Process philosophy

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

Process philosophy, also ontology of becoming, or processism, is an approach to philosophy that identifies processes, changes, or shifting relationships as the only true elements of the ordinary, everyday real world. It treats other real elements (examples: enduring physical objects, thoughts) as abstractions from, or ontological dependents on, processes. In opposition to the classical view of change as illusory (as argued by Parmenides) or accidental (as argued by Aristotle), process philosophy posits transient occasions of change or becoming as the only fundamental things of the ordinary everyday real world.

Since the time of Plato and Aristotle, classical ontology has posited ordinary world reality as constituted of enduring substances, to which transient processes are ontologically subordinate, if they are not denied. If Socrates changes, becoming sick, Socrates is still the same (the substance of Socrates being the same), and change (his sickness) only glides over his substance: change is accidental, and devoid of primary reality, whereas the substance is essential.

Philosophers who appeal to process rather than substance include Heraclitus, Friedrich Nietzsche, Henri Bergson, Martin Heidegger, Charles Sanders Peirce, William James, John Dewey, Alfred North Whitehead, Maurice Merleau-Ponty, Iain McGilchrist, Thomas Nail, Alfred Korzybski, R. G. Collingwood, Alan Watts, Robert M. Pirsig, Roberto Mangabeira Unger, Charles Hartshorne, Arran Gare, Nicholas Rescher, Colin Wilson, Tim Ingold, Bruno Latour, William E. Connolly, Gilles Deleuze, Alain Badiou, and Arthur M. Young. In physics, Ilya Prigogine distinguishes between the "physics of being" and the "physics of becoming". Process philosophy covers not just scientific intuitions and experiences, but can be used as a conceptual bridge to facilitate discussions among religion, philosophy, and science.

Process philosophy is sometimes classified as closer to Continental philosophy than analytic philosophy, because it is usually only taught in Continental departments. However, other sources state that process philosophy should be placed somewhere in the middle between the poles of analytic versus Continental methods in contemporary philosophy.

History

In ancient Greek thought

Heraclitus proclaimed that the basic nature of all things is change.

The quotation from Heraclitus appears in Plato's Cratylus twice; in 401d as:

τὰ ὄντα ἰέναι τε πάντα καὶ μένειν οὐδέν
Ta onta ienai te panta kai menein ouden
"All entities move and nothing remains still"

and in 402a

"πάντα χωρεῖ καὶ οὐδὲν μένει" καὶ "δὶς ἐς τὸν αὐτὸν ποταμὸν οὐκ ἂν ἐμβαίης"
Panta chōrei kai ouden menei kai dis es ton auton potamon ouk an embaies
"Everything changes and nothing remains still ... and ... you cannot step twice into the same stream"

Heraclitus considered fire as the most fundamental element.

"All things are an interchange for fire, and fire for all things, just like goods for gold and gold for goods."

The following is an interpretation of Heraclitus's concepts into modern terms by Nicholas Rescher.

"...reality is not a constellation of things at all, but one of processes. The fundamental "stuff" of the world is not material substance, but volatile flux, namely "fire", and all things are versions thereof (puros tropai). Process is fundamental: the river is not an object, but a continuing flow; the sun is not a thing, but an enduring fire. Everything is a matter of process, of activity, of change (panta rhei)."

An early expression of this viewpoint is in Heraclitus's fragments. He posits strife, ἡ ἔρις (strife, conflict), as the underlying basis of all reality defined by change. The balance and opposition in strife were the foundations of change and stability in the flux of existence.

Twentieth century

In early twentieth century, the philosophy of mathematics was undertaken to develop mathematics as an airtight, axiomatic system in which every truth could be derived logically from a set of axioms. In the foundations of mathematics, this project is variously understood as logicism or as part of the formalist program of David Hilbert. Alfred North Whitehead and Bertrand Russell attempted to complete, or at least facilitate, this program with their seminal book Principia Mathematica, which purported to build a logically consistent set theory on which to found mathematics. After this, Whitehead extended his interest to natural science, which he held needed a deeper philosophical basis. He intuited that natural science was struggling to overcome a traditional ontology of timeless material substances that does not suit natural phenomena. According to Whitehead, material is more properly understood as 'process'. In 1929, he produced the most famous work of process philosophy, Process and Reality, continuing the work begun by Hegel but describing a more complex and fluid dynamic ontology.

Process thought describes truth as "movement" in and through substance (Hegelian truth), rather than substances as fixed concepts or "things" (Aristotelian truth). Since Whitehead, process thought is distinguished from Hegel in that it describes entities that arise or coalesce in becoming, rather than being simply dialectically determined from prior posited determinates. These entities are referred to as complexes of occasions of experience. It is also distinguished in being not necessarily conflictual or oppositional in operation. Process may be integrative, destructive or both together, allowing for aspects of interdependence, influence, and confluence, and addressing coherence in universal as well as particular developments, i.e., those aspects not befitting Hegel's system. Additionally, instances of determinate occasions of experience, while always ephemeral, are nonetheless seen as important to define the type and continuity of those occasions of experience that flow from or relate to them.

Whitehead's Process and Reality

Alfred North Whitehead began teaching and writing on process and metaphysics when he joined Harvard University in 1924.

In his book Science and the Modern World (1925), Whitehead noted that the human intuitions and experiences of science, aesthetics, ethics, and religion influence the worldview of a community, but that in the last several centuries science dominates Western culture. Whitehead sought a holistic, comprehensive cosmology that provides a systematic descriptive theory of the world which can be used for the diverse human intuitions gained through ethical, aesthetic, religious, and scientific experiences, and not just the scientific.

Whitehead's influences were not restricted to philosophers or physicists or mathematicians. He was influenced by the French philosopher Henri Bergson (1859–1941), whom he credits along with William James and John Dewey in the preface to Process and Reality.

Process metaphysics

For Whitehead, metaphysics is about logical frameworks for the conduct of discussions of the character of the world. It is not directly and immediately about facts of nature, but only indirectly so, in that its task is to explicitly formulate the language and conceptual presuppositions that are used to describe the facts of nature. Whitehead thinks that discovery of previously unknown facts of nature can in principle call for reconstruction of metaphysics.

The process metaphysics elaborated in Process and Reality posits an ontology which is based on the two kinds of existence of an entity, that of actual entity and that of abstract entity or abstraction, also called 'object'.

Actual entity is a term coined by Whitehead to refer to the entities that really exist in the natural world. For Whitehead, actual entities are spatiotemporally extended events or processes. An actual entity is how something is happening, and how its happening is related to other actual entities. The actually existing world is a multiplicity of actual entities overlapping one another.

The ultimate abstract principle of actual existence for Whitehead is creativity. Creativity is a term coined by Whitehead to show a power in the world that allows the presence of an actual entity, a new actual entity, and multiple actual entities. Creativity is the principle of novelty. It is manifest in what can be called 'singular causality'. This term may be contrasted with the term 'nomic causality'. An example of singular causation is that I woke this morning because my alarm clock rang. An example of nomic causation is that alarm clocks generally wake people in the morning. Aristotle recognizes singular causality as efficient causality. For Whitehead, there are many contributory singular causes for an event. A further contributory singular cause of my being awoken by my alarm clock this morning was that I was lying asleep near it till it rang.

