nothing in the laws of nature forbids the numbered balls from falling in that combination. It is epistemically possible before the draw because you do not know the result. It may be practically possible to buy the ticket because the shop is open and you possess the required money. But none of those statements says that winning is probable. The winning event may remain vanishingly rare relative to the lottery’s outcome space. This is the simplest way to see the distinction: possibility is ordinarily a threshold concept—admitted or excluded under a particular set of constraints—whereas probability is graded. A door being unlocked does not tell you how many people will walk through it, and the metaphysical fact that a route exists does not provide traffic statistics.
The reverse confusion is just as common. Something can be impossible for you while perfectly possible in every broader physical sense. It is physically possible for someone to play Rachmaninoff competently on a concert grand; it may not be practically possible for you to do so this evening if your preparation has consisted chiefly of owning Spotify. It is physically possible to build a bridge across a particular river while economically infeasible under the current budget. It may be legally permissible to launch a project yet organisationally impossible because the staff, permissions, or supply chain cannot be assembled in time. Once the different modalities are named, a surprising amount of business language turns out to be the old trick of replacing “we cannot presently do this” with “theoretically it can be done,” as though theory were obliged to arrive with contractors.
2. Logical possibility is a very low bar, which is why unicorns clear it so comfortably
A unicorn is useful because it exposes the poverty of possibility as evidence. Define the creature modestly: a horse-like animal with a single horn projecting from its forehead. Nothing in that description is logically contradictory. Nature has already produced rhinoceroses, narwhals, deer with elaborate antlers, and countless structures stranger than the one-horned horse of heraldry. The minimal unicorn is therefore not a round square or a married bachelor; it is a coherent biological description. From that fact one may reasonably say that such a creature is logically conceivable and perhaps physically possible in a broad evolutionary sense. One may not reasonably proceed to the conclusion that unicorns actually inhabit Scotland, Patagonia, or the less supervised regions of LinkedIn. Logical coherence is an admission ticket to inquiry, not a certificate of residence.
Elves and fairies improve the example because their definitions reveal how much work a noun can hide. If an elf means a long-lived humanoid with pointed ears, the description may be biologically exotic but not transparently contradictory. If an elf is immortal, weightless, invisible at will, capable of violating conservation laws, and able to move instantaneously between forests, the question changes as each property is added. Some features may conflict with established physical theory; others may simply lack a known mechanism. The same creature can therefore migrate from logical possibility to physical implausibility depending on the specification. “Fairies are possible” is not a proposition until we know what a fairy is and which kind of possibility is intended. Without that discipline, mythology becomes impossible to falsify because every inconvenient feature can be revised after the evidence arrives.
Here the philosophy of science offers a useful correction to the habit of treating survival as support. A hypothesis can remain consistent with observed data without being confirmed by those data. The literature on confirmation, including the Stanford Encyclopedia’s overview, exists precisely because empirical evidence usually underdetermines theory: many hypotheses can fit the same finite observations. If no fairies are seen in a garden, the fairy enthusiast can say that they were hiding. If photographs are blurred, the blur is attributed to magical interference. If the photographs are clear and show moths, the fairy is said to resemble a moth. A hypothesis that can absorb every result may be possible, but it has purchased its survival by surrendering evidential discrimination. The ability to avoid contradiction is not the same as the ability to explain the world better than alternatives.
3. Conceivable, possible, and actual are three different social classes
People often move from “I can imagine it” to “it is possible,” but conceivability is itself an unreliable guide. We can imagine descriptions whose hidden implications are inconsistent, just as earlier generations could imagine mathematical structures later shown impossible under their assumptions. Conversely, some truths are difficult or impossible for human beings to visualise while remaining mathematically coherent. The philosophical literature on modal epistemology therefore asks how we know modal facts at all: what licenses movement from conceivability to possibility, and when is imagination misleading? The Stanford Encyclopedia’s treatment of modal epistemology surveys precisely this problem. Human imaginative comfort is evidence about the human imagination; it is not a universal court of appeal for reality.
This matters because speculative claims often arrive dressed as thought experiments. Thought experiments are immensely valuable when they isolate assumptions and expose contradictions, but they are not laboratories merely because the prose is vivid. “Imagine a civilisation that can rearrange galaxies” may help analyse a concept of technological power, yet the ease with which a sentence is written does not settle the energy, causality, coordination, or timescale required. “Imagine a perfectly informed voter” may illuminate a model while describing nobody who has ever queued at a polling station. “Imagine a frictionless market” may clarify equilibrium while being a deliberate abstraction from the institutions that make markets possible. The intellectual error occurs only when the convenience of the thought experiment is mistaken for evidence that the corresponding state is available in reality.
