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Proving a negative

Proving a negative refers to the logical and philosophical endeavor of demonstrating the falsity or non-existence of a claim, proposition, or entity, which contrasts with proving a positive by requiring evidence of absence rather than presence.^[1] This task is frequently associated with the adage "you can't prove a negative," a notion rooted in the practical challenges of verifying universal absences, such as confirming that no instance of a phenomenon exists anywhere in the universe.^[2] However, this saying is a myth, as negatives can be established through deductive reasoning, empirical testing within defined scopes, or inductive inference, provided the claim is not universally quantified without limits.^[3] In logic, proving a negative often employs modus tollens, a valid deductive form where one shows that if a hypothesis were true, certain observable consequences would follow, but since those consequences are absent, the hypothesis must be false—for instance, "If unicorns exist, fossil evidence would be present; no such evidence exists; therefore, unicorns do not exist."^[4] Specific negatives, like "there are no apples in this drawer," can be directly verified by exhaustive search within a bounded domain, rendering them empirically provable.^[1] Universal negatives, such as "no gods exist anywhere," pose greater difficulty due to the impossibility of omnipresent observation, but they remain defeasible through contradictory evidence or incoherence in the positive claim.^[2] The concept is central to discussions of the burden of proof, which typically falls on the affirmative claimant rather than the skeptic demanding disproof, as shifting it to prove a negative constitutes the fallacy of argumentum ad ignorantiam (appeal to ignorance).^[1] In philosophy and law, this principle ensures that extraordinary claims require supporting evidence, while negatives are presumed unless positives are substantiated— for example, in legal contexts, the prosecution must prove guilt beyond reasonable doubt, not the defense prove innocence.^[3] Applications extend to scientific skepticism, where absence of expected evidence (e.g., no detectable signals from extraterrestrial intelligence) supports negative conclusions probabilistically, though absolute certainty remains elusive due to inductive limitations.^[4]

Definition and Meaning

Core Definition

Proving a negative refers to the logical endeavor of establishing the non-existence, falsity, or absence of something, such as claiming "there are no black swans" in contrast to the positive assertion "there is a black swan."^[5] This process involves demonstrating that a predicate does not apply to any instance of a subject, often requiring evidence that rules out all potential cases.^[6] The key logical structure of negative claims typically demands an exhaustive search or universal negation, rendering them inherently challenging because the domain of possibilities may be infinite or practically unbounded.^[5] For instance, to verify "no black swans exist," one would need to confirm the absence across every conceivable location and time, which exceeds human capabilities without additional constraints like a defined finite set.^[5] This difficulty arises from the need to affirm a negation universally, unlike positive claims that can be supported by a single confirming instance.^[7] Negative claims can be distinguished into existential negatives, which assert that no entity exists satisfying a certain property (e.g., "there exists no unicorn"), and universal negatives, which state that no instance of a class possesses a given attribute (e.g., "no squares are circles").^[7] Existential negatives often translate to universal negations over the entire domain (∀x ¬P(x)), amplifying the proof burden, while universal negatives within logic, such as Aristotelian "E" propositions ("No S is P"), can sometimes be derived syllogistically but still face empirical verification hurdles.^[6] Relatedly, the burden of proof principle typically places the responsibility on the claimant making the affirmative assertion, since proving a negative can be particularly challenging and often relies on the absence of expected evidence.^[1]