An actual entity is a general philosophical term for an utterly determinate and completely concrete individual particular of the actually existing world or universe of changeable entities considered in terms of singular causality, about which categorical statements can be made. Whitehead's most far-reaching and radical contribution to metaphysics is his invention of a better way of choosing the actual entities. Whitehead chooses a way of defining the actual entities that makes them all alike, qua actual entities, with a single exception.

For example, for Aristotle, the actual entities were the substances, such as Socrates. Besides Aristotle's ontology of substances, another example of an ontology that posits actual entities is in the monads of Leibniz, which are said to be 'windowless'.

Whitehead's actual entities

For Whitehead's ontology of processes as defining the world, the actual entities exist as the only fundamental elements of reality.

The actual entities are of two kinds, temporal and atemporal.

With one exception, all actual entities for Whitehead are temporal and are occasions of experience (which are not to be confused with consciousness). An entity that people commonly think of as a simple concrete object, or that Aristotle would think of as a substance, is, in this ontology, considered to be a temporally serial composite of indefinitely many overlapping occasions of experience. A human being is thus composed of indefinitely many occasions of experience.

The one exceptional actual entity is at once both temporal and atemporal: God. He is objectively immortal, as well as being immanent in the world. He is objectified in each temporal actual entity; but He is not an eternal object.

The occasions of experience are of four grades. The first grade comprises processes in a physical vacuum such as the propagation of an electromagnetic wave or gravitational influence across empty space. The occasions of experience of the second grade involve just inanimate matter; "matter" being the composite overlapping of occasions of experience from the previous grade. The occasions of experience of the third grade involve living organisms. Occasions of experience of the fourth grade involve experience in the mode of presentational immediacy, which means more or less what are often called the qualia of subjective experience. So far as we know, experience in the mode of presentational immediacy occurs in only more evolved animals. That some occasions of experience involve experience in the mode of presentational immediacy is the one and only reason why Whitehead makes the occasions of experience his actual entities; for the actual entities must be of the ultimately general kind. Consequently, it is inessential that an occasion of experience have an aspect in the mode of presentational immediacy; occasions of the grades one, two, and three, lack that aspect.

There is no mind-matter duality in this ontology, because "mind" is simply seen as an abstraction from an occasion of experience which has also a material aspect, which is of course simply another abstraction from it; thus the mental aspect and the material aspect are abstractions from one and the same concrete occasion of experience. The brain is part of the body, both being abstractions of a kind known as persistent physical objects, neither being actual entities. Though not recognized by Aristotle, there is biological evidence, written about by Galen, that the human brain is an essential seat of human experience in the mode of presentational immediacy. We may say that the brain has a material and a mental aspect, all three being abstractions from their indefinitely many constitutive occasions of experience, which are actual entities.

Time, causality, and process

Inherent in each actual entity is its respective dimension of time. Potentially, each Whiteheadean occasion of experience is causally consequential on every other occasion of experience that precedes it in time, and has as its causal consequences every other occasion of experience that follows it in time; thus it has been said that Whitehead's occasions of experience are 'all window', in contrast to Leibniz's 'windowless' monads. In time defined relative to it, each occasion of experience is causally influenced by prior occasions of experiences, and causally influences future occasions of experience. An occasion of experience consists of a process of prehending other occasions of experience, reacting to them. This is the process in process philosophy.

Such process is never deterministic. Consequently, free will is essential and inherent to the universe.

The causal outcomes obey the usual well respected rule that the causes precede the effects in time. Some pairs of processes cannot be connected by cause-and-effect relations, and they are said to be spatially separated. This is in perfect agreement with the viewpoint of the Einstein theory of special relativity and with the Minkowski geometry of spacetime. It is clear that Whitehead respected these ideas, as may be seen for example in his 1919 book An Enquiry concerning the Principles of Natural Knowledge as well as in Process and Reality. In this view, time is relative to an inertial reference frame, different reference frames defining different versions of time.

Atomicity

The actual entities, the occasions of experience, are logically atomic in the sense that an occasion of experience cannot be cut and separated into two other occasions of experience. This kind of logical atomicity is perfectly compatible with indefinitely many spatio-temporal overlaps of occasions of experience. One can explain this kind of atomicity by saying that an occasion of experience has an internal causal structure that could not be reproduced in each of the two complementary sections into which it might be cut. Nevertheless, an actual entity can completely contain each of indefinitely many other actual entities.

Another aspect of the atomicity of occasions of experience is that they do not change. An actual entity is what it is. An occasion of experience can be described as a process of change, but it is itself unchangeable.

The reader should bear in mind that the atomicity of the actual entities is of a simply logical or philosophical kind, thoroughly different in concept from the natural kind of atomicity that describes the atoms of physics and chemistry.

Topology

Whitehead's theory of extension was concerned with the spatio-temporal features of his occasions of experience. Fundamental to both Newtonian and to quantum theoretical mechanics is the concept of momentum. The measurement of a momentum requires a finite spatiotemporal extent. Because it has no finite spatiotemporal extent, a single point of Minkowski space cannot be an occasion of experience, but is an abstraction from an infinite set of overlapping or contained occasions of experience, as explained in Process and Reality. Though the occasions of experience are atomic, they are not necessarily separate in extension, spatiotemporally, from one another. Indefinitely many occasions of experience can overlap in Minkowski space.

Nexus is a term coined by Whitehead to show the network actual entity from the universe. In the universe of actual entities spread actual entity. Actual entities are clashing with each other and form other actual entities. The birth of an actual entity based on an actual entity, actual entities around him referred to as nexus.

An example of a nexus of temporally overlapping occasions of experience is what Whitehead calls an enduring physical object, which corresponds closely with an Aristotelian substance. An enduring physical object has a temporally earliest and a temporally last member. Every member (apart from the earliest) of such a nexus is a causal consequence of the earliest member of the nexus, and every member (apart from the last) of such a nexus is a causal antecedent of the last member of the nexus. There are indefinitely many other causal antecedents and consequences of the enduring physical object, which overlap, but are not members, of the nexus. No member of the nexus is spatially separate from any other member. Within the nexus are indefinitely many continuous streams of overlapping nexūs, each stream including the earliest and the last member of the enduring physical object. Thus an enduring physical object, like an Aristotelian substance, undergoes changes and adventures during the course of its existence.

In some contexts, especially in the theory of relativity in physics, the word 'event' refers to a single point in Minkowski or in Riemannian space-time. A point event is not a process in the sense of Whitehead's metaphysics. Neither is a countable sequence or array of points. A Whiteheadian process is most importantly characterized by extension in space-time, marked by a continuum of uncountably many points in a Minkowski or a Riemannian space-time. The word 'event', indicating a Whiteheadian actual entity, is not being used in the sense of a point event.

Whitehead's abstractions

Whitehead's abstractions are conceptual entities that are abstracted from or derived from and founded upon his actual entities. Abstractions are themselves not actual entities. They are the only entities that can be real but are not actual entities. This statement is one form of Whitehead's 'ontological principle'.

