Privatization

From Wikipedia, the free encyclopedia

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

Privatization (rendered privatisation in British English) can mean several different things, most commonly referring to moving something from the public sector into the private sector. It is also sometimes used as a synonym for deregulation when a heavily regulated private company or industry becomes less regulated. Government functions and services may also be privatised (which may also be known as "franchising" or "out-sourcing"); in this case, private entities are tasked with the implementation of government programs or performance of government services that had previously been the purview of state-run agencies. Some examples include revenue collection, law enforcement, water supply, and prison management.

Another definition is that privatization is the sale of a state-owned enterprise or municipally owned corporation to private investors; in this case shares may be traded in the public market for the first time, or for the first time since an enterprise's previous nationalization. This type of privatization can include the demutualization of a mutual organization, cooperative, or public-private partnership in order to form a joint-stock company.

Separately, privatization can refer to the purchase of all outstanding shares of a publicly traded company by private equity investors, which is more often called "going private". Before and after this process the company is privately owned, but after the buyout its shares are withdrawn from being traded at a public stock exchange.

Etymology

The term privatizing first appeared in English, with quotation marks, in the New York Times, in April 1923, in a translation of a German speech referring to the potential for German state railroads to be bought by American companies. In German, the word Privatisierung has been used since at least the 19th century. Ultimately, the word came to German through French from the Latin privatus.

The term reprivatization, again translated directly from German (Reprivatisierung), was used frequently in the mid-1930s as The Economist reported on Nazi Germany's sale of nationalized banks back to public shareholders following the 1931 economic crisis.

The word became common in the late 1970s and early 1980s as part of UK prime minister Margaret Thatcher's economic policies. She was drawing on the work of the pro-privatization Member of Parliament David Howell, who was himself drawing on the Austrian-American management expert Peter Drucker's 1969 book, The Age of Discontinuity.

Definition

The word privatization may mean different things depending on the context in which it is used. It can mean moving something from the public sphere into the private sphere, but it may also be used to describe something that was always private, but heavily regulated, which becomes less regulated through a process of deregulation. The term may also be used descriptively for something that has always been private, but could be public in other jurisdictions.

There are also private entities that may perform public functions. These entities could also be described as privatized. Privatization may mean the government sells state-owned businesses to private interests, but it may also be discussed in the context of the privatization of services or government functions, where private entities are tasked with the implementation of government programs or the performance of government services. Gillian E. Metzger has written that: "Private entities [in the US] provide a vast array of social services for the government; administer core aspects of government programs; and perform tasks that appear quintessentially governmental, such as promulgating standards or regulating third-party activities." Metzger mentions an expansion of privatization that includes health and welfare programs, public education, and prisons.

Privatization can also refer to the transfer of something out of other forms of collective or communal ownership besides state ownership, such as occurs in enclosure of manorial land.

History

See also: List of privatizations

Pre-20th century

The history of privatization dates from Ancient Greece, when governments contracted out almost everything to the private sector. In the Roman Republic private individuals and companies performed the majority of services including tax collection (tax farming), army supplies (military contractors), religious sacrifices and construction. However, the Roman Empire also created state-owned enterprises—for example, much of the grain was eventually produced on estates owned by the Emperor. David Parker and David S. Saal suggest that the cost of bureaucracy was one of the reasons for the fall of the Roman Empire.

Perhaps one of the first ideological movements towards privatization came during China's golden age of the Han dynasty. Taoism came into prominence for the first time at a state level, and it advocated the laissez-faire principle of Wu wei (無為), literally meaning "do nothing". The rulers were counseled by the Taoist clergy that a strong ruler was virtually invisible.

During the Renaissance, most of Europe was still by and large following the feudal economic model. By contrast, the Ming dynasty in China began once more to practice privatization, especially with regards to their manufacturing industries. This was a reversal of the earlier Song dynasty policies, which had themselves overturned earlier policies in favor of more rigorous state control.

In Britain, the privatization of common lands is referred to as enclosure (in Scotland as the Lowland Clearances and the Highland Clearances). Significant privatizations of this nature occurred from 1760 to 1820, preceding the Industrial Revolution in that country.

20th century onwards

The first mass privatization of state property occurred in Nazi Germany between 1933 and 1937: "It is a fact that the government of the National Socialist Party sold off public ownership in several state-owned firms in the middle of the 1930s. The firms belonged to a wide range of sectors: steel, mining, banking, local public utilities, shipyard, ship-lines, railways, etc. In addition to this, delivery of some public services produced by public administrations prior to the 1930s, especially social services and services related to work, was transferred to the private sector, mainly to several organizations within the Nazi Party."

Great Britain privatized its steel industry in the 1950s, and the West German government embarked on large-scale privatization, including sale of the majority stake in Volkswagen to small investors in public share offerings in 1961. However, it was in the 1980s under Margaret Thatcher in the United Kingdom and Ronald Reagan in the United States that privatization gained worldwide momentum. Notable privatizations in the UK included Britoil (1982), the radioactive-chemicals company Amersham International (1982), British Telecom (1984), Sealink ferries (1984), British Petroleum (gradually privatized between 1979 and 1987), British Aerospace (1985 to 1987), British Gas (1986), Rolls-Royce (1987), Rover Group (formerly British Leyland, 1988), British Steel Corporation (1988), Girobank (1989), and the regional water authorities of England and Wales (mostly in 1989). After 1979, council house tenants in the UK were given the right to buy their homes at a heavily discounted price; one million had purchased their residences by 1986.

Such efforts culminated in 1993 when British Rail was privatized under Thatcher's successor, John Major. British Rail had been formed by prior nationalization of private rail companies. The privatization was controversial, and its impact is still debated today, as doubling of passenger numbers and investment was balanced by an increase in rail subsidy.

These privatizations received mixed views from the public and the parliament. Even former Conservative prime minister Harold Macmillan was critical of the policy, likening it to "selling the family silver". There were around 3 million shareholders in Britain when Thatcher took office in 1979, but the subsequent sale of state-run firms saw the number of shareholders double by 1985. By the time of her resignation in 1990, there were more than 10 million shareholders in Britain.

Privatization in Latin America was extensive in the 1980s and 1990s, as a result of a Western liberal economic policy. Companies providing public services such as water management, transportation, and telecommunications were rapidly sold off to the private sector. In the 1990s, privatization revenue from 18 Latin American countries totaled 6% of gross domestic product. Private investment in infrastructure from 1990 and 2001 reached $360.5 billion, $150 billion more than in the next emerging economy.

