Majority Generalization Rule
Posted: Tue Aug 25, 2026 1:22 pm
Summary
The Majority Generalization Rule is here proposed to reduce confusion in scholarly and debate communications: A statement is generally true (without need for qualifiers) only when it is estimated to be confidently true more than half of the time. This applies to a subject domain with a defined scope for which the prevalence within an underlying statistical subject set can be estimated with certain confidence. This rule constrains generalizations in scholarly text by applying a confident majority threshold to generic statements. The constraint is a high-utility heuristic for distinguishing supported generalizations from speculation in scholarly text. In this proposal domain, true means 'supported as a majority generalization'; false means 'confidently false as a majority generalization'; unknown means the prevalence overlaps with the uncertainty range between true and false. So, minority truth generic statements are encouraged to be rejected as invalid without qualifiers. This rule reduces presumptions, assumptions, and implications of no utility to the goal of the language used. Generic statements can traditionally encode characteristic, dispositional, and striking-property information [2]. Those minority generics should be rejected as too imprecise or confusing in scholarly language and debate, instead primarily offering frequency information [1][2]. This writing offers a formula as a trial formula for more precise language.
Qualification for Minority Cases
In colloquial written language, one could truthfully say "Parasites carry disease.". But in scholarly language under Majority Generalization Rule, this should be considered a false statement or unknown (including speculation). Instead, one would say "Some parasites carry disease.". The qualification here is "some". Only if one did a field study that confidently showed that most parasites are currently carrying a disease, would it then be considered true. This statement should be treated as false or unknown unless evidence shows that disease-carrying is true of more than half of a broad global sample of many kinds of parasites with adequate certainty as confidence. Stated in another way, this is because a general statement is only licensed when both its estimated prevalence and the certainty in that estimate are sufficiently high. If the evidence does not establish majority prevalence, the statement must be qualified with terms such as 'may', 'can', 'often', or 'some'.
Categorical Options
Statements may be formed into more statistically objective statements using either two-value logic (true or false) or three-value logic (true or unknown or false). We here borrow the dominant border from fuzzy logic to use 50% as the main dividing line of categorical membership, as a point of unknown [4][9]. This is furthermore between majority and minority prevalence, which is also one of the potential semantic meanings from fuzzy logic of the category membership [3][4]. That 50% dividing line is also associated with majority and minority in democratic voting as collective decision-making graded support in some contexts such as generalizing a collective will.
"50% or more" as the minimum truth condition is avoided, because otherwise one could make confusing, paradoxical claims such as "It is true that the result of a coin flip is heads. It is also true that the result of a coin flip is tails.". Instead, only an unfair coin could correctly result in the statement "our unfair coin flips to heads" with 'more' being implied there as the generalization. Furthermore, by using such generalizations, one then has a much higher obligation to use qualifiers where they'd otherwise be implied. So, "our unfair coin always flips to heads" would then be the better language when using this language style in such a case. For full clarity, state the generalization style being used.
Therefore, 50% is the recommended minimum inflection point of generalization for truth, though not as a universal rule because higher is better in some contexts. Probability of truth in fuzzy math can furthermore incorporate subjective certainty in a given fact being categorically true or false [5][6][7][8]. That certainty estimate is an additional factor, expressed as a certainty margin. It should include statistical confidence intervals when they are available. Intuition may be a component of both prevalence and certainty (confidence) estimates, but this makes the generalization subjectively rather than objectively true to the degree that intuition is a component.
Contextual Definitions
Certainty: Used as a kind of confidence in this proposal, distinct from specificity.
Certainty Estimate: A subjective estimate of belief from 0% to 100% that a prevalence assertion holds true.
Certainty Margin: A converse measure of the certainty estimate as uncertainty and subjective estimate that an assertion won't hold true. This may be drawn from a statistical confidence interval.
Confidence Interval: An objective statistical uncertainty interval, such as the standard deviation of a subject set.
Subject Set: The population, class, kind, collection, (set), or domain of entities to which a general statement is intended to apply, under a stated or implied scope. A statistical population is a subject set that has been defined for sampling or quantitative inference.
Metametastudy: A study regarding metastudies or collating metastudies together.