Actuality, then, is not the inevitable destination of imagination. A proposition may be conceivable, logically possible, physically possible, epistemically open, and still false. This sounds elementary because it is elementary. It nevertheless defeats an extraordinary amount of speculative rhetoric because “could” is so often used to create emotional weight without evidential cost. The correct question after “could” is not “how exciting?” but “what follows?” Usually the answer is: very little. A coherent scenario becomes scientifically or practically interesting only when we add evidence, mechanisms, comparative predictions, probability, or actionable consequences. Until then, one has a story. Stories are among civilisation’s greatest achievements. They merely become insufferable when they demand to be called data.
4. Probability begins where possibility stops being enough
Probability introduces graded uncertainty. In the standard mathematical treatment associated with Kolmogorov, probabilities satisfy formal axioms over a set of events; in philosophy, however, the interpretation of those numbers remains contested. The Stanford Encyclopedia’s overview by Alan Hájek distinguishes several broad ideas: probability as physical chance, probability as evidential support, probability as long-run frequency, and probability as subjective or rational degree of belief. The mathematics can remain the same while the interpretation changes. A statement such as “there is a 30 percent chance of rain” may refer to calibrated forecast frequencies, a model-derived chance, or an epistemic assessment. The number is not merely a more sophisticated word for “possible”; it is a quantity whose meaning depends on how the model connects to the world.
This is why one cannot assign probabilities simply by listing possibilities. Suppose you invent four stories about tomorrow: it rains, it snows, unicorns appear, or the Moon becomes a tasteful shade of mauve. The existence of four verbally distinct possibilities does not make each event 25 percent probable. There may be infinitely many describable possibilities, they may overlap, they may differ in granularity, and there may be no natural symmetry between them. Probability requires a measure or a rational credence structure, not a headcount of sentences. The principle of indifference can be useful in carefully symmetrical problems, but its careless use is famously vulnerable to how the outcome space is parameterised. One does not obtain mathematics merely by numbering one’s imagination.
A sensible probability claim therefore demands context. What is the event? What is the reference class? What is the model? What data inform it? Are we speaking of objective physical chance, observed frequency, predictive probability, or personal credence? How sensitive is the result to assumptions? These questions are not bureaucratic ornaments; they determine whether the number has a meaning. “There is a 20 percent probability of elves” is not rendered scientific by the decimal. Without a specified inferential structure, it is simply a mood with arithmetic typography. Probability can quantify uncertainty magnificently, but it cannot manufacture evidence from the dignified placement of a percentage sign.
5. The unicorn can be possible while its probability remains extraordinarily low
Suppose we agree that a one-horned equine species is biologically coherent. We have therefore granted the unicorn a broad kind of physical possibility. We now ask a different question: given centuries of zoological collection, exploration, photography, satellite observation, genomic study, fossil evidence, ecological surveys, and the absence of an accepted specimen, how much credence should we place in the claim that a breeding population of large unicorns exists unnoticed today? No exact number drops from philosophy. Different priors and models can produce different credences. But the evidence is plainly not neutral. Large terrestrial mammals leave bodies, DNA, tracks, ecological effects, photographs, prey relationships, parasites, and local knowledge. The absence of reliable evidence does not make existence logically impossible, but it changes rational belief.
This is where the slogan “absence of evidence is not evidence of absence” is routinely abused. Sometimes absence of evidence is indeed weak or irrelevant—for example, when nobody has looked in a place where evidence would be expected. But when a hypothesis predicts detectable traces and serious observation repeatedly fails to find them, the absence becomes evidence against the hypothesis. Bayesian reasoning captures this neatly because what matters is how expected the observed evidence would be under competing hypotheses. If unicorns are abundant and conspicuous, decades of non-detection would be surprising; if they do not exist, non-detection is unsurprising. The evidential force depends on detection probability, search effort, and alternative explanations, not on a slogan beloved by people whose preferred creature has become inconveniently shy.
Notice how disciplined scepticism differs from dogmatic impossibility. One need not say that unicorns are metaphysically impossible or assign literal probability zero. A rational position can instead be: the concept is coherent, existence is not logically excluded, but current evidence warrants extremely low credence in a surviving hidden population. This position keeps possibility open without allowing possibility to impersonate probability. It is a more adult posture than either “unicorns are impossible because I have not seen one” or “you cannot prove they do not exist, therefore my belief is respectable.” The first confuses evidence with logic; the second confuses logic with evidence. Between them lies the unfashionable country called reasoning.
6. Probability zero is not always the same thing as impossibility
The confusion between possibility and probability becomes particularly obvious in continuous probability models. Imagine choosing a real number uniformly from the interval between zero and one. In the mathematical model, any single exact value—say exactly one half—has probability zero because a single point has zero measure relative to the continuum. Yet the experiment must return some value, and whichever exact value appears was individually assigned probability zero. Probability zero therefore does not automatically mean logical or model-theoretic impossibility. It can mean that an event occupies no positive measure even though it belongs to the admissible outcome space. Measure theory has the unpleasant habit of ruining conversational shortcuts with impeccable manners.