An abstraction is a conceptual entity that refers to more than one single actual entity. Whitehead's ontology refers to importantly structured collections of actual entities as nexuses of actual entities. Collection of actual entities into a nexus emphasizes some aspect of those entities, and that emphasis is an abstraction, because it means that some aspects of the actual entities are emphasized or dragged away from their actuality, while other aspects are de-emphasized or left out or left behind.

'Eternal object' is a term coined by Whitehead. It is an abstraction, a possibility, or pure potential. It can be ingredient into some actual entity. It is a principle that can give a particular form to an actual entity.

Whitehead admitted indefinitely many eternal objects. An example of an eternal object is a number, such as the number 'two'. Whitehead held that eternal objects are abstractions of a very high degree of abstraction. Many abstractions, including eternal objects, are potential ingredients of processes.

Relation between actual entities and abstractions stated in the ontological principle

For Whitehead, besides its temporal generation by the actual entities which are its contributory causes, a process may be considered as a concrescence of abstract ingredient eternal objects. God enters into every temporal actual entity.

Whitehead's ontological principle is that whatever reality pertains to an abstraction is derived from the actual entities upon which it is founded or of which it is comprised.

Causation and concrescence of a process

Concrescence is a term coined by Whitehead to show the process of jointly forming an actual entity that was without form, but about to manifest itself into an entity Actual full (satisfaction) based on datums or for information on the universe. The process of forming an actual entity is the case based on the existing datums. Concretion process can be regarded as subjectification process.

Datum is a term coined by Whitehead to show the different variants of information possessed by actual entity. In process philosophy, datum is obtained through the events of concrescence. Every actual entity has a variety of datum.

Commentary on Whitehead and on process philosophy

Whitehead is not an idealist in the strict sense. Whitehead's thought may be regarded as related to the idea of panpsychism (also known as panexperientialism, because of Whitehead's emphasis on experience).

On God

Whitehead's philosophy is very complex, subtle and nuanced and in order to comprehend his thinking regarding what is commonly referred to by many religions as "God", it is recommended that one read from Process and Reality Corrected Edition, wherein regarding "God" the authors elaborate Whitehead's conception.

"He is the unconditioned actuality of conceptual feeling at the base of things; so that by reason of this primordial actuality, there is an order in the relevance of eternal objects to the process of creation (343 of 413) (Location 7624 of 9706 Kindle ed.) Whitehead continues later with, "The particularities of the actual world presuppose it; while it merely presupposes the general metaphysical character of creative advance, of which it is the primordial exemplification (344 of 413) (Location 7634 of 9706 Kindle Edition)."

Process philosophy, might be considered according to some theistic forms of religion to give God a special place in the universe of occasions of experience. Regarding Whitehead's use of the term "occasions" in reference to "God", it is explained in Process and Reality Corrected Edition that

"'Actual entities'-also termed 'actual occasions'-are the final real things of which the world is made up. There is no going behind actual entities to find anything [28] more real. They differ among themselves: God is an actual entity, and so is the most trivial puff of existence in far-off empty space. But, though there are gradations of importance, and diversities of function, yet in the principles which actuality exemplifies all are on the same level. The final facts are, all alike, actual entities; and these actual entities are drops of experience, complex and interdependent.

It also can be assumed within some forms of theology that a God encompasses all the other occasions of experience but also transcends them and this might lead to it being argued that Whitehead endorses some form of panentheism. Since, it is argued theologically, that "free will" is inherent to the nature of the universe, Whitehead's God is not omnipotent in Whitehead's metaphysics. God's role is to offer enhanced occasions of experience. God participates in the evolution of the universe by offering possibilities, which may be accepted or rejected. Whitehead's thinking here has given rise to process theology, whose prominent advocates include Charles Hartshorne, John B. Cobb, Jr., and Hans Jonas, who was also influenced by the non-theological philosopher Martin Heidegger. However, other process philosophers have questioned Whitehead's theology, seeing it as a regressive Platonism.

Whitehead enumerated three essential natures of God. The primordial nature of God consists of all potentialities of existence for actual occasions, which Whitehead dubbed eternal objects. God can offer possibilities by ordering the relevance of eternal objects. The consequent nature of God prehends everything that happens in reality. As such, God experiences all of reality in a sentient manner. The last nature is the superjective. This is the way in which God's synthesis becomes a sense-datum for other actual entities. In some sense, God is prehended by existing actual entities.

Legacy and applications

Biology

In plant morphology, Rolf Sattler developed a process morphology (dynamic morphology) that overcomes the structure/process (or structure/function) dualism that is commonly taken for granted in biology. According to process morphology, structures such as leaves of plants do not have processes, they are processes.

In evolution and in development, the nature of the changes of biological objects are considered by many authors to be more radical than in physical systems. In biology, changes are not just changes of state in a pre-given space, instead the space and more generally the mathematical structures required to understand object change over time.

Ecology

With its perspective that everything is interconnected, that all life has value, and that non-human entities are also experiencing subjects, process philosophy has played an important role in discourse on ecology and sustainability. The first book to connect process philosophy with environmental ethics was John B. Cobb, Jr.'s 1971 work, Is It Too Late: A Theology of Ecology. In a more recent book (2018) edited by John B. Cobb, Jr. and Wm. Andrew Schwartz, Putting Philosophy to Work: Toward an Ecological Civilization contributors explicitly explore the ways in which process philosophy can be put to work to address the most urgent issues facing our world today, by contributing to a transition toward an ecological civilization. That book emerged from the largest international conference held on the theme of ecological civilization (Seizing an Alternative: Toward an Ecological Civilization) which was organized by the Center for Process Studies in June 2015. The conference brought together roughly 2,000 participants from around the world and featured such leaders in the environmental movement as Bill McKibben, Vandana Shiva, John B. Cobb, Jr., Wes Jackson, and Sheri Liao. The notion of ecological civilization is often affiliated with the process philosophy of Alfred North Whitehead—especially in China.

Mathematics

In the philosophy of mathematics, some of Whitehead's ideas re-emerged in combination with cognitivism as the cognitive science of mathematics and embodied mind theses.

Somewhat earlier, exploration of mathematical practice and quasi-empiricism in mathematics from the 1950s to 1980s had sought alternatives to metamathematics in social behaviours around mathematics itself: for instance, Paul Erdős's simultaneous belief in Platonism and a single "big book" in which all proofs existed, combined with his personal obsessive need or decision to collaborate with the widest possible number of other mathematicians. The process, rather than the outcomes, seemed to drive his explicit behaviour and odd use of language, as if the synthesis of Erdős and collaborators in seeking proofs, creating sense-datum for other mathematicians, was itself the expression of a divine will. Certainly, Erdős behaved as if nothing else in the world mattered, including money or love, as emphasized in his biography The Man Who Loved Only Numbers.

Medicine

Several fields of science and especially medicine seem to make liberal use of ideas in process philosophy, notably the theory of pain and healing of the late 20th century. The philosophy of medicine began to deviate somewhat from scientific method and an emphasis on repeatable results in the very late 20th century by embracing population thinking, and a more pragmatic approach to issues in public health, environmental health and especially mental health. In this latter field, R. D. Laing, Thomas Szasz and Michel Foucault were instrumental in moving medicine away from emphasis on "cures" and towards concepts of individuals in balance with their society, both of which are changing, and against which no benchmarks or finished "cures" were very likely to be measurable.

