Uncertainty and Risk in Civil Engineering Practice

Life is short and the art long; the occasion instant, experiment perilous, decision difficult. (Hippocrates, as quoted in Fox)

See also Project Definition, Project Life Cycle and Phase Models, Project Delivery Methods

See also Technical Outcomes for Civil Engineering: Risk and Uncertainty

Basic considerations

Civil engineering practice requires decision-making in environments characterized by incomplete, imperfect, and evolving knowledge. Engineers routinely act in the presence of uncertainty related to physical phenomena, economic constraints, regulatory requirements, and human behavior. The professional problem is not the elimination of uncertainty, which is impossible in practice, but its reduction, partitioning, and management to a degree sufficient to support responsible action.

In this chapter, risk is treated as a bounded and decision-relevant subset of broader uncertainty. Uncertainty that can be reduced, structured, prioritized, and acted upon using professional knowledge, models, and experience is termed risk. Related but distinct is the concept of hazard, which denotes a source or condition with the potential to cause harm, independent of likelihood or consequence.

This framing aligns with interdisciplinary treatments of risk analysis, which emphasize the role of uncertainty in shaping social, technical, and governance systems over long historical periods. McDaniels and Small describe risk analysis as emerging from the need to act despite imperfect knowledge and argue that societies have progressively shifted from fatalistic acceptance of uncertainty toward its active management through structured analysis and decision frameworks.

Within civil engineering, this evolution has produced specialized practices focused on managing uncertainty through modeling, design standards, safety factors, reliability analysis, and project controls. These practices rely not only on domain knowledge of materials, loads, and systems, but also on meta-knowledge: knowledge about the limitations, reliability, and appropriate use of engineering knowledge itself.

Semantic, epistemic, and logical frameworks

Clear terminology is a prerequisite for coherent risk analysis. In professional practice, inconsistent or interchangeable use of terms such as uncertainty, risk, hazard, error, and bias can obscure decision responsibility and complicate communication among stakeholders.

The semantic framework adopted in this chapter proceeds in stages:

1. distinction between certainty and uncertainty;
2. refinement of uncertainty into epistemic, aleatory, and behavioral components;
3. use of error and bias as operational proxies for uncertainty;
4. transition from general uncertainty to discipline-specific risk taxonomies.

Beyond this point, the discussion shifts from semantics to logical frameworks, where uncertainty is partitioned into structured taxonomies suitable for analysis, communication, and decision-making in engineering projects.

Semantics framework

Certainty and uncertainty

Certainty is commonly described as an epistemic property of belief, implying immunity from doubt or criticism. In practical contexts, however, absolute certainty is neither attainable nor desirable. As Mill observed, assurance sufficient for the purposes of human life does not require freedom from all doubt, but rather justified confidence grounded in experience and evidence.

Uncertainty arises whenever beliefs or propositions are subject to doubt, criticism, or incomplete support. Contemporary treatments emphasize that uncertainty is not merely a feature of the external world, but also a cognitive condition reflecting awareness of incomplete knowledge. Han et al. characterize uncertainty as a meta-cognitive state: knowing that one does not know, or knowing that one’s knowledge may be incomplete or unreliable.

In professional reasoning, uncertainty manifests as the potential for error in observation, modeling, judgment, and action. Reasoning under uncertainty therefore differs fundamentally from reasoning under certainty: conclusions are provisional, contingent, and subject to revision as new information becomes available.

Error, bias, and uncertainty

In applied disciplines, uncertainty is rarely managed directly. Instead, it is addressed through its observable proxies: error and bias. Error refers to deviation between an estimate, measurement, or prediction and a reference value or observed outcome. Bias denotes systematic or directional error arising from model structure, data selection, judgment, or institutional practice.

Error can often be quantified, propagated, and reduced using analytical and statistical methods. Bias is more difficult to detect and correct, frequently requiring theoretical analysis, validation studies, or comparison across independent methods. The management of uncertainty in civil engineering therefore depends critically on understanding how error and bias arise, how they propagate through sequential decisions, and how they can be bounded through professional controls.