Detailed Definition
The Majority Generalization Rule: An unqualified empirical generalization about a defined subject set is licensed as true only when sufficient certainty as confidence supports a proposition that more than half of that subject set has the relevant attribute at a specific threshold prevalence [1][2]. Inversely a statement is licensed false when the proposition is certainly false for more than half of that subject set. Otherwise, the claim should be qualified, narrowed in scope, or stated as unknown. This constraint on generic statements is proposed as a heuristic to clarify scholarly and debate language. So, this heuristic is a normative evidence constraint. There is a layer of complexity that is easy to miss. When it relies on scientific studies, it operates at the metametastudy level rather than merely at the metastudy level, because any metastudy relied on is done so in further context. This additional layer is subjective rather than objective to some degree, but can be minimal, so objectivity could survive to be the dominant quality of the generalization. However, the degree to which the generalization is objective is likewise subjective as a personal perspective.
Trial Formula
In fuzzy logic truth, the degree that an object belongs to a vague category is measured [3][4]. For the Majority Generalization Rule, we have the chosen category of truth regarding a subject set, as an analogy. Prevalence is how often a property is true of a study subject or population. First, we use prevalence as an inference. Secondly, the certainty estimate measures certainty or uncertainty about whether that prevalence estimate is true. So, a certainty measure is also proposed. We may assign a certainty estimate using a percentage from 0% to 100% as a certainty margin, then invert it and divide it in half as an uncertainty margin. This establishes thresholds of belief at 50% +- that uncertainty margin. A belief metric past the uncertainty margin adjustment corresponds to scholarly generic truth under the Majority Generalization Rule, while anything below the uncertainty margin corresponds with a scholarly generic falsehood [5][6][7].
Certainty estimate in the formula here means subjective intuition as to the probability of being wrong, based on an experience perspective, which is different from the underlying statistic, but statistical margin of error can still be used as a certainty estimate basis. Scientific statistical study confidence intervals are fundamentally different in meaning from a certainty estimate, but may be used as part of this process if available because there is some to a near complete overlap in meaning depending on the application. Specifically, it is rendered subjective to some variable degree. If someone found a fundamental flaw in the study prevalence metric, then there would be near zero certainty estimate despite the confidence interval metric. As a starting point, a value based on about one standard deviation is suggested as a trial value, which corresponds with a 68% certainty estimate. It is important to understand that the choice of sigma and corresponding standard deviation value is subjective and any study can be wrong "by bad luck". As already stated, while using a confidence interval from a statistical study is fair, it carries a different meaning. One could easily be tempted to directly apply a statistical confidence interval to the prevalence metric as the adjustment, and one could arrive at a more objective measure doing something like that, but the formula suggested here would have to change. This proposed method is designed as a subjective metric in part because statistics might be entirely unavailable or incomplete, so the suggestion here works without a completed statistical study with a statistical confidence interval. Furthermore, a more rigorous method of estimation could be further developed later by metastudy analysis for better optimization and objectivity. So, this would require metametastudy as a further point of exploration, which is one reason why a mere metastudy confidence interval value isn't being suggested yet as a starting point.
Here is a three-valued truth logic scenario as true, false, or unknown, as a proposition belief calculation [9]. First, assign a prevalence estimate metric p as statistical probability for a prevalence domain, ranging from 0% to 100%, where p represents how frequently a property is true for a defined subject set or population. Second, estimate and assign a (subjective) evidential certainty estimate value c, represented on a belief strength scale from 0% to 100%. Calculate the uncertainty margin u as (100% - c)/2. The division is because the value shall be applicable equally and separately to both positive and negative margins. Let T, the truth threshold, be 50% + u. Let F, the false threshold, be 50% - u. The metric as a belief generalization can then be determined. If p > T, then the belief is qualified as a scholarly generic truth. If p < F, then the belief is qualified as a scholarly (generalized) falsehood. Otherwise, the statement is unknown when p >= F and p <= T, and considered an overlap with the uncertainty margin. Three-valued truth logic is expected to be useful for a broader range than two-valued truth logic [9]. This equation operates symmetrically to provide a symmetric definition of opposites for true and false as a metric of belief.