Psychology

See also: Process psychology

In psychology, the subject of imagination was again explored more extensively since Whitehead, and the question of feasibility or "eternal objects" of thought became central to the impaired theory of mind explorations that framed postmodern cognitive science. A biological understanding of the most eternal object, that being the emerging of similar but independent cognitive apparatus, led to an obsession with the process "embodiment", that being, the emergence of these cognitions. Like Whitehead's God, especially as elaborated in J. J. Gibson's perceptual psychology emphasizing affordances, by ordering the relevance of eternal objects (especially the cognitions of other such actors), the world becomes. Or, it becomes simple enough for human beings to begin to make choices, and to prehend what happens as a result. These experiences may be summed in some sense but can only approximately be shared, even among very similar cognitions with identical DNA. An early explorer of this view was Alan Turing who sought to prove the limits of expressive complexity of human genes in the late 1940s, to put bounds on the complexity of human intelligence and so assess the feasibility of artificial intelligence emerging. Since 2000, Process Psychology has progressed as an independent academic and therapeutic discipline: In 2000, Michel Weber created the Whitehead Psychology Nexus: an open forum dedicated to the cross-examination of Alfred North Whitehead's process philosophy and the various facets of the contemporary psychological field.

Philosophy of movement

The philosophy of movement is a sub-area within process philosophy that treats processes as movements. It studies processes as flows, folds, and fields in historical patterns of centripetal, centrifugal, tensional, and elastic motion. See Thomas Nail's philosophy of movement and process materialism.

Tree of knowledge system

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

 

Gregg Henriques' Tree of Knowledge System

The Tree of knowledge (ToK) system is a new map of Big History that traces cosmic evolution across four different planes of existence, identified as Matter, Life, Mind and Culture that are mapped respectively by the physical, biological, psychological and social domains of science. The Tree of Knowledge (ToK) System was developed by Gregg Henriques, who is a professor and core faculty member in the Combined-Integrated Doctoral Program in Clinical and School Psychology at James Madison University. The ToK System is part of a larger Unified Theory of Knowledge that Henriques describes as a consilient scientific humanistic philosophy for the 21st Century.

The official Unified Theory of Knowledge website describes the ToK System as:

"[A] theory of scientific knowledge that defines the human knower in relation to the known. It achieves this novel accomplishment by solving the problem of psychology and giving rise to a truly consilient view of the scientific landscape. It accomplishes this via dividing the evolution of behavioral complexity into four different planes of existence....The ToK also characterizes modern empirical natural science as a kind of justification system that functions to map complexity and change.

The outline of the ToK System was first published in 2003 in Review of General Psychology. Two special issues of the Journal of Clinical Psychology in December 2004 and January 2005 were devoted to the elaboration and evaluation of the model. In 2008 special issue of Theory & Psychology'' was devoted to the ToK System. In 2011, Henriques published A New Unified Theory of Psychology. That same year he also launched the Theory of Knowledge: A Unified Approach to Psychology and Philosophy on Psychology Today, which remains active. There is also a Theory Of Knowledge Society and discussion listserve that is devoted to discussing Henriques' work and other big picture viewpoints.

In some ways, the ToK System reflects a fairly common hierarchy of nature and of the sciences that has been represented in one way or another since the time of Auguste Comte, who in the 19th century used a hierarchical conception of nature to argue for the existence of sociology. It also has clear parallels with Aristotle's conception of the scales of nature and the first four levels of the Great Chain of Being.

Despite some overlap with a number of traditional schemes, the ToK System is properly thought of as a new theory of both ontic reality and our scientific knowledge of that reality. One of the most important and salient features of the Tree of Knowledge is how it represents reality as consisting of four different planes of existence. The theory is that, following Matter, Life, Mind and Culture each represent complex adaptive landscapes that are organized and mediated by novel emergent information processing and communication systems. Specifically, DNA/RNA store information that is processed by cells which then engage in intercellular communication to create the plane of existence called Life. Similarly, the brain and nervous system store and process information in animals which then engage in communication networks on the complex adaptive plane called Mind. Finally, linguistic storage and processing and communication between human beings generates the emergence of the Culture-Person plane of existence.

The separable planes of existence or dimension of complexity argument is one of the most crucial aspects of the system. Many have argued nature is hierarchically leveled; for example, a list of such levels might be subatomic particles, atoms, molecules, cells, organ structures, multi-celled organisms, consciousness, and society is common. The ToK System embraces a view of nature as levels, but adds the notion that there are also separable dimensions of complexity. The difference becomes particularly clear in the extension of the ToK System into the Periodic Table of Behavior. The Periodic Table of Behavior (PTB) shows that natural science can be arranged in terms of the four fundamental dimensions (i.e., matter, life, mind, and culture) and three fundamental levels of analysis (i.e., part, whole, group). The PTB also demonstrates that behavior is a central concept in science. Epistemologically, natural scientists view the world via a third person behavioral lens. Ontologically, science is about mapping different kinds of behaviors that take place in nature at various levels and dimensions of analysis.

The second central insight of the ToK System is that it shows how natural science is a particular kind of justification system that emerges out of Culture based on novel methods and specific epistemological commitments and assumptions (i.e., an exterior view point, quantification and experimentation). This epistemology and methodology functions to justify scientific ontology, which in turn maps the ontic reality. Specifically, the domains of the physical, biological, (basic) psychological and social sciences map the ontic dimensions of matter, life, mind and culture. The Periodic Table of Behavior further shows how science is a justification system that is arranged to map behavioral frequencies at different dimensions of complexity and levels of analysis.

The Matter/Object plane of existence is mapped by the Physical Sciences

The dimension of matter refers to the set of material objects and their behaviors through time. In accordance with modern cosmology, matter is theorized to have emerged from a pure energy singularity at the Big Bang. Space and time were also born at such a point. Nonliving material objects range in complexity from subatomic particles to large organic molecules. The physical sciences (i.e., physics, chemistry, geology, astronomy) describe the behavior of material objects.

The Life/Organism plane of existence is mapped by the Biological Sciences

The dimension of life refers to organisms and their behaviors through time. Living objects are considered a unique subset of material objects. Just as quantum particles form the fundamental units of material complexity, genes are the fundamental units of living information. Although many questions about the emergence of life remain unanswered, in accordance with modern biology, the ToK posits that natural selection operating on genetic combinations through time is the unified theory of biology and forms the foundational understanding for the emergence of organic complexity.