Variations in Usage

In logic, proving a negative is formally equivalent to asserting a universal negative proposition, symbolized as \forall x \neg P(x), which states that no element in the domain satisfies the predicate P. This contrasts sharply with existential positive claims, denoted \exists x P(x), which can be established by identifying even a single instance where P holds. The perceived challenge in such proofs stems not from the negation itself but from the universal quantifier, requiring exhaustive verification across an infinite or unbounded domain, whereas positives often suffice with one counterexample to the contrary.^[3] In rhetoric and debate, the concept of proving a negative frequently serves as a rhetorical device to deflect accountability, particularly when one party demands evidence against an unsubstantiated affirmative claim, such as challenging an opponent to "prove God doesn't exist" amid a discussion on divine existence. This maneuver exploits the asymmetry in evidentiary demands, positioning the negative as unduly burdensome to unfairly preserve the status quo for the positive assertion without providing supporting arguments. Such usage highlights how the phrase can manipulate discourse by implying impossibility, thereby shifting the onus away from the claimant.^[8] Within epistemology, proving a negative aligns closely with skeptical methodologies, where the absence of evidence for a claim fosters reasonable doubt about its truth, reinforcing the principle that unverified positives warrant suspension of belief. David Hume, for instance, invoked this in his critique of miracles, recognizing the inherent difficulty in conclusively proving that no such events have ever transpired, which bolsters skeptical restraint against extraordinary assertions lacking empirical backing. This interpretive lens positions negatives as foundational to epistemic humility, enabling the critique of knowledge claims through the lens of unverifiable possibilities rather than affirmative certainties.^[9] In mathematics, establishing a negative—such as the non-existence of solutions to an equation—is routinely accomplished via proof by contradiction, assuming a solution exists and deriving an inconsistency, or leveraging completeness theorems that guarantee all valid statements within a formal system are provable, including those asserting impossibility. A classic example is demonstrating no integers a and b satisfy $18a + 6b = 1: supposing such values exist leads to a contradiction, as dividing by 6 yields $3a + b = \frac{1}{6}, impossible for integers. These methods underscore the rigor of mathematical negatives, transforming apparent absences into demonstrable truths through logical closure.^[10]

Philosophical and Logical Foundations

Burden of Proof

The burden of proof refers to the obligation on a party in a dispute to provide sufficient evidence or warrant for their position, determining who must demonstrate the truth of an assertion to meet a specified standard, such as preponderance of evidence or beyond reasonable doubt.^[11] This principle allocates responsibility based on the nature of the claim, with the party failing to meet the burden typically losing the issue by default.^[11] In philosophical and logical contexts, it ensures that assertions are not accepted without justification, promoting rational discourse.^[12] The concept originates from Roman law, encapsulated in the maxim ei incumbit probatio qui dicit, non qui negat ("proof is incumbent upon the party who affirms a fact, not upon him who denies it"), as stated in Justinian's Digest (22.3.2).^[13] This principle placed the onus on the accuser or affirmative claimant, rather than requiring the denier to disprove the claim.^[13] It was later adapted into philosophical argumentation, where the burden typically rests on the positive claimant to substantiate their assertion before demanding rebuttal from opponents.^[12] When applied to negative claims—such as asserting that "no aliens exist"—the burden becomes particularly onerous, as it requires comprehensive evidence of absence across all possible domains, which is often impractical or impossible without exhaustive search.^[2] Positive claims, by contrast, can often be supported through targeted evidence of existence, making the default burden allocation favor them unless the negative can be reframed or proven via contradiction.^[2] In such cases, the negative claimant must provide strong inductive or evidential grounds, like systematic absence in empirical data, to shift or meet the burden effectively.^[2] In debates, demanding proof of a negative from an opponent constitutes a fallacy—specifically, an argument from ignorance—if the initial positive claim lacks supporting evidence, as it unjustifiably shifts the burden without warrant.^[14] This misuse exploits the inherent difficulty of negative proofs, undermining fair argumentation.

Logical Impossibility

In formal logic, certain negative claims—assertions that something does not exist, occur, or hold true—cannot be proven due to inherent limitations in axiomatic systems. Gödel's first incompleteness theorem establishes that any consistent formal system capable of expressing elementary arithmetic is incomplete, meaning there are true statements within its language that neither it nor their negations can prove. A paradigmatic example is the Gödel sentence G, which self-referentially states "This sentence is not provable in the system." If the system is consistent, G is true but unprovable, making the negative claim that G is provable undecidable within the system itself.^[15] Gödel's second incompleteness theorem extends this by showing that such a system cannot prove its own consistency, a statement equivalent to the universal negative "No contradiction can be derived from the axioms." Proving this negative internally would require demonstrating the absence of any derivable inconsistency across all possible proofs, which the theorem renders impossible without appealing to a stronger external system. These results underscore how self-referential negatives about provability and consistency evade resolution in sufficiently powerful formal frameworks.^[15] Another barrier arises from the infinite regress problem in domains without finite bounds. To prove a universal negative like "No element satisfies property P" over an infinite set, one must verify the absence of P for every possible element, leading to an unending chain of checks where each verification demands justification from the next without termination. This regress is vicious because it precludes a foundational basis for the proof, rendering the negative claim logically unattainable in open, unbounded contexts.^[16] The halting problem in computability theory provides a concrete illustration of undecidability for negatives. Formulated by Alan Turing, it asks whether there exists an algorithm to determine, for any program and input, if the program halts (terminates) or loops indefinitely. Turing proved no such general algorithm exists; specifically, deciding the negative—"This program does not halt"—is undecidable because the set of non-halting computations is not recursively enumerable, meaning no procedure can systematically confirm non-halting for all cases without risking infinite execution.^[17] Despite these challenges, some negatives are provable in closed or structured systems where exhaustive analysis is feasible. For example, in elementary number theory, the statement "There is no even prime greater than 2" can be proven by contradiction: assume an even integer n > 2 is prime; then n = 2k for integer k > 1, so n has divisors 1, 2, k, and n, contradicting primality. This holds because the natural numbers allow finite verification via the definition of primes and divisibility.