Likewise, probability one need not amount to logical necessity. Under the same continuous model, the probability of selecting an irrational number is one because the rational numbers form a countable set of measure zero, yet rational outcomes are still members of the interval and are not logically prohibited by the setup. Events of probability one are often called “almost sure,” a phrase invented by mathematicians who understood that “certain” would encourage the wrong kind of confidence. This distinction is important because ordinary language likes to arrange impossible, improbable, probable, and certain on a single intuitive ruler. Continuous probability demonstrates that the ruler is not general enough: modal status and measure are distinct structures.
In practical finite problems the distinction may not matter. If a quality-control model assigns probability zero to a component exceeding an impossible physical dimension, “zero” and “impossible” may function together. But intellectual hygiene requires knowing when the equivalence is a property of the particular model rather than a theorem of language. A probability measure is not a metaphysics machine. It tells us how weight is distributed across events under a formal structure; it does not automatically pronounce which propositions are contradictory, physically forbidden, or metaphysically impossible. The fact that the same number—zero—can appear in these different roles is a notational convenience, not a philosophical merger.
7. “Likelihood” is where ordinary English and statistics stop speaking to one another politely
In ordinary conversation, likelihood often means chance: “the likelihood of rain,” “the likelihood of failure,” “the likelihood that he is late again.” Risk standards frequently use the term this way; the NIST glossary, for example, includes definitions of likelihood as the chance of something happening. In mathematical statistics, however, likelihood is a more specialised object. With observed data held fixed, a likelihood function compares how compatible different parameter values or models are with those data. NIST’s engineering statistics handbook puts the idea plainly: the likelihood of the observed sample is based on the probability of obtaining that data under the chosen probability model, and maximum-likelihood estimation selects the parameter values that make the observed sample most supported within that model.
The direction matters. Probability asks something like: if the model were fixed, how often or with what chance would various data occur? Likelihood turns the same expression around for inference: the data are now fixed because we already observed them, and we ask which parameter values or hypotheses make those data comparatively less or more expected. In plain text, one can write: probability examines P(data | model); likelihood treats that same quantity as a function of the model after the data are fixed. This does not make likelihood a probability distribution over models. It need not sum or integrate to one across parameter values, and in density problems numerical likelihood values can even exceed one. Anyone who says “the likelihood of the hypothesis is 80 percent” without specifying a Bayesian posterior or another probability framework may simply be giving a technical word the wrong job.
The distinction becomes vivid with a medical-test example. Suppose a test result is very common among people with a rare disease. That gives a high value to P(positive test | disease). It does not follow that P(disease | positive test) is equally high, because the disease may be extraordinarily rare and the test may produce false positives among the enormous healthy population. This is the familiar base-rate problem. The same error appears in courtrooms, fraud detection, scientific anomalies, intelligence analysis, and every television programme in which an unusual photograph is declared “unlikely unless the paranormal explanation is true.” Evidence being likely under a hypothesis does not tell you the probability of the hypothesis until alternatives and prior information enter the analysis. The vertical bar in conditional probability is a small piece of punctuation that has ruined many large claims.
8. Bayes’ theorem is less mystical than the people who invoke it
Bayesian reasoning is useful here because it makes explicit what ordinary argument tends to hide. Posterior belief depends on prior belief and on how strongly the evidence discriminates between hypotheses. In plain text, the familiar relationship is: P(H | E) = [P(E | H) × P(H)] / P(E). More usefully for comparing two hypotheses, one may write: posterior odds = prior odds × likelihood ratio. Nothing supernatural occurs. Evidence changes belief in proportion to how much more expected that evidence is under one hypothesis than another, while the starting plausibility of the hypotheses still matters. The Stanford Encyclopedia’s entry on Bayesian epistemology describes this framework in terms of rational credences and conditionalisation.
The fairy photograph again makes itself useful. Suppose an image contains a luminous blur. The blur may be unsurprising if fairies exist, but it may also be unsurprising if insects reflect flash photography, if the lens contains dust, if the shutter captures motion, or if the image has been edited. A likelihood comparison therefore requires alternative hypotheses. Even if the evidence is ten times more expected under “fairy” than under one selected alternative, the fairy hypothesis can remain improbable if its prior credibility is vastly lower than that of the mundane explanations. This is not prejudice against fairies; it is arithmetic refusing to let the hypothesis choose its own opponent.
The lesson generalises. People routinely present a single explanation for evidence and ask whether the evidence “fits.” Nearly anything can be made to fit after the fact if enough flexibility is allowed. Proper inference asks not only whether the evidence is compatible with the hypothesis but how the evidence compares across serious alternatives, including mundane ones. Likelihood without alternatives is flattery. Probability without priors is incomplete. Possibility without either is merely hospitable. There is nothing wrong with hospitality, provided one does not appoint every guest to the board.