The Mind/Animal plane of existence is mapped by the (basic) Psychological Sciences

Mind/cognition in the ToK system refers to the set of mental behaviors. Mental behaviors are behaviors of animals mediated by the nervous system that produce a functional effect on the animal-environment relationship. As such, Mind/cognition is essentially synonymous with what behavioral psychologists have meant when they use the term behavior. Thus, a fly avoiding a fly swatter, a rat pushing a bar or a human getting a drink of water are all mental behaviors. Mind is not synonymous with sentience or the capacity for mental experience, although such processes are presumed to emerge in the mental/cognitive dimension. Cognition, in the broad sense of the term is meaning bodily-neuro-social information processing, as in EEEE Cognition: Embodied, Embedded, Enactive, Extended. While cognitive science stands for naturalist study of mind, psychology is an approach grounded in the tradition of humanities, especially philosophy. Thus, by defining mind as mental behavior, Henriques argues that the ToK System provides a way to bridge the epistemological differences between cognitive and behavioral science. Henriques argues that comparative psychology, ethology, and (animal) cognitive behavioral neuroscience should all be thought of as parts of the discipline that maps the animal-mental domain.

The Culture/Person plane of existence is mapped by the Human Social Sciences

Culture in the ToK system refers to the set of sociolinguistic behaviors, which range from large scale nation states to individual human justifications for particular actions. Just as genetic information processing is associated with the Life dimension and neuronal information processing associated with the Mind dimension, symbolic information processing emerges with the Cultural dimension. Henriques argues that human cognitive science, human psychology and the social sciences (i.e., anthropology, sociology, political science, and economics) work to map this domain.

Theoretical joint points

Quantum gravity

Main article: Quantum gravity

Quantum gravity refers to the imagined merger between the twin pillars of physical science which are quantum mechanics, the study of the microscopic (e.g., electrons), and general relativity, the science of the macroscopic (e.g., galaxies). Currently, these two great domains of science cannot be effectively interwoven into a single, physical Theory of Everything. Yet progress is being made, most notably through string theory, loop quantum gravity, black hole thermodynamics and the study of the early universe. Some of the difficulties combining these two pillars of physical science are philosophical in nature and it is possible that the macro view of knowledge offered by the ToK may eventually aid in the construction of a coherent theory of quantum gravity. The reason the ToK might help is that it locates scientific knowledge in relationship to the physical universe.

The modern synthesis

Main article: The Modern Synthesis

The modern synthesis refers to the merger of genetics with natural selection which occurred in the 1930s and 1940s and offers a reasonably complete framework for understanding the emergence of biological complexity. Although there remain significant gaps in biological knowledge surrounding questions such as the origin of life and the emergence of sexual reproduction, the modern synthesis represents the most complete and well-substantiated joint point.

Behavioral investment theory

Behavioral Investment Theory (BIT) is metatheoretical formulation for the mind, brain and animal behavioral sciences. Henriques proposes that it enables the merger of the selection science of behaviorism with the information science of cognitive neuroscience that has conceptual parallels with the modern synthesis. BIT posits that the nervous system evolved as an increasingly flexible computational control system that coordinates the behavioral expenditure of energy of the animal as a whole. Expenditure of behavioral energy is theorized to be computed on an investment value system built evolutionarily through natural selection operating on genetic combinations and ontogenetically through behavioral selection operating on neural combinations. As such, the current behavioral investments of the animal are conceptualized as the joint product of the two vectors of phylogeny and ontogeny. A unique element of BIT is that it finds a core of agreement and builds bridges between five brain-behavior paradigms: (1) cognitive science; (2) behavioral science; (3) evolutionary theory and genetics; (4) neuroscience; and (5) cybernetics/systems theory.

David C. Geary noted the similarities between his "motive-to-control" hypothesis and Henriques' Behavioral Investment Theory, which were developed independently of each other. Furthermore, Geary suggested that his model "seem[ed] to fill in many of the proximate mechanisms and evolutionary pressures that define the life-mind joint point, and provided a framework for further development of the mind-culture joint point."

Justification Systems Theory

Justification Systems Theory (JUST; formerly known as the Justification Hypothesis) posits that the evolution of language reached a tipping point with emergence of propositional claims. Specifically, propositional claims can be questioned, which generates the "question-answer" dynamic. This creates the problem of justification, which Henriques argues drives both the design of the human self-consciousness system as a mental organ of justification and gives rise to the evolution of the Culture-Person plane of existence. JUST is a novel proposal that allows for both the understanding of the evolution of culture and for identifying what makes humans distinct animals. A basic initial claim of JUST is that the process of justification is a crucial component of human mental behavior at both the individual and societal level. Unlike all other animals, humans everywhere ask for and give explanations for their actions. Arguments, debates, moral dictates, rationalizations, and excuses all involve the process of explaining why one's claims, thoughts or actions are warranted. In virtually every form of social exchange, from warfare to politics to family struggles to science, humans are constantly justifying their behavioral investments to themselves and others.

JUST consists of three key postulates:

• The first is that the evolution of propositional language must have created the problem of justification, which involves three interlocking problems of deciphering what is (1) analytically true and what is (2) good for the group and (3) good for the individual.
• The second postulate is that the structure and functional design of human consciousness can be understood as a solution to the problem of justification. Specifically, the three domains of human consciousness that Henriques identifies in the Updated Tripartite Model of the (1) experiential; (2) private narrator; and (3) public narrator are directly consistent with adaptive pressures that arise from the logic of the problem of justification. This analysis deepens when one considers the dynamic relationships and filtering that takes place between these three domains.
• The third postulate is that culture can be understood as large scale justification systems that coordinate the behavior of human populations. Cultural systems are seen to evolve much in the same way as organisms do in biological evolution: there is a process of variation, selection and retention of belief systems.

The Tree of Knowledge System and the "Problem of Psychology"

The ToK System emerged as a consequence of Henriques wrestling with what he calls "the problem of psychology". Henriques argues that the most difficult problem in psychology as a discipline is that while there is incredible diversity offered by different approaches to psychology, and there is no consensus model of what psychology actually is. Specifically, Henriques argues that the field lacks a clear definition, an agreed upon subject matter, and a coherent conceptual framework. The problem has been long standing, identified as the "crisis" by Lev Vygotsky in the mid 1920s.

Henriques further argues that the patent tendency of psychology has been toward theoretical and substantial fragmentation and increasing insularity among the "specialties." In other words, the discipline has fragmented into different schools of thought and methodology, with no overall framework to interpret and integrate the research of different areas. At its best, the different approaches are a strength of psychology; different approaches lead to novel ideas, and prevent psychologists from clinging to a paradigm that fails to explain a phenomenon. At its worst, adherents of one particular school cling to their beliefs concerning the relative importance of their research and disregard or are ignorant of different approaches. In most cases, individual psychologists have to determine for themselves which elements of which perspective to apply, and how to integrate them into their overall understanding.

Henriques argues that the problem of psychology is a central feature of modern knowledge systems. In A New Unified Theory of Psychology, he described it as follows:

The problem of psychology is the joint observation that the field cannot be coherently defined and yet it connects more deeply than any other discipline to the three great branches of learning. Taken together, these observations suggest that the problem of psychology is a profound problem in academia at large. This conclusion is bolstered by the fact that as psychology has lumbered along acquiring findings but not foundational clarity, the fragmentation of human knowledge has grown exponentially. All of this suggests that the question, "What is psychology?" is profoundly important, one of the central questions in all of philosophy. Asking the right questions is often the most important step in getting the right answer. My interest in psychotherapy integration ultimately led me to ask the question, "What is psychology?”. Although I had no idea at the time, it turns out that this is the right question. And, as startling as it sounds, because psychology connects to so many different domains, the correct answer to it opens up a whole new vision for integrating human knowledge.

