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Sunday, August 14, 2022

Rights of nature

From Wikipedia, the free encyclopedia

Rights of nature or Earth rights is a legal and jurisprudential theory that describes inherent rights as associated with ecosystems and species, similar to the concept of fundamental human rights. The rights of nature concept challenges twentieth-century laws as generally grounded in a flawed frame of nature as "resource" to be owned, used, and degraded. Proponents argue that laws grounded in rights of nature direct humanity to act appropriately and in a way consistent with modern, system-based science, which demonstrates that humans and the natural world are fundamentally interconnected.

This school of thought is underpinned by two basic lines of reasoning. First, since the recognition of human rights is based in part on the philosophical belief that those rights emanate from humanity's own existence, logically, so too do inherent rights of the natural world arise from the natural world's own existence. A second and more pragmatic argument asserts that the survival of humans depends on healthy ecosystems, and so protection of nature's rights in turn, advances human rights and well-being.

From a rights of nature perspective, most environmental laws of the twentieth century are based on an outmoded framework that considers nature to be composed of separate and independent parts, rather than components of a larger whole. A more significant criticism is that those laws tend to be subordinate to economic interests, and aim at reacting to and just partially mitigating economics-driven degradation, rather than placing nature's right to thrive as the primary goal of those laws. This critique of existing environmental laws is an important component of tactics such as climate change litigation that seeks to force societal action to mitigate climate change.

As of June 2021, rights of nature laws exist at the local to national levels in at least 39 countries, including dozens of cities and counties throughout the United States. They take the form of constitutional provisions, treaty agreements, statutes, local ordinances, and court decisions. A state constitutional provision is being sought in Florida.

Basic tenets

Proponents of rights of nature argue that, just as human rights have been recognized increasingly in law, so should nature's rights be recognized and incorporated into human ethics and laws. This claim is underpinned by two lines of reasoning: that the same ethics that justify human rights, also justify nature's rights, and, that humans' own survival depend on healthy ecosystems.

Thomas Berry - a U.S. cultural historian who introduced the legal concept of Earth Jurisprudence who proposed that society's laws should derive from the laws of nature, explaining that "the universe is a communion of subjects, not a collection of objects"

First, it is argued that if inherent human rights arise from human existence, so too logically do inherent rights of the natural world arise from the natural world's own existence. Human rights, and associated duties to protect those rights, have expanded over time. Most notably, the 1948 adoption by the United Nations, of the Universal Declaration of Human Rights (UDHR) that formalized recognition of broad categories of inalienable human rights. Drafters of the UDHR stated their belief that the concept of fundamental human rights arose not from "the decision of a worldly power, but rather in the fact of existing."

Some scholars have contended thereafter that, given that basic human rights emanate from humans' own existence, nature's rights similarly arise from the similar existence of nature, and so humans' legal systems should continue to expand to recognize the rights of nature.

Some notable proponents of this approach include U.S. cultural historian Thomas Berry, South African attorney Cormac Cullinan, Indian physicist and eco-social advocate Vandana Shiva, and Canadian law professor and U.N. Special Rapporteur for Human Rights and the Environment David R. Boyd.

Vandana Shiva - an Indian scholar and activist who has written extensively on Earth Jurisprudence and Earth Democracy that she describes as based on "local communities – organized on principles of inclusion, diversity, and ecological and social responsibility"

Thomas Berry introduced a philosophy and ethics of law concept called Earth jurisprudence that identifies the earth's laws as primary and reasons that everything by the fact of its existence, therefore, has an intrinsic right to be and evolve. Earth Jurisprudence has been increasingly recognized and promoted worldwide by legal scholars, the United Nations, lawmakers, philosophers, ecological economists, and other experts as a foundation for Earth-centered governance, including laws and economic systems that protect the fundamental rights of nature.

Second, support for rights of nature also is supported through the utilitarian argument that humanity can only thrive in the long term by accepting integrated co-existence of humans with the natural world. Berry noted that the concept of human well-being derived from natural systems with no fundamental right to exist is inherently illogical, and that by protecting nature's rights, humans advance their own self-interest.

The legal and philosophical concept of rights of nature offers a shift from a frame of nature as property or resource, to nature as an interconnected Earth community partner. This school of thought aims at following the same path that human rights movements have followed, where at first recognition of rights in the rightless appeared "unthinkable", but later matured into a broadly-espoused worldview.

Christopher Stone, a law professor at the University of Southern California, wrote extensively on this topic in his seminal essay, "Should Trees Have Standing", cited by a U.S. Supreme Court dissent in Sierra Club v. Morton for the position that "environmental issues should be tendered by [nature] itself." As described by Stone and others, human rights have increasingly been "found" over time and declared "self-evident", as in the U.S. Declaration of Independence, even where essentially non-existent in the law. The successes of past and current human rights movements provide lessons for the current movement to widen the circle of Earth community to include natural systems and species populations as rights-bearing entities.

Underpinnings and development

Critique of anthropocentric legal systems

Proponents of a shift to a more environmentally protective system of law contend that current legal and economic systems fail because they consider nature fundamentally as property, which can be degraded for profit and human desire. They point out that the perspective of nature as primarily an economic resource already has degraded some ecosystems and species so significantly that now, prominent policy experts are examining "endangered species triage" strategies to decide which species will be let go, rather than re-examine the economics driving the degradation. While twentieth and twenty-first century environmental laws do afford some level of protection to ecosystems and species, it is argued that such protections fail to stop, let alone reverse, overall environmental decline, because nature is by definition subordinated to anthropogenic and economic interests, rather than biocentric well-being.

Rights of nature proponents contend that re-envisioning current environmental laws from a nature's rights frame demonstrates the limitations of current legal systems. For example, the U.S. Endangered Species Act prioritizes protection of existing economic interests by activating only when species populations are headed toward extinction. By contrast, a "Healthy Species Act" would prioritize achievement of thriving species populations and facilitate economic systems that drive conservation of species.

As another example, the European Union's Water Framework Directive of 2000, "widely accepted as the most substantial and ambitious piece of European environmental legislation to date", relies on a target of "good status" of all EU waters, which includes consideration of needed "ecological flows". However, decades after the Directive's adoption, despite scientific advances in identifying flow-ecology relationships, there remains no EU definition of "ecological flow", nor a common understanding of how it should be calculated. A nature's rights frame would recognize not only the existing human right to water for basic needs, but would also recognize the rights of waterways to adequate, timely, clean water flows, and would define such basic ecological flow needs accordingly.

Underlying science and ethics

Modern environmental laws began to arise in the 1960s out of a foundational perspective of the environment as best managed in discrete pieces. For example, United States laws such as the Clean Water Act, Clean Air Act, Endangered Species Act, Marine Mammal Protection Act, and numerous others began to be adopted in the early 1970s to address various elements of the natural world, separately from other elements. Some laws, such as the U.S. National Environmental Policy Act, called for a more holistic analysis of proposed infrastructure projects and required the disclosure of expected negative environmental impacts. However, it did not require that actions be taken to address those impacts in order to ensure ecosystem and species health.

These laws reflected the science of the time, which was grounded in a reductionist analysis of the natural world; the modern, system-based understanding of the natural world, and the integrated place of humans with it, was still in development. The first major textbook on ecological science that described the natural world as a system rather than a collection of different parts, was not written until 1983. The Gaia Hypothesis, which offered a scientific vision of the world as a self-regulating, complex system, first arose in the 1970s. Systems dynamics similarly began to evolve from a business focus to include socioeconomic and natural systems starting in the 1970s. Since then, scientific disciplines have been converging and advancing on the concept that humans live in a dynamic, relationship-based world that "den[ies] the possibility of isolation".

While science in the late twentieth century shifted to a systems-based perspective, describing natural systems and human populations as fundamentally interconnected on a shared planet, environmental laws generally did not evolve with this shift. Reductionist U.S. environmental laws passed in the early 1970s remained largely unchanged, and other national and international environmental law regimes similarly stopped short of embracing the modern science of systems.

Nineteenth century linguist and scholar Edward Payson Evans, an early rights of nature theorist and author of "the first extensive American statement of (...) environmental ethics”, wrote that each human is “truly a part and product of Nature as any other animal, and [the] attempt to set him up on an isolated point outside of it is philosophically false and morally pernicious”.

Thomas Berry proposed that society's laws should derive from the laws of nature, explaining that "the universe is a communion of subjects, not a collection of objects". From the scientific perspective that all life arose from the context of the universe, Berry offered the ethical perspective that it is flawed to view humans as the universe's only subjects, with all other beings merely a collection of objects to be owned and used. Rather, consideration of life as a web of relationships extending back to a shared ancestry confers subject status to all, including the inherent rights associated with that status. Laws based on a recognition of the intrinsic moral value of the natural world, create a new societal moral compass that directs society's interactions with the natural world more effectively toward well-being for all.

Aldo Leopold - a scientist and forester who advocated to "see land as a community to which we belong" rather than as "a commodity belonging to us" (1946 photograph)

Scientists who similarly wrote in support of expanded human moral development and ethical obligation include naturalist John Muir and scientist and forester Aldo Leopold. Leopold expressed that "[w]hen we see land as a community to which we belong", rather than "a commodity belonging to us", we can "begin to use it with love and respect". Leopold offered implementation guidance for his position, stating that a "thing is right when it tends to preserve the integrity, stability, and beauty of the biotic community. It is wrong when it tends otherwise." Berry similarly observed that "whatever preserves and enhances this meadow in the natural cycles of its transformation is good; what is opposed to this meadow or negates it is not good." Physician and philosopher Albert Schweizer defined right actions as those that recognize a reverence for life and the "will to live".

The outgrowth of scientific and ethical advances around natural systems and species is a proposed new frame for legal and governance systems, one grounded in an ethic and a language that guide behavior away from ecological and social practices that ignore or minimize human-nature interconnections. Rather than a vision of merely "sustainable development", which reflects a frame of nature maintained as economic feedstock, scholars supporting rights of nature suggest that society is beginning to consider visions such as "thriving communities", where "communities" includes nature as a full subject, rather than simply an object to be used.

History

Common roots with Indigenous worldviews

The ethical and philosophical foundation of a nature's rights legal theory and movement is a worldview of respect for nature, as contrasted with the "nature domination" worldview that underlies the concept of nature as object and property. Indigenous law professor John Borrows observed that "[w]ithin indigenous legal traditions, creation stories... give guidance about how to live with the world", rather than live at odds with it. A 2012 international Declaration of Indigenous Peoples found that modern laws destroy the earth because they do not respect the "natural order of Creation". The Declaration observed that humans "have our place and our responsibilities within Creation's sacred order" and benefit from "sustaining joy as things occur in harmony with the Earth and with all life that it creates and sustains".

