Default logic

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Default_logic 

Default logic is a non-monotonic logic proposed by Raymond Reiter to formalize reasoning with default assumptions.

Default logic can express facts like “by default, something is true”; by contrast, standard logic can only express that something is true or that something is false. This is a problem because reasoning often involves facts that are true in the majority of cases but not always. A classical example is: “birds typically fly”. This rule can be expressed in standard logic either by “all birds fly”, which is inconsistent with the fact that penguins do not fly, or by “all birds that are not penguins and not ostriches and ... fly”, which requires all exceptions to the rule to be specified. Default logic aims at formalizing inference rules like this one without explicitly mentioning all their exceptions.

Syntax of default logic

A default theory is a pair ${\displaystyle \langle W,D\rangle }$. W is a set of logical formulas, called the background theory, that formalize the facts that are known for sure. D is a set of default rules, each one being of the form:

${\displaystyle {\frac {\mathrm {Prerequisite:Justification} _{1},\dots ,\mathrm {Justification} _{n}}{\mathrm {Conclusion} }}}$

According to this default, if we believe that Prerequisite is true, and each of ${\displaystyle \mathrm {Justification} _{n}}$ is consistent with our current beliefs, we are led to believe that Conclusion is true.

The logical formulae in W and all formulae in a default were originally assumed to be first-order logic formulae, but they can potentially be formulae in an arbitrary formal logic. The case in which they are formulae in propositional logic is one of the most studied.

Examples

The default rule “birds typically fly” is formalized by the following default:

${\displaystyle D=\left\{{\frac {\mathrm {Bird} (X):\mathrm {Flies} (X)}{\mathrm {Flies} (X)}}\right\}}$

This rule means that, "if X is a bird, and it can be assumed that it flies, then we can conclude that it flies". A background theory containing some facts about birds is the following one:

${\displaystyle W=\{\mathrm {Bird} (\mathrm {Condor} ),\mathrm {Bird} (\mathrm {Penguin} ),\neg \mathrm {Flies} (\mathrm {Penguin} ),\mathrm {Flies} (\mathrm {Bee} )\}}$.

According to this default rule, a condor flies because the precondition Bird(Condor) is true and the justification Flies(Condor) is not inconsistent with what is currently known. On the contrary, Bird(Penguin) does not allow concluding Flies(Penguin): even if the precondition of the default Bird(Penguin) is true, the justification Flies(Penguin) is inconsistent with what is known. From this background theory and this default, Bird(Bee) cannot be concluded because the default rule only allows deriving Flies(X) from Bird(X), but not vice versa. Deriving the antecedents of an inference rule from the consequences is a form of explanation of the consequences, and is the aim of abductive reasoning.

A common default assumption is that what is not known to be true is believed to be false. This is known as the Closed-World Assumption, and is formalized in default logic using a default like the following one for every fact F.

${\displaystyle {\frac {:{\neg }F}{{\neg }F}}}$

For example, the computer language Prolog uses a sort of default assumption when dealing with negation: if a negative atom cannot be proved to be true, then it is assumed to be false. Note, however, that Prolog uses the so-called negation as failure: when the interpreter has to evaluate the atom ${\displaystyle \neg F}$, it tries to prove that F is true, and conclude that ${\displaystyle \neg F}$ is true if it fails. In default logic, instead, a default having ${\displaystyle \neg F}$ as a justification can only be applied if ${\displaystyle \neg F}$ is consistent with the current knowledge.

Restrictions

A default is categorical or prerequisite-free if it has no prerequisite (or, equivalently, its prerequisite is tautological). A default is normal if it has a single justification that is equivalent to its conclusion. A default is supernormal if it is both categorical and normal. A default is seminormal if all its justifications entail its conclusion. A default theory is called categorical, normal, supernormal, or seminormal if all defaults it contains are categorical, normal, supernormal, or seminormal, respectively.

Semantics of default logic

A default rule can be applied to a theory if its precondition is entailed by the theory and its justifications are all consistent with the theory. The application of a default rule leads to the addition of its consequence to the theory. Other default rules may then be applied to the resulting theory. When the theory is such that no other default can be applied, the theory is called an extension of the default theory. The default rules may be applied in different order, and this may lead to different extensions. The Nixon diamond example is a default theory with two extensions:

${\displaystyle \left\langle \left\{{\frac {\mathrm {Republican} (X):\neg \mathrm {Pacifist} (X)}{\neg \mathrm {Pacifist} (X)}},{\frac {\mathrm {Quaker} (X):\mathrm {Pacifist} (X)}{\mathrm {Pacifist} (X)}}\right\},\left\{\mathrm {Republican} (\mathrm {Nixon} ),\mathrm {Quaker} (\mathrm {Nixon} )\right\}\right\rangle }$

Since Nixon is both a Republican and a Quaker, both defaults can be applied. However, applying the first default leads to the conclusion that Nixon is not a pacifist, which makes the second default not applicable. In the same way, applying the second default we obtain that Nixon is a pacifist, thus making the first default not applicable. This particular default theory has therefore two extensions, one in which Pacifist(Nixon) is true, and one in which Pacifist(Nixon) is false.