The reason for psychology's fragmentation, according to the ToK System, is that there has been no meta-theoretical frame that allows scholars to agree on the basic questions that need to be addressed. As such, the different schools of thought in psychology are like the blind men who each grab a part of the elephant and proclaim they have discovered its true nature. With its novel depiction of evolving dimensions of complexity, the ToK allows scholars finally to see the elephant. In his 2003 Review of General Psychology paper, Henriques used the ToK System with the attempt to clarify and align the views of B.F. Skinner and Sigmund Freud. These luminaries were chosen because when one considers their influence and historical opposition, it can readily be argued that they represent two schools of thought that are the most difficult to integrate. Henriques used the meta-perspective offered by the ToK to argue how one can retain the key insights from each school of thought, identify errors and points of confusion, and integrate the insights into a coherent whole.

Cultural and personality psychologist, Michael Katzko, however critiques Henriques' position on "the problem of psychology":

There is a very good reason for skepticism regarding the repeated claims that the one unique problem of psychology, applicable across the entire discipline, has been identified and that the ToK System solves it. The reason is given by the detail with which alternatives have been worked out, be they historical studies of institutional development or critical commentaries on the rhetorical structure of psychology's literature.

How the ToK solves the problem of psychology

The problem of psychology, according to the ToK, is its conceptual incoherence, which Henriques identifies by the following:

(1) There is no agreed upon definition.

(2) There is no agreed upon subject matter.

(3) There is a proliferation of overlapping and redundant concepts.

(4) There are a large number of paradigms with fundamentally different epistemological assumptions.

(5) Specialization continues to be increasingly emphasized at the expense of generalization and thus the problem of fragmentation only grows.

When the various conceptions of psychology (e.g., behavioral, humanistic, cognitive) are viewed through the lens of the ToK System, psychology spans two different dimensions of complexity: the mental and the cultural. In other words, the discipline has historically spanned two fundamentally separate problems:

(1) the problem of animal behavior in general, and

(2) the problem of human behavior at the individual level.

If, as previously thought, nature simply consisted of levels of complexity, psychology would not be crisply defined in relationship to biology or the social sciences. And, indeed, it is frequently suggested that psychology exists in an amorphous space between biology and the social sciences. However, with its dimension of complexity depiction, the ToK System suggests that psychology can be crisply defined as the science of mind, which is the third dimension of complexity. Furthermore, because human behavior exists in the fourth dimension, psychology must be divided into two broad scientific domains of

(1) psychological formalism and

(2) human psychology.

Psychological formalism is defined as the science of mind and corresponds to the behavior of animal objects. Human psychology is considered to be a unique subset of psychological formalism that deals with human behavior at the level of the individual. Because human behavior is immersed in the larger socio-cultural context (level four in the ToK System), human psychology is considered a hybrid discipline that merges the pure science of psychology with the social sciences. It is important to point out that there are other disciplines the ToK System would classify as “hybrids.” Molecular genetics, for example, is a hybrid between chemistry and biology and neuroscience is a hybrid between biology and psychology. As with Henriques' proposed conception of human psychology, both of these disciplines adopt an object level perspective (molecular and cellular, respectively) on phenomena that simultaneously exist as part of meta-level system processes (life and mind, respectively).

Though David A. F. Haaga "congratulate[d] Dr. Henriques' ambitious, scholarly, provocative paper", and "found the Tree of Knowledge taxonomy, the theoretical joint points, the evolutionary history, and the levels of emergent properties highly illuminating", he asks the rhetorical questions,

If it is so difficult to define terms such as 'psychology' with such precision, why bother? Why not just agree that we all have at least a rough idea of what psychology is, and take the rest of the afternoon off? After all, if theoretical or empirical work improves our understanding of some aspect of the world or our fellow people, or improves our ability to help people enhance their physical or emotional well being, what difference does it make whether this work is considered a part of psychology, of cognitive science, of behavioral neuroscience, of public health, or what have you? This raises the question of what definitions in general are good for.

In a similar vein, Scott O. Lilienfeld, who described Henriques' effort as "thoughtful", contended that psychology is "an inherently fuzzy concept that resists precise definition" and that "attempts to define psychology [would be] likely to hamper rather than foster consilience across disciplines". Lilienfield went on further to suggest that the scientist-practitioner gap in psychology lies not in definitional issues, but in different "epistemic attitudes" between these two groups. He stated that scientists have an epistemic attitude of empiricism, (where questions regarding human nature are settled by scientific evidence), and that practitioners have an epistemic attitude of romanticism, (where questions of human nature are settled by intuition). Lilienfeld suggested that the solution to the scientist-practitioner gulf isn't definitional, but in "train[ing] future clinical scientists to appreciate the proper places of romanticism and empiricism within science".

Consciousness and human behavior

A frequent question and point of confusion in the ToK System is the definition and meaning of consciousness. As mentioned above, mind is not synonymous with consciousness. And, to understand consciousness from a ToK vantage point, it is crucial to recognize that the term is often ambiguous in its meaning. Two primary meanings are sentience, which is the capacity for mental experience and self-awareness, which is the capacity to be aware of one's awareness. Sentience is conceptualized as a "level 3" phenomenon, possessed by many animals other than humans and is defined as a "perceived" electro-neuro-chemical representation of animal-environment relations. The ingredient of neurological behavior that allows for the emergence of mental experience is considered the "hard" problem of consciousness and the ToK System does not address this question explicitly. In contrast, through the Justification Hypothesis (see below), the ToK System involves a very direct analysis of the other issue of consciousness, that of self-awareness.

Another frequent question that is raised is "Where does individual human behavior fall on the ToK?" To analyze human behavior from the context of the ToK, one uses the ToK like a prism to separate the dimensions of behavior into physiochemical, biogenetic, neuropsychological and sociolinguistic. Thus if we imagine a conversation between a husband and wife as follows:

Wife: “You are late again.”

Husband: “Please not now. It was a stressful day and traffic was bad and you know if work needs to be done, I can’t just leave it.”

The words represent the sociolinguistic dimension and are understood as a function of justification. Justification systems are seen both at the level of individual, micro-social and societal (i.e., the context of justification in which men work and women stay at home). The actions of the husband and wife in terms of facial expression, body movement, etc. are seen as the mental dimension and are understood as a function of behavioral investment. The physiological make up of the organ systems and cells of each body is seen as the biogenetic dimension. Finally, the position, temperature, molecular make up is seen as the physiochemical dimension. Each of the more basic dimensions represent conditions of possibility that allow for the emergence of the higher dimension of process. Thus, insufficient oxygen disrupts organic processes which in turn renders neuropsychological and sociolinguistic processes impossible.

Toward the integration of human knowledge

As stated above, the ToK System proposes a new epistemology with the goal of moving academic knowledge toward what E.O. Wilson termed consilience. Consilience is the interlocking of fact and theory into a coherent, holistic view of knowledge. Henriques argues that the ToK affords new perspectives on how knowledge is obtained because it depicts how science emerges from culture and that the four dimensions of complexity correspond to four broad classes of science: the physical, biological, psychological and social sciences.