Propagation of error and uncertainty

Engineering decisions typically involve multi-step processes: observation, interpretation, modeling, design, construction, and operation. Errors introduced early in this chain can propagate and amplify downstream, while later errors may have limited impact depending on system sensitivity and redundancy.

Professional practice therefore emphasizes identifying where uncertainty enters the decision sequence and prioritizing error and bias reduction where it has the greatest effect on outcomes. This ordering principle underlies practices such as conservative design assumptions, staged investigations, peer review, and independent verification.

Epistemic, aleatory, and behavioral uncertainty

For practical purposes, uncertainty in civil engineering can be partitioned into three broad components:

• Epistemic uncertainty, arising from incomplete knowledge, limited data, or imperfect models;
• Aleatory uncertainty, reflecting inherent variability in physical processes;
• Behavioral uncertainty, associated with human judgment, decision-making, and organizational interaction.

This partitioning supports targeted mitigation strategies. Epistemic uncertainty may be reduced through investigation and analysis; aleatory uncertainty may be managed through design margins and reliability methods; behavioral uncertainty requires institutional controls, contractual clarity, and governance mechanisms.

From uncertainty to risk and hazard

Risk and hazard are not synonymous with uncertainty, but structured subsets of it. Risk refers to uncertainty that has been framed in terms of potential consequences and that is sufficiently bounded to support decision-making. Hazard denotes a source or condition with the potential to cause harm, independent of likelihood.

In this chapter, general risk is defined as:

a bounded subset of uncertainty associated with the potential for material error or bias when acting or choosing in physical, economic, or social environments, and that can be represented using coherent analytical or probabilistic frameworks and subjected to testing or validation.

This definition is discipline-neutral and applies across engineering, economics, medicine, and policy. Subsequent sections introduce discipline-specific taxonomies that adapt this general concept to the particular decision contexts of civil engineering projects.

Logical framework for specific risk taxonomies

Once uncertainty has been reduced to bounded, decision-relevant forms, it becomes possible to develop taxonomies that support analysis, communication, and management. Taxonomies are not required to be epistemically complete; rather, they must be logically distinct, practically exhaustive for their purpose, and aligned with decision needs.

The following sections examine how scientific, legal, economic, and engineering disciplines construct uncertainty and risk taxonomies, and how these perspectives inform civil engineering practice.

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Scientific risk taxonomies

Scientific inquiry proceeds by observing phenomena, proposing explanatory models, and testing those models against experience. At every stage of this process, uncertainty is present. Scientific risk taxonomies arise from the need to identify, distinguish, and manage different sources of uncertainty that affect observation, modeling, inference, and prediction.

Most scientific taxonomies of uncertainty have focused on uncertainties associated with data, parameters, and models, particularly those that can be represented statistically. While multiple taxonomies are possible, a common organizing principle is that uncertainty is environment-specific, phenomenon-oriented, and model-centric.

Environment-specific, phenomenon-oriented, model-centric taxonomy of uncertainty

A widely used scientific taxonomy distinguishes uncertainty according to its relationship to models and data:

• Parameter uncertainty

Parameter uncertainty arises from uncertainty in the numerical values of model inputs. These parameters may be physically unobservable, indirectly inferred, or only partially constrained by available data.

• • Experimental (observational) uncertainty results from variability in repeated measurements under nominally identical conditions.
  • Parametric variability reflects natural or contextual variability in input variables that are treated as fixed values within a model.

• Structural uncertainty

Structural uncertainty, also referred to as model inadequacy or model discrepancy, arises when a model fails to capture relevant features of the underlying phenomenon. Because all models are simplifications of reality, structural uncertainty is unavoidable and depends on model assumptions, scope, and fitness for purpose.

• • Structural uncertainty is distinct from parameter uncertainty and cannot be reduced solely by collecting additional data.
  • Inadequate model structure can lead to systematically misleading conclusions even when parameter estimates appear precise.