9. Plausibility is the middle child nobody defines and everybody uses
Plausibility is less mathematically regimented than probability, which makes it both useful and dangerous. In ordinary scientific reasoning, a plausible hypothesis is one that fits relevant background knowledge, possesses a credible mechanism, coheres with established results, and is not merely invented to save itself from contrary evidence. Philosophers disagree about how plausibility relates to confirmation, explanation, simplicity, and prior probability, but the concept generally does more work than bare possibility. A hypothesis can be possible yet wildly implausible because accepting it would require abandoning a vast network of better-supported beliefs without compensating evidence. Conversely, a hypothesis can become plausible before anyone can attach a defensible numerical probability to it. This is why serious researchers often speak qualitatively before they speak quantitatively.
Plausibility also depends on the state of knowledge. Continental drift was once regarded as implausible because Wegener lacked a convincing mechanism; plate tectonics later supplied the missing structure and transformed the evidential landscape. Meteorites were once dismissed by some scholars despite reports because stones falling from the sky conflicted with prevailing assumptions; accumulating physical evidence changed that assessment. These examples are often abused as invitations to believe any current fringe proposition, but they teach the opposite lesson. Extraordinary claims become respectable when mechanisms, predictions, and evidence improve, not because somebody reminds us that experts have been wrong before. Past error establishes human fallibility; it does not provide a scholarship fund for present nonsense.
A useful definition of plausibility is therefore comparative rather than ceremonial. Ask: compared with serious alternatives, how well does this hypothesis fit what we already know, explain the evidence, avoid ad hoc rescue devices, and generate informative predictions? This does not eliminate judgement, but it makes judgement accountable. “It is plausible” should mean more than “I can tell a coherent story about it.” Human beings can tell coherent stories about almost anything; indeed, this ability explains both literature and conspiracy theory. The difference is not narrative quality. It is evidential discipline.
10. Feasibility is possibility after the bill arrives
Feasibility asks whether something can actually be done under relevant constraints. This is an action-centred concept, which is why it belongs as much to engineering, economics, operations research, and practical philosophy as to abstract logic. The philosophical literature on practical reason begins from the problem of deciding what one is to do, not merely what states of the world can be imagined. A proposal can be physically possible while technically infeasible, technically feasible while economically absurd, economically feasible while legally prohibited, legally permitted while operationally impossible, or operationally feasible while strategically foolish. Feasibility therefore comes in layers because real action is constrained by several systems at once.
Consider a city that proposes a tunnel. The tunnel may be physically possible because geology and materials science permit it. Engineering feasibility asks whether the design can be constructed with available methods and tolerances. Economic feasibility asks whether the expected benefits justify costs and financing. Environmental feasibility asks whether impacts can be mitigated within regulation. Political feasibility asks whether institutions can approve and sustain the project. Schedule feasibility asks whether the work can finish before the need disappears. None of these questions is answered by the sentence “humans have built tunnels before.” The proposition has crossed the modal border but not the project gate.
Feasibility is also agent-relative. Climbing Everest is feasible for some trained climbers under suitable conditions and infeasible for many other people. Landing a spacecraft on Mars is feasible for a well-resourced space agency and infeasible for a household, despite the laws of physics being indifferent to the distinction. Writing a novel this afternoon may be physically possible in the sense that typing is permitted by nature, yet infeasible if one insists on quality, research, editing, and lunch. Once feasibility enters the discussion, “possible” must acquire a subject, resources, constraints, and timeframe. The sentence “it can be done” is incomplete until we know by whom, with what, by when, and at what cost.
11. The quantum wall is the perfect insult to careless language
Quantum tunnelling supplies perhaps the most entertaining example of why physical possibility and practical feasibility should not be confused. In quantum mechanics, a particle can have a nonzero probability of appearing beyond a finite potential barrier even when classical mechanics says it lacks sufficient energy to cross. This is not speculative metaphysics; tunnelling is experimentally observed and technologically exploited in phenomena ranging from radioactive decay to scanning tunnelling microscopy. OpenStax’s university physics treatment explains that transmission probability depends strongly on barrier height, barrier width, particle energy, and mass, and that the suppression with barrier width is exponential. Microscopic tunnelling is therefore both real and useful.
The party trick begins when someone says, “Quantum mechanics means a person could walk through a wall.” There is a defensible kernel inside the sentence: quantum theory does not treat barrier penetration as categorically forbidden in the way classical mechanics does for a particle below a finite barrier. But a warm human body is not a single isolated electron confronting a neat one-dimensional potential barrier. It is a macroscopic, strongly interacting many-body system continually coupled to an environment. For an intact body to “tunnel through” an ordinary wall in any meaningful everyday sense, an astronomical collection of constituents and correlations would have to participate in an extraordinarily suppressed process while preserving the organisation we call the person. The elementary tunnelling result does not provide an engineering route; it provides the opposite lesson, because the same theory that permits tunnelling also tells us how violently the probability collapses with scale and barrier properties.