While economists generally give favorable evaluations of the impact of privatization in Latin America, opinion polls and public protests across the countries suggest that a large segment of the public is dissatisfied with or have negative views of privatization in the region.

In the 1990s, the governments in Eastern and Central Europe engaged in extensive privatization of state-owned enterprises in Eastern and Central Europe and Russia, with assistance from the World Bank, the U.S. Agency for International Development, the German Treuhand, and other governmental and non-governmental organization.

Nippon Telegraph and Telephone's privatization in 1987 involved the largest share offering in financial history at the time. 15 of the world's 20 largest public share offerings have been privatizations of telecoms.

In 1988, the perestroika policy of Mikhail Gorbachev started allowing privatization of the centrally planned economy. Large privatization of the Soviet economy occurred over the next few years as the country dissolved. Other Eastern Bloc countries followed suit after the Revolutions of 1989 introduced non-communist governments.

Freedom House's privatization index, 1998 and 2002
Freedom House's privatization index rated transition countries from 1 (maximum progress) to 7 (no progress). The table below shows the privatization index for various Eastern European countries in 1998 and 2002:

Privatization index	1998	2002
Bulgaria	4.0	3.0
Czech Republic	2.0	1.75
Hungary	1.5	1.5
Poland	2.25	2.25
Romania	4.5	3.75
Slovakia	3.25	2.0
Slovenia	2.5	2.5
Russia	3.0	3.5

The largest public shares offering in France involved France Télécom.

Egypt undertook widespread privatization under Hosni Mubarak. Following his overthrow in the 2011 revolution, most of the public began to call for re-nationalization, citing allegations of the privatized firms practicing crony capitalism under the old regime.

Reasons for privatization

There are various reasons why a government may decide to privatize; commonly due to economic reasons. The economic factors that influence a government's decision to privatize assume this will lower government debt. Studies have shown that governments are more likely to privatise with higher public debt, typically because governments do not have the needed time to wait for a return. Another economic factor that influences this area is the resulting efficiency of SOEs once privatised. Commonly, governments aren’t able to provide the required investments required to ensure profitability for various reasons. These factors may lead to a government deciding to privatize.

Forms of privatization

There are several main methods of privatization:

1. Share issue privatization: shares sale on the stock market.
2. Asset sale privatization: asset divestiture to a strategic investor, usually by auction or through the Treuhand model.
3. Voucher privatization: distribution of vouchers, which represent part ownership of a corporation, to all citizens, usually for free or at a very low price.
4. Privatization from below: start of new private businesses in formerly socialist countries.
5. Management buyout: purchase of public shares by management of the company, sometimes by borrowing from external lenders
6. Employee buyout: distribution of shares for free or at a very low price to workers or management of the organization.

The choice of sale method is influenced by the capital market and the political and firm-specific factors. Privatization through the stock market is more likely to be the method used when there is an established capital market capable of absorbing the shares. A market with high liquidity can facilitate the privatization. If the capital markets are insufficiently developed, however, it would be difficult to find enough buyers. The shares may have to be underpriced, and the sales may not raise as much capital as would be justified by the fair value of the company being privatized. Many governments, therefore, elect for listings in more sophisticated markets, for example, Euronext, and the London, New York and Hong Kong stock exchanges.

Governments in developing countries and transition countries more often resort to direct asset sales to a few investors, partly because those countries do not yet have a stock market with high capital.

Voucher privatization occurred mainly in the transition economies in Central and Eastern Europe, such as Russia, Poland, the Czech Republic, and Slovakia. Additionally, privatization from below had made important contribution to economic growth in transition economies.

In one study assimilating some of the literature on "privatization" that occurred in Russian and Czech Republic transition economies, the authors identified three methods of privatization: "privatization by sale", "mass privatization", and "mixed privatization". Their calculations showed that "mass privatization" was the most effective method.

However, in economies "characterized by shortages" and maintained by the state bureaucracy, wealth was accumulated and concentrated by "gray/black market" operators. Privatizing industries by sale to these individuals did not mean a transition to "effective private sector owners [of former] state assets". Rather than mainly participating in a market economy, these individuals could prefer elevating their personal status or prefer accumulating political power. Instead, outside foreign investment led to the efficient conduct of former state assets in the private sector and market economy.

Through privatization by direct asset sale or the stock market, bidders compete to offer higher prices, generating more revenue for the state. Voucher privatization, on the other hand, could represent a genuine transfer of assets to the general population, creating a sense of participation and inclusion. A market could be created if the government permits transfer of vouchers among voucher holders.

Secured borrowing

Some privatization transactions can be interpreted as a form of a secured loan and are criticized as a "particularly noxious form of governmental debt". In this interpretation, the upfront payment from the privatization sale corresponds to the principal amount of the loan, while the proceeds from the underlying asset correspond to secured interest payments—the transaction can be considered substantively the same as a secured loan, though it is structured as a sale. This interpretation is particularly argued to apply to recent municipal transactions in the United States, particularly for fixed term, such as the 2008 sale of the proceeds from Chicago parking meters for 75 years. It is argued that this is motivated by "politicians' desires to borrow money surreptitiously", due to legal restrictions on and political resistance to alternative sources of revenue, viz, raising taxes or issuing debt.

Results of privatization

Privatization had different outcomes around the world. Results of privatization may vary depending on the privatization model employed. According to Irwin Stelzer, "it is somewhere between difficult and impossible to separate the effects of privatisation from the effects of such things as trends in the economy".

According to research performed by the World Bank and William L. Megginson in the early 2000s, privatization in competitive industries with well-informed consumers, consistently improved efficiency. According to APEC, the more competitive the industry, the greater the improvement in output, profitability, and efficiency. Such efficiency gains mean a one-off increase in GDP, but through improved incentives to innovate and reduce costs also tend to raise the rate of economic growth.

More recent research and literature review performed by Professor Saul Estrin and Adeline Pelletier concluded that "the literature now reflects a more cautious and nuanced evaluation of privatization" and that "private ownership alone is no longer argued to automatically generate economic gains in developing economies". According to a 2008 study published in Annals of Public and Cooperative Economics, liberalization and privatization have produced mixed results.