Clarifications and Closing Thoughts
This rule does not imply any specific domain or scope of the subject matter. The subjective domain and range may have many unstated implications such as kind-level properties, context, exceptions, and temporal conditions. For example, the statement "Dogs bark" implies a domain of all dogs, with the implication of a domain of dogs alive today on Earth within the time span of their full lives as the scope. The domain and scope may change a truth value such as with "We ran a study on dogs in Springfield. We are about 99% certain that less than 49% of those dogs barked during a randomly selected observation. Therefore, Springfield dogs don't bark." That subject is dogs, with a domain of dogs in Springfield within the time range of the observation. Furthermore, notice that this means for scholarly writing that any statements with less than 50% certainty estimate or margin of error require (explicit) qualifiers. Presuming, assuming, or implying that these non-barking dogs could bark would be a belief about ability rather than frequency, so is ignored for these generalizations. This may not have been a study to find mute dogs; and certainly enables a conclusion about frequency regardless of capability. The study purpose and reasoning without further information would be speculative.
References to material in papers that don't follow this rule such as "mosquitoes cause malaria" should be notated when citing the material.
Finally, there are ways to provide assertion strength for any generic statement. These are beyond the scope of this rule. This includes:
Define the subject set. Extending the parasite example above: parasite species, individual parasites, parasite populations, etc.
Define the subject attribute. Extending the parasite example above: transmit disease, currently carries a pathogen, commonly transmits disease, and so on.
Define the certainty confidence threshold both as subjective experience confidence and objective numeric confidence.
When a confidence qualifier is used, a "Consistent Treatment of Uncertainties" paper by the Intergovernmental Panel on Climate Change guides helpful qualifiers that assign specific numeric values to confidence and likelihood [10]. This also advances the concept of connecting statistics to improve precision and reduce confusing language choices.
Citations
[1] "Similar to the gradable adjectives model, the generics model adopts a threshold semantics where the generic is true just in case the prevalence of the feature f within the category k exceeds the relevant threshold"
Scontras, Gregory, Michael Henry Tessler, and Michael Franke.
A practical introduction to the Rational Speech Act modeling framework. arXiv:2105.09867 [cs.CL]. 2021.
https://arxiv.org/abs/2105.09867
[2] "a very similar—but not identical—distinction among different types of generics: 'characteristic,' 'majority,' and 'striking.' Characteristic generics can be true even when they apply only to a minority of members, e.g., 'Ducks lay eggs,' whereas 'majority' generics can be understood in terms of statistical frequency."
Thomas Marré
Biological Functions, Generics, and Explanation. Synthese 206, no. 1 (2025): 1–25. 2025.
https://doi.org/10.1007/s11229-025-05126-z
[3] "We present mathematical fuzzy logic as a set of logical tools that can be used to model reasoning with graded predicates, and discuss a philosophical account of vagueness that makes use of these tools."
Petr Cintula, Carles Noguera, and Nicholas J. J. Smith
A Logical Framework for Graded Predicates. 2017.
https://link.springer.com/chapter/10.10 ... -49130-1_1
[4] "There is wide agreement that a term is vague to the extent that it has borderline cases. This makes the notion of a borderline case crucial in accounts of vagueness."
Sorensen, Roy.
*Vagueness.* The Stanford Encyclopedia of Philosophy (Winter 2023 Edition), Edward N. Zalta & Uri Nodelman (eds.). 2023.
https://plato.stanford.edu/archives/win ... vagueness/
[5] "The aim of this paper is to show a way of applying Bayesian inference on interval fuzzy data assuming that all the time we work with interval probabilities."
Juan Miguel León-Rojas and Montaña Morales Morgado
Interval Fuzzy Bayesian Inference. 2003.
https://link.springer.com/content/pdf/1 ... 44465-7_69
[6] "It is necessary and challenging to represent the probabilities of fuzzy events and make inferences between them based on a Bayesian network. Motivated by such real applications, in this paper, we first define the interval probabilities of type-2 fuzzy events."