Indigenous worldviews align with and have accelerated the development of rights of nature law, including in Ecuador and Bolivia. Ecuador amended its constitution in 2008 to recognize the rights of nature in light of the perceived need to better protect and respect Pachamama, a term that embodies both the physical and the spiritual aspects of the natural world. Bolivia similarly amended its constitution and enacted nature's rights statutes to reflect traditional Indigenous respect for Pachamama, and a worldview of natural systems and humans as part of one family.

New Zealand law professor Catherine Iorns Magallanes observed that traditional Indigenous worldviews embody a connection with nature is so deep that nature is regarded as a living ancestor. From this worldview arises responsibilities to protect nature as one would a family member, and the need for a legal structure that reflects a primary frame of responsibilities to the natural world as kin.

Common roots with world religions

Many of the world's other religious and spiritual traditions offer insights consistent with a nature's rights worldview. Eastern religious and philosophical traditions embrace a holistic conception of spirituality that includes the Earth. Chinese Daoism and Neo-Confucianism, as well as Japanese Buddhism, teach that the world is a dynamic force field of energies known as bussho (Buddha nature or qi), the material force that flows through humans, nature, and universe. As the eleventh century pioneering Neo-Confucianist philosopher Zhang Zai explained, "that which extends throughout the universe I regard as my body and that which directs the universe I consider as my nature".

In both Hinduism and Buddhism, karma ("action" or "declaration" in Sanskrit) reflects the reality of humanity's networked interrelations with Earth and universe. Buddhist concepts of “co-dependent arising” similarly hold that all phenomena are intimately connected. Mahayana Buddhism's "Indra's Net" symbolizes a universe of infinitely repeated mutual relations, with no one thing dominating.

Western religious and philosophical traditions have recognized the context of Earth and universe in providing spiritual guidance as well. From the Neolithic through the Bronze ages, the societies of "Old Europe" revered numerous female deities as incarnations of Mother Earth. In early Greece, the earth goddess Gaia was worshipped as a supreme deity. In the Philebus and Timaeus, Plato asserted that the "world is indeed a living being endowed with a soul and intelligence (...) a single visible living entity containing all other living entities, which by their nature are all related". Medieval theologian St. Thomas Aquinas later wrote of the place of humans, not at the center of being, but as one part of an integrated whole with the universe as primary, stating that “The order of the universe is the ultimate and noblest perfection in things."

More recently, Pope Benedict XVI, head of the Catholic church, reflected that, "[t]he obedience to the voice of Earth is more important for our future happiness... than the desires of the moment. Our Earth is talking to us and we must listen to it and decipher its message if we want to survive." His successor, Pope Francis, has been particularly vocal on humanity's relationship with the Earth, describing how humans must change their current actions in light of the fact that "a true 'right of the environment' does exist". He warned against humanity's current path, stating that "the deepest roots of our present failures" lie in the direction and meaning of economic growth, and the overarching rule of a "deified market".

The Qur’an, Islam's primary authority in all matters of individual and communal life, reflects that "the whole creation praises God by its very being". Scholars describe the "ultimate purpose of the Shari'ah" as "the universal common good, the welfare of the entire creation," and note that "not a single creature, present or future, may be excluded from consideration in deciding a course of action."

Bringing together Western and Indigenous traditions, Archbishop Desmond Tutu spoke of "Ubuntu", an African ethical concept that translates roughly to "I am because you are", observing that: "Ubuntu speaks particularly about the fact that you can't exist as a human being in isolation. It speaks about our interconnectedness... We think of ourselves far too frequently as just individuals, separated from one another, whereas you are connected and what you do affects the whole world."

Common roots with human rights

Human rights have been developing over centuries, with the most notable outgrowth being the adoption of Universal Declaration of Human Rights (UDHR) by the United Nations in 1948. Key to the development of those rights are the concepts of natural rights, and rights of humans emanating from the existence of humanity.

Roderick Fraser Nash, professor of history and environmental studies at the University of California, Santa Barbara, traced the history of rights for species and the natural world back to the thirteenth century Magna Carta's launch of the concept of "natural rights" that underlies modern rights discourse.

Peter Burdon, professor at the University of Adelaide Law School and an Earth Jurisprudence scholar, has expanded upon Nash's analysis, offering that seventeenth century English philosopher and physician John Locke's transformative natural rights thesis led to the American Revolution, through the concept that the British monarchy was denying colonists their natural rights. Building on that concept, U.S. President, attorney, and philosopher Thomas Jefferson argued that the "laws of nature and of nature's God" reveal "self-evident" truths that "all Men are created equal" in their possession of "certain unalienable rights", particularly "life, liberty, and the pursuit of happiness". The 1789 French Declaration of the Rights of Man and of the Citizen later recognized as well the "natural, inalienable and sacred rights of Man", adding that the "final end of every political institution is the preservation of the natural and imprescriptible rights of Man."

The expansion of rights continued out to animals, with eighteenth-nineteenth century English philosopher and legal theorist Jeremy Bentham claiming that the “day may come when the rest of the animal creation may acquire those rights which never could have been withholden from them but by the hand of tyranny”. Nineteenth century linguist and scholar Edward Payson Evans observed that:

"[i]n tracing the history of the evolution of ethics we find the recognition of mutual rights and duties confined at first to members of the same horde or tribe, then extended to worshippers of the same gods, and gradually enlarged so as to include every civilized nation, until at length all races of men are at least theoretically conceived as being united in a common bond of brotherhood and benevolent sympathy, which is now slowly expanding so as to comprise not only the higher species of animals, but also every sensitive embodiment of organic life."

The 1948 adoption of the Universal Declaration of Human Rights (UDHR) by the United Nations was another milestone, underpinned by the belief that fundamental human rights arise from "the fact of existing". The movement for rights of nature built on this belief, arguing that if "existence" is the defining condition for fundamental rights, this defining condition could not be limited to the rights of only one form of existence, and that all forms of existence should enjoy fundamental rights. For example, Aldo Leopold's land ethic explicitly recognized nature's "right to continued existence" and sought to "change the role of Homo sapiens from conqueror of the land-community to plain member and citizen of it".

Proponents of the rights of nature also contend that from the abolition of slavery, to the granting of the right to vote to women, to the civil rights movement, and the recognition of other fundamental rights, societies have continued to expand rights in parallel with a growing acceptance of the inherent moral worth of the potential new rights holders. And, that this expansion of the circle of community ought to continue to grow to encompass the natural world, a position that has seen growing acceptance in the late twentieth century and early twenty-first.

Proponents suggest that rights derived from existence in nature do not confer human rights to all beings, but rather confer unique rights to different kinds of beings. Thomas Berry put forth the theory that rights "are species specific and limited"; that is, "rivers have river rights", "birds have bird rights", and "humans have human rights". In his view, the difference is "qualitative, not quantitative".

Extending this point, the common ethical and moral grounding of human rights and the rights of nature gives rise to the concept of "co-violations" of rights, defined as a "situation in which governments, industries, or others violate both the rights of nature and human rights, including indigenous rights, with the same action". For example, in the Ecuadorian Amazon, pollution from Texaco's (now Chevron) oil drilling operations from 1967 to 1992 resulted in an epidemic of birth defects, miscarriages, and an estimated 1,400 cancer deaths, that were particularly devastating to indigenous communities. These operations further caused more than one million acres of deforestation and polluted local waterways with 18 billion gallons of toxic wastewater and contaminants, severely damaging a formerly pristine rainforest of extraordinary biodiversity. Asserting that the same human actions that created such impacts violated the fundamental rights of both people and natural systems, it is argued that ethical and legal theories that recognize both sets of rights will better guide human behavior to recognize and respect humans' interconnected relationships with each other and the natural world.

As with the recognition of human rights, legal scholars find that recognition of the rights of nature alters the framework of human laws and practices. Harvard Law professor Laurence Tribe theorized further that "choosing to accord nature a fraternal rather than an exploited role... might well make us different persons from the manipulators and subjugators we are in danger of becoming".

20th and 21st century developments

The adoption of the UDHR in 1948 formalized recognition of broad categories of inalienable human rights globally. These include recognition that "[a]ll human beings are born free and equal in dignity and rights", that "[e]veryone has the right to life, liberty and security of person", and that "[e]veryone has the right to an effective remedy by the competent national tribunals for acts violating the fundamental rights granted". Recognition of fundamental rights in "soft law" instruments such as the UDHR provided guidance to nations around the world, who have since developed constitutional provisions, statutes, court decisions, regulations, and other bodies of law based on the UDHR and the human rights it champions.

Decades later, USC law professor Christopher Stone called for recognition of the legal standing and associated rights of the natural world as well, consistent with the "successive extension of rights" throughout legal history. As was done in the UDHR, Stone outlined the necessary elements of nature's participation in human legal systems, describing such a legal system as necessarily including: recognition of injuries as subject to redress by public body, standing to institute legal actions (with guardians acting on behalf of the natural entity), redress calculated for natural entity's own damages, and relief running to the benefit of the injured natural entity.

In addition to Stone's legal work, other late twentieth and early twenty-first century drivers of the rights of nature movement include indigenous perspectives and the work of the indigenous rights movement; the writings of Arne Naess and the Deep Ecology movement; Thomas Berry's 2001 jurisprudential call for recognizing the laws of nature as the "primary text"; the publication of Cormac Cullinan's Wild Law book in 2003, followed by the creation of an eponymous legal association in the UK; growing concern about corporate power through the expansion of legal personhood for corporations; adoption by U.S. communities of local laws addressing rights of nature; the creation of the Global Alliance of the Rights of Nature in 2010 (GARN; a nonprofit advancing rights on nature worldwide); and mounting global concerns with species losses, ecosystem destruction, and the existential threat of climate change.

These and other factors supported the development of the 2010 Universal Declaration of the Rights of Mother Earth (UDRME). The UDRME was adopted by representatives of 130 nations at the World People's Conference on Climate Change and the Rights of Mother Earth, convened in Bolivia following the concerns of many regarding the disappointing results of the 2009 Copenhagen climate negotiations. Just as the U.N. recognized human rights as arising from existence, so did the UDRME find that the "inherent rights of Mother Earth are inalienable in that they arise from the same source as existence". Like the UDHR, the UDRME defends the rights-bearing entity (nature and her elements) from the excesses of governing authorities. These rights include, among others, the recognition that "Mother Earth and all beings of which she is composed have... the right to life and to exist" as well as the "right to integral health". The UDRME adds that "[e]ach being has the right to a place and to play its role in Mother Earth for her harmonious functioning".