Although typically there are many costs associated with these efficiency gains, many economists argue that these can be dealt with by appropriate government support through redistribution and perhaps retraining. Yet, some empirical literature suggests that privatization could also have very modest effects on efficiency and quite regressive distributive impact. In the first attempt at a social welfare analysis of the British privatization program under the Conservative governments of Margaret Thatcher and John Major during the 1980s and 1990s, Massimo Florio points to the absence of any productivity shock resulting strictly from ownership change. Instead, the impact on the previously nationalized companies of the UK productivity leap under the Conservatives varied in different industries. In some cases, it occurred prior to privatization, and in other cases, it occurred upon privatization or several years afterward.

A 2012 study published by the European Commission argues that privatisation in Europe had mixed effects on service quality and has achieved only minor productivity gains, driven mainly by lower labour input combined with other cost cutting strategies that led to a deterioration of employment and working conditions. Meanwhile, a different study by the commission found that the UK rail network (which was privatized from 1994 to 1997) was most improved out of all the 27 EU nations from 1997 to 2012. The report examined a range of 14 different factors and the UK came top in four of the factors, second and third in another two and fourth in three, coming top overall. Nonetheless, the impact of the privatisation of British Rail has been the subject of much debate, with the stated benefits including improved customer service, and more investment; and stated drawbacks including higher fares, lower punctuality and increased rail subsidies.

Privatizations in Russia and Latin America were accompanied by large-scale corruption during the sale of the state-owned companies. Those with political connections unfairly gained large wealth, which has discredited privatization in these regions. While media have widely reported the grand corruption that accompanied those sales, according to research released by the World Bank there has been increased operating efficiency, daily petty corruption is, or would be, larger without privatization, and that corruption is more prevalent in non-privatized sectors. Furthermore, according to the World Bank extralegal and unofficial activities are more prevalent in countries that privatized less. Other research suggests that privatization in Russia resulted in a dramatic rise in the level of economic inequality and a collapse in GDP and industrial output.

Russian President Boris Yeltsin's IMF-backed rapid privatization schemes saw half the Russian population fall into destitution in just several years as unemployment climbed to double digits by the early to mid 1990s. A 2009 study published in The Lancet medical journal has found that as many as a million working men died as a result of economic shocks associated with mass privatization in the former Soviet Union and in Eastern Europe during the 1990s, although a further study suggested that there were errors in their method and "correlations reported in the original article are simply not robust." A subsequent body of scholarship, while still controversial, demonstrates that rapid privatization schemes associated with neoliberal economic reforms did result in poorer health outcomes in former Eastern Bloc countries during the transition to markets economies, with the World Health Organization contributing to the debate by stating "IMF economic reform programs are associated with significantly worsened tuberculosis incidence, prevalence, and mortality rates in post-communist Eastern European and former Soviet countries." Historian Walter Scheidel, a specialist in ancient history, posits that economic inequality and wealth concentration in the top percentile "had been made possible by the transfer of state assets to private owners."

In Latin America, on the one hand, according to John Nellis's research for Center for Global Development, economic indicators, including firm profitability, productivity, and growth, project positive microeconomic results. On the other hand, however, privatisation has been largely met with a negative criticism and citizen coalitions. This neoliberal criticism highlights the ongoing conflict between varying visions of economic development. Karl Polanyi emphasizes the societal concerns of self-regulating markets through a concept known as a "double movement". In essence, whenever societies move towards increasingly unrestrained, free-market rule, a natural and inevitable societal correction emerges to undermine the contradictions of capitalism. This was the case in the 2000 Cochabamba protests.

Privatization in Latin America has invariably experienced increasing push-back from the public. Mary Shirley from The Ronald Coase Institute suggests that implementing a less efficient but more politically mindful approach could be more sustainable.

In India, a survey by the National Commission for Protection of Child Rights (NCPCR) – Utilization of Free Medical Services by Children Belonging to the Economically Weaker Section (EWS) in Private Hospitals in New Delhi, 2011–12: A Rapid Appraisal – indicates under-utilization of the free beds available for EWS category in private hospitals in Delhi, though they were allotted land at subsidized rates.

In Australia a "People's Inquiry into Privatisation" (2016/17) found that the impact of privatisation on communities was negative. The report from the inquiry "Taking Back Control" made a range of recommendations to provide accountability and transparency in the process. The report highlighted privatisation in healthcare, aged care, child care, social services, government departments, electricity, prisons and vocational education featuring the voices of workers, community members and academics.

Some reports show that the results of privatization are experienced differently between men and women for numerous reasons: when public services are privatized women are expected to take on the health and social care of dependents, women have less access to privatized goods, public sector employs a larger proportion of women than does the private sector, and the women in the public sector are more likely to be unionized than those in the private sector. In Chile, women are disproportionately affected by the privatization of the pension system because factors such as "women's longer life expectancy, earlier retirement age, and lower rates of labor-force participation, lower salaries" affect their ability to accumulate funds for retirement which leads to lower pensions. Low-income women face an even greater burden; Anjela Taneja, of Oxfam India says "The privatization of public services...implies limited or no access to essential services for women living in poverty, who are often the ones more in need of these services."

The increase in privatization since the 1980s has been a factor in rising income and wealth inequality in the United States.

Foreign privatization

Due to low levels of native capital accumulation in the former Central and Eastern Europe, the rapid privatization preferred by international institutions (EBRD, IMF, World Bank) and other foreign banks was a de facto call for international bidding, reflecting the assumption that foreign investment would play a major role.

Contrasting cases in Eastern Europe: Romania and East Germany

In post-reunification East Germany, by the end of June 1992, the Treuhandanstalt had privatized 8,175 companies, with 5,950 left on hand (4,340 remaining to be sold and the remainder to be liquidated). June 1992 was also when the last East German on the board of the Treuhand left. By the end of 1994, Treuhand had sold almost everything, having only 65 firms left to privatize as of December 1994. More than 80% of the privatized businesses were bought by foreigners (chiefly West Germans – 75%).

Romania's first privatization took place on 3 August 1992. There was "very little" privatization during 1992: only 22 state-owned enterprises were privatized. The pace picked up throughout the following year, with more than 260 companies privatized. Four of the 22 enterprises privatized in 1992 were sold to foreign investors. In 1993, 265 companies were privatized, followed by 604 in 1994. Two companies were sold to foreign investors during this period, one each in 1993 and 1994. At the start of 1999, 4,330 companies were left to be privatized, with 5,476 having been sold during 1993–1998. At the end of 1998, only 2.4% of privatized companies had foreign participation.