• Algorithmic (numerical) uncertainty

Algorithmic uncertainty arises from numerical approximations used to implement models computationally. Examples include discretization error, convergence error, and truncation error associated with numerical solution methods such as finite difference or finite element approximations.

• • These uncertainties depend**

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Uncertainty taxonomies in a legal environment

Legal decision-making operates under conditions of uncertainty that differ in purpose and structure from those of science or engineering. From a legal perspective, a taxonomy is a systematic scheme for distinguishing, ordering, and naming types within a subject field. Such taxonomies are not required to be epistemically complete; rather, they must be sufficient to organize the kinds of descriptive information commonly encountered by decision-makers and to support fair, effective, and defensible judgments.

Walker characterizes legal taxonomies of scientific and technical uncertainty as tools for cataloging uncertainty associated with descriptive assertions, particularly those involving cause-and-effect claims. These taxonomies emphasize clarity, logical distinction, and practical exhaustiveness over theoretical completeness.

Core components of legal uncertainty

Legal frameworks distinguish multiple, logically distinct sources of uncertainty that may affect descriptive assertions and decisions:

• Linguistic uncertainty

Linguistic uncertainty, often described as vagueness, arises from limitations of language rather than from data or models.

• • Context dependence occurs when the applicability of a statement depends on unstated or shifting contextual conditions.
  • Ambiguity arises when terms admit multiple meanings that cannot be resolved within the given context.
  • Underspecification occurs when descriptions lack sufficient detail to support a unique interpretation.

Linguistic uncertainty underlies all other components of legal uncertainty, as it affects how evidence, concepts, and models are interpreted.

• Conceptual uncertainty

Conceptual uncertainty arises from the choice and definition of concepts used to describe phenomena. Whenever predicates are selected to characterize an object, event, or condition, alternative conceptualizations are excluded. Conceptualization generates useful structure, but it can also constrain understanding by privileging certain interpretations over others. Poorly chosen or inadequately defined concepts can therefore introduce systematic distortion into legal and technical reasoning.

Conceptual uncertainty is closely related to issues of validity. A concept is valid to the extent that it accurately represents the phenomenon it is intended to describe. Conceptual bias, like systematic error in measurement, is difficult to detect and often recognizable only through theoretical analysis or comparison across alternative frameworks.

• Measurement and classification uncertainty

Measurement uncertainty arises when concepts are applied to specific cases through observation, assessment, or classification. Even when conceptual uncertainty has been minimized, misclassification may occur due to procedural limitations, observer judgment, or data quality. Measurement uncertainty affects the reliability and precision of descriptive assertions. Errors at this stage can propagate through subsequent reasoning and reduce the quality of causal inference or decision outcomes.

• Sampling uncertainty

Sampling uncertainty arises when conclusions about a population are inferred from a limited subset of observations. The uncertainty reflects the extent to which the sample represents the relevant population. In legal and regulatory contexts, sampling uncertainty affects the credibility of generalizations, forecasts, and risk characterizations based on empirical evidence.

• Modeling uncertainty

Modeling uncertainty arises when descriptive or causal claims rely on simplified representations of complex systems. This includes uncertainty associated with variable selection, functional relationships, and parameter specification.

• • Causal uncertainty arises when relationships between variables are inferred rather than directly observed. Legal decision-makers often rely on causal models to explain why an outcome occurred or to predict the consequences of alternative actions.
  • Epistemic choice refers to uncertainty introduced by selecting among competing interpretive frameworks or explanatory models.

Aggregation and interaction of uncertainties

The components of legal uncertainty do not operate independently. Linguistic, conceptual, measurement, sampling, and modeling uncertainties can combine in different ways to produce aggregate uncertainty affecting the credibility, relevance, and weight of evidence.

Legal taxonomies therefore focus not on eliminating uncertainty, but on making its sources explicit so that decision-makers can assess reliability, assign weight, and determine appropriate standards of proof or precaution.