So is walking through a wall “possible under quantum mechanics”? As a piece of pedagogical shorthand, one can say that quantum theory does not impose the same classical zero-probability barrier for idealised finite-barrier systems, and macroscopic quantum tunnelling exists in carefully controlled collective systems. But the phrase becomes misleading the moment “possible” is heard as “practical option.” There is no door-replacement industry waiting for a sufficiently patient customer. The intellectually correct formulation is that nonzero amplitude or nonzero theoretical probability does not imply feasibility, and for a macroscopic object under ordinary conditions the relevant probability is so suppressed that the event is, for practical purposes, unavailable. Quantum mechanics has done nothing wrong. English has simply tried to expense a miracle.
12. “Nonzero” is not a synonym for “worth considering”
This distinction matters because many decisions concern events with tiny probabilities. A probability greater than zero can still be irrelevant for action if the expected consequences are negligible or if other risks dominate by many orders of magnitude. Conversely, a very small probability can matter enormously when the consequences are catastrophic and mitigation is cheap. Decision theory exists because probability alone does not tell us what to do. The Stanford Encyclopedia’s treatment of decision theory describes the standard structure: beliefs about uncertain outcomes are combined with preferences, utilities, or losses to evaluate actions. In ordinary risk language, one often uses some form of “probability times consequence,” though real decision models can be considerably richer.
The crucial point is that possibility has almost no decision value by itself. Every large organisation can imagine thousands of possible disasters: asteroid impact, database corruption, supplier insolvency, sabotage, plague, flood, alien invasion, executive incompetence, and the coffee machine becoming sentient. Resources are finite, so risks must be ranked. A risk register that lists everything conceivable without probability, severity, exposure, detectability, mitigation cost, and feasibility is not prudent; it is a catalogue. The fact that an outcome is possible can justify asking a second question. It does not, on its own, justify spending money on the answer.
This is why tail-risk arguments need precision. Saying that an event is “low probability but high impact” can be entirely rational when the probability is grounded and the consequence is severe. Saying “we cannot rule it out, therefore precaution requires action” is weaker because the space of unruled-out catastrophes is effectively unlimited. Precaution must itself be subject to proportionality, opportunity cost, and competing risks. Protecting against one imaginative possibility can increase exposure to another more probable danger. Rational caution is not the art of being frightened by every noun. It is the art of allocating scarce attention where uncertainty and consequence together warrant it.
13. Frequency, chance, and credence are not identical either
Even after we agree to speak numerically, the word probability remains philosophically plural. A frequentist may connect probability to long-run relative frequencies under repeated trials; a propensity theorist may treat probability as a physical tendency of a setup; an objective-chance view seeks mind-independent chances; a Bayesian may represent degrees of belief or rational credences. Hájek’s survey of probability interpretations shows how persistent these disagreements are. The point is not that probability is hopelessly confused; the formal theory is extraordinarily successful. The point is that the same number can enter discourse under different philosophical interpretations, and careless arguments often switch between them without permission.
A weather forecast illustrates the subtlety. “Thirty percent chance of rain” does not mean that exactly 30 percent of your garden will become wet, nor that rain will occur for 30 percent of the day. In a calibrated forecasting system it may mean, roughly, that among situations assigned similar forecast probabilities, rain occurs at the advertised long-run frequency, though operational definitions vary. A bettor may then convert that forecast into a personal credence after considering local information. A physical scientist may ask about atmospheric stochasticity and model uncertainty. Each perspective can be legitimate, but none is captured adequately by saying that rain is “possible.” Probability earns its usefulness by forcing us to say how uncertain.
Subjective probability deserves special care because “subjective” is often mistaken for “arbitrary.” Bayesian epistemology does not say that any number chosen before lunch becomes rational by being called a prior. Coherence constraints, evidence, calibration, exchangeability, model checking, and sensitivity analysis all discipline credences. Priors can be debated; indeed, they should be. But the existence of judgement does not abolish standards. A surgeon uses judgement, a judge uses judgement, and a wine critic uses judgement; only one of these should be permitted to determine anaesthetic dosage, and even there we insist on evidence. Subjectivity is not permission to become decorative.
14. Evidence can increase probability without making a proposition probable
One of the most useful distinctions in confirmation theory is between raising a probability and making that probability high. Evidence can strongly confirm a hypothesis relative to its prior state while leaving the hypothesis unlikely overall. Suppose a rare disease has prior probability one in one hundred thousand, and a test result is one hundred times more likely under the disease than without it. The evidence is strongly favourable in a likelihood-ratio sense, yet the posterior probability can remain modest because the starting odds were so small. This is a simple consequence of Bayesian updating, but it is rhetorically devastating because it separates “important evidence” from “probable conclusion.”