Historical Development

Origins in Philosophy

The concept of proving a negative traces its philosophical origins to ancient Greece, particularly in Aristotle's development of syllogistic logic as detailed in his Prior Analytics. Aristotle distinguished between universal affirmative and universal negative propositions, such as "No S is P," which form the basis for valid deductions in the first figure of syllogisms when combined with particular affirmatives. These universal negatives allowed for conclusive reasoning about absences or exclusions, establishing a framework where negatives could be proven through logical structure rather than empirical enumeration alone.^[6] In this system, the square of opposition further illustrated how universal negatives contradict particular affirmatives, enabling proofs of non-inclusion via contradictory opposition.^[18] During the medieval period, Thomas Aquinas advanced discussions of negatives through apophatic theology, or the via negativa, which emphasizes describing God by what He is not, thereby addressing the limits of affirmative knowledge about divine existence and essence. In the Summa Theologica, Aquinas argues that while God's existence can be demonstrated via causal arguments, His essence remains unknowable in positive terms, leading to negative affirmations like "God is not composed" or "God is not limited by matter." This approach, influenced by Pseudo-Dionysius, does not seek to prove God's non-existence but highlights the inadequacy of human concepts for divine reality, using negation to purify theological language and avoid anthropomorphism.^[19] Aquinas integrated this with cataphatic elements, but the apophatic method underscored the philosophical challenge of substantiating negatives about transcendent entities. In the Enlightenment era, David Hume's empiricism brought negatives to the forefront by questioning unobserved necessities, particularly in causation, as explored in An Enquiry Concerning Human Understanding. Hume contended that no necessary connection between cause and effect is observable; instead, we infer only constant conjunction from experience, leading to the negative claim that causal necessity lacks empirical foundation beyond habitual association.^[20] This skepticism limited knowledge to sensory limits, arguing that claims of hidden necessities cannot be verified, thus proving a negative about the scope of human understanding.^[21] Eighteenth-century skeptical debates, exemplified by Hume's essay "Of Miracles" in the same Enquiry, centered negatives in challenges to supernatural claims, asserting that uniform human experience proves the non-occurrence of miracles. Hume argued that testimony for miracles is outweighed by the consistent evidence against violations of natural laws, rendering belief in them irrational without extraordinary counter-evidence.^[22] These discussions, amid broader Enlightenment critiques of revelation, positioned proving negatives as essential to empirical philosophy, influencing ongoing rational inquiries into improbable events.^[23]

