Kurt Friedrich Gödel was an Austrian, American, logician and philosopher. Considered along with Aristotle, Alfred Tarski and Gottlob Frege to be one of the most significant logicians in history, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, Alfred North Whitehead, David Hilbert were analyzing the use of logic and set theory to understand the foundations of mathematics pioneered by Georg Cantor. Gödel published his two incompleteness theorems in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna; the first incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers, there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.
He showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs, he made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, modal logic. Gödel was born April 28, 1906, in Brünn, Austria-Hungary into the German family of Rudolf Gödel, the manager of a textile factory, Marianne Gödel. Throughout his life, Gödel would remain close to his mother. At the time of his birth the city had a German-speaking majority, his father was Catholic and his mother was Protestant and the children were raised Protestant. The ancestors of Kurt Gödel were active in Brünn's cultural life. For example, his grandfather Joseph Gödel was a famous singer of that time and for some years a member of the Brünner Männergesangverein. Gödel automatically became a Czechoslovak citizen at age 12 when the Austro-Hungarian Empire broke up at the end of World War I.
In his family, young Kurt was known as Herr Warum because of his insatiable curiosity. According to his brother Rudolf, at the age of six or seven Kurt suffered from rheumatic fever. Beginning at age four, Gödel suffered from "frequent episodes of poor health," which would continue for his entire life. Gödel attended the Evangelische Volksschule, a Lutheran school in Brünn from 1912 to 1916, was enrolled in the Deutsches Staats-Realgymnasium from 1916 to 1924, excelling with honors in all his subjects in mathematics and religion. Although Kurt had first excelled in languages, he became more interested in history and mathematics, his interest in mathematics increased when in 1920 his older brother Rudolf left for Vienna to go to medical school at the University of Vienna. During his teens, Kurt studied Gabelsberger shorthand, Goethe's Theory of Colours and criticisms of Isaac Newton, the writings of Immanuel Kant. At the age of 18, Gödel entered the University of Vienna. By that time, he had mastered university-level mathematics.
Although intending to study theoretical physics, he attended courses on mathematics and philosophy. During this time, he adopted ideas of mathematical realism, he read Kant's Metaphysische Anfangsgründe der Naturwissenschaft, participated in the Vienna Circle with Moritz Schlick, Hans Hahn, Rudolf Carnap. Gödel studied number theory, but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell's book Introduction to Mathematical Philosophy, he became interested in mathematical logic. According to Gödel, mathematical logic was "a science prior to all others, which contains the ideas and principles underlying all sciences."Attending a lecture by David Hilbert in Bologna on completeness and consistency of mathematical systems may have set Gödel's life course. In 1928, Hilbert and Wilhelm Ackermann published Grundzüge der theoretischen Logik, an introduction to first-order logic in which the problem of completeness was posed: Are the axioms of a formal system sufficient to derive every statement, true in all models of the system?
This problem became the topic. In 1929, at the age of 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, he established his eponymous completeness theorem regarding the first-order predicate calculus, he was awarded his doctorate in 1930, his thesis was published by the Vienna Academy of Science. Kurt Gödel's achievement in modern logic is singular and monumental—indeed it is more than a monument, it is a landmark which will remain visible far in space and time.... The subject of logic has completely changed its nature and possibilities with Gödel's achievement. In 1931 and while still in Vienna, Gödel published
The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of sized spheres filling space has a greater average density than that of the cubic close packing and hexagonal close packing arrangements; the density of these arrangements is around 74.05%. In 1998 Thomas Hales, following an approach suggested by Fejes Tóth, announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving the checking of many individual cases using complex computer calculations. Referees said that they were "99% certain" of the correctness of Hales' proof, the Kepler conjecture was accepted as a theorem. In 2014, the Flyspeck project team, headed by Hales, announced the completion of a formal proof of the Kepler conjecture using a combination of the Isabelle and HOL Light proof assistants. In 2017, the formal proof was accepted by the journal Forum of Pi.
Imagine filling a large container with small equal-sized spheres. The density of the arrangement is equal to the collective volume of the spheres divided by the volume of the container. To maximize the number of spheres in the container means to create an arrangement with the highest possible density, so that the spheres are packed together as as possible. Experiment shows that dropping the spheres in randomly will achieve a density of around 65%. However, a higher density can be achieved by arranging the spheres as follows. Start with a layer of spheres in a hexagonal lattice put the next layer of spheres in the lowest points you can find above the first layer, so on. At each step there are two choices of where to put the next layer, so this natural method of stacking the spheres creates an uncountably infinite number of dense packings, the best known of which are called cubic close packing and hexagonal close packing; each of these arrangements has an average density of π 3 2 = 0.740480489 … The Kepler conjecture says that this is the best that can be done—no other arrangement of spheres has a higher average density.