The original semantics of default logic was based on the fixed point of a function. The following is an equivalent algorithmic definition. If a default contains formulae with free variables, it is considered to represent the set of all defaults obtained by giving a value to all these variables. A default ${\displaystyle {\frac {\alpha :\beta _{1},\ldots ,\beta _{n}}{\gamma }}}$ is applicable to a propositional theory T if ${\displaystyle T\models \alpha }$ and all theories ${\displaystyle T\cup \{\beta _{i}\}}$ are consistent. The application of this default to T leads to the theory ${\displaystyle T\cup \{\gamma \}}$. An extension can be generated by applying the following algorithm:

T = W         /* current theory */
A = 0         /* set of defaults applied so far */
 
              /* apply a sequence of defaults */
while there is a default d that is not in A and is applicable to T
    add the consequence of d to T
    add d to A
 
              /* final consistency check */
if 
    for every default d in A
        T is consistent with all justifications of d
then
    output T

This algorithm is non-deterministic, as several defaults can alternatively be applied to a given theory T. In the Nixon diamond example, the application of the first default leads to a theory to which the second default cannot be applied and vice versa. As a result, two extensions are generated: one in which Nixon is a pacifist and one in which Nixon is not a pacifist.

The final check of consistency of the justifications of all defaults that have been applied implies that some theories do not have any extensions. In particular, this happens whenever this check fails for every possible sequence of applicable defaults. The following default theory has no extension:

${\displaystyle \left\langle \left\{{\frac {:A(b)}{\neg A(b)}}\right\},\emptyset \right\rangle }$

Since ${\displaystyle A(b)}$ is consistent with the background theory, the default can be applied, thus leading to the conclusion that ${\displaystyle A(b)}$ is false. This result however undermines the assumption that has been made for applying the first default. Consequently, this theory has no extensions.

In a normal default theory, all defaults are normal: each default has the form ${\displaystyle {\frac {\phi :\psi }{\psi }}}$. A normal default theory is guaranteed to have at least one extension. Furthermore, the extensions of a normal default theory are mutually inconsistent, i.e., inconsistent with each other.

Entailment

A default theory can have zero, one, or more extensions. Entailment of a formula from a default theory can be defined in two ways:

Skeptical

a formula is entailed by a default theory if it is entailed by all its extensions;

Credulous

a formula is entailed by a default theory if it is entailed by at least one of its extensions.

Thus, the Nixon diamond example theory has two extensions, one in which Nixon is a pacifist and one in which he is not a pacifist. Consequently, neither Pacifist(Nixon) nor ¬Pacifist(Nixon) are skeptically entailed, while both of them are credulously entailed. As this example shows, the credulous consequences of a default theory may be inconsistent with each other.

Alternative default inference rules

The following alternative inference rules for default logic are all based on the same syntax as the original system.

Justified

differs from the original one in that a default is not applied if thereby the set T becomes inconsistent with a justification of an applied default;

Concise

a default is applied only if its consequence is not already entailed by T (the exact definition is more complicated than this one; this is only the main idea behind it);

Constrained

a default is applied only if the set composed of the background theory, the justifications of all applied defaults, and the consequences of all applied defaults (including this one) is consistent;

Rational

similar to constrained default logic, but the consequence of the default to add is not considered in the consistency check;

Cautious

defaults that can be applied but are conflicting with each other (like the ones of the Nixon diamond example) are not applied.

The justified and constrained versions of the inference rule assign at least an extension to every default theory.

Variants of default logic

The following variants of default logic differ from the original one on both syntax and semantics.

Assertional variants

An assertion is a pair ${\displaystyle \langle p:\{r_{1},\ldots ,r_{n}\}\rangle }$ composed of a formula and a set of formulae. Such a pair indicates that p is true while the formulae ${\displaystyle r_{1},\ldots ,r_{n}}$ have been assumed consistent to prove that p is true. An assertional default theory is composed of an assertional theory (a set of assertional formulae) called the background theory and a set of defaults defined as in the original syntax. Whenever a default is applied to an assertional theory, the pair composed of its consequence and its set of justifications is added to the theory. The following semantics use assertional theories:

• Cumulative default logic
• Commitment to assumptions default logic
• Quasi-default logic

Weak extensions

rather than checking whether the preconditions are valid in the theory composed of the background theory and the consequences of the applied defaults, the preconditions are checked for validity in the extension that will be generated; in other words, the algorithm for generating extensions starts by guessing a theory and using it in place of the background theory; what results from the process of extension generation is actually an extension only if it is equivalent to the theory guessed at the beginning. This variant of default logic is related in principle to autoepistemic logic, where a theory ${\displaystyle \Box x\rightarrow x}$ has the model in which x is true just because, assuming ${\displaystyle \Box x}$ true, the formula ${\displaystyle \Box x\rightarrow x}$ supports the initial assumption.

Disjunctive default logic

the consequence of a default is a set of formulae instead of a single formula. Whenever the default is applied, at least one of its consequences is nondeterministically chosen and made true.

Priorities on defaults

the relative priority of defaults can be explicitly specified; among the defaults that are applicable to a theory, only one of the most preferred ones can be applied. Some semantics of default logic do not require priorities to be explicitly specified; rather, more specific defaults (those that are applicable in fewer cases) are preferred over less specific ones.