Relationship to scientific and engineering practice

While legal taxonomies do not require quantitative completeness, they depend critically on validated scientific and engineering methods to support descriptive and causal claims. Scientific and engineering disciplines provide procedures and models capable of integrating qualitative and quantitative information into structured representations of uncertainty.

However, the legal use of such information remains distinct. The objective is not optimization or prediction per se, but defensible judgment under uncertainty. Validation, transparency, and clarity of assumptions are therefore central concerns at the interface between legal, scientific, and engineering domains.

These distinctions inform how uncertainty is framed, communicated, and relied upon in civil engineering projects subject to regulatory review, contractual interpretation, or dispute resolution.

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Economics uncertainty taxonomies

Like scientific and engineering treatments of uncertainty, economic approaches are environment-specific, phenomenon-oriented, and model-centric. In economics, the relevant environment is the market or sector of economic activity, and the phenomena of interest are human actions, organizational behavior, and manufactured constructs such as firms, contracts, and markets.

Economics does not address all forms of uncertainty. Instead, it focuses on subsets of uncertainty that possess causal connections to material outcomes, particularly those affecting prices, profits, losses, and allocation of resources. These subsets are commonly labeled as risks.

A foundational contribution to the economic treatment of uncertainty is associated with Frank Knight, who distinguished between measurable risk and unmeasurable uncertainty. Knight characterized uncertainty as arising from defects or limitations in knowledge, particularly managerial and organizational knowledge, rather than from inherent randomness alone. In this framework, risk refers to situations in which probabilities can be assigned to outcomes, while uncertainty refers to situations in which such assignment is not possible.

Knight defined risk as the class of uncertainties that can be reduced or managed through probabilistic reasoning and mechanisms such as insurance. By contrast, uncertainty in the strict sense reflects imperfect knowledge that cannot be reduced to numerical probabilities. For Knight, measuring uncertainty involved assessing both the likelihood of occurrence and the severity of economic consequences, while recognizing that some consequences remain fundamentally indeterminate.

Knight emphasized that economic decision-making is rarely conducted under conditions of complete ignorance or perfect information. Instead, action is typically based on partial knowledge, judgment, and opinion formed under uncertainty. Economic behavior therefore reflects degrees of belief and confidence rather than certainty, and the practical problem lies in managing variation in knowledge rather than eliminating uncertainty altogether.

In his analysis of insurance markets, Knight illustrated the difficulty of classifying certain uncertainties. Events such as accidental death could be treated probabilistically, while sickness or business failure resisted objective classification due to heterogeneity, judgment, and contextual dependence. Uncertainties that could not be classified or grouped were therefore not insurable and remained outside the domain of measurable risk.

Core distinctions in economic uncertainty

Economic treatments of uncertainty commonly rely on the following distinctions:

• Uncertainty about the economic environment, including prices, demand, and external conditions, is distinct from uncertainty about the behavior or choices of other actors.
• Risk is a subset of uncertainty in which the decision-maker can enumerate possible actions, identify possible outcomes, and assign probabilities to those outcomes.
• Uncertainty arises when such enumeration or probability assignment is not feasible due to incomplete knowledge, structural change, or reliance on judgment.

These distinctions form an idealized classification that has been influential but remains philosophically and empirically contested. Nevertheless, it provides a useful framework for distinguishing calculable exposure from judgment-dependent uncertainty in economic analysis.

Uncertainty, knowledge, and economic models

Like the theory of the firm, an economic theory of uncertainty seeks to explain and predict the behavior of specific subsets of uncertainty that affect economic outcomes. These theories examine how the presence, distribution, and variability of knowledge influence decision-making, pricing, and allocation.

Economic models often assume certainty or perfect information for analytical convenience. The economics of uncertainty exists as an alternative framework that explicitly incorporates imperfect information, unknown preferences, and judgment-based decisions when doing so improves explanatory or predictive power.