The conjecture was first stated by Johannes Kepler in his paper'On the six-cornered snowflake'. He had started to study arrangements of spheres as a result of his correspondence with the English mathematician and astronomer Thomas Harriot in 1606. Harriot was a friend and assistant of Sir Walter Raleigh, who had set Harriot the problem of determining how best to stack cannonballs on the decks of his ships. Harriot published a study of various stacking patterns in 1591, went on to develop an early version of atomic theory. Kepler did not have a proof of the conjecture, the next step was taken by Carl Friedrich Gauss, who proved that the Kepler conjecture is true if the spheres have to be arranged in a regular lattice; this meant that any packing arrangement that disproved the Kepler conjecture would have to be an irregular one. But eliminating all possible irregular arrangements is difficult, this is what made the Kepler conjecture so hard to prove. In fact, there are irregular arrangements that are denser than the cubic close packing arrangement over a small enough volume, but any attempt to extend these arrangements to fill a larger volume is now known to always reduce their density.
After Gauss, no further progress was made towards proving the Kepler conjecture in the nineteenth century. In 1900 David Hilbert included it in his list of twenty three unsolved problems of mathematics—it forms part of Hilbert's eighteenth problem; the next step toward a solution was taken by László Fejes Tóth. Fejes Tóth showed that the problem of determining the maximum density of all arrangements could be reduced to a finite number of calculations; this meant. As Fejes Tóth realised, a fast enough computer could turn this theoretical result into a practical approach to the problem. Meanwhile, attempts were made to find an upper bound for the maximum density of any possible arrangement of spheres. English mathematician Claude Ambrose Rogers established an upper bound value of about 78%, subsequent efforts by other mathematicians reduced this value but this was still much larger than the cubic close packing density of about 74%. In 1990, Wu-Yi Hsiang claimed to have proven the Kepler conjecture.
The proof was praised by Encyclopædia Britannica and Science and Hsiang was honored at joint meetings of AMS-MAA. Wu-Yi Hsiang claimed to prove the Kepler conjecture using geometric methods; however Gábor Fejes Tóth stated in his review of the paper "As far as details are concerned, my opinion is that many of the key statements have no acceptable proofs." Hales gave a detailed criticism of Hsiang's work. The current consensus is. Following the approach suggested by Fejes Tóth, Thomas Hales at the University of Michigan, determined that the maximum density of all arrangements could be found by minimizing a function with 150 variables. In 1992, assisted by his graduate student Samuel Ferguson, he embarked on a research program to systematically apply linear programming methods to find a lower bound on the value of this function for each one of a set of over 5,000 different configurations of spheres. If a lower bound could be found for every one of these configurations, greater than the value of the function for the cubic
In the history of artificial intelligence, an AI winter is a period of reduced funding and interest in artificial intelligence research. The term was coined by analogy to the idea of a nuclear winter; the field has experienced several hype cycles, followed by disappointment and criticism, followed by funding cuts, followed by renewed interest years or decades later. The term first appeared in 1984 as the topic of a public debate at the annual meeting of AAAI, it is a chain reaction that begins with pessimism in the AI community, followed by pessimism in the press, followed by a severe cutback in funding, followed by the end of serious research. At the meeting, Roger Schank and Marvin Minsky—two leading AI researchers who had survived the "winter" of the 1970s—warned the business community that enthusiasm for AI had spiraled out of control in the 1980s and that disappointment would follow. Three years the billion-dollar AI industry began to collapse. Hypes are common in many emerging technologies, such as the dot-com bubble.
The AI winter is a collapse in the perception of AI by government bureaucrats and venture capitalists. Despite the rise and fall of AI's reputation, it has continued to develop new and successful technologies. AI researcher Rodney Brooks would complain in 2002 that "there's this stupid myth out there that AI has failed, but AI is around you every second of the day." In 2005, Ray Kurzweil agreed: "Many observers still think that the AI winter was the end of the story and that nothing since has come of the AI field. Yet today many thousands of AI applications are embedded in the infrastructure of every industry."Enthusiasm and optimism about AI has increased since its low point in 1990, by the 2010s artificial intelligence became used and well-funded. There were two major winters in 1974–1980 and 1987–1993 and several smaller episodes, including the following: 1966: failure of machine translation 1970: abandonment of connectionism Period of overlapping trends: 1971–75: DARPA's frustration with the Speech Understanding Research program at Carnegie Mellon University 1973: large decrease in AI research in the United Kingdom in response to the Lighthill report 1973–74: DARPA's cutbacks to academic AI research in general 1987: collapse of the Lisp machine market 1988: cancellation of new spending on AI by the Strategic Computing Initiative 1993: expert systems reaching the bottom 1990s: quiet disappearance of the fifth-generation computer project's original goals During the Cold War, the US government was interested in the automatic, instant translation of Russian documents and scientific reports.