Henriques further argues that developing such a system for integrating knowledge is not just an academic enterprise. He suggests that in an increasingly complex world, the fragmented state of knowledge can be seen as one of the most pressing social problems of our time. Henriques also believes that history seems to attest that the absence of a collective worldview ostensibly condemns humanity to an endless series of conflicts that inevitably stem from incompatible, partially correct, locally situated justification systems. Thus, from Henriques' perspective, there are good reasons for believing that if there was a shared, general background of explanation, humanity might be able to achieve much greater levels of harmonious relations.

In a 2008 article on the ToK, Henriques cites Oliver Reiser's 1958 call for unifying scientific knowledge that Henriques implies is similar in theme to the ToK:

In this time of divisive tendencies within and between the nations, races, religions, sciences and humanities, synthesis must become the great magnet which orients us all…[Yet] scientists have not done what is possible toward integrating bodies of knowledge created by science into a unified interpretation of man, his place in nature, and his potentialities for creating the good society. Instead, they are entombing us in dark and meaningless catacombs of learning.

With its depiction of the dimensions of complexity and interlocking theoretical joint points, Henriques' believes that his ToK System offers new avenues that might allow scholars to meet Reiser’s call for academic synthesis. Henriques, like Reiser, believes that with a shared sense of purpose and a common background of explanation, people might yet be able to integrate bodies of knowledge into a unified interpretation of humanity, with humanity's place in nature and its potentialities for creating the good society.

Inductive logic programming

From Wikipedia, the free encyclopedia

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

 

Programming paradigms
Action Agent-oriented Array-oriented Automata-based Concurrent computing Choreographic programming Relativistic programming Data-driven Declarative (contrast: Imperative) Functional Functional logic Purely functional Logic Abductive logic Answer set Concurrent logic Functional logic Inductive logic Constraint Constraint logic Concurrent constraint logic Dataflow Flow-based Reactive Functional reactive Ontology Query language Differentiable Dynamic/scripting Event-driven Function-level (contrast: Value-level) Point-free style Concatenative Generic Imperative (contrast: Declarative) Procedural Object-oriented Polymorphic Intentional Language-oriented Domain-specific Literate Natural-language programming Metaprogramming Automatic Inductive programming Reflective Attribute-oriented Macro Template Non-structured (contrast: Structured) Array Nondeterministic Parallel computing Process-oriented Probabilistic Quantum Set-theoretic Stack-based Structured (contrast: Non-structured) Block-structured Structured concurrency Object-oriented Actor-based Class-based Concurrent Prototype-based By separation of concerns: Aspect-oriented Role-oriented Subject-oriented Recursive Symbolic Value-level (contrast: Function-level)

Inductive logic programming (ILP) is a subfield of symbolic artificial intelligence which uses logic programming as a uniform representation for examples, background knowledge and hypotheses. Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesised logic program which entails all the positive and none of the negative examples.

• Schema: positive examples + negative examples + background knowledge ⇒ hypothesis.

Inductive logic programming is particularly useful in bioinformatics and natural language processing. Gordon Plotkin and Ehud Shapiro laid the initial theoretical foundation for inductive machine learning in a logical setting. Shapiro built their first implementation (Model Inference System) in 1981: a Prolog program that inductively inferred logic programs from positive and negative examples. The term Inductive Logic Programming was first introduced in a paper by Stephen Muggleton in 1991. Muggleton also founded the annual international conference on Inductive Logic Programming, introduced the theoretical ideas of Predicate Invention, Inverse resolution, and Inverse entailment. Muggleton implemented Inverse entailment first in the PROGOL system. The term "inductive" here refers to philosophical (i.e. suggesting a theory to explain observed facts) rather than mathematical (i.e. proving a property for all members of a well-ordered set) induction.

Formal definition

The background knowledge is given as a logic theory B, commonly in the form of Horn clauses used in logic programming. The positive and negative examples are given as a conjunction ${\displaystyle E^{+}}$ and ${\displaystyle E^{-}}$ of unnegated and negated ground literals, respectively. A correct hypothesis h is a logic proposition satisfying the following requirements.

${\displaystyle {\begin{array}{llll}{\text{Necessity:}}&B&\not \models &E^{+}\\{\text{Sufficiency:}}&B\land h&\color {blue}{\models }&E^{+}\\{\text{Weak consistency:}}&B\land h&\not \models &{\textit {false}}\\{\text{Strong consistency:}}&B\land h\land E^{-}&\not \models &{\textit {false}}\end{array}}}$

"Necessity" does not impose a restriction on h, but forbids any generation of a hypothesis as long as the positive facts are explainable without it. "Sufficiency" requires any generated hypothesis h to explain all positive examples ${\displaystyle E^{+}}$. "Weak consistency" forbids generation of any hypothesis h that contradicts the background knowledge B. "Strong consistency" also forbids generation of any hypothesis h that is inconsistent with the negative examples ${\displaystyle E^{-}}$, given the background knowledge B; it implies "Weak consistency"; if no negative examples are given, both requirements coincide. Džeroski  requires only "Sufficiency" (called "Completeness" there) and "Strong consistency".

Example

Assumed family relations in section "Example"

The following well-known example about learning definitions of family relations uses the abbreviations

par: parent, fem: female, dau: daughter, g: George, h: Helen, m: Mary, t: Tom, n: Nancy, and e: Eve.

It starts from the background knowledge (cf. picture)

${\displaystyle {\textit {par}}(h,m)\land {\textit {par}}(h,t)\land {\textit {par}}(g,m)\land {\textit {par}}(t,e)\land {\textit {par}}(n,e)\land {\textit {fem}}(h)\land {\textit {fem}}(m)\land {\textit {fem}}(n)\land {\textit {fem}}(e)}$,

the positive examples

${\displaystyle {\textit {dau}}(m,h)\land {\textit {dau}}(e,t)}$,

and the trivial proposition true to denote the absence of negative examples.

Plotkin's  "relative least general generalization (rlgg)" approach to inductive logic programming shall be used to obtain a suggestion about how to formally define the daughter relation dau.

This approach uses the following steps.