Evolution in Modern Thought

In the 19th century, John Stuart Mill advanced the role of negative instances in inductive logic through his canonical methods of causal inquiry, particularly emphasizing their utility in falsifying alternative explanations and strengthening causal attributions over mere verification. In A System of Logic (1843), Mill described the Joint Method of Agreement and Difference, where negative instances—cases in which the phenomenon is absent despite the presence of potential causes—help eliminate competing factors, thus confirming a hypothesis by ruling out plurality of causes more effectively than positive instances alone.^[24] This approach highlighted the asymmetry in induction, where negatives provide critical disconfirmatory power, influencing later empiricist views on evidence. The 20th century saw Karl Popper formalize this emphasis on negation with his falsifiability criterion, positing that scientific theories gain demarcation from non-science precisely through their vulnerability to empirical disproof. In The Logic of Scientific Discovery (1934), Popper argued that theories must entail testable predictions that, if contradicted by observation, lead to refutation via modus tollens, rejecting inductivist verification in favor of bold conjectures subjected to severe negative tests.^[25] Falsifiability, measured by the theory's capacity to be overthrown by singular counterinstances, underscores the logical impossibility of proving universal positives conclusively while enabling decisive negative disproofs, as in the example of a single black swan refuting "all swans are white."^[25] Post-1950s analytic philosophy intensified debates on absence of evidence through Bertrand Russell's teapot analogy, which illustrated the impropriety of shifting the burden of proof to disprove unfalsifiable claims. In his 1952 essay "Is There a God?", Russell posited an undetectable orbiting teapot whose existence cannot be refuted yet warrants disbelief absent positive evidence, critiquing the fallacy of equating undetectability with plausibility in existential assertions.^[26] This analogy, influential in subsequent discussions, reinforced that negative evidence—or its absence—does not affirm existence but demands evidential support from claimants. In contemporary epistemology, Bayesian frameworks quantify negative evidence through probabilistic updates, treating absences as data that lower posteriors for existence claims under reasonable priors and likelihoods.^[27] Applying Bayes' theorem, the failure to observe expected evidence (¬E) given hypothesis H reduces Pr(H|¬E) when Pr(¬E|H) < Pr(¬E|¬H), providing graded disconfirmation rather than binary proof, as explored in analyses of "absence of evidence" mottos. For instance, exhaustive searches yielding no traces (e.g., for hidden entities) rationally diminish belief in their presence, integrating Mill's and Popper's insights into a probabilistic model of evidential weight.^[27]

Applications and Examples

In Science and Empiricism

In scientific methodology, proving a negative plays a central role in hypothesis testing through the framework of null hypothesis significance testing (NHST). The null hypothesis (H₀) typically asserts the absence of an effect, relationship, or difference in the population, such as no difference between treatment and control groups. Researchers collect data and compute a p-value, which represents the probability of observing the sample results (or more extreme) assuming H₀ is true. If the p-value falls below a predetermined significance level (commonly 0.05), the null hypothesis is rejected, providing evidence against the negative claim (e.g., no effect exists) and in favor of the alternative hypothesis (e.g., an effect does exist). However, failure to reject H₀ does not confirm the negative; it merely indicates insufficient evidence to rule it out, as this could stem from low statistical power, small sample sizes, or weak effects.^[28] A seminal example of attempting to prove a negative in empiricism is the Michelson-Morley experiment of 1887, which sought to detect the Earth's motion relative to the hypothetical luminiferous ether—a medium posited to propagate light waves. Using an interferometer to measure differences in light speed along perpendicular paths aligned with and against the Earth's orbital motion, the experimenters expected a fringe shift if the ether existed. Instead, they observed a null result, with no detectable variation, to within 1/40th of the expected magnitude. This failure to detect the ether wind provided strong empirical evidence against the stationary ether hypothesis, paving the way for Einstein's special relativity, which eliminated the need for such a medium.^[29] Despite these successes, proving negatives in science faces significant challenges, particularly with absence of evidence. For instance, direct detection experiments for dark matter—such as those using cryogenic detectors or liquid xenon targets—have yielded predominantly null results over decades, setting stringent upper limits on weakly interacting massive particle (WIMP) interactions. These non-detections rule out certain parameter spaces but do not prove dark matter's non-existence, as the particles may interact too feebly for current sensitivities, evade detection due to unknown properties, or require larger-scale or novel experimental designs. Observational limits, background noise, and theoretical uncertainties underscore that absence of evidence remains inconclusive without exhaustive searches across all plausible regimes.^[30] A foundational concept in this domain is Karl Popper's principle of falsification, which prioritizes hypotheses that are empirically disprovable over those that are merely verifiable. Scientific theories must make bold predictions that, if contradicted by observation, falsify them—such as a universal law negated by a single counterinstance—rather than relying on unverifiable positive claims that can accommodate any data. This asymmetry favors testable negatives, as verification cannot conclusively prove a general proposition (e.g., "all swans are white" survives white swans but falls to one black swan), while falsification rigorously advances knowledge by eliminating untenable ideas. Popper argued this demarcates science from pseudoscience, emphasizing critical testing to prefer disprovable conjectures.^[31]