Statistical variant

a statistical default is a default with an attached upper bound on its frequency of error; in other words, the default is assumed to be an incorrect inference rule in at most that fraction of times it is applied.

Translations

Default theories can be translated into theories in other logics and vice versa. The following conditions on translations have been considered:

Consequence-Preserving

the original and the translated theories have the same (propositional) consequences;

Faithful

this condition only makes sense when translating between two variants of default logic or between default logic and a logic in which a concept similar to extension exists, e.g., models in modal logic; a translation is faithful if there exists a mapping (typically, a bijection) between the extensions (or models) of the original and translated theories;

Modular

a translation from default logic to another logic is modular if the defaults and the background theory can be translated separately; moreover, the addition of formulae to the background theory only leads to adding the new formulae to the result of the translation;

Same-Alphabet

the original and translated theories are built on the same alphabet;

Polynomial

the running time of the translation or the size of the generated theory are required to be polynomial in the size of the original theory.

Translations are typically required to be faithful or at least consequence-preserving, while the conditions of modularity and same alphabet are sometimes ignored.

The translatability between propositional default logic and the following logics have been studied:

• classical propositional logic;
• autoepistemic logic;
• propositional default logic restricted to seminormal theories;
• alternative semantics of default logic;
• circumscription.

Translations exist or not depending on which conditions are imposed. Translations from propositional default logic to classical propositional logic cannot always generate a polynomially sized propositional theory, unless the polynomial hierarchy collapses. Translations to autoepistemic logic exist or not depending on whether modularity or the use of the same alphabet is required.

Complexity

The computational complexity of the following problems about default logic is known:

Existence of extensions

deciding whether a propositional default theory has at least one extension is ${\displaystyle \Sigma _{2}^{P}}$-complete;

Skeptical entailment

deciding whether a propositional default theory skeptically entails a propositional formula is ${\displaystyle \Pi _{2}^{P}}$-complete;

Credulous entailment

deciding whether a propositional default theory credulously entails a propositional formula is ${\displaystyle \Sigma _{2}^{P}}$-complete;

Extension checking

deciding whether a propositional formula is equivalent to an extension of a propositional default theory is ${\displaystyle \Delta _{2}^{P[\log ]}}$-complete;

Model checking

deciding whether a propositional interpretation is a model of an extension of a propositional default theory is ${\displaystyle \Sigma _{2}^{P}}$-complete.

Implementations

Four systems implementing default logics are DeReS, XRay, GADeL, and Catala.

Distributive justice

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Distributive justice concerns the socially just allocation of resources. Often contrasted with just process, which is concerned with the administration of law, distributive justice concentrates on outcomes. This subject has been given considerable attention in philosophy and the social sciences.

In social psychology, distributive justice is defined as perceived fairness of how rewards and costs are shared by (distributed across) group members. For example, when some workers work more hours but receive the same pay, group members may feel that distributive justice has not occurred. To determine whether distributive justice has taken place, individuals often turn to the behavioral expectations of their group. If rewards and costs are allocated according to the designated distributive norms of the group, distributive justice has occurred.

Types of distributive norms

Five types of distributive norm are defined by Donelson R. Forsyth:

1. Equality: Regardless of their inputs, all group members should be given an equal share of the rewards/costs. Equality supports that someone who contributes 20% of the group's resources should receive as much as someone who contributes 60%.
2. Equity: Members' outcomes should be based upon their inputs. Therefore, an individual who has invested a large amount of input (e.g. time, money, energy) should receive more from the group than someone who has contributed very little. Members of large groups prefer to base allocations of rewards and costs on equity
3. Power: Those with more authority, status, or control over the group should receive more than those in lower level positions.
4. Need: Those in greatest needs should be provided with resources needed to meet those needs. These individuals should be given more resources than those who already possess them, regardless of their input.
5. Responsibility: Group members who have the most should share their resources with those who have less.

Theories of distributive justice

To create a list of the theories of distributive justice will inevitably come with its implications. It is important to take into consideration the various nuances within each theory, as well as the development and variations in interpretations that exist for the theories presented in this article. The listed theories below are three of the most prominent Anglo-American theories within the field. With this in mind, the list is in no way to be considered exhaustive for distributive justice theory.

Justice as fairness

In his book A Theory of Justice, John Rawls outlines his famous theory about justice as fairness. The theory consists of three core components:

1. the equality of people in rights and liberties;
2. the equality of opportunities for all; and
3. an arrangement of economic inequalities focused on benefit maximisation for those who are least advantaged.

The just 'basic structure'

Building a modern view on social contract theory, Rawls bases his work on an idea of justice being rooted in the basic structure, constituting the fundamental rules in society, which shape the social and economic institutions, as well as the governance. This basic structure is what shapes the citizens’ life opportunities. According to Rawls, the structure is based on principles about basic rights and duties that any self-interested, rational individual would accept in order to further his/her own interests in a context of social cooperation.