Yue Liu, Y. Su, and Y. Yao
Probabilistic representation and approximate inference of type-2 fuzzy events in Bayesian networks with interval probability parameters. 2009.
https://www.sciencedirect.com/science/a ... 7408007513
[7] "We propose fuzzy Bayesian inference on the basis of interval Bayesian inference ... [addressing] normalized fuzzy Bayesian inference."
Yin Pan, G.J. Klir, Bo Yuan.
Bayesian Inference Based on Fuzzy Probabilities. 1996.
https://ieeexplore.ieee.org/document/552625
[8] “A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one.”
Lotfi A. Zadeh
Fuzzy Sets. Information and Control, 8(3), 338–353. 1965.
https://doi.org/10.1016/S0019-9958(65)90241-X
[9] “For example, in three-valued logic three truth values have been employed. These are TRUTH, FALSE, and UNKNOWN represented by 1, 0 and 0.5 respectively. [...] The element of a fuzzy set that has a grade of membership equal to 0.5 is known as the crossover point.”
M. Perkowski. Citing Dimitris Tsaptsino.
Fuzzy Sets and Logic. Portland State University, Department of Electrical and Computer Engineering, Lecture Slides FL001. n.d.
http://web.cecs.pdx.edu/~mperkows/CLASS ... /FL001.PDF
https://web.cecs.pdx.edu/~mperkows/ (author data)
[10] “The qualifiers used to express a level of confidence are very low, low, medium, high, and very high. [...] describing quantified uncertainties through the likelihood scale [...]
Virtually certain 99–100% probability
Very likely 90–100% probability
Likely 66–100% probability About as likely as not
33 to 66% probability
Unlikely 0–33% probability
Very unlikely 0–10% probability
Exceptionally unlikely 0–1% probability"
Michael D. Mastrandrea, Katharine J. Mach, et al. IPCC.
Consistent Treatment of Uncertainties. Climatic Change 108(4), 675–691. 2011.
https://link.springer.com/article/10.10 ... 011-0178-6
https://link.springer.com/article/10.10 ... 6/tables/1 (Likelihood Qualifiers)
https://link.springer.com/article/10.10 ... /figures/2 (Confidences Qualifiers)
The Majority Generalization Rule is here proposed to reduce confusion in scholarly and debate communications: A statement is generally true (without need for qualifiers) only when it is estimated to be confidently true more than half of the time. This applies to a subject domain with a defined scope for which the prevalence within an underlying statistical subject set can be estimated with certain confidence. This rule constrains generalizations in scholarly text by applying a confident majority threshold to generic statements. The constraint is a high-utility heuristic for distinguishing supported generalizations from speculation in scholarly text. In this proposal domain, true means 'supported as a majority generalization'; false means 'confidently false as a majority generalization'; unknown means the prevalence overlaps with the uncertainty range between true and false. So, minority truth generic statements are encouraged to be rejected as invalid without qualifiers. This rule reduces presumptions, assumptions, and implications of no utility to the goal of the language used. Generic statements can traditionally encode characteristic, dispositional, and striking-property information [2]. Those minority generics should be rejected as too imprecise or confusing in scholarly language and debate, instead primarily offering frequency information [1][2]. This writing offers a formula as a trial formula for more precise language.
Qualification for Minority Cases
In colloquial written language, one could truthfully say "Parasites carry disease.". But in scholarly language under Majority Generalization Rule, this should be considered a false statement or unknown (including speculation). Instead, one would say "Some parasites carry disease.". The qualification here is "some". Only if one did a field study that confidently showed that most parasites are currently carrying a disease, would it then be considered true. This statement should be treated as false or unknown unless evidence shows that disease-carrying is true of more than half of a broad global sample of many kinds of parasites with adequate certainty as confidence. Stated in another way, this is because a general statement is only licensed when both its estimated prevalence and the certainty in that estimate are sufficiently high. If the evidence does not establish majority prevalence, the statement must be qualified with terms such as 'may', 'can', 'often', or 'some'.