• Relativize each positive example literal with the complete background knowledge:

  ${\displaystyle {\begin{aligned}{\textit {dau}}(m,h)\leftarrow {\textit {par}}(h,m)\land {\textit {par}}(h,t)\land {\textit {par}}(g,m)\land {\textit {par}}(t,e)\land {\textit {par}}(n,e)\land {\textit {fem}}(h)\land {\textit {fem}}(m)\land {\textit {fem}}(n)\land {\textit {fem}}(e)\\{\textit {dau}}(e,t)\leftarrow {\textit {par}}(h,m)\land {\textit {par}}(h,t)\land {\textit {par}}(g,m)\land {\textit {par}}(t,e)\land {\textit {par}}(n,e)\land {\textit {fem}}(h)\land {\textit {fem}}(m)\land {\textit {fem}}(n)\land {\textit {fem}}(e)\end{aligned}}}$,

• Convert into clause normal form:

  ${\displaystyle {\begin{aligned}{\textit {dau}}(m,h)\lor \lnot {\textit {par}}(h,m)\lor \lnot {\textit {par}}(h,t)\lor \lnot {\textit {par}}(g,m)\lor \lnot {\textit {par}}(t,e)\lor \lnot {\textit {par}}(n,e)\lor \lnot {\textit {fem}}(h)\lor \lnot {\textit {fem}}(m)\lor \lnot {\textit {fem}}(n)\lor \lnot {\textit {fem}}(e)\\{\textit {dau}}(e,t)\lor \lnot {\textit {par}}(h,m)\lor \lnot {\textit {par}}(h,t)\lor \lnot {\textit {par}}(g,m)\lor \lnot {\textit {par}}(t,e)\lor \lnot {\textit {par}}(n,e)\lor \lnot {\textit {fem}}(h)\lor \lnot {\textit {fem}}(m)\lor \lnot {\textit {fem}}(n)\lor \lnot {\textit {fem}}(e)\end{aligned}}}$,

• Anti-unify each compatible  pair  of literals:
  • ${\displaystyle {\textit {dau}}(x_{me},x_{ht})}$ from ${\displaystyle {\textit {dau}}(m,h)}$ and ${\displaystyle {\textit {dau}}(e,t)}$,
  • ${\displaystyle \lnot {\textit {par}}(x_{ht},x_{me})}$ from ${\displaystyle \lnot {\textit {par}}(h,m)}$ and ${\displaystyle \lnot {\textit {par}}(t,e)}$,
  • ${\displaystyle \lnot {\textit {fem}}(x_{me})}$ from ${\displaystyle \lnot {\textit {fem}}(m)}$ and ${\displaystyle \lnot {\textit {fem}}(e)}$,
  • ${\displaystyle \lnot {\textit {par}}(g,m)}$ from ${\displaystyle \lnot {\textit {par}}(g,m)}$ and ${\displaystyle \lnot {\textit {par}}(g,m)}$, similar for all other background-knowledge literals
  • ${\displaystyle \lnot {\textit {par}}(x_{gt},x_{me})}$ from ${\displaystyle \lnot {\textit {par}}(g,m)}$ and ${\displaystyle \lnot {\textit {par}}(t,e)}$, and many more negated literals
• Delete all negated literals containing variables that don't occur in a positive literal:
  • after deleting all negated literals containing other variables than ${\displaystyle x_{me},x_{ht}}$, only ${\displaystyle {\textit {dau}}(x_{me},x_{ht})\lor \lnot {\textit {par}}(x_{ht},x_{me})\lor \lnot {\textit {fem}}(x_{me})}$ remains, together with all ground literals from the background knowledge
• Convert clauses back to Horn form:
  • ${\displaystyle {\textit {dau}}(x_{me},x_{ht})\leftarrow {\textit {par}}(x_{ht},x_{me})\land {\textit {fem}}(x_{me})\land ({\text{all background knowledge facts}})}$

The resulting Horn clause is the hypothesis h obtained by the rlgg approach. Ignoring the background knowledge facts, the clause informally reads "${\displaystyle x_{me}}$ is called a daughter of ${\displaystyle x_{ht}}$ if ${\displaystyle x_{ht}}$ is the parent of ${\displaystyle x_{me}}$ and ${\displaystyle x_{me}}$ is female", which is a commonly accepted definition.

Concerning the above requirements, "Necessity" was satisfied because the predicate dau doesn't appear in the background knowledge, which hence cannot imply any property containing this predicate, such as the positive examples are. "Sufficiency" is satisfied by the computed hypothesis h, since it, together with ${\displaystyle {\textit {par}}(h,m)\land {\textit {fem}}(m)}$ from the background knowledge, implies the first positive example ${\displaystyle {\textit {dau}}(m,h)}$, and similarly h and ${\displaystyle {\textit {par}}(t,e)\land {\textit {fem}}(e)}$ from the background knowledge implies the second positive example ${\displaystyle {\textit {dau}}(e,t)}$. "Weak consistency" is satisfied by h, since h holds in the (finite) Herbrand structure described by the background knowledge; similar for "Strong consistency".

The common definition of the grandmother relation, viz. ${\displaystyle {\textit {gra}}(x,z)\leftarrow {\textit {fem}}(x)\land {\textit {par}}(x,y)\land {\textit {par}}(y,z)}$, cannot be learned using the above approach, since the variable y occurs in the clause body only; the corresponding literals would have been deleted in the 4th step of the approach. To overcome this flaw, that step has to be modified such that it can be parametrized with different literal post-selection heuristics. Historically, the GOLEM implementation is based on the rlgg approach.

Inductive Logic Programming system

Inductive Logic Programming system is a program that takes as an input logic theories ${\displaystyle B,E^{+},E^{-}}$ and outputs a correct hypothesis H wrt theories ${\displaystyle B,E^{+},E^{-}}$ An algorithm of an ILP system consists of two parts: hypothesis search and hypothesis selection. First a hypothesis is searched with an inductive logic programming procedure, then a subset of the found hypotheses (in most systems one hypothesis) is chosen by a selection algorithm. A selection algorithm scores each of the found hypotheses and returns the ones with the highest score. An example of score function include minimal compression length where a hypothesis with a lowest Kolmogorov complexity has the highest score and is returned. An ILP system is complete iff for any input logic theories ${\displaystyle B,E^{+},E^{-}}$ any correct hypothesis H wrt to these input theories can be found with its hypothesis search procedure.

Hypothesis search

Modern ILP systems like Progol, Hail  and Imparo  find a hypothesis H using the principle of the inverse entailment for theories B, E, H: ${\displaystyle B\land H\models E\iff B\land \neg E\models \neg H}$. First they construct an intermediate theory F called a bridge theory satisfying the conditions ${\displaystyle B\land \neg E\models F}$ and ${\displaystyle F\models \neg H}$. Then as ${\displaystyle H\models \neg F}$, they generalize the negation of the bridge theory F with the anti-entailment. However, the operation of the anti-entailment since being highly non-deterministic is computationally more expensive. Therefore, an alternative hypothesis search can be conducted using the operation of the inverse subsumption (anti-subsumption) instead which is less non-deterministic than anti-entailment.

Questions of completeness of a hypothesis search procedure of specific ILP system arise. For example, Progol's hypothesis search procedure based on the inverse entailment inference rule is not complete by Yamamoto's example. On the other hand, Imparo is complete by both anti-entailment procedure  and its extended inverse subsumption  procedure.

Implementations

• 1BC and 1BC2: first-order naive Bayesian classifiers:
• ACE (A Combined Engine)
• Aleph
• Atom
• Claudien
• DL-Learner
• DMax
• FastLAS (Fast Learning from Answer Sets)
• FOIL (First Order Inductive Learner)
• Golem
• ILASP (Inductive Learning of Answer Set Programs)
• Imparo
• Inthelex (INcremental THEory Learner from EXamples)
• Lime
• Metagol
• Mio
• MIS (Model Inference System) by Ehud Shapiro
• PROGOL
• RSD
• Warmr (now included in ACE)
• ProGolem