Opinion

Arguments for and against the controversial subject of privatization are presented here.

Support

Proponents of privatization argue that, over time, this can lead to lower prices, improved quality, more choices, less corruption, less red tape, and/or quicker delivery. Many proponents do not argue that everything should be privatized. According to them, market failures and natural monopolies could be problematic. However, anarcho-capitalists prefer that every function of the state be privatized, including defense and dispute resolution.

Proponents of privatization make the following arguments:

• Performance: state-run industries tend to be bureaucratic. A political government may only be motivated to improve a function when its poor performance becomes politically sensitive.
• Increased efficiency: private companies and firms have a greater incentive to produce goods and services more efficiently to increase profits.
• Specialization: a private business has the ability to focus all relevant human and financial resources onto specific functions. A state-owned firm does not have the necessary resources to specialize its goods and services as a result of the general products provided to the greatest number of people in the population.
• Improvements: conversely, the government may put off improvements due to political sensitivity and special interests—even in cases of companies that are run well and better serve their customers' needs.
• Corruption: a state-monopolized function is prone to corruption; decisions are made primarily for political reasons, personal gain of the decision-maker (i.e. "graft"), rather than economic ones. Corruption (or principal–agent issues) in a state-run corporation affects the ongoing asset stream and company performance, whereas any corruption that may occur during the privatization process is a one-time event and does not affect ongoing cash flow or performance of the company.
• Accountability: managers of privately owned companies are accountable to their owners/shareholders and to the consumer, and can only exist and thrive where needs are met. Managers of publicly owned companies are required to be more accountable to the broader community and to political "stakeholders". This can reduce their ability to directly and specifically serve the needs of their customers, and can bias investment decisions away from otherwise profitable areas.
• Civil-liberty concerns: a company controlled by the state may have access to information or assets which may be used against dissidents or any individuals who disagree with their policies.
• Goals: a political government tends to run an industry or company for political goals rather than economic ones.
• Capital: a privately held companies can sometimes more easily raise investment capital in the financial markets when such local markets exist and are suitably liquid. While interest rates for private companies are often higher than for government debt, this can serve as a useful constraint to promote efficient investments by private companies, instead of cross-subsidizing them with the overall credit-risk of the country. Investment decisions are then governed by market interest rates. State-owned industries have to compete with demands from other government departments and special interests. In either case, for smaller markets, political risk may add substantially to the cost of capital.
• Security: governments have had the tendency to "bail out" poorly run businesses, often due to the sensitivity of job losses, when economically, it may be better to let the business fold.
• Lack of market discipline: poorly managed state companies are insulated from the same discipline as private companies, which could go bankrupt, have their management removed, or be taken over by competitors. Private companies are also able to take greater risks and then seek bankruptcy protection against creditors if those risks turn sour.
• Natural monopolies: the existence of natural monopolies does not mean that these sectors must be state owned. Governments can enact or are armed with anti-trust legislation and bodies to deal with anti-competitive behavior of all companies public or private.
• Concentration of wealth: ownership of and profits from successful enterprises tend to be dispersed and diversified—particularly in voucher privatization. The availability of more investment vehicles stimulates capital markets and promotes liquidity and job creation.
• Political influence: nationalized industries are prone to interference from politicians for political or populist reasons. Examples include making an industry buy supplies from local producers (when that may be more expensive than buying from abroad), forcing an industry to freeze its prices/fares to satisfy the electorate or control inflation, increasing its staffing to reduce unemployment, or moving its operations to marginal constituencies.
• Profits: corporations exist to generate profits for their shareholders. Private companies make a profit by enticing consumers to buy their products in preference to their competitors' (or by increasing primary demand for their products, or by reducing costs). Private corporations typically profit more if they serve the needs of their clients well. Corporations of different sizes may target different market niches in order to focus on marginal groups and satisfy their demand. A company with good corporate governance will therefore be incentivized to meet the needs of its customers efficiently.
• Job gains: as the economy becomes more efficient, more profits are obtained and no government subsidies and less taxes are needed, there will be more private money available for investments and consumption and more profitable and better-paid jobs will be created than in the case of a more regulated economy.

Opposition

Opponents of privatization in general—or of certain privatizations in particular—believe that public goods and services should remain primarily in the hands of government in order to ensure that everyone in society has access to them (such as law enforcement, basic health care, and basic education). There is a positive externality when the government provides society at large with public goods and services such as defense and disease control. Some national constitutions in effect define their governments' "core businesses" as being the provision of such things as justice, tranquility, defense, and general welfare. These governments' direct provision of security, stability, and safety, is intended to be done for the common good (in the public interest) with a long-term (for posterity) perspective. As for natural monopolies, opponents of privatization claim that they aren't subject to fair competition, and better administrated by the state.

Although private companies may provide a similar good or service alongside the government, opponents of privatization are critical about completely transferring the provision of public goods, services and assets into private hands for the following reasons:

• Performance: a democratically elected government is accountable to the people through a legislature, Congress or Parliament, and is motivated to safeguarding the assets of the nation. The profit motive may be subordinated to social objectives.
• Improvements: the government is motivated to performance improvements as well run businesses contribute to the State's revenues.
• Corruption: government ministers and civil servants are bound to uphold the highest ethical standards, and standards of probity are guaranteed through codes of conduct and declarations of interest. However, the selling process could lack transparency, allowing the purchaser and civil servants controlling the sale to gain personally.
• Accountability: the public has less control and oversight of private companies although these remain answerable to various stakeholders, including shareholders, clients, suppliers, regulators, employees and collaborators.
• Civil-liberty concerns: a democratically elected government is accountable to the people through a parliament, and can intervene when civil liberties are threatened.
• Goals: the government may seek to use state companies as instruments to further social goals for the benefit of the nation as a whole.
• Capital: governments can raise money in the financial markets most cheaply to re-lend to state-owned enterprises, although this preferential access to capital markets risks undermining financial discipline because of the assurance of a bailout from the government.
• Cuts in essential services: if a government-owned company providing an essential service (such as the water supply) to all citizens is privatized, its new owner(s) could lead to the abandoning of the social obligation to those who are less able to pay, or to regions where this service is unprofitable.
• Natural monopolies: privatization will not result in true competition if a natural monopoly exists.
• Concentration of wealth: profits from successful enterprises end up in private hands instead of being available for public use.
• Political influence: governments may more easily exert pressure on state-owned firms to help implement government policy.
• Profit: private companies do not have any goal other than to maximize profits.
• Privatization and poverty: it is acknowledged by many studies that there are winners and losers with privatization. The number of losers—which may add up to the size and severity of poverty—can be unexpectedly large if the method and process of privatization and how it is implemented are seriously flawed (e.g. lack of transparency leading to state-owned assets being appropriated at minuscule amounts by those with political connections, absence of regulatory institutions leading to transfer of monopoly rents from public to private sector, improper design and inadequate control of the privatization process leading to asset stripping).
• Job loss: due to the additional financial burden placed on privatized companies to succeed without any government help, unlike the public companies, jobs could be lost to keep more money in the company.
• Reduced wages and benefits: a 2014 report by In the Public Interest, a resource center on privatization, argues that "outsourcing public services sets off a downward spiral in which reduced worker wages and benefits can hurt the local economy and overall stability of middle and working class communities."
• Inferior quality products: private, for-profit companies cut corners on providing quality goods and services in order to maximize profit.