The same point applies to exotic hypotheses. Discovering an unusual track in a remote forest may raise the probability of an unknown animal. Finding hair with unfamiliar DNA would raise it much more. Obtaining a verified carcass would transform the matter. Evidence comes in strengths because different observations discriminate differently among alternatives. The correct response to new evidence is neither to dismiss it because the hypothesis began implausibly nor to treat any upward update as final vindication. Rational belief moves by degrees. A single anomaly can deserve investigation without deserving belief. Science would be much calmer if headlines understood this sentence.
This gradualism is one reason probability is superior to the crude binary vocabulary of “possible/impossible” for empirical reasoning. Most hypotheses are not deductively destroyed by one contrary observation, nor are they deductively established by one favourable observation. Models contain noise, measurement error, auxiliary assumptions, and scope conditions. Probability and likelihood let evidence accumulate without pretending that every result is a verdict. The courtroom metaphor is tempting but often misleading: science is less like a trial ending in guilty or not guilty and more like an account whose balance changes with each transaction. Possibility merely tells us the account exists.
15. Improbable does not mean surprising, because there are far too many ways to be improbable
Another recurring mistake is to observe that a specific event had tiny probability and infer that the event therefore requires explanation. Shuffle a deck of cards properly and the exact order produced has probability one divided by 52 factorial, an absurdly small number. Yet some order had to occur, and after the shuffle every realised order is equally improbable under the ideal model. The improbability of a fully specified outcome is therefore not itself evidence of manipulation. What matters is whether the observed outcome belongs to a class that was specified in advance or is unusually probable under an alternative hypothesis. Four aces on top after a casino shuffle is suspicious because the class “all four aces together at the top” was salient before inspection and is much more expected under certain forms of interference than under honest randomisation.
This is the same reason post hoc pattern hunting generates miracles. Given enough data, some sequence will appear astonishing. Birthdays cluster, stock charts resemble animals, sports records break in oddly specific categories, and clouds look like historical figures. If the pattern is defined after it is seen, its tiny point probability is usually irrelevant because innumerable other patterns could have been selected. Statistical reasoning therefore insists on reference classes, pre-specification, multiplicity correction, and model comparison. The universe produces rare configurations constantly because it produces one configuration at a time from enormous spaces. “What are the odds?” is often the opening line of an error rather than the closing line of an analysis.
Unicorn reasoning can fall into the same trap. Suppose one sees a white horse with a branch aligned behind its forehead and takes a photograph. The exact arrangement of horse, branch, light, camera angle, and observer is extraordinarily improbable if specified beforehand. Yet that does not make “unicorn” the better hypothesis because countless equally improbable visual arrangements happen continuously. The evidential question is comparative: how much more likely is this image under an actual unicorn than under ordinary perspective and chance alignment? Probability without an alternative model is merely astonishment wearing numbers.
16. Feasibility has at least five dimensions, all of which optimism prefers to forget
A serious feasibility claim should name its constraints. Physical feasibility asks whether natural laws permit the outcome. Technical feasibility asks whether existing or credibly developable technology can achieve it. Economic feasibility asks whether resources and expected benefits justify the cost. Operational feasibility asks whether organisations, people, supply chains, and processes can execute it. Temporal feasibility asks whether it can be completed within the required time. One may add legal, political, ethical, and computational feasibility depending on the domain. These categories are not philosophical embroidery; projects fail every day because an idea that was feasible in one dimension was assumed feasible in all.
The wall-tunnelling example fails across almost every practical dimension despite its usefulness as a quantum thought experiment. The microscopic phenomenon is physically real, but there is no technical mechanism for arranging a warm human body into a controlled coherent tunnelling event through an ordinary wall. There is no operational procedure, no tolerable waiting time, and no economic programme that converts the formal nonzero of an idealised model into transport. This is precisely what feasibility contributes: it asks not whether the universe has signed a categorical prohibition, but whether the agent possesses a route from present state to desired outcome. A map of possible worlds is not a project plan.
The distinction also explains why technological forecasting is so often wrong in both directions. Forecasters sometimes declare an innovation impossible because current implementations are poor, confusing present technical infeasibility with physical impossibility. Others predict imminent transformation because a laboratory demonstration exists, confusing physical or technical proof-of-concept with economic and operational feasibility at scale. Between those errors lies the less glamorous work of engineering: yield, reliability, maintenance, training, supply, regulation, integration, cost, and time. Possibility supplies the prototype with applause. Feasibility asks whether anyone can manufacture ten thousand of them on Tuesday.