In Law and Jurisprudence

In legal systems adhering to common law traditions, the presumption of innocence requires the prosecution to affirmatively prove the defendant's guilt beyond a reasonable doubt, thereby shifting away from any obligation on the defense to prove the negative assertion of innocence.^[32] This foundational principle ensures that the accused is not compelled to disprove allegations, as doing so would impose an impractical burden given the infinite possibilities for absence of guilt.^[33] For instance, the "beyond a reasonable doubt" standard explicitly avoids requiring proof of non-guilt, allowing defenses like alibis to function as circumstantial evidence that negates the prosecution's case by demonstrating the defendant's presence elsewhere at the time of the alleged crime.^[34] Historically, this modern approach contrasts sharply with earlier injustices, such as those in the 16th- and 17th-century European witch trials, where the accused bore the burden of proving their non-involvement in witchcraft—a negative claim nearly impossible to substantiate without exhaustive or supernatural evidence.^[35] In these proceedings, evidentiary rules often reversed the presumption, relying on spectral testimony or confessions obtained under duress, which led to widespread convictions and executions viewed today as profound miscarriages of justice due to the unfair demand for negative proof.^[36] Such practices highlighted the dangers of inverting the burden, prompting reforms that entrenched the prosecution's responsibility to provide positive evidence of culpability. In international law, particularly within tribunals like the International Criminal Tribunal for the former Yugoslavia (ICTY), proving the absence of certain conducts—such as a commander's failure to prevent or punish subordinates' war crimes—demands exhaustive documentation from the defense to rebut the prosecution's inferences.^[37] Under doctrines like command responsibility, the accused must often produce detailed records of orders issued, investigations conducted, or preventive measures taken, illustrating the evidentiary challenges in negating liability for omissions in conflict zones.^[37] The logical difficulties inherent in gathering such negative evidence further emphasize the tribunals' reliance on the prosecution's burden to establish crimes beyond reasonable doubt, while allowing defenses to introduce affirmative rebuttals through comprehensive records.^[38]

In Everyday Debates

In everyday debates, proving a negative often arises in discussions of conspiracy theories, where proponents demand that skeptics disprove unsubstantiated claims, thereby unfairly shifting the burden of proof. For instance, in arguments asserting that the Apollo moon landing was faked, believers may insist on evidence that no hoax occurred, exploiting the inherent difficulty of falsifying secretive plots. This tactic leverages the non-falsifiable nature of many conspiracy theories, as it is challenging to demonstrate the absence of hidden actions or cover-ups, allowing such claims to persist without supporting evidence.^[39] Rhetorical strategies in public discourse frequently involve demanding proof of a negative to undermine established positions, such as in climate change denial where skeptics set "impossible expectations" for evidence, like requiring irrefutable proof that human emissions have no impact despite overwhelming data to the contrary. This approach resembles the argument from ignorance fallacy, where the absence of conclusive disproof is misconstrued as validation for denial, often combined with straw man tactics that caricature scientific consensus as unproven alarmism. In structured debates, this improperly shifts the burden of proof away from the claimant, violating principles of rational argumentation.^[40]^[41] On social media, proving a negative becomes a common demand in debates over misinformation and hoaxes, such as challenges to show "no evidence exists" for viral claims like election fraud or vaccine dangers, which amplifies falsehoods by requiring exhaustive refutation. These platforms facilitate rapid spread of such demands, where users resist corrections due to the continued influence of initial misinformation, even after fact-checks. Psychological factors exacerbate this, as confirmation bias leads individuals to favor negative claims that align with preexisting doubts, making disconfirming evidence less persuasive and sustaining belief in unproven narratives.^[42]^[43]