Economic theory

In economic theory, privatization has been studied in the field of contract theory. When contracts are complete, institutions such as (private or public) property are difficult to explain, since every desired incentive structure can be achieved with sufficiently complex contractual arrangements, regardless of the institutional structure. All that matters is who are the decision makers and what is their available information. In contrast, when contracts are incomplete, institutions matter. A leading application of the incomplete contract paradigm in the context of privatization is the model by Hart, Shleifer, and Vishny (1997). In their model, a manager can make investments to increase quality (but they may also increase costs) and investments to decrease costs (but they may also reduce quality). It turns out that it depends on the particular situation whether private ownership or public ownership is desirable. The Hart-Shleifer-Vishny model has been further developed in various directions, e.g. to allow for mixed public-private ownership and endogenous assignments of the investment tasks.

Privatization of private companies

Privatization can also refer to the purchase of all outstanding shares of a publicly traded private company by private equity investors, which is more often called "going private". The buyout withdraws the company's shares from being traded at a public stock exchange. Depending on the involvement of internal and external investors, it may occur through a leveraged buyout or a management buyout, tender offer, or hostile takeover.

Law of excluded middle

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Law_of_excluded_middle

In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and the law of identity; however, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law/principle of the excluded third, in Latin principium tertii exclusi. Another Latin designation for this law is tertium non datur or "no third [possibility] is given". In classical logic, the law is a tautology.

In contemporary logic the principle is distinguished from the semantical principle of bivalence, which states that every proposition is either true or false. The principle of bivalence always implies the law of excluded middle, while the converse is not always true. A commonly cited counterexample uses statements unprovable now, but provable in the future to show that the law of excluded middle may apply when the principle of bivalence fails.

History

Aristotle

The earliest known formulation is in Aristotle's discussion of the principle of non-contradiction, first proposed in On Interpretation, where he says that of two contradictory propositions (i.e. where one proposition is the negation of the other) one must be true, and the other false. He also states it as a principle in the Metaphysics book 4, saying that it is necessary in every case to affirm or deny, and that it is impossible that there should be anything between the two parts of a contradiction.

Aristotle wrote that ambiguity can arise from the use of ambiguous names, but cannot exist in the facts themselves:

It is impossible, then, that "being a man" should mean precisely "not being a man", if "man" not only signifies something about one subject but also has one significance. … And it will not be possible to be and not to be the same thing, except in virtue of an ambiguity, just as if one whom we call "man", and others were to call "not-man"; but the point in question is not this, whether the same thing can at the same time be and not be a man in name, but whether it can be in fact. (Metaphysics 4.4, W. D. Ross (trans.), GBWW 8, 525–526).

Aristotle's assertion that "it will not be possible to be and not to be the same thing" would be written in propositional logic as ~(P ∧ ~P). In modern so called classical logic, this statement is equivalent to the law of excluded middle (P ∨ ~P), through distribution of the negation in Aristotle's assertion. The former claims that no statement is both true and false, while the latter requires that any statement is either true or false. (Refer to the List of logic symbols for the meaning of symbols used in this article).

But Aristotle also writes, "since it is impossible that contradictories should be at the same time true of the same thing, obviously contraries also cannot belong at the same time to the same thing" (Book IV, CH 6, p. 531). He then proposes that "there cannot be an intermediate between contradictories, but of one subject we must either affirm or deny any one predicate" (Book IV, CH 7, p. 531). In the context of Aristotle's traditional logic, this is a remarkably precise statement of the law of excluded middle, P ∨ ~P.

Yet in On Interpretation Aristotle seems to deny the law of excluded middle in the case of future contingents, in his discussion on the sea battle.

Leibniz

Its usual form, "Every judgment is either true or false" [footnote 9] …"(from Kolmogorov in van Heijenoort, p. 421) footnote 9: "This is Leibniz's very simple formulation (see Nouveaux Essais, IV,2)" (ibid p 421)

Bertrand Russell and Principia Mathematica

The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:

${\displaystyle \mathbf{\ast }\mathbf{2}\cdot \mathbf{11}.⊢.p\vee \sim p}$.^[8]

So just what is "truth" and "falsehood"? At the opening PM quickly announces some definitions:

Truth-values. The "truth-value" of a proposition is truth if it is true and falsehood if it is false* [*This phrase is due to Frege] … the truth-value of "p ∨ q" is truth if the truth-value of either p or q is truth, and is falsehood otherwise … that of "~ p" is the opposite of that of p …" (pp. 7–8)

This is not much help. But later, in a much deeper discussion ("Definition and systematic ambiguity of Truth and Falsehood" Chapter II part III, p. 41 ff), PM defines truth and falsehood in terms of a relationship between the "a" and the "b" and the "percipient". For example "This 'a' is 'b'" (e.g. "This 'object a' is 'red'") really means "'object a' is a sense-datum" and "'red' is a sense-datum", and they "stand in relation" to one another and in relation to "I". Thus what we really mean is: "I perceive that 'This object a is red'" and this is an undeniable-by-3rd-party "truth".