17. “It could happen” is not an argument until the reference class appears
Every probability statement refers, explicitly or implicitly, to a structure of alternatives. “The probability that a 45-year-old survives another year” depends on sex, country, health status, socioeconomic conditions, and the life table used. “The probability that a startup succeeds” is meaningless until success is defined and a relevant comparison group selected. “The likelihood of war” depends on time horizon, actors, thresholds, and information. Reference-class choice is therefore one of the central hidden decisions in applied probability. Choose the wrong class and a beautifully calculated number answers the wrong question.
This is especially important for one-off events. There is no literal long-run frequency of “this exact election under these exact conditions” or “this exact company surviving this exact crisis,” yet forecasters can still use probabilistic models by drawing on analogous cases, structural variables, expert beliefs, simulation, market prices, or ensembles. The resulting probabilities are model-dependent and should be judged by calibration and predictive performance where possible. What one should not do is evade the difficulty by reverting to “possible.” The fact that a unique event cannot be repeated does not make probability meaningless; it makes the assumptions behind the probability more important.
A useful question is therefore: possible relative to what comparison set? If someone says it is possible that a hidden civilisation has built a base beneath the ocean, the statement may be logically coherent. If they say it is probable, they must locate the claim within evidence and alternatives. If they say sonar anomalies make it likely, they must specify a likelihood model comparing natural geology, artefact, instrument error, and engineered structure. If they say an expedition is feasible, they must provide technology, cost, location, safety, and expected information gain. Every upgrade in language increases the debt. Most speculative arguments default before the second payment.
18. Uncertainty is not ignorance, and ignorance is not permission to use any number
Risk discussions often confuse uncertainty with ignorance. Uncertainty means that relevant outcomes or parameters are not known with certainty but can be represented, bounded, ranked, or modelled. Ignorance may mean that the model itself is poorly known, key variables are missing, or the space of possibilities is incomplete. Economists sometimes distinguish risk, where probabilities are known or estimable, from Knightian uncertainty, where they are not. Philosophers and statisticians have developed imprecise probabilities, interval probabilities, robust Bayesian methods, scenario analysis, and sensitivity analysis to handle cases in which a single precise number would overstate knowledge. The correct response to poor information is therefore not always to guess more confidently.
This matters for possibility because “we do not know” is frequently converted into “anything could happen,” and then into “all possibilities deserve serious weight.” They do not. Ignorance may widen the range of reasonable models, but background knowledge still constrains it. If I know little about tomorrow’s lunch, many menus are open; the uncertainty does not make cannibalism, lunar cheese, and photosynthesis equally live options. Rational ignorance has structure. The world we do not know remains connected to the world we already know.
A mature forecast can therefore say: this event is physically possible; evidence is weak; available data do not support a precise probability; plausible bounds are wide; decision consequences are asymmetric; and a low-cost information-gathering step is feasible. That sentence contains more uncertainty than “there is a 17 percent chance,” but it contains vastly more information about the uncertainty itself. Precision is not rigour when the third decimal place was purchased from imagination.
19. The modal fallacy of modern futurism
Futurism is particularly fond of treating technological possibility as temporal probability. A laboratory result shows that a phenomenon can occur, and within several paragraphs the writer is forecasting mass adoption within five years. But “can be demonstrated” and “will become common” are separated by adoption curves, cost decline, infrastructure, regulation, competing technologies, human behaviour, and organisational inertia. A technology may be possible and never become economical; economical and never become socially accepted; socially desirable and blocked by institutions; technically mature and superseded before diffusion. Temporal prediction therefore requires a model of transition, not merely a photograph of a prototype.
The same error appears in catastrophe forecasting. If a failure mode can be described, it becomes a “risk”; because it is a risk, it becomes “likely enough to prepare for”; because preparation is discussed, the scenario acquires institutional reality. None of this means rare risks should be ignored. It means the pathway from possibility to policy should be explicit. What probability or range is credible? What evidence supports it? What consequences follow? What mitigation is effective? What does mitigation cost? What other risks compete for the same resources? A rational institution can act under deep uncertainty, but it should know whether it is buying insurance, information, resilience, or theatre.
Possibility is therefore not the enemy. Unexamined promotion is the enemy. Every major innovation began as something merely possible in somebody’s mind. Every failed fantasy did too. The distinction between them was not imagination; it was evidence, mechanism, iteration, resources, and execution. To worship possibility because successful inventions were once possible is like worshipping childhood because every Nobel laureate was once five years old.
20. A short field guide to the words people misuse
Possible: not excluded under a specified set of rules, laws, assumptions, or information. Always ask: possible in what sense? Impossible: excluded under those same constraints; do not confuse physical impossibility with present inability. Necessary: true in all relevant possible worlds or under all admissible states in the specified modal framework. Contingent: true in some relevant possibilities and false in others. Plausible: supported enough by evidence, mechanism, or background knowledge to deserve serious consideration, though not necessarily quantifiable. Probable: assigned comparatively high probability under a specified interpretation or model. Likelihood: in ordinary language, often a synonym for chance; in statistics, a function that evaluates fixed observed data across alternative parameter values or models. Feasible: achievable by an identified agent under relevant resource and constraint conditions.
