Inductive reasoning in the context of Discursive reasoning

Reason is the capacity of consciously applying logic by drawing valid conclusions from new or existing information, with the aim of seeking truth. It is associated with such characteristically human activities as philosophy, religion, science, language, and mathematics, and is normally considered to be a distinguishing ability possessed by humans. Reason is sometimes referred to as rationality, although the latter is more about its application.

Reasoning involves using more-or-less rational processes of thinking and cognition to extrapolate from one's existing knowledge to generate new knowledge, and involves the use of one's intellect. The field of logic studies the ways in which humans can use formal reasoning to produce logically valid arguments and true conclusions. Reasoning may be subdivided into forms of logical reasoning, such as deductive reasoning, inductive reasoning, and abductive reasoning.

David Hume (/hjuːm/; born David Home; 7 May 1711 – 25 August 1776) was a Scottish philosopher, historian, economist and essayist who is known for his highly influential system of empiricism, philosophical scepticism and metaphysical naturalism. Beginning with A Treatise of Human Nature (1739–40), Hume strove to create a naturalistic science of man that examined the psychological basis of human nature. Hume followed John Locke in rejecting the existence of innate ideas, concluding that all human knowledge derives solely from experience; this places him amongst such empiricists as Francis Bacon, Thomas Hobbes, John Locke, and George Berkeley.

Hume argued that inductive reasoning and belief in causality cannot be justified rationally; instead, they result from custom and mental habit. People never actually perceive that one event causes another but only experience the "constant conjunction" of events. This problem of induction means that to draw any causal inferences from past experience, it is necessary to presuppose that the future will resemble the past; this metaphysical presupposition cannot itself be grounded in prior experience.

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning that establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.

Proofs employ logic expressed in mathematical symbols, along with natural language that usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

Abductive reasoning (also called abduction, abductive inference, or retroduction) is a form of logical inference that seeks the simplest and most likely conclusion from a set of observations. It was formulated and advanced by the American philosopher and logician Charles Sanders Peirce beginning in the latter half of the 19th century.

Abductive reasoning, unlike deductive reasoning, yields a plausible conclusion but does not definitively verify it. Abductive conclusions do not eliminate uncertainty or doubt, which is expressed in terms such as "best available" or "most likely". While inductive reasoning draws general conclusions that apply to many situations, abductive conclusions are confined to the particular observations in question.

The scientific method is an empirical method for acquiring knowledge that has been referred to while doing science since at least the 17th century. Historically, it was developed through the centuries from the ancient and medieval world. The scientific method involves careful observation coupled with rigorous skepticism, because cognitive assumptions can distort the interpretation of the observation. Scientific inquiry includes creating a testable hypothesis through inductive reasoning, testing it through experiments and statistical analysis, and adjusting or discarding the hypothesis based on the results.

Although procedures vary across fields, the underlying process is often similar. In more detail: the scientific method involves making conjectures (hypothetical explanations), predicting the logical consequences of hypothesis, then carrying out experiments or empirical observations based on those predictions. A hypothesis is a conjecture based on knowledge obtained while seeking answers to the question. Hypotheses can be very specific or broad but must be falsifiable, implying that it is possible to identify a possible outcome of an experiment or observation that conflicts with predictions deduced from the hypothesis; otherwise, the hypothesis cannot be meaningfully tested.

Logos (UK: /ˈloʊɡɒs, ˈlɒɡɒs/, US: /ˈloʊɡoʊs/; Ancient Greek: λόγος, romanized: lógos, lit. 'word, discourse, or reason') is a term used in Western philosophy, psychology and rhetoric, as well as religion (notably Christianity), that most broadly means reason, logic, order, or understanding. Among its connotations is that of a rational form of discourse that relies on inductive and deductive reasoning.

Aristotle first systematized the usage of the word, making it one of the three principles of rhetoric alongside ethos and pathos. This original use identifies the word closely to the structure and content of language or text. Both Plato and Aristotle used the term logos (along with rhema) to refer to sentences and propositions.