Just as the rights protected by the UDHR are enforceable by the "right to an effective remedy by the competent national tribunals", so too does the UDRME specifically require humans and their institutions to "recognize and promote the full implementation and enforcement of the rights and obligations recognized in this Declaration". The UDRME addresses enforcement by requiring "damages caused by human violations of the inherent rights" to be "rectified", with those responsible "held accountable".[88] Moreover, it calls on states to "empower human beings and institutions to defend the rights of Mother Earth and of all beings".

Bolivian President Evo Morales urged then-U.N. Secretary-General Ban Ki-Moon to make U.N. adoption of the UDRME a priority. While that recommendation remains to be addressed, since then the UDRME has served to inform other international and national efforts, such as a 2012 Resolution by the International Union for Conservation of Nature (IUCN) proposing a Universal Declaration of the Rights of Nature. The Incorporation of the Rights of Nature was adopted at the IUCN World Conservation Congress in Hawaii (2016).

As of 2021 rights of nature has been reflected in treaties, constitutions, court decisions, and statutory and administrative law at all levels of government. Craig Kauffman, political science professor at the University of Oregon, and scholar of nature's rights and global governance, contends that evolving rights of nature initiatives and networks represent an "important new global movement" arising from "an informal global governance system... being constructed by citizens disillusioned by the failure of governments to take stronger actions to address the dual crises of climate change and biodiversity loss".

Rights of nature law

The early 2000s saw a significant expansion of rights of nature law, in the form of constitutional provisions, treaty agreements, national and subnational statutes, local laws, and court decisions. As of 2021, nature's rights laws exist in 17 countries, including in Canada, seven Tribal Nations in the U.S. and Canada, and dozens of cities and counties throughout the United States. The total number of countries with either existing or pending rights of nature legal provisions was 28 as of 2019.

New Zealand - in 2012 a treaty agreement between the government and the indigenous group Maori iwi established the Whanganui River (top image), and its tributaries as a legal entity with its own standing.Similarly, Mount Taranaki (bottom image) was recognized in 2014 as "a legal personality, in its own right".

Treaties

New Zealand

Legal standing for natural systems in New Zealand arose alongside new attention paid to long-ignored treaty agreements with the Indigenous Maori. In August 2012, a treaty agreement signed with the Maori iwi recognized the Whanganui River and tributaries as a legal entity, an "indivisible and living whole" with its own standing. The national Te Awa Tupua Act was enacted in March 2017 to further formalize this status.

In 2013, the Te Urewera Forest treaty agreement similarly recognized the legal personhood of the Forest, with the Te Urewera Act signed into law in 2014 to formalize this status. In 2017 a treaty settlement with the Maori was signed that recognized Mount Taranaki as "a legal personality, in its own right".

Each of these developments advanced the indigenous principle that the ecosystems are living, spiritual beings with intrinsic value, incapable of being owned in an absolute sense.

Constitutional law

Ecuador

See also: Rights of nature in Ecuador
 
Yasuní National Park, Ecuador

In 2008, the people of Ecuador amended their Constitution to recognize the inherent rights of nature, or Pachamama. The new text arose in large part as a result of cosmologies of the indigenous rights movement and actions to protect the Amazon, consistent with the concept of sumak kawsay ("buen vivir" in Spanish, "good living" in English), or encapsulating a life in harmony with nature with humans as part of the ecosystem. Among other provisions, Article 71 states that "Nature or Pachamama, where life is reproduced and exists, has the right to exist, persist, maintain itself and regenerate its own vital cycles, structure, functions and its evolutionary processes." The Article adds enforcement language as well, stating that "Any person... may demand the observance of the rights of the natural environment before public bodies", and echoing Christopher Stone, Article 72 adds that “Nature has the right to be completely restored... independent of the obligation... to compensate people”.

Ho-Chunk Nation of Wisconsin

In 2015 the Ho-Chunk Nation of Wisconsin passed a resolution amending their constitution to include the rights of nature. By 2020 a working group was determining how to integrate the resolution into their constitution, laws, regulations, and processes.

Judicial decisions

Turag River, near Dhaka, Bangladesh

Bangladesh

In 2019, the High Court of Bangladesh ruled on a case addressing pollution of and illegal development along the Turag River, an upper tributary of the Buriganga.

Among its findings, the high court recognized the river as a living entity with legal rights, and it further held that the same would apply to all rivers in Bangladesh. The court ordered the National River Protection Commission to serve as the guardian for the Turag and other rivers.

Colombia

Atrato River in Colombia - in a 2016 ruling by the Constitutional Court involving the river's pollution, the court stated that the river is a subject of rights, and that humans are "only one more event within a long evolutionary chain [and] in no way... owner of other species, biodiversity or natural resources, or the fate of the planet".

Colombia has not adopted statutes or constitutional provisions addressing nature's rights (as of 2019). However, this has not prevented Colombian courts from finding nature's rights as inherent. In a 2016 case, the Colombia Constitutional Court ordered cleanup of the polluted Atrato River, stating that nature is a "true subject of rights that must be recognized by states and exercised... for example, by the communities that inhabit it or have a special relationship with it”. The court added that humans are “only one more event within a long evolutionary chain [and] in no way... owner of other species, biodiversity or natural resources, or the fate of the planet".

In 2018, the Colombia Supreme Court took up a climate change case by a group of children and young adults that also raised fundamental rights issues. In addition to making legal findings related to human rights, the court found that the Colombian Amazon is a "'subject of rights', entitled to protection, conservation, maintenance and restoration". It recognized the special role of Amazon deforestation in creating greenhouse gas emissions in Colombia, and as a remedy ordered the nation and its administrative agencies to ensure a halt to all deforestation by 2020. The court further allocated enforcement power to the plaintiffs and affected communities, requiring the agencies to report to the communities and empowering them to inform the court if the agencies were not meeting their deforestation targets.

Ecuador

A significant body of case law has been expanding in Ecuador to implement the nation's constitutional provisions regarding the rights of nature. Examples include lawsuits in the areas of biodigestor pollution, impaired flow in the Vilcabamba River, and hydropower.

India

Gangotri Glacier, a source of the Ganga river
 
The river Yamuna at Yamunotri Glacier

As in Colombia, as of 2019 no statutes or constitutional provisions in India specifically identified rights of nature. Nevertheless, the India Supreme Court in 2012 set the stage for cases to come before it on rights of nature, finding that "Environmental justice could be achieved only if we drift away from the principle of anthropocentric to ecocentric... humans are part of nature and non-human has intrinsic value."

The Uttarakhand High Court applied the principle of ecocentric law in 2017, recognizing the legal personhood of the Ganga and Yamuna rivers and ecosystems, and calling them "living human entities" and juridical and moral persons. The court quickly followed with similar judgments for the glaciers associated with the rivers, including the Gangotri and Yamunotri, and other natural systems. While the India Supreme Court stayed the Ganga and Yamuna judgment at the request of local authorities, those authorities supported the proposed legal status in concept, but were seeking "implementation guidance".

National, sub-national, and local law

Bolivia

Following adoption of nature's rights language in its 2009 Constitution, in 2010 Bolivia's Legislature passed the Law of the Rights of Mother Earth, Act No. 071. Bolivia followed this broad outline of nature's rights with the 2012 Law of Mother Earth and Integral Development for Living Well, Act. No. 300, which provided some implementation details consistent with nature's rights. It states in part that the "violation of the rights of Mother Earth, as part of comprehensive development for Living Well, is a violation of public law and the collective and individual rights". While a step forward, this enforcement piece has not yet risen to the level of a specific enforcement mechanism.

Mexico

State, regional, and local laws and local constitutional provisions have been arising in Mexico, including adoption in the constitutions of the Mexican states of Colima and Guerrero, and that of Mexico City.

Ponca

In 2017, the Ponca Nation enacted a rights of nature law which is a resolution that gives the Ponca Tribal Court the power to punish crimes against nature with prison and fines.

Uganda

Part 1, Section 4 of Uganda's 2019 National Environment Act addresses the Rights of Nature, stating in part that "Nature has the right to exist, persist, maintain and regenerate its vital cycles, structure, functions and its processes in evolution." Advocates who had sought inclusion of such language observed that "Ugandans' right to a healthy environment cannot be realised unless the health of Nature herself is protected," and that the language adoption reflected "recent gains for the growing African movement for Earth Jurisprudence".

United States

At the local level dozens of ordinances with rights of nature provisions have been passed as of 2019 throughout the United States, and in tribal lands located within the U.S. boundaries. Most were passed in reaction to a specific threat to local well-being, such as threats posed by hydrofracking, groundwater extraction, gravel mining, and fossil fuel extraction. For example, Pittsburgh, Pennsylvania passed an anti-fracking law that included the following provision to buttress protections: "Natural communities and ecosystems... possess inalienable and fundamental rights to exist and flourish." The ordinance continues that "Residents... shall possess legal standing to enforce those rights."

Santa Monica State Beach - in 2013 the city adopted a "Sustainability Rights Ordinance", recognizing the "fundamental and inalienable rights" of "natural communities and ecosystems"

Residents in Santa Monica, California proactively sought to recognize nature's rights in local law after the U.S. Supreme Court's expansion of corporate rights in Citizens United v. FEC. In 2013 the Santa Monica City Council adopted a "Sustainability Rights Ordinance", recognizing the "fundamental and inalienable rights" of "natural communities and ecosystems" in the city to "exist and flourish". The ordinance emphasized that "[c]orporate entities... do not enjoy special privileges or powers under the law that subordinate the community's rights to their private interests". It specifically defined "natural communities and ecosystems" to include "groundwater aquifers, atmospheric systems, marine waters, and native species". Santa Monica updated its Sustainable City Plan in 2014 to reinforce its codified commitment to nature's rights. In 2018, the city council adopted a Sustainable Groundwater Management Ordinance that specifically referenced the inherent rights of the local aquifer to flourish.

In November, 2020, voters in Orange County, Florida passed a charter amendment for the "right to clean water" by a margin of 89% that protects waterways in the county from pollution and enables citizens to bring lawsuits to defend against such pollution, becoming the largest community in the country to enact such a rights of nature initiative. It has prompted the Florida Right To Clean Water direct initiative to incorporate the principle into the state constitution, which is gathering petition signatures to have an amendment put onto the 2024 ballot for consideration by all Florida voters. In his January 2022 monthly newsletter, Jim Hightower identified the Florida initiative as, "the epicenter of today’s Rights of Nature political movement".