The government aggressively supported efforts at machine translation starting in 1954. At the outset, the researchers were optimistic. Noam Chomsky's new work in grammar was streamlining the translation process and there were "many predictions of imminent'breakthroughs'". However, researchers had underestimated the profound difficulty of word-sense disambiguation. In order to translate a sentence, a machine needed to have some idea what the sentence was about, otherwise it made mistakes. An apocryphal example is "the spirit is willing but the flesh is weak." Translated back and forth with Russian, it became "the vodka is good but the meat is rotten." "out of sight, out of mind" became "blind idiot". Researchers would call this the commonsense knowledge problem. By 1964, the National Research Council had become concerned about the lack of progress and formed the Automatic Language Processing Advisory Committee to look into the problem, they concluded, in a famous 1966 report, that machine translation was more expensive, less accurate and slower than human translation.
After spending some 20 million dollars, the NRC ended all support. Careers were destroyed and research ended. Machine translation is still an open research problem in the 21st century, met with some success. See also: Perceptrons and Frank Rosenblatt Some of the earliest work in AI used networks or circuits of connected units to simulate intelligent behavior. Examples of this kind of work, called "connectionism", include Walter Pitts and Warren McCullough's first description of a neural network for logic and Marvin Minsky's work on the SNARC system. In the late 1950s, most of these approaches were abandoned when researchers began to explore symbolic reasoning as the essence of intelligence, following the success of programs like the Logic Theorist and the General Problem Solver. However, one type of connectionist work continued: the study of perceptrons, invented by Frank Rosenblatt, who kept the field alive with his salesmanship and the sheer force of his personality, he optimistically predicted that the perceptron "may be able to learn, make decisions, translate languages".
Mainstream research into perceptrons came to an abrupt end in 1969, when Marvin Minsky and Seymour Papert published the book Perceptrons, perceived as outlining the limits of what perceptrons could do. Connectionist approaches were abandoned for the next decade or so. While important work, such as Paul Werbos' discovery of backpropagation, continued in a limited way, major funding for connectionist projects was difficult to find in the 1970s and early 1980s; the "winter" of connectionist research came to an end in the middle 1980s, when the work of John Hopfield, David Rumelhart and others revived large scale interest in neural networks. Rosenblatt did not live to see this, however, as he died in a boating accident shortly after Perceptrons was published. In 1973, professor Sir James Lighthill was asked by the UK Parliament to evaluate t
Ferdinand Gotthold Max Eisenstein was a German mathematician. He specialized in number theory and analysis, proved several results that eluded Gauss. Like Galois and Abel before him, Eisenstein died before the age of 30, he was died in Berlin, Prussia. His parents, Johan Konstantin Eisenstein and Helene Pollack, were of Jewish descent and converted to Protestantism prior to his birth. From an early age, he demonstrated talent in music; as a young child he learned to play piano, he continued to play and compose for piano throughout his life. He suffered various health problems throughout his life, including meningitis as an infant, a disease that took the lives of all five of his brothers and sisters. In 1837, at the age of 14, he enrolled at Friedrich Wilhelm Gymnasium, soon thereafter at Friedrich Werder Gymnasium in Berlin, his teachers recognized his talents in mathematics, but by 15 years of age he had learned all the material taught at the school. He began to study differential calculus from the works of Leonhard Euler and Joseph-Louis Lagrange.
At 17, still a student, Eisenstein began to attend classes given by Peter Gustav Lejeune Dirichlet and others at the University of Berlin. In 1842, before taking his final exams, he traveled with his mother to England, to search for his father. In 1843 he met William Rowan Hamilton in Dublin, who gave him a copy of his book on Niels Henrik Abel's proof of the impossibility of solving fifth-degree polynomials, a work that would stimulate Eisenstein's interest in mathematical research. In 1843 Eisenstein returned to Berlin, where he passed his graduation exams and enrolled in the University the following autumn. In January 1844 he had presented his first work to the Berlin Academy, on cubic forms in two variables; the same year he met for the first time with Alexander von Humboldt, who would become Eisenstein's patron. Humboldt managed to find grants from the King, the government of Prussia, the Berlin academy to compensate for Eisenstein's extreme poverty; the monies, always late and grudgingly given, were earned in full measure by Eisenstein: in 1844 alone he published over 23 papers and two problems in Crelle's Journal, including two proofs of the law of quadratic reciprocity, the analogous laws of cubic reciprocity and quartic reciprocity.