Francis Bacon, 1st Viscount St Alban PC (/ˈbeɪkən/; 22 January 1561 – 9 April 1626) was an English philosopher and statesman who served as Attorney General and Lord Chancellor of England under King James I. Bacon argued for the importance of natural philosophy, guided by the scientific method, and his works remained influential throughout the Scientific Revolution.

Bacon has been called the father of empiricism. He argued for the possibility of scientific knowledge based only upon inductive reasoning and careful observation of events in nature. He believed that science could be achieved by the use of a sceptical and methodical approach whereby scientists aim to avoid misleading themselves. Although his most specific proposals about such a method, the Baconian method, did not have long-lasting influence, the general idea of the importance and possibility of a sceptical methodology makes Bacon one of the founders of the scientific method. His portion of the method based in scepticism was a new rhetorical and theoretical framework for science, whose practical details are still central to debates on science and methodology. He is famous for his role in the scientific revolution, promoting scientific experimentation as a way of glorifying God and fulfilling scripture.

Analogy is a comparison or correspondence between two things (or two groups of things) because of a third element that they are considered to share.

Logically, it is an inference or an argument from one particular to another particular, as opposed to deduction, induction, and abduction. It is also used where at least one of the premises, or the conclusion, is general rather than particular in nature. It has the general form A is to B as C is to D.

A priori ('from the earlier') and a posteriori ('from the later') are Latin phrases used in philosophy & linguistics to distinguish types of knowledge, justification, or argument by their reliance on experience. A priori knowledge is independent of any experience. Examples include mathematics, tautologies and deduction from pure reason. A posteriori knowledge depends on empirical evidence. Examples include most fields of science and aspects of personal knowledge.

The terms originate from the analytic methods found in Organon, a collection of works by Aristotle. Prior analytics (a priori) is about deductive logic, which comes from definitions and first principles. Posterior analytics (a posteriori) is about inductive logic, which comes from observational evidence.

Inferences are steps in logical reasoning, moving from premises to logical consequences; etymologically, the word infer means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in Europe dates at least to Aristotle (300s BC). Deduction is inference deriving logical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic. Induction is inference from particular evidence to a universal conclusion. A third type of inference is sometimes distinguished, notably by Charles Sanders Peirce, contradistinguishing abduction from induction.

Various fields study how inference is done in practice. Human inference (i.e. how humans draw conclusions) is traditionally studied within the fields of logic, argumentation studies, and cognitive psychology; artificial intelligence researchers develop automated inference systems to emulate human inference. Statistical inference uses mathematics to draw conclusions in the presence of uncertainty. This generalizes deterministic reasoning, with the absence of uncertainty as a special case. Statistical inference uses quantitative or qualitative (categorical) data which may be subject to random variations.

Sir John Frederick William Herschel, 1st Baronet KH FRS (/ˈhɜːrʃəl, ˈhɛər-/; 7 March 1792 – 11 May 1871) was an English polymath active as a mathematician, astronomer, chemist, inventor and experimental photographer who invented the blueprint and did botanical work.

Herschel originated the use of the Julian day system in astronomy. He named seven moons of Saturn and four moons of Uranus – the seventh planet, discovered by his father Sir William Herschel. He made many contributions to the science of photography, and investigated colour blindness and the chemical power of ultraviolet rays. His Preliminary Discourse (1831), which advocated an inductive approach to scientific experiment and theory-building, was an important contribution to the philosophy of science.

Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers of mass. Separated objects attract and are attracted as if all their mass were concentrated at their centers. The publication of the law has become known as the "first great unification", as it marked the unification of the previously described phenomena of gravity on Earth with known astronomical behaviors.

This is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning. It is a part of classical mechanics and was formulated in Newton's work Philosophiæ Naturalis Principia Mathematica (Latin for 'Mathematical Principles of Natural Philosophy' (the Principia)), first published on 5 July 1687.