Toledo, Ohio passed the “Lake Erie Bill of Rights” (LEBOR). In 2019 it was struck down by the Supreme Court of Ohio in 2020. BP North America spent almost $300,000 fighting the bill through a front group.

International bodies and soft law

United Nations

Advancements during the early twenty-firstst century in international "soft law" (quasi-legal instruments generally without legally binding force) have precipitated broader discussions about the potential for integrating nature's rights into legal systems. The United Nations has held nine "Harmony with Nature" General Assembly Dialogues as of 2019 on Earth-centered governance systems and philosophies, including discussions of rights of nature specifically. The companion United Nations Harmony with Nature initiative compiles rights of nature laws globally and offers a U.N. "Knowledge Network" of Earth Jurisprudence practitioners across disciplines. These U.N. Dialogues and the Harmony with Nature initiative may provide a foundation for development of a United Nations-adopted Universal Declaration of the Rights of Nature which, like the U.N.'s Universal Declaration of Human Rights, could form the foundation for rights-based laws worldwide. A model could be the 2010 UDRME, an informal, but widely-supported nature's rights agreement based on the UDHR.

International Union for Conservation of Nature

In 2012, the International Union for Conservation of Nature (IUCN, the only international observer organization to the U.N. General Assembly with expertise in the environment) adopted a resolution specifically calling for a Universal Declaration of the Rights of Nature. The IUCN reaffirmed its commitment to nature's rights at its next meeting in 2016, where the body voted to build rights of nature implementation into the upcoming, four-year IUCN Workplan. The IUCN's subgroup of legal experts, the World Commission on Environmental Law, later issued an "IUCN World Declaration on the Environmental Rule of Law" recognizing that "Nature has the inherent right to exist, thrive, and evolve".

Related initiatives

The development of stronger and more active transnational rights of nature networks during the early 2000s, is a likely cause for the greater adoption of those championed principles into law. This has occurred in close integration with other, system-changing initiatives and movements for rights, including: development and implementation of new economic and finance models that seek to better reflect human rights and nature's rights; indigenous leadership to advance both the rights of indigenous peoples and nature's rights; international social movements such as the human right to water; advancement of practical solutions consistent with a nature's rights frame, such as rewilding; and rights of nature movement capacity building, including through development of nature's rights movement hubs globally.

To illustrate implementation of nature's rights laws, the Global Alliance for the Rights of Nature has established International Rights of Nature Tribunals. These tribunals are a civil society initiative and they issue non-binding recommendations. The tribunals bring together advocates of rights of nature, human rights, and rights of indigenous peoples into a process similar to the Permanent Peoples' Tribunals. The goal of the tribunals is to provide formal public recognition, visibility, and voice to the people and natural systems injured by alleged violations of fundamental rights and marginalized in current law, and to offer a model for redress for such injuries.

As awareness of rights of nature law and jurisprudence has spread, a new field of academic research is developing, where legal scholars and other scholars have begun to offer strategies and analysis to drive broader application of such laws, particularly in the face of early implementation successes and challenges.

In popular culture

The 2018 documentary Rights of Nature: A Global Movement, directed by Isaac Goeckeritz, Hal Crimmel and Valeria Berros explores the challenges of creating new legal structures in relation to Rights of Nature.

A documentary film entitled Invisible Hand about the rights of nature movement, directed by Joshua Boaz Pribanic and Melissa Troutman of Public Herald, was released in 2020, executive-produced and narrated by actor Mark Ruffalo. It won four Best Documentary Awards.

The Overstory, which won the 2019 Pulitzer Prize for Fiction and spent over a year on the New York Times bestseller list, examined relationships with and rights of trees.

The podcast Damages explores the concept of the rights of nature in different contexts.

The Daily Show covered the concept of the rights of nature in an episode.

Notable documents

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Logicism

From Wikipedia, the free encyclopedia

In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and Alfred North Whitehead championed this programme, initiated by Gottlob Frege and subsequently developed by Richard Dedekind and Giuseppe Peano.

Overview

Dedekind's path to logicism had a turning point when he was able to construct a model satisfying the axioms characterizing the real numbers using certain sets of rational numbers. This and related ideas convinced him that arithmetic, algebra and analysis were reducible to the natural numbers plus a "logic" of classes. Furthermore by 1872 he had concluded that the naturals themselves were reducible to sets and mappings. It is likely that other logicists, most importantly Frege, were also guided by the new theories of the real numbers published in the year 1872.

The philosophical impetus behind Frege's logicist programme from the Grundlagen der Arithmetik onwards was in part his dissatisfaction with the epistemological and ontological commitments of then-extant accounts of the natural numbers, and his conviction that Kant's use of truths about the natural numbers as examples of synthetic a priori truth was incorrect.

This started a period of expansion for logicism, with Dedekind and Frege as its main exponents. However, this initial phase of the logicist programme was brought into crisis with the discovery of the classical paradoxes of set theory (Cantor 1896, Zermelo and Russell 1900–1901). Frege gave up on the project after Russell recognized and communicated his paradox identifying an inconsistency in Frege's system set out in the Grundgesetze der Arithmetik. Note that naive set theory also suffers from this difficulty.

On the other hand, Russell wrote The Principles of Mathematics in 1903 using the paradox and developments of Giuseppe Peano's school of geometry. Since he treated the subject of primitive notions in geometry and set theory, this text is a watershed in the development of logicism. Evidence of the assertion of logicism was collected by Russell and Whitehead in their Principia Mathematica.

Today, the bulk of extant mathematics is believed to be derivable logically from a small number of extralogical axioms, such as the axioms of Zermelo–Fraenkel set theory (or its extension ZFC), from which no inconsistencies have as yet been derived. Thus, elements of the logicist programmes have proved viable, but in the process theories of classes, sets and mappings, and higher-order logics other than with Henkin semantics, have come to be regarded as extralogical in nature, in part under the influence of Quine's later thought.

Kurt Gödel's incompleteness theorems show that no formal system from which the Peano axioms for the natural numbers may be derived — such as Russell's systems in PM — can decide all the well-formed sentences of that system. This result damaged David Hilbert's programme for foundations of mathematics whereby 'infinitary' theories — such as that of PM — were to be proved consistent from finitary theories, with the aim that those uneasy about 'infinitary methods' could be reassured that their use should provably not result in the derivation of a contradiction. Gödel's result suggests that in order to maintain a logicist position, while still retaining as much as possible of classical mathematics, one must accept some axiom of infinity as part of logic. On the face of it, this damages the logicist programme also, albeit only for those already doubtful concerning 'infinitary methods'. Nonetheless, positions deriving from both logicism and from Hilbertian finitism have continued to be propounded since the publication of Gödel's result.

One argument that programmes derived from logicism remain valid might be that the incompleteness theorems are 'proved with logic just like any other theorems'. However, that argument appears not to acknowledge the distinction between theorems of first-order logic and theorems of higher-order logic. The former can be proven using finistic methods, while the latter — in general — cannot. Tarski's undefinability theorem shows that Gödel numbering can be used to prove syntactical constructs, but not semantic assertions. Therefore, the claim that logicism remains a valid programme may commit one to holding that a system of proof based on the existence and properties of the natural numbers is less convincing than one based on some particular formal system.

Logicism — especially through the influence of Frege on Russell and Wittgenstein and later Dummett — was a significant contributor to the development of analytic philosophy during the twentieth century.

Origin of the name 'logicism'

Ivor Grattan-Guinness states that the French word 'Logistique' was "introduced by Couturat and others at the 1904 International Congress of Philosophy, and was used by Russell and others from then on, in versions appropriate for various languages." (G-G 2000:501).

Apparently the first (and only) usage by Russell appeared in his 1919: "Russell referred several time [sic] to Frege, introducing him as one 'who first succeeded in "logicising" mathematics' (p. 7). Apart from the misrepresentation (which Russell partly rectified by explaining his own view of the role of arithmetic in mathematics), the passage is notable for the word which he put in quotation marks, but their presence suggests nervousness, and he never used the word again, so that 'logicism' did not emerge until the later 1920s" (G-G 2002:434).

About the same time as Rudolf Carnap (1929), but apparently independently, Fraenkel (1928) used the word: "Without comment he used the name 'logicism' to characterise the Whitehead/Russell position (in the title of the section on p. 244, explanation on p. 263)" (G-G 2002:269). Carnap used a slightly different word 'Logistik'; Behmann complained about its use in Carnap's manuscript so Carnap proposed the word 'Logizismus', but he finally stuck to his word-choice 'Logistik' (G-G 2002:501). Ultimately "the spread was mainly due to Carnap, from 1930 onwards." (G-G 2000:502).

Intent, or goal, of logicism

Symbolic logic: The overt intent of Logicism is to derive all of mathematics from symbolic logic (Frege, Dedekind, Peano, Russell.) As contrasted with algebraic logic (Boolean logic) that employs arithmetic concepts, symbolic logic begins with a very reduced set of marks (non-arithmetic symbols), a few "logical" axioms that embody the "laws of thought", and rules of inference that dictate how the marks are to be assembled and manipulated. Logicism also adopts from Frege's groundwork the reduction of natural language statements from "subject|predicate" into either propositional "atoms" or the "argument|function" of "generalization"—the notions "all", "some", "class" (collection, aggregate) and "relation".

In a logicist derivation of the natural numbers and their properties, no "intuition" of number should "sneak in" either as an axiom or by accident. The goal is to derive all of mathematics, starting with the counting numbers and then the real numbers, from some chosen "laws of thought" alone, without any tacit assumptions of "before" and "after" or "less" and "more" or to the point: "successor" and "predecessor". Gödel 1944 summarized Russell's logicistic "constructions", when compared to "constructions" in the foundational systems of Intuitionism and Formalism ("the Hilbert School") as follows: "Both of these schools base their constructions on a mathematical intuition whose avoidance is exactly one of the principal aims of Russell's constructivism" (Gödel 1944 in Collected Works 1990:119).

History: Gödel 1944 summarized the historical background from Leibniz's in Characteristica universalis, through Frege and Peano to Russell: "Frege was chiefly interested in the analysis of thought and used his calculus in the first place for deriving arithmetic from pure logic", whereas Peano "was more interested in its applications within mathematics". But "It was only [Russell's] Principia Mathematica that full use was made of the new method for actually deriving large parts of mathematics from a very few logical concepts and axioms. In addition, the young science was enriched by a new instrument, the abstract theory of relations" (p. 120-121).