PM further defines a distinction between a "sense-datum" and a "sensation":

That is, when we judge (say) "this is red", what occurs is a relation of three terms, the mind, and "this", and "red". On the other hand, when we perceive "the redness of this", there is a relation of two terms, namely the mind and the complex object "the redness of this" (pp. 43–44).

Russell reiterated his distinction between "sense-datum" and "sensation" in his book The Problems of Philosophy (1912), published at the same time as PM (1910–1913):

Let us give the name of "sense-data" to the things that are immediately known in sensation: such things as colours, sounds, smells, hardnesses, roughnesses, and so on. We shall give the name "sensation" to the experience of being immediately aware of these things … The colour itself is a sense-datum, not a sensation. (p. 12)

Russell further described his reasoning behind his definitions of "truth" and "falsehood" in the same book (Chapter XII, Truth and Falsehood).

Consequences of the law of excluded middle in Principia Mathematica

From the law of excluded middle, formula ✸2.1 in Principia Mathematica, Whitehead and Russell derive some of the most powerful tools in the logician's argumentation toolkit. (In Principia Mathematica, formulas and propositions are identified by a leading asterisk and two numbers, such as "✸2.1".)

✸2.1 ~p ∨ p "This is the Law of excluded middle" (PM, p. 101).

The proof of ✸2.1 is roughly as follows: "primitive idea" 1.08 defines p → q = ~p ∨ q. Substituting p for q in this rule yields p → p = ~p ∨ p. Since p → p is true (this is Theorem 2.08, which is proved separately), then ~p ∨ p must be true.

✸2.11 p ∨ ~p (Permutation of the assertions is allowed by axiom 1.4)
✸2.12 p → ~(~p) (Principle of double negation, part 1: if "this rose is red" is true then it's not true that "'this rose is not-red' is true".)
✸2.13 p ∨ ~{~(~p)} (Lemma together with 2.12 used to derive 2.14)
✸2.14 ~(~p) → p (Principle of double negation, part 2)
✸2.15 (~p → q) → (~q → p) (One of the four "Principles of transposition". Similar to 1.03, 1.16 and 1.17. A very long demonstration was required here.)
✸2.16 (p → q) → (~q → ~p) (If it's true that "If this rose is red then this pig flies" then it's true that "If this pig doesn't fly then this rose isn't red.")
✸2.17 ( ~p → ~q ) → (q → p) (Another of the "Principles of transposition".)
✸2.18 (~p → p) → p (Called "The complement of reductio ad absurdum. It states that a proposition which follows from the hypothesis of its own falsehood is true" (PM, pp. 103–104).)

Most of these theorems—in particular ✸2.1, ✸2.11, and ✸2.14—are rejected by intuitionism. These tools are recast into another form that Kolmogorov cites as "Hilbert's four axioms of implication" and "Hilbert's two axioms of negation" (Kolmogorov in van Heijenoort, p. 335).

Propositions ✸2.12 and ✸2.14, "double negation": The intuitionist writings of L. E. J. Brouwer refer to what he calls "the principle of the reciprocity of the multiple species, that is, the principle that for every system the correctness of a property follows from the impossibility of the impossibility of this property" (Brouwer, ibid, p. 335).

This principle is commonly called "the principle of double negation" (PM, pp. 101–102). From the law of excluded middle (✸2.1 and ✸2.11), PM derives principle ✸2.12 immediately. We substitute ~p for p in 2.11 to yield ~p ∨ ~(~p), and by the definition of implication (i.e. 1.01 p → q = ~p ∨ q) then ~p ∨ ~(~p)= p → ~(~p). QED (The derivation of 2.14 is a bit more involved.)

Reichenbach

It is correct, at least for bivalent logic—i.e. it can be seen with a Karnaugh map—that this law removes "the middle" of the inclusive-or used in his law (3). And this is the point of Reichenbach's demonstration that some believe the exclusive-or should take the place of the inclusive-or.

About this issue (in admittedly very technical terms) Reichenbach observes:

The tertium non datur

29. (x)[f(x) ∨ ~f(x)]

is not exhaustive in its major terms and is therefore an inflated formula. This fact may perhaps explain why some people consider it unreasonable to write (29) with the inclusive-'or', and want to have it written with the sign of the exclusive-'or'

30. (x)[f(x) ⊕ ~f(x)], where the symbol "⊕" signifies exclusive-or

in which form it would be fully exhaustive and therefore nomological in the narrower sense. (Reichenbach, p. 376)

In line (30) the "(x)" means "for all" or "for every", a form used by Russell and Reichenbach; today the symbolism is usually ${\displaystyle \mathrm{\forall }}$ x. Thus an example of the expression would look like this:

• (pig): (Flies(pig) ⊕ ~Flies(pig))
• (For all instances of "pig" seen and unseen): ("Pig does fly" or "Pig does not fly" but not both simultaneously)

Formalists versus Intuitionists

From the late 1800s through the 1930s, a bitter, persistent debate raged between Hilbert and his followers versus Hermann Weyl and L. E. J. Brouwer. Brouwer's philosophy, called intuitionism, started in earnest with Leopold Kronecker in the late 1800s.

Hilbert intensely disliked Kronecker's ideas:

Kronecker insisted that there could be no existence without construction. For him, as for Paul Gordan [another elderly mathematician], Hilbert's proof of the finiteness of the basis of the invariant system was simply not mathematics. Hilbert, on the other hand, throughout his life was to insist that if one can prove that the attributes assigned to a concept will never lead to a contradiction, the mathematical existence of the concept is thereby established (Reid p. 34)

It was his [Kronecker's] contention that nothing could be said to have mathematical existence unless it could actually be constructed with a finite number of positive integers (Reid p. 26)

The debate had a profound effect on Hilbert. Reid indicates that Hilbert's second problem (one of Hilbert's problems from the Second International Conference in Paris in 1900) evolved from this debate (italics in the original):

In his second problem, [Hilbert] had asked for a mathematical proof of the consistency of the axioms of the arithmetic of real numbers.

To show the significance of this problem, he added the following observation:

"If contradictory attributes be assigned to a concept, I say that mathematically the concept does not exist" (Reid p. 71)

Thus, Hilbert was saying: "If p and ~p are both shown to be true, then p does not exist", and was thereby invoking the law of excluded middle cast into the form of the law of contradiction.