The original position

Main article: Original position

Rawls presents the concept of an original position as a hypothetical idea of how to establish "a fair procedure so that any principles agreed on will be just." In his envisioning of the original position, it is created from a judgement made through negotiations between a group of people who will decide on what a just distribution of primary goods is (according to Rawls, the primary goods include freedoms, opportunities, and control over resources). These people are assumed to be guided by self-interest, while also having a basic idea of morality and justice, and thus capable of understanding and evaluating a moral argument. Rawls then argues that procedural justice in the process of negotiation will be possible via a nullification of temptations for these people to exploit circumstances so as to favor their own position in society.

Veil of ignorance

Main article: Veil of ignorance

This nullification of temptations is realised through a veil of ignorance, which these people will be behind. The veil prevents the people from knowing what particular preferences they will have by concealing their talents, objectives, and, most importantly, where in society they themselves will end up. The veil, on the other hand, does not conceal general information about the society, and the people are assumed to possess societal and economic knowledge beyond the personal level. Thereby, such veil creates an environment for negotiations where the evaluation of the distribution of goods is based on general considerations, regardless of place in society, rather than biased considerations based on personal gains for specific citizen positions. By this logic, the negotiations will be sensitive to both those who are worst off, given that a risk of being in that category yourself will incentivize protection of these people, but also the rest of society, as one would not wish to hinder maximal utilisation for these in case you would end up in higher classes.

Basic principles of a just distribution

In this original position, the main concern will be to secure the goods that are most essential for pursuing the goals of each individual, regardless of what this specific goal might be. With this in mind, Rawls theorizes two basic principles of just distribution.

The first principle, the liberty principle, is the equal access to basic rights and liberties for all. With this, each person should be able to access the most extensive set of liberties that is compatible with similar schemes of access by other citizens. Thereby, it is not only a question of positive individual access but also of negative restrictions so as to respect others’ basic rights and liberties.

The second principle, the difference principle, addresses how the arrangement of social and economic inequalities, and thus the just distribution should look. Firstly, Rawls argues that such distribution should be based on a reasonable expectation of advantage for all, but also to the greatest benefit of the least advantaged in society. Secondly, the offices and positions attached to this arrangement should be open to all.

These principles of justice are then prioritised according to two additional principles:

1. the principles of the priority of liberty, wherein basic liberties only can be restricted if this is done for the sake of protecting liberty either:
   1. by strengthening “the total system of liberties shared by all;” or
   2. if a less than equal liberty is acceptable to those who are subject to this same lesser liberty.
2. inequality of opportunity, and the priority of efficiency & welfare, can only be acceptable if:
   1. it enhances “the opportunities of those with lesser opportunities” in society; and/or
   2. excessive saving either balances out or lessens the gravity of hardship for those who do not traditionally benefit.

Utilitarianism

Main article: Utilitarianism

In 1789, Jeremy Bentham published his book An Introduction to the Principles of Morals and Legislation. Centred on individual utility and welfare, utilitarianism builds on the notion that any action which increases the overall welfare in society is good, and any action that decreases welfare is bad. By this notion, utilitarianism's focus lies with its outcomes and pays little attention to how these outcomes are shaped. This idea of utilisation maximisation, while being a much broader philosophical consideration, also translates into a theory of justice.

Conceptualising welfare

While the basic notion that utilitarianism builds on seems simple, one major dispute within the school of utilitarianism revolved around the conceptualisation and measurement of welfare. With disputes over this fundamental aspect, utilitarianism is evidently a broad term embracing many different sub-theories under its umbrella, and while much of the theoretical framework transects across these conceptualisations, using the different conceptualisation have clear implications for how we understand the more practical side of utilitarianism in distributive justice.

Bentham originally conceptualised this according to the hedonistic calculus, which also became the foundation for John Stuart Mill's focus on intellectual pleasures as the most beneficial contribution to societal welfare. Another path has been painted by Aristotle, based on an attempt to create a more universal list of conditions required for human prosperity. Opposite this, another path focuses on a subjective evaluation of happiness and satisfaction in human lives.

Egalitarianism

Main article: Egalitarianism

Based on a fundamental notion of equal worth and moral status of human beings, egalitarianism is concerned with equal treatment of all citizens in both respect and in concern, and in relation to the state as well as one another. Egalitarianism focuses more on the process through which distribution takes place, egalitarianism evaluates the justification for a certain distribution based on how the society and its institutions have been shaped, rather than what the outcome is. Attention is mainly given to ways in which unchosen person circumstances affect and hinder individuals and their life opportunities. As Elizabeth Anderson defines it, "the positive aim of egalitarian justice is...to create a community in which people stand in relation of equality to others."

While much academic work distinguishes between luck egalitarianism and social egalitarianism, Roland Pierik presents a synthesis combining the two branches. In his synthesis, he argues that instead of focusing on compensations for unjust inequalities in society via redistribution of primary goods, egalitarianism scholars should instead, given the fundamental notion upon which the theory is built, strive to create institutions that creates and promotes meaningful equal opportunities from the get-go. Pierik thus moves egalitarianism's otherwise reactive nature by emphasising a need for attention to the development of fundamentally different institutions that would eradicate the need for redistribution and instead focus on the initial equal distribution of opportunities from which people then themselves be able to shape their lives.