Kleene 1952 states it this way: "Leibniz (1666) first conceived of logic as a science containing the ideas and principles underlying all other sciences. Dedekind (1888) and Frege (1884, 1893, 1903) were engaged in defining mathematical notions in terms of logical ones, and Peano (1889, 1894–1908) in expressing mathematical theorems in a logical symbolism" (p. 43); in the previous paragraph he includes Russell and Whitehead as exemplars of the "logicistic school", the other two "foundational" schools being the intuitionistic and the "formalistic or axiomatic school" (p. 43).

Frege 1879 describes his intent in the Preface to his 1879 Begriffsschrift: He started with a consideration of arithmetic: did it derive from "logic" or from "facts of experience"?

"I first had to ascertain how far one could proceed in arithmetic by means of inferences alone, with the sole support of those laws of thought that transcend all particulars. My initial step was to attempt to reduce the concept of ordering in a sequence to that of logical consequence, so as to proceed from there to the concept of number. To prevent anything intuitive from penetrating here unnoticed I had to bend every effort to keep the chain of inferences free of gaps . . . I found the inadequacy of language to be an obstacle; no matter how unwieldy the expressions I was ready to accept, I was less and less able, as the relations became more and more complex, to attain the precision that my purpose required. This deficiency led me to the idea of the present ideography. Its first purpose, therefore, is to provide us with the most reliable test of the validity of a chain of inferences and to point out every presupposition that tries to sneak in unnoticed" (Frege 1879 in van Heijenoort 1967:5).

Dedekind 1887 describes his intent in the 1887 Preface to the First Edition of his The Nature and Meaning of Numbers. He believed that in the "foundations of the simplest science; viz., that part of logic which deals with the theory of numbers" had not been properly argued — "nothing capable of proof ought to be accepted without proof":

In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept entirely independent of the notions of intuitions of space and time, that I consider it an immediate result from the laws of thought . . . numbers are free creations of the human mind . . . [and] only through the purely logical process of building up the science of numbers . . . are we prepared accurately to investigate our notions of space and time by bringing them into relation with this number-domain created in our mind" (Dedekind 1887 Dover republication 1963 :31).

Peano 1889 states his intent in his Preface to his 1889 Principles of Arithmetic:

Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. The difficulty has its main source in the ambiguity of language. ¶ That is why it is of the utmost importance to examine attentively the very words we use. My goal has been to undertake this examination" (Peano 1889 in van Heijenoort 1967:85).

Russell 1903 describes his intent in the Preface to his 1903 Principles of Mathematics:

"THE present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles" (Preface 1903:vi).
"A few words as to the origin of the present work may serve to show the importance of the questions discussed. About six years ago, I began an investigation into the philosophy of Dynamics. . . . [From two questions — acceleration and absolute motion in a "relational theory of space"] I was led to a re-examination of the principles of Geometry, thence to the philosophy of continuity and infinity, and then, with a view to discovering the meaning of the word any, to Symbolic Logic" (Preface 1903:vi-vii).

Epistemology, ontology and logicism

Dedekind and Frege: The epistemologies of Dedekind and of Frege seem less well-defined than that of Russell, but both seem accepting as a priori the customary "laws of thought" concerning simple propositional statements (usually of belief); these laws would be sufficient in themselves if augmented with theory of classes and relations (e.g. x R y) between individuals x and y linked by the generalization R.

Dedekind's "free formations of the human mind" in contrast to the "strictures" of Kronecker: Dedekind's argument begins with "1. In what follows I understand by thing every object of our thought"; we humans use symbols to discuss these "things" of our minds; "A thing is completely determined by all that can be affirmed or thought concerning it" (p. 44). In a subsequent paragraph Dedekind discusses what a "system S is: it is an aggregate, a manifold, a totality of associated elements (things) a, b, c"; he asserts that "such a system S . . . as an object of our thought is likewise a thing (1); it is completely determined when with respect to every thing it is determined whether it is an element of S or not.*" (p. 45, italics added). The * indicates a footnote where he states that:

"Kronecker not long ago (Crelle's Journal, Vol. 99, pp. 334-336) has endeavored to impose certain limitations upon the free formation of concepts in mathematics which I do not believe to be justified" (p. 45).

Indeed he awaits Kronecker's "publishing his reasons for the necessity or merely the expediency of these limitations" (p. 45).

Leopold Kronecker, famous for his assertion that "God made the integers, all else is the work of man" had his foes, among them Hilbert. Hilbert called Kronecker a "dogmatist, to the extent that he accepts the integer with its essential properties as a dogma and does not look back" and equated his extreme constructivist stance with that of Brouwer's intuitionism, accusing both of "subjectivism": "It is part of the task of science to liberate us from arbitrariness, sentiment and habit and to protect us from the subjectivism that already made itself felt in Kronecker's views and, it seems to me, finds its culmination in intuitionism". Hilbert then states that "mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker . . ." (p. 479).

Russell as realist: Russell's Realism served him as an antidote to British Idealism, with portions borrowed from European Rationalism and British empiricism. To begin with, "Russell was a realist about two key issues: universals and material objects" (Russell 1912:xi). For Russell, tables are real things that exist independent of Russell the observer. Rationalism would contribute the notion of a priori knowledge, while empiricism would contribute the role of experiential knowledge (induction from experience). Russell would credit Kant with the idea of "a priori" knowledge, but he offers an objection to Kant he deems "fatal": "The facts [of the world] must always conform to logic and arithmetic. To say that logic and arithmetic are contributed by us does not account for this" (1912:87); Russell concludes that the a priori knowledge that we possess is "about things, and not merely about thoughts" (1912:89). And in this Russell's epistemology seems different from that of Dedekind's belief that "numbers are free creations of the human mind" (Dedekind 1887:31)

But his epistemology about the innate (he prefers the word a priori when applied to logical principles, cf. 1912:74) is intricate. He would strongly, unambiguously express support for the Platonic "universals" (cf. 1912:91-118) and he would conclude that truth and falsity are "out there"; minds create beliefs and what makes a belief true is a fact, "and this fact does not (except in exceptional cases) involve the mind of the person who has the belief" (1912:130).

Where did Russell derive these epistemic notions? He tells us in the Preface to his 1903 Principles of Mathematics. Note that he asserts that the belief: "Emily is a rabbit" is non-existent, and yet the truth of this non-existent proposition is independent of any knowing mind; if Emily really is a rabbit, the fact of this truth exists whether or not Russell or any other mind is alive or dead, and the relation of Emily to rabbit-hood is "ultimate" :

"On fundamental questions of philosophy, my position, in all its chief features, is derived from Mr G. E. Moore. I have accepted from him the non-existential nature of propositions (except such as happen to assert existence) and their independence of any knowing mind; also the pluralism which regards the world, both that of existents and that of entities, as composed of an infinite number of mutually independent entities, with relations which are ultimate, and not reducible to adjectives of their terms or of the whole which these compose. . . . The doctrines just mentioned are, in my opinion, quite indispensable to any even tolerably satisfactory philosophy of mathematics, as I hope the following pages will show. . . . Formally, my premisses are simply assumed; but the fact that they allow mathematics to be true, which most current philosophies do not, is surely a powerful argument in their favour." (Preface 1903:viii)

Russell's paradox: In 1902 Russell discovered a "vicious circle" (Russell's paradox) in Frege's Grundgesetze der Arithmetik, derived from Frege's Basic Law V and he was determined not to repeat it in his 1903 Principles of Mathematics. In two Appendices added at the last minute he devoted 28 pages to both a detailed analysis of Frege's theory contrasted against his own, and a fix for the paradox. But he was not optimistic about the outcome:

"In the case of classes, I must confess, I have failed to perceive any concept fulfilling the conditions requisite for the notion of class. And the contradiction discussed in Chapter x. proves that something is amiss, but what this is I have hitherto failed to discover. (Preface to Russell 1903:vi)"

"Fictionalism" and Russell's no-class theory: Gödel in his 1944 would disagree with the young Russell of 1903 ("[my premisses] allow mathematics to be true") but would probably agree with Russell's statement quoted above ("something is amiss"); Russell's theory had failed to arrive at a satisfactory foundation of mathematics: the result was "essentially negative; i.e. the classes and concepts introduced this way do not have all the properties required for the use of mathematics" (Gödel 1944:132).

How did Russell arrive in this situation? Gödel observes that Russell is a surprising "realist" with a twist: he cites Russell's 1919:169 "Logic is concerned with the real world just as truly as zoology" (Gödel 1944:120). But he observes that "when he started on a concrete problem, the objects to be analyzed (e.g. the classes or propositions) soon for the most part turned into "logical fictions" . . . [meaning] only that we have no direct perception of them." (Gödel 1944:120)

In an observation pertinent to Russell's brand of logicism, Perry remarks that Russell went through three phases of realism: extreme, moderate and constructive (Perry 1997:xxv). In 1903 he was in his extreme phase; by 1905 he would be in his moderate phase. In a few years he would "dispense with physical or material objects as basic bits of the furniture of the world. He would attempt to construct them out of sense-data" in his next book Our knowledge of the External World [1914]" (Perry 1997:xxvi).

These constructions in what Gödel 1944 would call "nominalistic constructivism . . . which might better be called fictionalism" derived from Russell's "more radical idea, the no-class theory" (p. 125):

"according to which classes or concepts never exist as real objects, and sentences containing these terms are meaningful only as they can be interpreted as . . . a manner of speaking about other things" (p. 125).

See more in the Criticism sections, below.

An example of a logicist construction of the natural numbers: Russell's construction in the Principia

The logicism of Frege and Dedekind is similar to that of Russell, but with differences in the particulars (see Criticisms, below). Overall, the logicist derivations of the natural numbers are different from derivations from, for example, Zermelo's axioms for set theory ('Z'). Whereas, in derivations from Z, one definition of "number" uses an axiom of that system — the axiom of pairing — that leads to the definition of "ordered pair" — no overt number axiom exists in the various logicist axiom systems allowing the derivation of the natural numbers. Note that the axioms needed to derive the definition of a number may differ between axiom systems for set theory in any case. For instance, in ZF and ZFC, the axiom of pairing, and hence ultimately the notion of an ordered pair is derivable from the Axiom of Infinity and the Axiom of Replacement and is required in the definition of the Von Neumann numerals (but not the Zermelo numerals), whereas in NFU the Frege numerals may be derived in an analogous way to their derivation in the Grundgesetze.