Categorical Options
Statements may be formed into more statistically objective statements using either two-value logic (true or false) or three-value logic (true or unknown or false). We here borrow the dominant border from fuzzy logic to use 50% as the main dividing line of categorical membership, as a point of unknown [4][9]. This is furthermore between majority and minority prevalence, which is also one of the potential semantic meanings from fuzzy logic of the category membership [3][4]. That 50% dividing line is also associated with majority and minority in democratic voting as collective decision-making graded support in some contexts such as generalizing a collective will.
"50% or more" as the minimum truth condition is avoided, because otherwise one could make confusing, paradoxical claims such as "It is true that the result of a coin flip is heads. It is also true that the result of a coin flip is tails.". Instead, only an unfair coin could correctly result in the statement "our unfair coin flips to heads" with 'more' being implied there as the generalization. Furthermore, by using such generalizations, one then has a much higher obligation to use qualifiers where they'd otherwise be implied. So, "our unfair coin always flips to heads" would then be the better language when using this language style in such a case. For full clarity, state the generalization style being used.
Therefore, 50% is the recommended minimum inflection point of generalization for truth, though not as a universal rule because higher is better in some contexts. Probability of truth in fuzzy math can furthermore incorporate subjective certainty in a given fact being categorically true or false [5][6][7][8]. That certainty estimate is an additional factor, expressed as a certainty margin. It should include statistical confidence intervals when they are available. Intuition may be a component of both prevalence and certainty (confidence) estimates, but this makes the generalization subjectively rather than objectively true to the degree that intuition is a component.
Contextual Definitions
Certainty: Used as a kind of confidence in this proposal, distinct from specificity.
Certainty Estimate: A subjective estimate of belief from 0% to 100% that a prevalence assertion holds true.
Certainty Margin: A converse measure of the certainty estimate as uncertainty and subjective estimate that an assertion won't hold true. This may be drawn from a statistical confidence interval.
Confidence Interval: An objective statistical uncertainty interval, such as the standard deviation of a subject set.
Subject Set: The population, class, kind, collection, (set), or domain of entities to which a general statement is intended to apply, under a stated or implied scope. A statistical population is a subject set that has been defined for sampling or quantitative inference.
Metametastudy: A study regarding metastudies or collating metastudies together.
Detailed Definition
The Majority Generalization Rule: An unqualified empirical generalization about a defined subject set is licensed as true only when sufficient certainty as confidence supports a proposition that more than half of that subject set has the relevant attribute at a specific threshold prevalence [1][2]. Inversely a statement is licensed false when the proposition is certainly false for more than half of that subject set. Otherwise, the claim should be qualified, narrowed in scope, or stated as unknown. This constraint on generic statements is proposed as a heuristic to clarify scholarly and debate language. So, this heuristic is a normative evidence constraint. There is a layer of complexity that is easy to miss. When it relies on scientific studies, it operates at the metametastudy level rather than merely at the metastudy level, because any metastudy relied on is done so in further context. This additional layer is subjective rather than objective to some degree, but can be minimal, so objectivity could survive to be the dominant quality of the generalization. However, the degree to which the generalization is objective is likewise subjective as a personal perspective.
Trial Formula
In fuzzy logic truth, the degree that an object belongs to a vague category is measured [3][4]. For the Majority Generalization Rule, we have the chosen category of truth regarding a subject set, as an analogy. Prevalence is how often a property is true of a study subject or population. First, we use prevalence as an inference. Secondly, the certainty estimate measures certainty or uncertainty about whether that prevalence estimate is true. So, a certainty measure is also proposed. We may assign a certainty estimate using a percentage from 0% to 100% as a certainty margin, then invert it and divide it in half as an uncertainty margin. This establishes thresholds of belief at 50% +- that uncertainty margin. A belief metric past the uncertainty margin adjustment corresponds to scholarly generic truth under the Majority Generalization Rule, while anything below the uncertainty margin corresponds with a scholarly generic falsehood [5][6][7].