Several neighbouring words deserve similar care. Conceivable means imaginable or cognitively representable and does not guarantee genuine possibility. Credible normally concerns warranted belief or trustworthy evidence. Viable often means capable of surviving or functioning over time, which is stronger than mere feasibility at launch. Tractable in mathematics or computation concerns whether a problem can be handled with acceptable resources, not whether a solution exists in principle. Expected has both ordinary and technical meanings; an expected value is a probability-weighted average and need not itself be a possible outcome. Risk generally combines uncertainty with consequence rather than being a synonym for possibility. Language is not the problem here. Using all of these words as though they were “maybe” is.
A competent argument should therefore survive a simple substitution test. Replace “possible” with the exact claim intended. “It is logically consistent that…” “Current evidence does not rule out…” “Our model assigns a five percent probability…” “The observed data are three times more likely under model A than model B…” “The project is technically feasible within two years…” If the sentence suddenly becomes difficult to complete, the original word was probably hiding the missing analysis. Precision often feels restrictive only because vagueness had been doing unpaid labour.
21. The five-question test
When confronted with a dramatic claim, begin with modality: what kind of possibility is being asserted? Logical, metaphysical, physical, epistemic, legal, practical? Then move to evidence: what observations distinguish this proposition from alternatives? Then ask about probability: is there a model, reference class, chance mechanism, calibrated forecast, or defensible credence? Then ask about likelihood: how expected are the observed data under the favoured hypothesis compared with serious alternatives? Finally ask about feasibility: what agent, resources, technology, time, cost, and constraints turn the proposition into an actionable option? This five-question sequence is not a universal philosophy of science. It is simply an excellent way to stop the word “could” from stealing several promotions at once.
Apply it to the unicorn. Logical possibility: yes, under a modest definition. Physical possibility: arguably, because a one-horned equine-like mammal violates no obvious law of nature. Evidence: poor for an extant hidden population. Probability: no defensible exact number, but rational credence should be low given the evidence. Likelihood: any claimed evidence must be compared with mundane alternatives, not merely shown compatible with unicorns. Feasibility: a search may be feasible in a bounded habitat, but proving universal nonexistence is not. The result is neither mystical belief nor dogmatic denial. It is a structured statement about what we know.
Apply it to walking through a wall. Logical possibility: the phrase is coherent. Classical mechanical feasibility for an intact person: no, absent breaking or going around the wall. Quantum physics: tunnelling through finite barriers is real, and simple quantum models assign nonzero transmission under suitable conditions. Probability for an intact macroscopic human under ordinary conditions: so profoundly suppressed and model-dependent that treating it as an everyday chance would be absurd. Feasibility: effectively none. Decision value: buy a door. The example is funny because the mathematics is serious and the practical conclusion is obvious; it becomes confusing only when “nonzero in a physical theory” is translated into “an option available to us.”
22. Conclusion: the unicorn may stay; the inference must leave
Possibility is indispensable because reasoning without alternatives becomes dogma. Modal logic lets us distinguish what must be true from what could be true; counterfactuals let us examine causation and responsibility; imagination lets science ask questions before evidence is available. The mistake is not in entertaining possibilities. It is in rewarding them for merely having been entertained. A proposition does not become probable because it survived contradiction, plausible because it inspired an analogy, statistically likely because one piece of evidence fits, or feasible because physics neglected to issue a formal prohibition. Each promotion requires new work.
Unicorns, elves, and fairies are therefore entitled to a respectable philosophical residence. They help demonstrate that coherence is not evidence and that absence of contradiction is not presence of probability. Quantum tunnelling is entitled to even greater respect because it is experimentally real, mathematically precise, and a delightful reminder that physical law is stranger than common sense. Yet its existence does not make walking through walls a feasible transport policy. Lottery wins remain possible and improbable. Exact values in continuous models can have probability zero and still occur. Evidence can be likely under a hypothesis without making the hypothesis probable. A project can be physically possible and financially impossible. These are not semantic curiosities; they are the grammar of rational uncertainty.
The next time somebody announces that “it is possible,” there is no need to disagree. Merely ask the question that politeness has postponed: and what exactly does that buy us? If the answer is a model, evidence, probability, likelihood ratio, mechanism, or feasibility analysis, a conversation has begun. If the answer is another repetition of “but you cannot rule it out,” then the proposition has remained exactly where it started: admitted to the waiting room, splendidly dressed, with no appointment. Possibility is cheap because reality produces an infinity of ways things could have been. Knowledge begins when we learn which of them deserve belief, and wisdom begins rather later, when we stop confusing belief with a project plan.
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