The Principia, like its forerunner the Grundgesetze, begins its construction of the numbers from primitive propositions such as "class", "propositional function", and in particular, relations of "similarity" ("equinumerosity": placing the elements of collections in one-to-one correspondence) and "ordering" (using "the successor of" relation to order the collections of the equinumerous classes)". The logicistic derivation equates the cardinal numbers constructed this way to the natural numbers, and these numbers end up all of the same "type" — as classes of classes — whereas in some set theoretical constructions — for instance the von Neumman and the Zermelo numerals — each number has its predecessor as a subset. Kleene observes the following. (Kleene's assumptions (1) and (2) state that 0 has property P and n+1 has property P whenever n has property P.)

"The viewpoint here is very different from that of [Kronecker]'s maxim that 'God made the integers' plus Peano's axioms of number and mathematical induction], where we presupposed an intuitive conception of the natural number sequence, and elicited from it the principle that, whenever a particular property P of natural numbers is given such that (1) and (2), then any given natural number must have the property P." (Kleene 1952:44).

The importance to the logicist programme of the construction of the natural numbers derives from Russell's contention that "That all traditional pure mathematics can be derived from the natural numbers is a fairly recent discovery, though it had long been suspected" (1919:4). One derivation of the real numbers derives from the theory of Dedekind cuts on the rational numbers, rational numbers in turn being derived from the naturals. While an example of how this is done is useful, it relies first on the derivation of the natural numbers. So, if philosophical difficulties appear in a logicist derivation of the natural numbers, these problems should be sufficient to stop the program until these are resolved (see Criticisms, below).

One attempt to construct the natural numbers is summarized by Bernays 1930–1931. But rather than use Bernays' précis, which is incomplete in some details, an attempt at a paraphrase of Russell's construction, incorporating some finite illustrations, is set out below:

Preliminaries

For Russell, collections (classes) are aggregates of "things" specified by proper names, that come about as the result of propositions (assertions of fact about a thing or things). Russell analysed this general notion. He begins with "terms" in sentences, which he analysed as follows:

Terms: For Russell, "terms" are either "things" or "concepts": "Whatever may be an object of thought, or may occur in any true or false proposition, or can be counted as one, I call a term. This, then, is the widest word in the philosophical vocabulary. I shall use as synonymous with it the words, unit, individual, and entity. The first two emphasize the fact that every term is one, while the third is derived from the fact that every term has being, i.e. is in some sense. A man, a moment, a number, a class, a relation, a chimaera, or anything else that can be mentioned, is sure to be a term; and to deny that such and such a thing is a term must always be false" (Russell 1903:43)

Things are indicated by proper names; concepts are indicated by adjectives or verbs: "Among terms, it is possible to distinguish two kinds, which I shall call respectively things and concepts; the former are the terms indicated by proper names, the latter those indicated by all other words . . . Among concepts, again, two kinds at least must be distinguished, namely those indicated by adjectives and those indicated by verbs" (1903:44).

Concept-adjectives are "predicates"; concept-verbs are "relations": "The former kind will often be called predicates or class-concepts; the latter are always or almost always relations." (1903:44)

The notion of a "variable" subject appearing in a proposition: "I shall speak of the terms of a proposition as those terms, however numerous, which occur in a proposition and may be regarded as subjects about which the proposition is. It is a characteristic of the terms of a proposition that anyone of them may be replaced by any other entity without our ceasing to have a proposition. Thus we shall say that "Socrates is human" is a proposition having only one term; of the remaining component of the proposition, one is the verb, the other is a predicate.. . . Predicates, then, are concepts, other than verbs, which occur in propositions having only one term or subject." (1903:45)

Truth and falsehood: Suppose one were to point to an object and say: "This object in front of me named 'Emily' is a woman." This is a proposition, an assertion of the speaker's belief, which is to be tested against the "facts" of the outer world: "Minds do not create truth or falsehood. They create beliefs . . . what makes a belief true is a fact, and this fact does not (except in exceptional cases) in any way involve the mind of the person who has the belief" (1912:130). If by investigation of the utterance and correspondence with "fact", Russell discovers that Emily is a rabbit, then his utterance is considered "false"; if Emily is a female human (a female "featherless biped" as Russell likes to call humans, following Diogenes Laërtius's anecdote about Plato), then his utterance is considered "true".

Classes (aggregates, complexes): "The class, as opposed to the class-concept, is the sum or conjunction of all the terms which have the given predicate" (1903 p. 55). Classes can be specified by extension (listing their members) or by intension, i.e. by a "propositional function" such as "x is a u" or "x is v". But "if we take extension pure, our class is defined by enumeration of its terms, and this method will not allow us to deal, as Symbolic Logic does, with infinite classes. Thus our classes must in general be regarded as objects denoted by concepts, and to this extent the point of view of intension is essential." (1909 p. 66)

Propositional functions: "The characteristic of a class concept, as distinguished from terms in general, is that "x is a u" is a propositional function when, and only when, u is a class-concept." (1903:56)

Extensional versus intensional definition of a class: "71. Class may be defined either extensionally or intensionally. That is to say, we may define the kind of object which is a class, or the kind of concept which denotes a class: this is the precise meaning of the opposition of extension and intension in this connection. But although the general notion can be defined in this two-fold manner, particular classes, except when they happen to be finite, can only be defined intensionally, i.e. as the objects denoted by such and such concepts. . . logically; the extensional definition appears to be equally applicable to infinite classes, but practically, if we were to attempt it, Death would cut short our laudable endeavour before it had attained its goal."(1903:69)

The definition of the natural numbers

In the Prinicipia, the natural numbers derive from all propositions that can be asserted about any collection of entities. Russell makes this clear in the second (italicized) sentence below.

"In the first place, numbers themselves form an infinite collection, and cannot therefore be defined by enumeration. In the second place, the collections having a given number of terms themselves presumably form an infinite collection: it is to be presumed, for example, that there are an infinite collection of trios in the world, for if this were not the case the total number of things in the world would be finite, which, though possible, seems unlikely. In the third place, we wish to define "number" in such a way that infinite numbers may be possible; thus we must be able to speak of the number of terms in an infinite collection, and such a collection must be defined by intension, i.e. by a property common to all its members and peculiar to them." (1919:13)

To illustrate, consider the following finite example: Suppose there are 12 families on a street. Some have children, some do not. To discuss the names of the children in these households requires 12 propositions asserting "childname is the name of a child in family Fn" applied to this collection of households on the particular street of families with names F1, F2, . . . F12. Each of the 12 propositions regards whether or not the "argument" childname applies to a child in a particular household. The children's names (childname) can be thought of as the x in a propositional function f(x), where the function is "name of a child in the family with name Fn".

Step 1: Assemble all the classes: Whereas the preceding example is finite over the finite propositional function "childnames of the children in family Fn'" on the finite street of a finite number of families, Russell apparently intended the following to extend to all propositional functions extending over an infinite domain so as to allow the creation of all the numbers.

Kleene considers that Russell has set out an impredicative definition that he will have to resolve, or risk deriving something like the Russell paradox. "Here instead we presuppose the totality of all properties of cardinal numbers, as existing in logic, prior to the definition of the natural number sequence" (Kleene 1952:44). The problem will appear, even in the finite example presented here, when Russell deals with the unit class (cf. Russell 1903:517).

The question arises what precisely a "class" is or should be. For Dedekind and Frege, a class is a distinct entity in its own right, a 'unity' that can be identified with all those entities x that satisfy some propositional function F. (This symbolism appears in Russell, attributed there to Frege: "The essence of a function is what is left when the x is taken away, i.e in the above instance, 2( )3 + ( ). The argument x does not belong to the function, but the two together make a whole (ib. p. 6 [i.e. Frege's 1891 Function und Begriff]" (Russell 1903:505).) For example, a particular "unity" could be given a name; suppose a family Fα has the children with the names Annie, Barbie and Charles:

{ a, b, c }Fα

This notion of collection or class as object, when used without restriction, results in Russell's paradox; see more below about impredicative definitions. Russell's solution was to define the notion of a class to be only those elements that satisfy the proposition, his argument being that, indeed, the arguments x do not belong to the propositional function aka "class" created by the function. The class itself is not to be regarded as a unitary object in its own right, it exists only as a kind of useful fiction: "We have avoided the decision as to whether a class of things has in any sense an existence as one object. A decision of this question in either way is indifferent to our logic" (First edition of Principia Mathematica 1927:24).

Russell continues to hold this opinion in his 1919; observe the words "symbolic fictions":

"When we have decided that classes cannot be things of the same sort as their members, that they cannot be just heaps or aggregates, and also that they cannot be identified with propositional functions, it becomes very difficult to see what they can be, if they are to be more than symbolic fictions. And if we can find any way of dealing with them as symbolic fictions, we increase the logical security of our position, since we avoid the need of assuming that there are classes without being compelled to make the opposite assumption that there are no classes. We merely abstain from both assumptions. . . . But when we refuse to assert that there are classes, we must not be supposed to be asserting dogmatically that there are none. We are merely agnostic as regards them . . .." (1919:184)

And in the second edition of PM (1927) Russell holds that "functions occur only through their values, . . . all functions of functions are extensional, . . . [and] consequently there is no reason to distinguish between functions and classes . . . Thus classes, as distinct from functions, lose even that shadowy being which they retain in *20" (p. xxxix). In other words, classes as a separate notion have vanished altogether.

Step 2: Collect "similar" classes into 'bundles' : These above collections can be put into a "binary relation" (comparing for) similarity by "equinumerosity", symbolized here by ≈, i.e. one-one correspondence of the elements, and thereby create Russellian classes of classes or what Russell called "bundles". "We can suppose all couples in one bundle, all trios in another, and so on. In this way we obtain various bundles of collections, each bundle consisting of all the collections that have a certain number of terms. Each bundle is a class whose members are collections, i.e. classes; thus each is a class of classes" (Russell 1919:14).

Step 3: Define the null class: Notice that a certain class of classes is special because its classes contain no elements, i.e. no elements satisfy the predicates whose assertion defined this particular class/collection.

The resulting entity may be called "the null class" or "the empty class". Russell symbolized the null/empty class with Λ. So what exactly is the Russellian null class? In PM Russell says that "A class is said to exist when it has at least one member . . . the class which has no members is called the "null class" . . . "α is the null-class" is equivalent to "α does not exist". The question naturally arises whether the null class itself 'exists'? Difficulties related to this question occur in Russell's 1903 work. After he discovered the paradox in Frege's Grundgesetze he added Appendix A to his 1903 where through the analysis of the nature of the null and unit classes, he discovered the need for a "doctrine of types"; see more about the unit class, the problem of impredicative definitions and Russell's "vicious circle principle" below.

Step 4: Assign a "numeral" to each bundle: For purposes of abbreviation and identification, to each bundle assign a unique symbol (aka a "numeral"). These symbols are arbitrary.