Unlike physical systems, economic activity produces artificial constructs such as firms, contracts, markets, and financial instruments. Uncertainty in economics therefore arises not only from external conditions but also from institutional design, incentives, and strategic interaction.

Risk, hazard, and moral hazard

Economic analysis also distinguishes between risk and hazard. While Knight used the term hazard in several contexts, the distinction becomes most explicit in the concept of moral hazard. Moral hazard arises when the presence of insurance or protection alters behavior, leading actors to take actions that increase exposure to loss.

In such cases, uncertainty is not merely a property of the environment but is generated endogenously through behavior and information asymmetry. This distinction is particularly relevant for engineering projects involving insurance, contracting, public–private partnerships, and regulatory oversight.

In summary, economic uncertainty taxonomies emphasize the role of partial knowledge, judgment, and incentives in shaping material outcomes. These frameworks inform how risk is identified, priced, transferred, or retained, and they provide an essential foundation for understanding cost, schedule, and financial uncertainty in civil engineering practice.

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Uncertainty and Risk in Policy and Regulatory Environment

"Hume's 'just reasoner', when faced with a difficult public policy decision in current times, would probably commission a risk analysis. But how could they incorporate 'a degree of doubt, caution, and modesty?" note-43 [36]

The problem of "decision-making in the face of uncertainty" is how to regulate based on incomplete information that has the potential to be materially inaccurate. A contextual assumption is that we need to evaluate decision rules for dealing with uncertainty. As a social enterprise, risk regulation, whatever its substantive objectives, should be as effective, efficient, and equitable as practically possible. These "three E's" form a set of "process objectives" or "meta-goals." From the uncertainty standpoint, causal information can be usefully divided into two major categories: information about groups and information about individuals.

Each category, group, and individual has its distinctive types of inherent uncertainty. These are logically distinct, generally independent, and cumulative, contributing to (?????)

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Engineering Uncertainty Taxonomies

Like scientific and economic concepts of uncertainty, engineering is environment-specific, phenomena-oriented, model-centric, external choice, and driven by decision-making in the face of uncertainty. Like economics and unlike science, engineering is not interested in all forms of uncertainty, only those with determinable and material consequences. Like economics, engineering conceives risk as a specific and unambiguous subset of uncertainty determined by measuring the probability of its occurrence and the severity of its consequences or economic impact, e.g., measurable risk of loss versus unmeasurable uncertainty consequences. Also, engineering recognizes the role information and, by extension, knowledge plays in constructing the built environment. note-44 [37]] Like economics, engineering differs from economics and scientific uncertainty in its interest in behavioral uncertainty.

• "Behavioral or interaction uncertainty is how individuals or organizations act or interact. Behavioral uncertainty arises from four sources: design uncertainty, requirement uncertainty, volitional uncertainty, and human errors." note-45 [38]]
  • A design uncertainty is a choice among alternatives over which an individual or group of individuals exercises direct control but has not yet decided upon.
  • Requirement uncertainty includes parameters of interest to and determined by the stakeholder, independent of the engineer or designer.
  • Volitional uncertainty is uncertainty about what the subject him/herself will decide. Other people’s future actions and conduct are not entirely predictable, particularly when dealing with other organizations.
  • Human errors occur during the development of a system or project due to blunders or mistakes by an individual or individuals.

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Practice Frameworks for Uncertainty and Risk in Civil Engineering Practice