Application and outcomes

Outcomes

Distributive justice affects performance when efficiency and productivity are involved. Improving perceptions of justice increases performance. Organizational citizenship behaviors (OCBs) are employee actions in support of the organization that are outside the scope of their job description. Such behaviors depend on the degree to which an organization is perceived to be distributively just. As organizational actions and decisions are perceived as more just, employees are more likely to engage in OCBs. Perceptions of distributive justice are also strongly related to the withdrawal of employees from the organization.

Wealth

See also: Redistribution (economics)

Distributive justice considers whether the distribution of goods among the members of society at a given time is subjectively acceptable.

Not all advocates of consequentialist theories are concerned with an equitable society. What unites them is the mutual interest in achieving the best possible results or, in terms of the example above, the best possible distribution of wealth.

Environmental justice

Main article: Environmental justice

Distributive justice in an environmental context is the equitable distribution of a society's technological and environmental risks, impacts, and benefits. These burdens include exposure to hazardous waste, land appropriation, armed violence, and murder. Distributive justice is an essential principle of environmental justice because there is evidence that shows that these burdens cause health problems, negatively affect quality of life, and drive down property value.

The potential negative social impacts of environmental degradation and regulatory policies have been at the center environmental discussions since the rise of environmental justice. Environmental burdens fall disproportionately upon the Global South, while benefits are primarily accrued to the Global North.

In politics

Distributive justice theory argues that societies have a duty to individuals in need and that all individuals have a duty to help others in need. Proponents of distributive justice link it to human rights. Many governments are known for dealing with issues of distributive justice, especially in countries with ethnic tensions and geographically distinctive minorities. Post-apartheid South Africa is an example of a country that deals with issues of re-allocating resources with respect to the distributive justice framework.

Influenced figures

Distributive justice is also fundamental to the Catholic Church's social teaching, inspiring such figures as Dorothy Day and Pope John Paul II.

Hayek's criticism

Within the context of Western liberal democracies in the post-WWII decades, Friedrich von Hayek was one of the most famous opposers of the idea of distributive justice. For him, social and distributive justice were meaningless and impossible to attain, on the grounds of being within a system where the outcomes are not determined deliberately by the people but contrarily spontaneity is the norm. Therefore, distributive justice, redistribution of wealth, and the demands for social justice in a society ruled by an impersonal process such as the market are in this sense incompatible with that system.

In his famous book Road to Serfdom, there can be found considerations about social assistance from the state. There, in talking about the importance of a restrictive kind of security (the one against physical privation) in front of one that necessarily needs to control or abolish the market, Hayek poses that "there can be no doubt that some minimum of food, shelter, and clothing, sufficient to preserve health and the capacity to work, can be assured to everybody". Providing this type of security is for Hayek compatible with individual freedom as it does not involve planning. But already in this early work, he acknowledges the fact that this provision must keep the incentives and the external pressure going and not select which group enjoys security and which does not, for under these conditions "the striving for security tends to become higher than the love of freedom". Therefore, fostering a certain kind of security (the one that for him socialist economic policies follow) can entail growing insecurity as the privilege increases social differences. Notwithstanding, he concludes that "adequate security against severe privation, and the reduction of the avoidable causes of misdirected effort and consequent disappointment, will have to be one of the main goals of policy".

Despite that vague social awareness (also a sign of the times as WWII's devastating consequences were obvious), Hayek dismisses an organizational view that ascribes certain outcomes to an intentional design, which would be contrary to his worshipped spontaneous order. For this, Hayek famously firstly regards the term social (or distributive) justice as meaningless when it is applied to the results of a liberal market system that should yield spontaneous outcomes. Justice has an individual component for Hayek, is only understood in the aggregation of individual actions which follow common rules, social and distributive justice are the negative opposite as they need a command economy. Secondly, following Tebble's (2009) view, the concept of social justice is for Hayek a reminiscence of an atavistic view towards society, that has been overcome by the survival capacity of the catallactic order and its values.

The third Hayekian critique is about the unfeasibility of attaining distributive justice in a free market order and this is defended on the basis of the determinate goal that all distributive justice aims to. In a catallactic order, the individual morality should freely determine what are distributive fairness and the values that govern economic activity, and the fact that it is impossible to gather all the individual information in a single pursuit for social and distributive justice results in realizing the fact that it cannot be pursued. Lastly, Hayek claims for the incompatibility between the free market and social justice, for, in essence, they are different kinds of inequalities. The former is one determined by the interaction of free individuals and the latter by the decision of an authority. Hayek will, on ethical grounds, choose the former.

Libertarian perspective

One of the major exponents of the libertarian outlook toward distributive justice is Robert Nozick. In his book Anarchy, State and Utopia he stresses how the term distributive justice is not a neutral one. In fact, there is no central distributor that can be regarded as such. What each person gets, he or she gets from the outcomes of Lockean self-ownership (a condition that implies one's labor mixed with the world), or others who give to him in exchange for something, or as a gift. For him, "there is no more a distributing or distribution of shares than there is a distribution of mates in a society in which persons choose whom they shall marry". This means that there can be no pattern to which to conform or aim. The market and the result of individual actions provided the conditions for libertarian principles of just acquisition and exchange (contained in his Entitlement Theory) will have as a result a distribution that will be just, without the need for considerations about the specific model or standard it should follow.