Step 5: Define "0" Following Frege, Russell picked the empty or null class of classes as the appropriate class to fill this role, this being the class of classes having no members. This null class of classes may be labelled "0"

Step 6: Define the notion of "successor": Russell defined a new characteristic "hereditary" (cf Frege's 'ancestral'), a property of certain classes with the ability to "inherit" a characteristic from another class (which may be a class of classes) i.e. "A property is said to be "hereditary" in the natural-number series if, whenever it belongs to a number n, it also belongs to n+1, the successor of n". (1903:21). He asserts that "the natural numbers are the posterity — the "children", the inheritors of the "successor" — of 0 with respect to the relation "the immediate predecessor of (which is the converse of "successor") (1919:23).

Note Russell has used a few words here without definition, in particular "number series", "number n", and "successor". He will define these in due course. Observe in particular that Russell does not use the unit class of classes "1" to construct the successor. The reason is that, in Russell's detailed analysis, if a unit class becomes an entity in its own right, then it too can be an element in its own proposition; this causes the proposition to become "impredicative" and result in a "vicious circle". Rather, he states: "We saw in Chapter II that a cardinal number is to be defined as a class of classes, and in Chapter III that the number 1 is to be defined as the class of all unit classes, of all that have just one member, as we should say but for the vicious circle. Of course, when the number 1 is defined as the class of all unit classes, unit classes must be defined so as not to assume that we know what is meant by one (1919:181).

For his definition of successor, Russell will use for his "unit" a single entity or "term" as follows:

"It remains to define "successor". Given any number n let α be a class which has n members, and let x be a term which is not a member of α. Then the class consisting of α with x added on will have +1 members. Thus we have the following definition:
the successor of the number of terms in the class α is the number of terms in the class consisting of α together with x where x is not any term belonging to the class." (1919:23)

Russell's definition requires a new "term" which is "added into" the collections inside the bundles.

Step 7: Construct the successor of the null class.

Step 8: For every class of equinumerous classes, create its successor.

Step 9: Order the numbers: The process of creating a successor requires the relation " . . . is the successor of . . .", which may be denoted "S", between the various "numerals". "We must now consider the serial character of the natural numbers in the order 0, 1, 2, 3, . . . We ordinarily think of the numbers as in this order, and it is an essential part of the work of analysing our data to seek a definition of "order" or "series " in logical terms. . . . The order lies, not in the class of terms, but in a relation among the members of the class, in respect of which some appear as earlier and some as later." (1919:31)

Russell applies to the notion of "ordering relation" three criteria: First, he defines the notion of "asymmetry" i.e. given the relation such as S (" . . . is the successor of . . . ") between two terms x, and y: x S y ≠ y S x. Second, he defines the notion of "transitivity" for three numerals x, y and z: if x S y and y S z then x S z. Third, he defines the notion of "connected": "Given any two terms of the class which is to be ordered, there must be one which precedes and the other which follows. . . . A relation is connected when, given any two different terms of its field [both domain and converse domain of a relation e.g. husbands versus wives in the relation of married] the relation holds between the first and the second or between the second and the first (not excluding the possibility that both may happen, though both cannot happen if the relation is asymmetrical).(1919:32)

He concludes: ". . . [natural] number m is said to be less than another number n when n possesses every hereditary property possessed by the successor of m. It is easy to see, and not difficult to prove, that the relation "less than", so defined, is asymmetrical, transitive, and connected, and has the [natural] numbers for its field [i.e. both domain and converse domain are the numbers]." (1919:35)

Criticism

The presumption of an 'extralogical' notion of iteration: Kleene notes that "the logicistic thesis can be questioned finally on the ground that logic already presupposes mathematical ideas in its formulation. In the Intuitionistic view, an essential mathematical kernel is contained in the idea of iteration" (Kleene 1952:46)

Bernays 1930–1931 observes that this notion "two things" already presupposes something, even without the claim of existence of two things, and also without reference to a predicate, which applies to the two things; it means, simply, "a thing and one more thing. . . . With respect to this simple definition, the Number concept turns out to be an elementary structural concept . . . the claim of the logicists that mathematics is purely logical knowledge turns out to be blurred and misleading upon closer observation of theoretical logic. . . . [one can extend the definition of "logical"] however, through this definition what is epistemologically essential is concealed, and what is peculiar to mathematics is overlooked" (in Mancosu 1998:243).

Hilbert 1931:266-7, like Bernays, considers there is "something extra-logical" in mathematics: "Besides experience and thought, there is yet a third source of knowledge. Even if today we can no longer agree with Kant in the details, nevertheless the most general and fundamental idea of the Kantian epistemology retains its significance: to ascertain the intuitive a priori mode of thought, and thereby to investigate the condition of the possibility of all knowledge. In my opinion this is essentially what happens in my investigations of the principles of mathematics. The a priori is here nothing more and nothing less than a fundamental mode of thought, which I also call the finite mode of thought: something is already given to us in advance in our faculty of representation: certain extra-logical concrete objects that exist intuitively as an immediate experience before all thought. If logical inference is to be certain, then these objects must be completely surveyable in all their parts, and their presentation, their differences, their succeeding one another or their being arrayed next to one another is immediately and intuitively given to us, along with the objects, as something that neither can be reduced to anything else, nor needs such a reduction." (Hilbert 1931 in Mancosu 1998: 266, 267).

In brief, according to Hilbert and Bernays, the notion of "sequence" or "successor" is an a priori notion that lies outside symbolic logic.

Hilbert dismissed logicism as a "false path": "Some tried to define the numbers purely logically; others simply took the usual number-theoretic modes of inference to be self-evident. On both paths they encountered obstacles that proved to be insuperable." (Hilbert 1931 in Mancoso 1998:267). The incompleteness theorems arguably constitute a similar obstacle for Hilbertian finitism.

Mancosu states that Brouwer concluded that: "the classical laws or principles of logic are part of [the] perceived regularity [in the symbolic representation]; they are derived from the post factum record of mathematical constructions . . . Theoretical logic . . . [is] an empirical science and an application of mathematics" (Brouwer quoted by Mancosu 1998:9).

Gödel 1944: With respect to the technical aspects of Russellian logicism as it appears in Principia Mathematica (either edition), Gödel was disappointed:

"It is to be regretted that this first comprehensive and thorough-going presentation of a mathematical logic and the derivation of mathematics from it [is?] so greatly lacking in formal precision in the foundations (contained in *1–*21 of Principia) that it presents in this respect a considerable step backwards as compared with Frege. What is missing, above all, is a precise statement of the syntax of the formalism" (cf. footnote 1 in Gödel 1944 Collected Works 1990:120).

In particular he pointed out that "The matter is especially doubtful for the rule of substitution and of replacing defined symbols by their definiens" (Russell 1944:120)

With respect to the philosophy that might underlie these foundations, Gödel considered Russell's "no-class theory" as embodying a "nominalistic kind of constructivism . . . which might better be called fictionalism" (cf. footnote 1 in Gödel 1944:119) — to be faulty. See more in "Gödel's criticism and suggestions" below.

Grattan-Guinness: A complicated theory of relations continued to strangle Russell's explanatory 1919 Introduction to Mathematical Philosophy and his 1927 second edition of Principia. Set theory, meanwhile had moved on with its reduction of relation to the ordered pair of sets. Grattan-Guinness observes that in the second edition of Principia Russell ignored this reduction that had been achieved by his own student Norbert Wiener (1914). Perhaps because of "residual annoyance, Russell did not react at all". By 1914 Hausdorff would provide another, equivalent definition, and Kuratowski in 1921 would provide the one in use today.

The unit class, impredicativity, and the vicious circle principle

A benign impredicative definition: Suppose a librarian wants to index her collection into a single book (call it Ι for "index"). Her index will list all the books and their locations in the library. As it turns out, there are only three books, and these have titles Ά, β, and Γ. To form her index I, she goes out and buys a book of 200 blank pages and labels it "I". Now she has four books: I, Ά, β, and Γ. Her task is not difficult. When completed, the contents of her index I are 4 pages, each with a unique title and unique location (each entry abbreviated as Title.LocationT):

I = { I.LI, Ά.LΆ, β.Lβ, Γ.LΓ}.

This sort of definition of I was deemed by Poincaré to be "impredicative". He seems to have considered that only predicative definitions can be allowed in mathematics:

"a definition is 'predicative' and logically admissible only if it excludes all objects that are dependent upon the notion defined, that is, that can in any way be determined by it".

By Poincaré's definition, the librarian's index book is "impredicative" because the definition of I is dependent upon the definition of the totality I, Ά, β, and Γ. As noted below, some commentators insist that impredicativity in commonsense versions is harmless, but as the examples show below there are versions which are not harmless. In response to these difficulties, Russell advocated a strong prohibition, his "vicious circle principle":

"No totality can contain members definable only in terms of this totality, or members involving or presupposing this totality" (vicious circle principle)" (Gödel 1944 appearing in Collected Works Vol. II 1990:125).

A pernicious impredicativity: α = NOT-α: To illustrate what a pernicious instance of impredicativity might be, consider the consequence of inputting argument α into the function f with output ω = 1 – α. This may be seen as the equivalent 'algebraic-logic' expression to the 'symbolic-logic' expression ω = NOT-α, with truth values 1 and 0. When input α = 0, output ω = 1; when input α = 1, output ω = 0.

To make the function "impredicative", identify the input with the output, yielding α = 1-α

Within the algebra of, say, rational numbers the equation is satisfied when α = 0.5. But within, for instance, a Boolean algebra, where only "truth values" 0 and 1 are permitted, then the equality cannot be satisfied.

Fatal impredicativity in the definition of the unit class: Some of the difficulties in the logicist programme may derive from the α = NOT-α paradox Russell discovered in Frege's 1879 Begriffsschrift that Frege had allowed a function to derive its input "functional" (value of its variable) not only from an object (thing, term), but also from the function's own output.

As described above, Both Frege's and Russell's constructions of the natural numbers begin with the formation of equinumerous classes of classes ("bundles"), followed by an assignment of a unique "numeral" to each bundle, and then by the placing of the bundles into an order via a relation S that is asymmetric: x S y ≠ y S x. But Frege, unlike Russell, allowed the class of unit classes to be identified as a unit itself:

But, since the class with numeral 1 is a single object or unit in its own right, it too must be included in the class of unit classes. This inclusion results in an infinite regress of increasing type and increasing content.