Criticisms and Misconceptions

The Myth of Proving a Negative

The notion that proving a negative is inherently impossible is a pervasive misconception in popular discourse, often invoked to dismiss claims of absence without further scrutiny. This myth arises from an overgeneralization of challenges in proving universal negatives in open or infinite domains, such as the impossibility of exhaustively searching an unbounded space for non-existence. In reality, the difficulty lies not in the negative form of the proposition but in the scope of quantification; existential positives (e.g., "there exists a black swan") are straightforward to affirm with one example, while their negations require broader evidence.^[3]^[44] Many negative claims are readily provable within finite or closed domains, where exhaustive inspection or verification is feasible. For instance, the statement "there are no elephants in this room" can be confirmed by a thorough visual and physical search of the enclosed space, establishing the absence beyond reasonable doubt. Similarly, in closed systems, formal methods can verify the absence of certain defects for specified conditions in well-defined models.^[45] This misconception originates from philosophical and logical traditions emphasizing open-world assumptions, where induction from finite observations cannot conclusively rule out possibilities in infinite realms, leading to an undue extension of such limitations to all negatives. Counterexamples abound in mathematics, where proofs routinely establish non-existence. Fermat's Last Theorem, proven by Andrew Wiles in 1995, demonstrates that no positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer n > 2, effectively disproving the existence of such solutions across the domain of positive integers. In infinite cases, logical impossibility may arise due to undecidability or incomplete evidence, but this does not render all negatives unprovable.^[46]

Strategies for Addressing Negative Claims

When direct proof of a negative claim—such as the non-existence of an entity or the absence of a relationship—is infeasible due to logical or practical constraints, several strategies can approximate or strengthen such assertions through indirect reasoning. These approaches draw from formal logic and probabilistic methods, allowing claimants to build a compelling case without requiring exhaustive universal verification.^[44] One primary strategy involves inductive reasoning, where accumulated evidence of absence bolsters the negative claim. For instance, repeated exhaustive searches in relevant domains that yield no confirming evidence can probabilistically support conclusions like the non-existence of mythical creatures such as Bigfoot, as each failed attempt reduces the likelihood of oversight. This method strengthens the negative case incrementally but remains probabilistic rather than deductive, relying on the comprehensiveness of the search efforts to minimize alternative explanations like undetected hiding.^[44] Bayesian updating provides another framework for assessing negative claims by incorporating prior probabilities and updating beliefs based on observed data, including absences. In this approach, the lack of expected evidence lowers the posterior probability of the positive hypothesis; for example, in evaluating fine-tuning arguments in cosmology, the absence of certain observational data can shift credence toward non-existence claims without demanding absolute proof. This method treats absence of evidence as potentially evidential of absence, depending on the reliability of the search process and background assumptions about detectability.^[47] Contraposition offers a deductive route by reformulating the negative claim into a positive one via logical equivalence. To prove "no A is B," one instead demonstrates "all B are not A," leveraging valid inferences like modus tollens; a classic example is arguing against unicorn existence by showing that if unicorns existed, fossil evidence would be present, but since no such evidence appears, unicorns do not exist. This transforms the burden into proving a feasible positive statement, preserving logical rigor.^[44] In policy contexts, these strategies can manifest through modeling to indirectly establish negatives, such as assessing low risks based on probabilistic models integrating likelihood and verification data to guide decisions on resource allocation without claiming certainty. This allocation of proof burden often falls on agencies conducting the assessments to justify such inferences.

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Jan 12, 2022 · Moreover, the term disinformation is often specifically used for the subset of misinformation that is spread intentionally. More research is ...
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We have reviewed the current literature on the psychological factors that appear to drive conspiracy belief. We conclude that conspiracy belief appears to stem ...Missing: proving | Show results with:proving
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[PDF] Thinking Tools: You Can Prove a Negative - departments.bloomu.edu
A principle of folk logic is that one can't prove a negative. Dr. Nelson L. Price, a Georgia minister, writes on his website.
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Sep 15, 2011 · "You can't prove a negative" means you can't prove beyond reasonable doubt that certain things don't exist, then the claim is just false.Missing: origins | Show results with:origins
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The Logic, Philosophy, and Science of Software Testing
Jun 17, 2025 · This handbook takes you on a journey from fundamental logical principles to their practical applications in software development, scientific reasoning, and ...
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Epistemic Paradoxes - Stanford Encyclopedia of Philosophy
Jun 21, 2006 · The logical myth that “You cannot prove a universal negative” is itself a universal negative. So it implies its own unprovability. This ...Missing: proving | Show results with:proving
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Nov 24, 2015 · Risk = Threat × Vulnerability × Consequence. Eq. 2 where the consequence again is some undesirable outcome, the threat is the probability of.