Calibration

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Calibration

In measurement technology and metrology, calibration is the comparison of measurement values delivered by a device under test with those of a calibration standard of known accuracy. Such a standard could be another measurement device of known accuracy, a device generating the quantity to be measured such as a voltage, a sound tone, or a physical artifact, such as a meter ruler.

The outcome of the comparison can result in one of the following:

• no significant error being noted on the device under test
• a significant error being noted but no adjustment made
• an adjustment made to correct the error to an acceptable level

Strictly speaking, the term "calibration" means just the act of comparison and does not include any subsequent adjustment.

The calibration standard is normally traceable to a national or international standard held by a metrology body.

BIPM Definition

The formal definition of calibration by the International Bureau of Weights and Measures (BIPM) is the following: "Operation that, under specified conditions, in a first step, establishes a relation between the quantity values with measurement uncertainties provided by measurement standards and corresponding indications with associated measurement uncertainties (of the calibrated instrument or secondary standard) and, in a second step, uses this information to establish a relation for obtaining a measurement result from an indication."

This definition states that the calibration process is purely a comparison, but introduces the concept of measurement uncertainty in relating the accuracies of the device under test and the standard.

Modern calibration processes

The increasing need for known accuracy and uncertainty and the need to have consistent and comparable standards internationally has led to the establishment of national laboratories. In many countries a National Metrology Institute (NMI) will exist which will maintain primary standards of measurement (the main SI units plus a number of derived units) which will be used to provide traceability to customer's instruments by calibration.

The NMI supports the metrological infrastructure in that country (and often others) by establishing an unbroken chain, from the top level of standards to an instrument used for measurement. Examples of National Metrology Institutes are NPL in the UK, NIST in the United States, PTB in Germany and many others. Since the Mutual Recognition Agreement was signed it is now straightforward to take traceability from any participating NMI and it is no longer necessary for a company to obtain traceability for measurements from the NMI of the country in which it is situated, such as the National Physical Laboratory in the UK.

Quality

To improve the quality of the calibration and have the results accepted by outside organizations it is desirable for the calibration and subsequent measurements to be "traceable" to the internationally defined measurement units. Establishing traceability is accomplished by a formal comparison to a standard which is directly or indirectly related to national standards (such as NIST in the USA), international standards, or certified reference materials. This may be done by national standards laboratories operated by the government or by private firms offering metrology services.

Quality management systems call for an effective metrology system which includes formal, periodic, and documented calibration of all measuring instruments. ISO 9000 and ISO 17025 standards require that these traceable actions are to a high level and set out how they can be quantified.

To communicate the quality of a calibration the calibration value is often accompanied by a traceable uncertainty statement to a stated confidence level. This is evaluated through careful uncertainty analysis. Some times a DFS (Departure From Spec) is required to operate machinery in a degraded state. Whenever this does happen, it must be in writing and authorized by a manager with the technical assistance of a calibration technician.

Measuring devices and instruments are categorized according to the physical quantities they are designed to measure. These vary internationally, e.g., NIST 150-2G in the U.S. and NABL-141 in India. Together, these standards cover instruments that measure various physical quantities such as electromagnetic radiation (RF probes), sound (sound level meter or noise dosimeter), time and frequency (intervalometer), ionizing radiation (Geiger counter), light (light meter), mechanical quantities (limit switch, pressure gauge, pressure switch), and, thermodynamic or thermal properties (thermometer, temperature controller). The standard instrument for each test device varies accordingly, e.g., a dead weight tester for pressure gauge calibration and a dry block temperature tester for temperature gauge calibration.

Instrument calibration prompts

Calibration may be required for the following reasons:

• a new instrument
• after an instrument has been repaired or modified
• moving from one location to another location
• when a specified time period has elapsed
• when a specified usage (operating hours) has elapsed
• before and/or after a critical measurement
• after an event, for example
  • after an instrument has been exposed to a shock, vibration, or physical damage, which might potentially have compromised the integrity of its calibration
  • sudden changes in weather
• whenever observations appear questionable or instrument indications do not match the output of surrogate instruments
• as specified by a requirement, e.g., customer specification, instrument manufacturer recommendation.

In general use, calibration is often regarded as including the process of adjusting the output or indication on a measurement instrument to agree with value of the applied standard, within a specified accuracy. For example, a thermometer could be calibrated so the error of indication or the correction is determined, and adjusted (e.g. via calibration constants) so that it shows the true temperature in Celsius at specific points on the scale. This is the perception of the instrument's end-user. However, very few instruments can be adjusted to exactly match the standards they are compared to. For the vast majority of calibrations, the calibration process is actually the comparison of an unknown to a known and recording the results.

Basic calibration process

Purpose and scope

The calibration process begins with the design of the measuring instrument that needs to be calibrated. The design has to be able to "hold a calibration" through its calibration interval. In other words, the design has to be capable of measurements that are "within engineering tolerance" when used within the stated environmental conditions over some reasonable period of time. Having a design with these characteristics increases the likelihood of the actual measuring instruments performing as expected. Basically, the purpose of calibration is for maintaining the quality of measurement as well as to ensure the proper working of particular instrument.