Certainty estimate in the formula here means subjective intuition as to the probability of being wrong, based on an experience perspective, which is different from the underlying statistic, but statistical margin of error can still be used as a certainty estimate basis. Scientific statistical study confidence intervals are fundamentally different in meaning from a certainty estimate, but may be used as part of this process if available because there is some to a near complete overlap in meaning depending on the application. Specifically, it is rendered subjective to some variable degree. If someone found a fundamental flaw in the study prevalence metric, then there would be near zero certainty estimate despite the confidence interval metric. As a starting point, a value based on about one standard deviation is suggested as a trial value, which corresponds with a 68% certainty estimate. It is important to understand that the choice of sigma and corresponding standard deviation value is subjective and any study can be wrong "by bad luck". As already stated, while using a confidence interval from a statistical study is fair, it carries a different meaning. One could easily be tempted to directly apply a statistical confidence interval to the prevalence metric as the adjustment, and one could arrive at a more objective measure doing something like that, but the formula suggested here would have to change. This proposed method is designed as a subjective metric in part because statistics might be entirely unavailable or incomplete, so the suggestion here works without a completed statistical study with a statistical confidence interval. Furthermore, a more rigorous method of estimation could be further developed later by metastudy analysis for better optimization and objectivity. So, this would require metametastudy as a further point of exploration, which is one reason why a mere metastudy confidence interval value isn't being suggested yet as a starting point.
Here is a three-valued truth logic scenario as true, false, or unknown, as a proposition belief calculation [9]. First, assign a prevalence estimate metric p as statistical probability for a prevalence domain, ranging from 0% to 100%, where p represents how frequently a property is true for a defined subject set or population. Second, estimate and assign a (subjective) evidential certainty estimate value c, represented on a belief strength scale from 0% to 100%. Calculate the uncertainty margin u as (100% - c)/2. The division is because the value shall be applicable equally and separately to both positive and negative margins. Let T, the truth threshold, be 50% + u. Let F, the false threshold, be 50% - u. The metric as a belief generalization can then be determined. If p > T, then the belief is qualified as a scholarly generic truth. If p < F, then the belief is qualified as a scholarly (generalized) falsehood. Otherwise, the statement is unknown when p >= F and p <= T, and considered an overlap with the uncertainty margin. Three-valued truth logic is expected to be useful for a broader range than two-valued truth logic [9]. This equation operates symmetrically to provide a symmetric definition of opposites for true and false as a metric of belief.
Clarifications and Closing Thoughts
This rule does not imply any specific domain or scope of the subject matter. The subjective domain and range may have many unstated implications such as kind-level properties, context, exceptions, and temporal conditions. For example, the statement "Dogs bark" implies a domain of all dogs, with the implication of a domain of dogs alive today on Earth within the time span of their full lives as the scope. The domain and scope may change a truth value such as with "We ran a study on dogs in Springfield. We are about 99% certain that less than 49% of those dogs barked during a randomly selected observation. Therefore, Springfield dogs don't bark." That subject is dogs, with a domain of dogs in Springfield within the time range of the observation. Furthermore, notice that this means for scholarly writing that any statements with less than 50% certainty estimate or margin of error require (explicit) qualifiers. Presuming, assuming, or implying that these non-barking dogs could bark would be a belief about ability rather than frequency, so is ignored for these generalizations. This may not have been a study to find mute dogs; and certainly enables a conclusion about frequency regardless of capability. The study purpose and reasoning without further information would be speculative.
References to material in papers that don't follow this rule such as "mosquitoes cause malaria" should be notated when citing the material.
Finally, there are ways to provide assertion strength for any generic statement. These are beyond the scope of this rule. This includes:
Define the subject set. Extending the parasite example above: parasite species, individual parasites, parasite populations, etc.
Define the subject attribute. Extending the parasite example above: transmit disease, currently carries a pathogen, commonly transmits disease, and so on.
Define the certainty confidence threshold both as subjective experience confidence and objective numeric confidence.
When a confidence qualifier is used, a "Consistent Treatment of Uncertainties" paper by the Intergovernmental Panel on Climate Change guides helpful qualifiers that assign specific numeric values to confidence and likelihood [10]. This also advances the concept of connecting statistics to improve precision and reduce confusing language choices.
Citations
[1] "Similar to the gradable adjectives model, the generics model adopts a threshold semantics where the generic is true just in case the prevalence of the feature f within the category k exceeds the relevant threshold"
Scontras, Gregory, Michael Henry Tessler, and Michael Franke.