And finally constructivists … restricted mathematics to the study of concrete operations on finite or potentially (but not actually) infinite structures; completed infinite totalities … were rejected, as were indirect proof based on the Law of Excluded Middle. Most radical among the constructivists were the intuitionists, led by the erstwhile topologist L. E. J. Brouwer (Dawson p. 49)

The rancorous debate continued through the early 1900s into the 1920s; in 1927 Brouwer complained about "polemicizing against it [intuitionism] in sneering tones" (Brouwer in van Heijenoort, p. 492). But the debate was fertile: it resulted in Principia Mathematica (1910–1913), and that work gave a precise definition to the law of excluded middle, and all this provided an intellectual setting and the tools necessary for the mathematicians of the early 20th century:

Out of the rancor, and spawned in part by it, there arose several important logical developments; Zermelo's axiomatization of set theory (1908a), that was followed two years later by the first volume of Principia Mathematica, in which Russell and Whitehead showed how, via the theory of types: much of arithmetic could be developed by logicist means (Dawson p. 49)

Brouwer reduced the debate to the use of proofs designed from "negative" or "non-existence" versus "constructive" proof:

According to Brouwer, a statement that an object exists having a given property means that, and is only proved, when a method is known which in principle at least will enable such an object to be found or constructed …

Hilbert naturally disagreed.

"pure existence proofs have been the most important landmarks in the historical development of our science," he maintained. (Reid p. 155)

Brouwer refused to accept the logical principle of the excluded middle, His argument was the following:

"Suppose that A is the statement "There exists a member of the set S having the property P." If the set is finite, it is possible—in principle—to examine each member of S and determine whether there is a member of S with the property P or that every member of S lacks the property P." For finite sets, therefore, Brouwer accepted the principle of the excluded middle as valid. He refused to accept it for infinite sets because if the set S is infinite, we cannot—even in principle—examine each member of the set. If, during the course of our examination, we find a member of the set with the property P, the first alternative is substantiated; but if we never find such a member, the second alternative is still not substantiated.

Since mathematical theorems are often proved by establishing that the negation would involve us in a contradiction, this third possibility which Brouwer suggested would throw into question many of the mathematical statements currently accepted.

"Taking the Principle of the Excluded Middle from the mathematician," Hilbert said, "is the same as … prohibiting the boxer the use of his fists."

"The possible loss did not seem to bother Weyl … Brouwer's program was the coming thing, he insisted to his friends in Zürich." (Reid, p. 149)

In his lecture in 1941 at Yale and the subsequent paper, Gödel proposed a solution: "that the negation of a universal proposition was to be understood as asserting the existence … of a counterexample" (Dawson, p. 157)

Gödel's approach to the law of excluded middle was to assert that objections against "the use of 'impredicative definitions'" had "carried more weight" than "the law of excluded middle and related theorems of the propositional calculus" (Dawson p. 156). He proposed his "system Σ … and he concluded by mentioning several applications of his interpretation. Among them were a proof of the consistency with intuitionistic logic of the principle ~ (∀A: (A ∨ ~A)) (despite the inconsistency of the assumption ∃ A: ~ (A ∨ ~A))" (Dawson, p. 157)

The debate seemed to weaken: mathematicians, logicians and engineers continue to use the law of excluded middle (and double negation) in their daily work.

Intuitionist definitions of the law (principle) of excluded middle

The following highlights the deep mathematical and philosophic problem behind what it means to "know", and also helps elucidate what the "law" implies (i.e. what the law really means). Their difficulties with the law emerge: that they do not want to accept as true implications drawn from that which is unverifiable (untestable, unknowable) or from the impossible or the false. (All quotes are from van Heijenoort, italics added).

Brouwer offers his definition of "principle of excluded middle"; we see here also the issue of "testability":

On the basis of the testability just mentioned, there hold, for properties conceived within a specific finite main system, the "principle of excluded middle", that is, the principle that for every system every property is either correct [richtig] or impossible, and in particular the principle of the reciprocity of the complementary species, that is, the principle that for every system the correctness of a property follows from the impossibility of the impossibility of this property. (335)

Kolmogorov's definition cites Hilbert's two axioms of negation

5. A → (~A → B)
6. (A → B) → { (~A → B) → B}

Hilbert's first axiom of negation, "anything follows from the false", made its appearance only with the rise of symbolic logic, as did the first axiom of implication … while … the axiom under consideration [axiom 5] asserts something about the consequences of something impossible: we have to accept B if the true judgment A is regarded as false …

Hilbert's second axiom of negation expresses the principle of excluded middle. The principle is expressed here in the form in which is it used for derivations: if B follows from A as well as from ~A, then B is true. Its usual form, "every judgment is either true or false" is equivalent to that given above".

From the first interpretation of negation, that is, the interdiction from regarding the judgment as true, it is impossible to obtain the certitude that the principle of excluded middle is true … Brouwer showed that in the case of such transfinite judgments the principle of excluded middle cannot be considered obvious

footnote 9: "This is Leibniz's very simple formulation (see Nouveaux Essais, IV,2). The formulation "A is either B or not-B" has nothing to do with the logic of judgments.

footnote 10: "Symbolically the second form is expressed thus

A ∨ ~A

where ∨ means "or". The equivalence of the two forms is easily proved (p. 421)

Examples

For example, if P is the proposition:

Socrates is mortal.

then the law of excluded middle holds that the logical disjunction:

Either Socrates is mortal, or it is not the case that Socrates is mortal.

is true by virtue of its form alone. That is, the "middle" position, that Socrates is neither mortal nor not-mortal, is excluded by logic, and therefore either the first possibility (Socrates is mortal) or its negation (it is not the case that Socrates is mortal) must be true.

An example of an argument that depends on the law of excluded middle follows. We seek to prove that

there exist two irrational numbers ${\displaystyle a}$ and ${\displaystyle b}$ such that ${\displaystyle a^b}$ is rational.

It is known that ${\displaystyle \sqrt{2}}$ is irrational (see proof). Consider the number

${\displaystyle {\sqrt{2}}^{\sqrt{2}}}$.

Clearly (excluded middle) this number is either rational or irrational. If it is rational, the proof is complete, and

${\displaystyle a=\sqrt{2}}$ and ${\displaystyle b=\sqrt{2}}$.

But if ${\displaystyle {\sqrt{2}}^{\sqrt{2}}}$ is irrational, then let

${\displaystyle a={\sqrt{2}}^{\sqrt{2}}}$ and ${\displaystyle b=\sqrt{2}}$.