PMIBoK

PMI does not define uncertainty, but the PMI Body of Knowledge uses the term in 45 places (uncertainty) and as describing properties in 8 places (uncertainties). [See Note 8] PMI does define "risk" and uses the term over 1,300 times in the same document. In its glossary, PMI also defines terms such as "threat" and "opportunity" but not events.[391 The BoK contextually defines uncertainty when it states that project risk has its origins in the uncertainty present in all projects. (Op. Cit., pg 309)
Uncertainty is contextually defined and taken as having the property of affecting project execution and the ability to meet stakeholder expectations. Uncertainty is more than the sum of all individual risks (known and unknown). The nature of this uncertainty is not defined, only its capacity to affect something else or its properties.
Risk is defined as " ... an uncertain event or condition that, if it occurs, has a positive or negative effect on one or more project objectives." (Ibid.) Beyond this formal definition, PMI adds context when, in its risk management section (Sec. 11 ), the BoK states that risk is a caused event or condition with multiple causes and impacts. The BoK even offers specific techniques for mapping cause-and-effect relationships and influence diagrams. (Sec. 11.2.2.5) Not as evident but equally important is the recognition that not only is risk caused but the impact is "triggered". The risk trigger is an event or situation that signals that it is about to occur (Glossary, pg. 566). Risk conditions are factors that affect or otherwise contribute to risk, such as project stakeholders. Risk implicitly retains some residual element of probability in it, or a value of less than 1.0 with risk that approaches a level of near certainty is termed "issues" or "realized risk" .(Op. Cit., pg 309). PMI also links risk to underlying variations in project outcomes. (Ibid.) Risk is based upon data and information which can be assessed as to its quality (Sec. 11.3.2.3). This analysis looks to determine the value of the information and examines the degree to which the risk information is understood as well as its " ... accuracy, quality, reliability and integrity ... " (Ibid.) Part of that is understanding the relationship between occurrence and impact as outlined in PMl's Figure 11-10 Risk Impact Matrix and developing a model for prioritizing risk products and setting the threshold for action.

Commentary:

For defining and identifying risk, PMI notes in Sec. 11.2.2.4 that " ... Every project and its plan is conceived and developed based on a set of hypotheses, scenarios, or assumptions." Part of the risk analysis process is identifying risks to the project, such as inaccuracy, instability, inconsistency, or incompleteness of assumptions. Similarly, the quality of management plans, as well as their consistency with others and the project objectives and assumptions are 11 •• .indicators of risk in the project." (Sec. 11.2.2.1)
Similarly, the role of engineering economics in risk is embedded or implied in the PMI definition of risk when the BoK discusses risk identification as a process of assessing which risks out of the total risk exposure for the project "may affect" the project and distilling their characteristics. Embedded in this statement is the concept that not all risks can cause economic impacts (positive and negative) to the project. Also embedded is the requirement to understand risk characteristics to identify project risk (Cost and schedule). Lastly, risk identification is an iterative process of incrementing project risk documentation, much like project scope, cost, and schedule documentation.
Lastly, PMI acknowledges the complex constitution of risk aggregates or the "sum" of all risks in its material, but is it a sum? Risk information has descriptive and explanatory content. These are knowledge and meta-knowledge components. Understanding project assumptions is project-specific knowledge, but identifying risks from assumption inaccuracy, instability, inconsistency, or incompleteness requires professional or program meta-knowledge.
Uncertainty exists in all projects, but risk, a subset of uncertainty with material economic impacts, is a key meta-knowledge element for civil engineering practice.

Software Development Process- Risk Focused

The Unified Process requires the project team to focus on addressing the most critical risks early in the project life cycle. The deliverables of each iteration, especially in the Elaboration phase, must be selected to ensure that the greatest risks are addressed first.

American Society of Civil Engineers (ASCE)

ASCE does not define uncertainty, but the ASCE CE Body of Knowledge (CEBoK) uses the term in 22 places (uncertainty) and describes properties in 13 places (uncertainties).
The CEBoK does not use the phrase "uncertainty and risk" as PMI and AACE do in their publication. ASCE reverses the two (Risk and uncertainty) in twelve places regarding technical outcomes (Outcome 12). A fundamental difference between PMI and ISO is that risk is considered a subset of uncertainty. ASCE does not define "risk" but uses the term 25 times in the same document. Almost all are in the context of variation of design parameters and not in the classical context of cost, schedule uncertainty, and risk.

Commentary:

...