A practical introduction to the Rational Speech Act modeling framework. arXiv:2105.09867 [cs.CL]. 2021.
https://arxiv.org/abs/2105.09867
[2] "a very similar—but not identical—distinction among different types of generics: 'characteristic,' 'majority,' and 'striking.' Characteristic generics can be true even when they apply only to a minority of members, e.g., 'Ducks lay eggs,' whereas 'majority' generics can be understood in terms of statistical frequency."
Thomas Marré
Biological Functions, Generics, and Explanation. Synthese 206, no. 1 (2025): 1–25. 2025.
https://doi.org/10.1007/s11229-025-05126-z
[3] "We present mathematical fuzzy logic as a set of logical tools that can be used to model reasoning with graded predicates, and discuss a philosophical account of vagueness that makes use of these tools."
Petr Cintula, Carles Noguera, and Nicholas J. J. Smith
A Logical Framework for Graded Predicates. 2017.
https://link.springer.com/chapter/10.10 ... -49130-1_1
[4] "There is wide agreement that a term is vague to the extent that it has borderline cases. This makes the notion of a borderline case crucial in accounts of vagueness."
Sorensen, Roy.
*Vagueness.* The Stanford Encyclopedia of Philosophy (Winter 2023 Edition), Edward N. Zalta & Uri Nodelman (eds.). 2023.
https://plato.stanford.edu/archives/win ... vagueness/
[5] "The aim of this paper is to show a way of applying Bayesian inference on interval fuzzy data assuming that all the time we work with interval probabilities."
Juan Miguel León-Rojas and Montaña Morales Morgado
Interval Fuzzy Bayesian Inference. 2003.
https://link.springer.com/content/pdf/1 ... 44465-7_69
[6] "It is necessary and challenging to represent the probabilities of fuzzy events and make inferences between them based on a Bayesian network. Motivated by such real applications, in this paper, we first define the interval probabilities of type-2 fuzzy events."
Yue Liu, Y. Su, and Y. Yao
Probabilistic representation and approximate inference of type-2 fuzzy events in Bayesian networks with interval probability parameters. 2009.
https://www.sciencedirect.com/science/a ... 7408007513
[7] "We propose fuzzy Bayesian inference on the basis of interval Bayesian inference ... [addressing] normalized fuzzy Bayesian inference."
Yin Pan, G.J. Klir, Bo Yuan.
Bayesian Inference Based on Fuzzy Probabilities. 1996.
https://ieeexplore.ieee.org/document/552625
[8] “A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one.”
Lotfi A. Zadeh
Fuzzy Sets. Information and Control, 8(3), 338–353. 1965.
https://doi.org/10.1016/S0019-9958(65)90241-X
[9] “For example, in three-valued logic three truth values have been employed. These are TRUTH, FALSE, and UNKNOWN represented by 1, 0 and 0.5 respectively. [...] The element of a fuzzy set that has a grade of membership equal to 0.5 is known as the crossover point.”
M. Perkowski. Citing Dimitris Tsaptsino.
Fuzzy Sets and Logic. Portland State University, Department of Electrical and Computer Engineering, Lecture Slides FL001. n.d.
http://web.cecs.pdx.edu/~mperkows/CLASS ... /FL001.PDF
https://web.cecs.pdx.edu/~mperkows/ (author data)
[10] “The qualifiers used to express a level of confidence are very low, low, medium, high, and very high. [...] describing quantified uncertainties through the likelihood scale [...]
Virtually certain 99–100% probability
Very likely 90–100% probability
Likely 66–100% probability About as likely as not
33 to 66% probability
Unlikely 0–33% probability
Very unlikely 0–10% probability
Exceptionally unlikely 0–1% probability"
Michael D. Mastrandrea, Katharine J. Mach, et al. IPCC.
Consistent Treatment of Uncertainties. Climatic Change 108(4), 675–691. 2011.
https://link.springer.com/article/10.10 ... 011-0178-6
https://link.springer.com/article/10.10 ... 6/tables/1 (Likelihood Qualifiers)
https://link.springer.com/article/10.10 ... /figures/2 (Confidences Qualifiers)