Frequency

The exact mechanism for assigning tolerance values varies by country and as per the industry type. The measuring of equipment is manufacturer generally assigns the measurement tolerance, suggests a calibration interval (CI) and specifies the environmental range of use and storage. The using organization generally assigns the actual calibration interval, which is dependent on this specific measuring equipment's likely usage level. The assignment of calibration intervals can be a formal process based on the results of previous calibrations. The standards themselves are not clear on recommended CI values:

ISO 17025

"A calibration certificate (or calibration label) shall not contain any recommendation on the calibration interval except where this has been agreed with the customer. This requirement may be superseded by legal regulations.”

ANSI/NCSL Z540

"...shall be calibrated or verified at periodic intervals established and maintained to assure acceptable reliability..."

ISO-9001

"Where necessary to ensure valid results, measuring equipment shall...be calibrated or verified at specified intervals, or prior to use...”

MIL-STD-45662A

"... shall be calibrated at periodic intervals established and maintained to assure acceptable accuracy and reliability...Intervals shall be shortened or may be lengthened, by the contractor, when the results of previous calibrations indicate that such action is appropriate to maintain acceptable reliability."

Standards required and accuracy

The next step is defining the calibration process. The selection of a standard or standards is the most visible part of the calibration process. Ideally, the standard has less than 1/4 of the measurement uncertainty of the device being calibrated. When this goal is met, the accumulated measurement uncertainty of all of the standards involved is considered to be insignificant when the final measurement is also made with the 4:1 ratio. This ratio was probably first formalized in Handbook 52 that accompanied MIL-STD-45662A, an early US Department of Defense metrology program specification. It was 10:1 from its inception in the 1950s until the 1970s, when advancing technology made 10:1 impossible for most electronic measurements.

Maintaining a 4:1 accuracy ratio with modern equipment is difficult. The test equipment being calibrated can be just as accurate as the working standard. If the accuracy ratio is less than 4:1, then the calibration tolerance can be reduced to compensate. When 1:1 is reached, only an exact match between the standard and the device being calibrated is a completely correct calibration. Another common method for dealing with this capability mismatch is to reduce the accuracy of the device being calibrated.

For example, a gauge with 3% manufacturer-stated accuracy can be changed to 4% so that a 1% accuracy standard can be used at 4:1. If the gauge is used in an application requiring 16% accuracy, having the gauge accuracy reduced to 4% will not affect the accuracy of the final measurements. This is called a limited calibration. But if the final measurement requires 10% accuracy, then the 3% gauge never can be better than 3.3:1. Then perhaps adjusting the calibration tolerance for the gauge would be a better solution. If the calibration is performed at 100 units, the 1% standard would actually be anywhere between 99 and 101 units. The acceptable values of calibrations where the test equipment is at the 4:1 ratio would be 96 to 104 units, inclusive. Changing the acceptable range to 97 to 103 units would remove the potential contribution of all of the standards and preserve a 3.3:1 ratio. Continuing, a further change to the acceptable range to 98 to 102 restores more than a 4:1 final ratio.

This is a simplified example. The mathematics of the example can be challenged. It is important that whatever thinking guided this process in an actual calibration be recorded and accessible. Informality contributes to tolerance stacks and other difficult to diagnose post calibration problems.

Also in the example above, ideally the calibration value of 100 units would be the best point in the gauge's range to perform a single-point calibration. It may be the manufacturer's recommendation or it may be the way similar devices are already being calibrated. Multiple point calibrations are also used. Depending on the device, a zero unit state, the absence of the phenomenon being measured, may also be a calibration point. Or zero may be resettable by the user-there are several variations possible. Again, the points to use during calibration should be recorded.

There may be specific connection techniques between the standard and the device being calibrated that may influence the calibration. For example, in electronic calibrations involving analog phenomena, the impedance of the cable connections can directly influence the result.

Manual and automatic calibrations

Calibration methods for modern devices can be manual or automatic.

Manual calibration - US serviceman calibrating a pressure gauge. The device under test is on his left and the test standard on his right.

As an example, a manual process may be used for calibration of a pressure gauge. The procedure requires multiple steps, to connect the gauge under test to a reference master gauge and an adjustable pressure source, to apply fluid pressure to both reference and test gauges at definite points over the span of the gauge, and to compare the readings of the two. The gauge under test may be adjusted to ensure its zero point and response to pressure comply as closely as possible to the intended accuracy. Each step of the process requires manual record keeping.

Automatic calibration - A U.S. serviceman using a 3666C auto pressure calibrator

An automatic pressure calibrator  is a device that combines an electronic control unit, a pressure intensifier used to compress a gas such as Nitrogen, a pressure transducer used to detect desired levels in a hydraulic accumulator, and accessories such as liquid traps and gauge fittings. An automatic system may also include data collection facilities to automate the gathering of data for record keeping.

Process description and documentation

All of the information above is collected in a calibration procedure, which is a specific test method. These procedures capture all of the steps needed to perform a successful calibration. The manufacturer may provide one or the organization may prepare one that also captures all of the organization's other requirements. There are clearinghouses for calibration procedures such as the Government-Industry Data Exchange Program (GIDEP) in the United States.