Russell avoided this problem by declaring a class to be more or a "fiction". By this he meant that a class could designate only those elements that satisfied its propositional function and nothing else. As a "fiction" a class cannot be considered to be a thing: an entity, a "term", a singularity, a "unit". It is an assemblage but is not in Russell's view "worthy of thing-hood":

"The class as many . . . is unobjectionable, but is many and not one. We may, if we choose, represent this by a single symbol: thus x ε u will mean " x is one of the u's." This must not be taken as a relation of two terms, x and u, because u as the numerical conjunction is not a single term . . . Thus a class of classes will be many many's; its constituents will each be only many, and cannot therefore in any sense, one might suppose, be single constituents.[etc]" (1903:516).

This supposes that "at the bottom" every single solitary "term" can be listed (specified by a "predicative" predicate) for any class, for any class of classes, for class of classes of classes, etc, but it introduces a new problem—a hierarchy of "types" of classes.

A solution to impredicativity: a hierarchy of types

Classes as non-objects, as useful fictions: Gödel 1944:131 observes that "Russell adduces two reasons against the extensional view of classes, namely the existence of (1) the null class, which cannot very well be a collection, and (2) the unit classes, which would have to be identical with their single elements." He suggests that Russell should have regarded these as fictitious, but not derive the further conclusion that all classes (such as the class-of-classes that define the numbers 2, 3, etc) are fictions.

But Russell did not do this. After a detailed analysis in Appendix A: The Logical and Arithmetical Doctrines of Frege in his 1903, Russell concludes:

"The logical doctrine which is thus forced upon us is this: The subject of a proposition may be not a single term, but essentially many terms; this is the case with all propositions asserting numbers other than 0 and 1" (1903:516).

In the following notice the wording "the class as many"—a class is an aggregate of those terms (things) that satisfy the propositional function, but a class is not a thing-in-itself:

"Thus the final conclusion is, that the correct theory of classes is even more extensional than that of Chapter VI; that the class as many is the only object always defined by a propositional function, and that this is adequate for formal purposes" (1903:518).

It is as if a rancher were to round up all his livestock (sheep, cows and horses) into three fictitious corrals (one for the sheep, one for the cows, and one for the horses) that are located in his fictitious ranch. What actually exist are the sheep, the cows and the horses (the extensions), but not the fictitious "concepts" corrals and ranch.

Ramified theory of types: function-orders and argument-types, predicative functions: When Russell proclaimed all classes are useful fictions he solved the problem of the "unit" class, but the overall problem did not go away; rather, it arrived in a new form: "It will now be necessary to distinguish (1) terms, (2) classes, (3) classes of classes, and so on ad infinitum; we shall have to hold that no member of one set is a member of any other set, and that x ε u requires that x should be of a set of a degree lower by one than the set to which u belongs. Thus x ε x will become a meaningless proposition; and in this way the contradiction is avoided" (1903:517).

This is Russell's "doctrine of types". To guarantee that impredicative expressions such as x ε x can be treated in his logic, Russell proposed, as a kind of working hypothesis, that all such impredicative definitions have predicative definitions. This supposition requires the notions of function-"orders" and argument-"types". First, functions (and their classes-as-extensions, i.e. "matrices") are to be classified by their "order", where functions of individuals are of order 1, functions of functions (classes of classes) are of order 2, and so forth. Next, he defines the "type" of a function's arguments (the function's "inputs") to be their "range of significance", i.e. what are those inputs α (individuals? classes? classes-of-classes? etc.) that, when plugged into f(x), yield a meaningful output ω. Note that this means that a "type" can be of mixed order, as the following example shows:

"Joe DiMaggio and the Yankees won the 1947 World Series".

This sentence can be decomposed into two clauses: "x won the 1947 World Series" + "y won the 1947 World Series". The first sentence takes for x an individual "Joe DiMaggio" as its input, the other takes for y an aggregate "Yankees" as its input. Thus the composite-sentence has a (mixed) type of 2, mixed as to order (1 and 2).

By "predicative", Russell meant that the function must be of an order higher than the "type" of its variable(s). Thus a function (of order 2) that creates a class of classes can only entertain arguments for its variable(s) that are classes (type 1) and individuals (type 0), as these are lower types. Type 3 can only entertain types 2, 1 or 0, and so forth. But these types can be mixed (for example, for this sentence to be (sort of) true: " z won the 1947 World Series " could accept the individual (type 0) "Joe DiMaggio" and/or the names of his other teammates, and it could accept the class (type 1) of individual players "The Yankees".

The axiom of reducibility: The axiom of reducibility is the hypothesis that any function of any order can be reduced to (or replaced by) an equivalent predicative function of the appropriate order. A careful reading of the first edition indicates that an nth order predicative function need not be expressed "all the way down" as a huge "matrix" or aggregate of individual atomic propositions. "For in practice only the relative types of variables are relevant; thus the lowest type occurring in a given context may be called that of individuals" (p. 161). But the axiom of reducibility proposes that in theory a reduction "all the way down" is possible.

Russell 1927 abandons the axiom of reducibility: By the 2nd edition of PM of 1927, though, Russell had given up on the axiom of reducibility and concluded he would indeed force any order of function "all the way down" to its elementary propositions, linked together with logical operators:

"All propositions, of whatever order, are derived from a matrix composed of elementary propositions combined by means of the stroke" (PM 1927 Appendix A, p. 385)

(The "stroke" is Sheffer's stroke — adopted for the 2nd edition of PM — a single two argument logical function from which all other logical functions may be defined.)

The net result, though, was a collapse of his theory. Russell arrived at this disheartening conclusion: that "the theory of ordinals and cardinals survives . . . but irrationals, and real numbers generally, can no longer be adequately dealt with. . . . Perhaps some further axiom, less objectionable than the axiom of reducibility, might give these results, but we have not succeeded in finding such an axiom" (PM 1927:xiv).

Gödel 1944 agrees that Russell's logicist project was stymied; he seems to disagree that even the integers survived:

"[In the second edition] The axiom of reducibility is dropped, and it is stated explicitly that all primitive predicates belong to the lowest type and that the only purpose of variables (and evidently also of constants) of higher orders and types is to make it possible to assert more complicated truth-functions of atomic propositions" (Gödel 1944 in Collected Works:134).

Gödel asserts, however, that this procedure seems to presuppose arithmetic in some form or other (p. 134). He deduces that "one obtains integers of different orders" (p. 134-135); the proof in Russell 1927 PM Appendix B that "the integers of any order higher than 5 are the same as those of order 5" is "not conclusive" and "the question whether (or to what extent) the theory of integers can be obtained on the basis of the ramified hierarchy [classes plus types] must be considered as unsolved at the present time". Gödel concluded that it wouldn't matter anyway because propositional functions of order n (any n) must be described by finite combinations of symbols (all quotes and content derived from page 135).

Gödel's criticism and suggestions

Gödel, in his 1944 work, identifies the place where he considers Russell's logicism to fail and offers suggestions to rectify the problems. He submits the "vicious circle principle" to re-examination, splitting it into three parts "definable only in terms of", "involving" and "presupposing". It is the first part that "makes impredicative definitions impossible and thereby destroys the derivation of mathematics from logic, effected by Dedekind and Frege, and a good deal of mathematics itself". Since, he argues, mathematics sees to rely on its inherent impredicativities (e.g. "real numbers defined by reference to all real numbers"), he concludes that what he has offered is "a proof that the vicious circle principle is false [rather] than that classical mathematics is false" (all quotes Gödel 1944:127).

Russell's no-class theory is the root of the problem: Gödel believes that impredicativity is not "absurd", as it appears throughout mathematics. Russell's problem derives from his "constructivistic (or nominalistic") standpoint toward the objects of logic and mathematics, in particular toward propositions, classes, and notions . . . a notion being a symbol . . . so that a separate object denoted by the symbol appears as a mere fiction" (p. 128).

Indeed, Russell's "no class" theory, Gödel concludes:

"is of great interest as one of the few examples, carried out in detail, of the tendency to eliminate assumptions about the existence of objects outside the "data" and to replace them by constructions on the basis of these data33. The "data" are to understand in a relative sense here; i.e. in our case as logic without the assumption of the existence of classes and concepts]. The result has been in this case essentially negative; i.e. the classes and concepts introduced in this way do not have all the properties required from their use in mathematics. . . . All this is only a verification of the view defended above that logic and mathematics (just as physics) are built up on axioms with a real content which cannot be explained away" (p. 132)

He concludes his essay with the following suggestions and observations:

"One should take a more conservative course, such as would consist in trying to make the meaning of terms "class" and "concept" clearer, and to set up a consistent theory of classes and concepts as objectively existing entities. This is the course which the actual development of mathematical logic has been taking and which Russell himself has been forced to enter upon in the more constructive parts of his work. Major among the attempts in this direction . . . are the simple theory of types . . . and axiomatic set theory, both of which have been successful at least to this extent, that they permit the derivation of modern mathematics and at the same time avoid all known paradoxes . . . ¶ It seems reasonable to suspect that it is this incomplete understanding of the foundations which is responsible for the fact that mathematical logic has up to now remained so far behind the high expectations of Peano and others . . .." (p. 140)

Neo-logicism

Neo-logicism describes a range of views considered by their proponents to be successors of the original logicist program. More narrowly, neo-logicism may be seen as the attempt to salvage some or all elements of Frege's programme through the use of a modified version of Frege's system in the Grundgesetze (which may be seen as a kind of second-order logic).

For instance, one might replace Basic Law V (analogous to the axiom schema of unrestricted comprehension in naive set theory) with some 'safer' axiom so as to prevent the derivation of the known paradoxes. The most cited candidate to replace BLV is Hume's principle, the contextual definition of '#' given by '#F = #G if and only if there is a bijection between F and G'. This kind of neo-logicism is often referred to as neo-Fregeanism. Proponents of neo-Fregeanism include Crispin Wright and Bob Hale, sometimes also called the Scottish School or abstractionist Platonism, who espouse a form of epistemic foundationalism.

Other major proponents of neo-logicism include Bernard Linsky and Edward N. Zalta, sometimes called the Stanford–Edmonton School, abstract structuralism or modal neo-logicism who espouse a form of axiomatic metaphysics. Modal neo-logicism derives the Peano axioms within second-order modal object theory.

Another quasi-neo-logicist approach has been suggested by M. Randall Holmes. In this kind of amendment to the Grundgesetze, BLV remains intact, save for a restriction to stratifiable formulae in the manner of Quine's NF and related systems. Essentially all of the Grundgesetze then 'goes through'. The resulting system has the same consistency strength as Jensen's NFU + Rosser's Axiom of Counting.

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