Then

${\displaystyle a^b={\left({\sqrt{2}}^{\sqrt{2}}\right)}^{\sqrt{2}}={\sqrt{2}}^{\left(\sqrt{2}\cdot \sqrt{2}\right)}={\sqrt{2}}^2=2}$,

and 2 is certainly rational. This concludes the proof.

In the above argument, the assertion "this number is either rational or irrational" invokes the law of excluded middle. An intuitionist, for example, would not accept this argument without further support for that statement. This might come in the form of a proof that the number in question is in fact irrational (or rational, as the case may be); or a finite algorithm that could determine whether the number is rational.

Non-constructive proofs over the infinite

The above proof is an example of a non-constructive proof disallowed by intuitionists:

The proof is non-constructive because it doesn't give specific numbers ${\displaystyle a}$ and ${\displaystyle b}$ that satisfy the theorem but only two separate possibilities, one of which must work. (Actually ${\displaystyle a={\sqrt{2}}^{\sqrt{2}}}$ is irrational but there is no known easy proof of that fact.) (Davis 2000:220)

(Constructive proofs of the specific example above are not hard to produce; for example ${\displaystyle a=\sqrt{2}}$ and ${\displaystyle b={\log }_{2}9}$ are both easily shown to be irrational, and ${\displaystyle a^b=3}$; a proof allowed by intuitionists).

By non-constructive Davis means that "a proof that there actually are mathematic entities satisfying certain conditions would not have to provide a method to exhibit explicitly the entities in question." (p. 85). Such proofs presume the existence of a totality that is complete, a notion disallowed by intuitionists when extended to the infinite—for them the infinite can never be completed:

In classical mathematics there occur non-constructive or indirect existence proofs, which intuitionists do not accept. For example, to prove there exists an n such that P(n), the classical mathematician may deduce a contradiction from the assumption for all n, not P(n). Under both the classical and the intuitionistic logic, by reductio ad absurdum this gives not for all n, not P(n). The classical logic allows this result to be transformed into there exists an n such that P(n), but not in general the intuitionistic … the classical meaning, that somewhere in the completed infinite totality of the natural numbers there occurs an n such that P(n), is not available to him, since he does not conceive the natural numbers as a completed totality. (Kleene 1952:49–50)

David Hilbert and Luitzen E. J. Brouwer both give examples of the law of excluded middle extended to the infinite. Hilbert's example: "the assertion that either there are only finitely many prime numbers or there are infinitely many" (quoted in Davis 2000:97); and Brouwer's: "Every mathematical species is either finite or infinite." (Brouwer 1923 in van Heijenoort 1967:336). In general, intuitionists allow the use of the law of excluded middle when it is confined to discourse over finite collections (sets), but not when it is used in discourse over infinite sets (e.g. the natural numbers). Thus intuitionists absolutely disallow the blanket assertion: "For all propositions P concerning infinite sets D: P or ~P" (Kleene 1952:48).

Putative counterexamples to the law of excluded middle include the liar paradox or Quine's paradox. Certain resolutions of these paradoxes, particularly Graham Priest's dialetheism as formalised in LP, have the law of excluded middle as a theorem, but resolve out the Liar as both true and false. In this way, the law of excluded middle is true, but because truth itself, and therefore disjunction, is not exclusive, it says next to nothing if one of the disjuncts is paradoxical, or both true and false.

Criticisms

The Catuṣkoṭi (tetralemma) is an ancient alternative to the law of excluded middle, which examines all four possible assignments of truth values to a proposition and its negation. It has been important in Indian logic and Buddhist logic as well as the ancient Greek philosophical school known as Pyrrhonism.

Many modern logic systems replace the law of excluded middle with the concept of negation as failure. Instead of a proposition's being either true or false, a proposition is either true or not able to be proved true. These two dichotomies only differ in logical systems that are not complete. The principle of negation as failure is used as a foundation for autoepistemic logic, and is widely used in logic programming. In these systems, the programmer is free to assert the law of excluded middle as a true fact, but it is not built-in a priori into these systems.

Mathematicians such as L. E. J. Brouwer and Arend Heyting have also contested the usefulness of the law of excluded middle in the context of modern mathematics.

In mathematical logic

In modern mathematical logic, the excluded middle has been argued to result in possible self-contradiction. It is possible in logic to make well-constructed propositions that can be neither true nor false; a common example of this is the "Liar's paradox", the statement "this statement is false", which is argued to itself be neither true nor false. Arthur Prior has argued that the Paradox is not an example of a statement that cannot be true or false. The law of excluded middle still holds here as the negation of this statement "This statement is not false", can be assigned true. In set theory, such a self-referential paradox can be constructed by examining the set "the set of all sets that do not contain themselves". This set is unambiguously defined, but leads to a Russell's paradox: does the set contain, as one of its elements, itself? However, in the modern Zermelo–Fraenkel set theory, this type of contradiction is no longer admitted. Furthermore, paradoxes of self reference can be constructed without even invoking negation at all, as in Curry's paradox.

Analogous laws

Some systems of logic have different but analogous laws. For some finite n-valued logics, there is an analogous law called the law of excluded n+1th. If negation is cyclic and "∨" is a "max operator", then the law can be expressed in the object language by (P ∨ ~P ∨ ~~P ∨ ... ∨ ~...~P), where "~...~" represents n−1 negation signs and "∨ ... ∨" n−1 disjunction signs. It is easy to check that the sentence must receive at least one of the n truth values (and not a value that is not one of the n).

Other systems reject the law entirely.

Law of the weak excluded middle

A particularly well-studied intermediate logic is given by De Morgan logic, which adds the axiom ${\displaystyle \mathrm{\neg }P\vee \mathrm{\neg }\mathrm{\neg }P}$ to intuitionistic logic, which is sometimes called the law of the weak excluded middle.

This is equivalent to a few other statements:

• Satisfying all of De Morgan's laws including ${\displaystyle \mathrm{\neg }(P\wedge Q)\leftrightarrow \mathrm{\neg }P\vee \mathrm{\neg }Q}$
• ${\displaystyle (P\to Q)\vee (\mathrm{\neg }P\to \mathrm{\neg }Q)}$
• ${\displaystyle (P\to (Q\vee \mathrm{\neg }R))\to ((P\to Q)\vee (P\to \mathrm{\neg }R))}$