This exact process is repeated for each of the standards used until transfer standards, certified reference materials and/or natural physical constants, the measurement standards with the least uncertainty in the laboratory, are reached. This establishes the traceability of the calibration.

See Metrology for other factors that are considered during calibration process development.

After all of this, individual instruments of the specific type discussed above can finally be calibrated. The process generally begins with a basic damage check. Some organizations such as nuclear power plants collect "as-found" calibration data before any routine maintenance is performed. After routine maintenance and deficiencies detected during calibration are addressed, an "as-left" calibration is performed.

More commonly, a calibration technician is entrusted with the entire process and signs the calibration certificate, which documents the completion of a successful calibration. The basic process outlined above is a difficult and expensive challenge. The cost for ordinary equipment support is generally about 10% of the original purchase price on a yearly basis, as a commonly accepted rule-of-thumb. Exotic devices such as scanning electron microscopes, gas chromatograph systems and laser interferometer devices can be even more costly to maintain.

The 'single measurement' device used in the basic calibration process description above does exist. But, depending on the organization, the majority of the devices that need calibration can have several ranges and many functionalities in a single instrument. A good example is a common modern oscilloscope. There easily could be 200,000 combinations of settings to completely calibrate and limitations on how much of an all-inclusive calibration can be automated.

An instrument rack with tamper-indicating seals

To prevent unauthorized access to an instrument tamper-proof seals are usually applied after calibration. The picture of the oscilloscope rack shows these, and prove that the instrument has not been removed since it was last calibrated as they will possible unauthorized to the adjusting elements of the instrument. There also are labels showing the date of the last calibration and when the calibration interval dictates when the next one is needed. Some organizations also assign unique identification to each instrument to standardize the record keeping and keep track of accessories that are integral to a specific calibration condition.

When the instruments being calibrated are integrated with computers, the integrated computer programs and any calibration corrections are also under control.

Historical development

Main article: History of measurement

Origins

The words "calibrate" and "calibration" entered the English language as recently as the American Civil War, in descriptions of artillery, thought to be derived from a measurement of the calibre of a gun.

Some of the earliest known systems of measurement and calibration seem to have been created between the ancient civilizations of Egypt, Mesopotamia and the Indus Valley, with excavations revealing the use of angular gradations for construction. The term "calibration" was likely first associated with the precise division of linear distance and angles using a dividing engine and the measurement of gravitational mass using a weighing scale. These two forms of measurement alone and their direct derivatives supported nearly all commerce and technology development from the earliest civilizations until about AD 1800.

Calibration of weights and distances (c. 1100 CE)

See also: Weights and Measures Act

 

An example of a weighing scale with a ½ ounce calibration error at zero. This is a "zeroing error" which is inherently indicated, and can normally be adjusted by the user, but may be due to the string and rubber band in this case

Early measurement devices were direct, i.e. they had the same units as the quantity being measured. Examples include length using a yardstick and mass using a weighing scale. At the beginning of the twelfth century, during the reign of Henry I (1100-1135), it was decreed that a yard be "the distance from the tip of the King's nose to the end of his outstretched thumb." However, it wasn't until the reign of Richard I (1197) that we find documented evidence.

Assize of Measures

"Throughout the realm there shall be the same yard of the same size and it should be of iron."

Other standardization attempts followed, such as the Magna Carta (1225) for liquid measures, until the Mètre des Archives from France and the establishment of the Metric system.

The early calibration of pressure instruments

Direct reading design of a U-tube manometer

One of the earliest pressure measurement devices was the Mercury barometer, credited to Torricelli (1643), which read atmospheric pressure using Mercury. Soon after, water-filled manometers were designed. All these would have linear calibrations using gravimetric principles, where the difference in levels was proportional to pressure. The normal units of measure would be the convenient inches of mercury or water.

In the direct reading hydrostatic manometer design on the right, applied pressure P_a pushes the liquid down the right side of the manometer U-tube, while a length scale next to the tube measures the difference of levels. The resulting height difference "H" is a direct measurement of the pressure or vacuum with respect to atmospheric pressure. In the absence of differential pressure both levels would be equal, and this would be used as the zero point.

The Industrial Revolution saw the adoption of "indirect" pressure measuring devices, which were more practical than the manometer. An example is in high pressure (up to 50 psi) steam engines, where mercury was used to reduce the scale length to about 60 inches, but such a manometer was expensive and prone to damage. This stimulated the development of indirect reading instruments, of which the Bourdon tube invented by Eugène Bourdon is a notable example.

Indirect reading design showing a Bourdon tube from the front (top) and the rear (bottom).

In the front and back views of a Bourdon gauge on the right, applied pressure at the bottom fitting reduces the curl on the flattened pipe proportionally to pressure. This moves the free end of the tube which is linked to the pointer. The instrument would be calibrated against a manometer, which would be the calibration standard. For measurement of indirect quantities of pressure per unit area, the calibration uncertainty would be dependent on the density of the manometer fluid, and the means of measuring the height difference. From this other units such as pounds per square inch could be inferred and marked on the scale.