American Association of Cost Engineers (AACE)

AACE defined uncertainty, risk, and other related terms in its Risk Management Dictionary. AACE's overall strategy was to define risk and other terms that stem from base uncertainty. AACE first defined uncertainty as .... "(t)he total range of events that may happen and produce risks (including both threats and opportunities) affecting a project (see opportunities, events, conditions, risk, and threats" where the following sub-definitions apply:

• • Biases--A lack of objectivity based on the individual's position or perspective. Systematic and predictable relationships between a person's opinion or statement and his/her underlying knowledge or circumstances. Note: there may be "system biases" as well as "individual biases."
  • Condition (Uncertain Condition)-Any specific identifiable circumstance (such as the rate of inflation or the quality of labor available) that might affect the outcome of the project
  • Event (Uncertain Condition)-is a specific identifiable action (such as a large government project being started in the same labor area as your project) or an act of nature that might happen and that (if it does happen) could affect the outcome of the project.
  • Opportunities are Uncertain events that could improve the results or improve the probability that the desired outcome will happen
  • Threats are Uncertain events that are potentially negative or reduce the probability that the desired outcome will happen.
    • AACE defines Risk as an "ambiguous term" (sic) that is synonymous with uncertainty or a negative subset such as "threats", or an overall negative impact of all possible uncertainties.

Commentary:

...

Limitations of the definition

...

Working definition

...

Beneficial outcomes

...

See also

Wiki articles on the project.

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Notes

Implicit in this definition of uncertainty as a "state of' is ...

" ... a conceptualization of uncertainty as a subjective, cognitive experience of people--a state of mind rather than a feature of the objective world. Furthermore, the defining feature of this state appears to be a lack of knowledge about some aspect of reality. Importantly, however, the concept of uncertainty also implies a subjective consciousness or awareness of one's lack of knowledge, without which one could not feel uncertain; uncertainty is a form of "meta-cognition" ... ( or alternatively, its main component, meta-knowledge) ... -a knowing about knowing." (Han, 2011, op. cit., Emphasis added)

5. Cox wanted his system to satisfy the following conditions:

Divisibility and comparability- The plausibility of a statement is a real number and depends on the information we have related to the statement.

Common sense - Plausibilities should vary sensibly with the assessment of plausibilities in the model.

Consistency - If the plausibility of a statement can be derived in many ways, all the results must be equal.

Cox's theorem has come to be used as one of the justifications for the use of Bayesian probability theory. (For example, in Jaynes Jayne, Probability Theory: The Logic of Science, Cambridge University Press (2003). -preprint version (1996) at http://omega.albany.edu:8008/JaynesBook.html; Chapters 1 to 3 of published version at http://bayes.wustl.edu/etj/prob/book.pdf

6. The utility of existing taxonomies of uncertainty in civil engineering and, by extension, "risk" has been unsatisfactory for this very reason. This is particularly true in cases where the source of the uncertainty is conflicting versus incomplete information. See Regan, et. al.,"A taxonomy and treatment of uncertainty for ecology and conservation biology" (2002) for a survey of scientific taxonomies for uncertainty.
7. Regan et al. argue that " ... genuine examples of this kind of uncertainty are hard to find. Even classic cases of random experiments like coin tosses and the throwing of dice are deterministic; it is just that we do not have enough information about the dynamic processes and initial conditions to make any sensible estimates about the outcomes. Such processes are, for all intents and purposes, inherently random, but they are not genuinely inherently random. For similar reasons, complex systems such as ecosystems and weather patterns are unlikely to be inherently random. Similarly, chaotic systems are entirely deterministic. They are unpredictable because the deterministic processes generating them and the relevant initial conditions are hard to fully specify (see Stewart 1989, Sugihara et al. 1990)." (Regan (2002)
8. The CEBoK uses the phrase "uncertainty and risk" twice in reference to scheduling (Section 6.5,2) and cost estimating (Section 7.2,2) as well as reversing the two (Risk and Uncertainty) also in two places in discussing project life cycle (both in Section 2.4.1)