In June 1844 Eisenstein visited Carl Friedrich Gauss in Göttingen. In 1845, Kummer saw to it. Jacobi encouraged the distinction, but relations between Jacobi and Eisenstein were always rocky, due to a disagreement over the order of discoveries made in 1846. In 1847 Eisenstein habilitated at the University of Berlin, he began to teach there. Bernhard Riemann attended his classes on elliptic functions. In 1848 Eisenstein was imprisoned by the Prussian army for his revolutionary activities in Berlin. Eisenstein always had republican sympathies, while he did not participate in the revolution of 1848, he was arrested on 19 March of that year. Although he was released just one day the harsh treatment he suffered damaged his delicate health, but his association with the Republican cause led to his official stipends being revoked, despite Humboldt tenaciously coming to his defense. Despite his health, Eisenstein continued writing papers on quadratic partitions of prime numbers and the reciprocity laws.
In 1851, at the instigation of Gauss, he was elected to the Academy of Göttingen. He died of tuberculosis at the age of 29. Humboldt 83, accompanied his remains to the cemetery, he had obtained, too late, as it turned out, the funding necessary to send Eisenstein on holiday to Sicily. E. T. Bell in his 1937 book Men of Mathematics claims that Gauss said "There have been but three epoch-making mathematicians, Archimedes and Eisenstein", this has been quoted in writings about Eisenstein; this is not a quote by Gauss, but is the end of a sentence from the biography of Eisenstein by Moritz Cantor, one of Gauss's last students and a historian of mathematics, summarizing his recollection of a remark made by Gauss about Eisenstein in a conversation many years earlier. Although it is doubtful that Gauss put Eisenstein in the same league as Newton, his writings show that Gauss thought highly of Eisenstein. For example, a letter from Gauss to Humboldt, dated the 14th of April in 1846, says that Eisenstein's talent is one that nature bestows only a few times a century.
Eisenstein, Mathematische Abhandlungen. Besonders aus dem Gebiete der höheren Arithmetik und der elliptischen Funktionen, Berlin Eisenstein, Mathematische Werke, New York: AMS Chelsea Publishing, ISBN 978-0-8284-1280-3, MR 0427029 Weil's review Eisenstein's criterion Eisenstein ideal Eisenstein integer Eisenstein prime Eisenstein reciprocity Eisenstein sum Eisenstein series Eisenstein's theorem Eisenstein–Kronecker number Elliptic Gauss sum Real analytic Eisenstein series Biermann, Kurt-R. "Eisenstein, Ferdinand Gotthold Max", Neue Deutsche Biographie, 4: 420 Cantor, Moritz, "Eisenstein, Gotthold", Allgemeine Deutsche Biographie, 5: 774–775, wikisource Dunnington, G. Waldo, Carl Friedrich Gauss: titan of science. A study of his life and work, New York: Exposition Press, ISBN 978-0-88385-547-8, MR 0072814 O'Connor, John J.. Schappacher, Norbert (199
Nqthm is a theorem prover sometimes referred to as the Boyer–Moore theorem prover. It was a precursor to ACL2; the system was developed by Robert S. Boyer and J Strother Moore, professors of computer science at the University of Texas, Austin, they began work on the system in 1971 in Scotland. Their goal was to make a automatic, logic-based theorem prover, they used a variant of Pure LISP as the working logic. Definitions are formed as recursive functions, the system makes extensive use of rewriting and an induction heuristic, used when rewriting and something that they called symbolic evaluation fails; the system was built on top of Lisp and had some basic knowledge in what was called "Ground-zero", the state of the machine after bootstrapping it onto a Common Lisp implementation. This is an example of the proof of a simple arithmetic theorem; the function TIMES is part of the BOOT-STRAP and is defined to be The formulation of the theorem is given in a Lisp-like syntax: Should the theorem prove to be true, it will be added to the knowledge basis of the system and can be used as a rewrite rule for future proofs.
The proof itself is given in a quasi-natural language manner. The authors randomly choose typical mathematical phrases for embedding the steps in the mathematical proof, which does make the proofs quite readable. There are macros for LaTeX that can transform the Lisp structure into more or less readable mathematical language; the proof of the commutativity of times continues: Give the conjecture the name *1. We will appeal to induction. Two inductions are suggested by terms in the conjecture. We limit our consideration to the two suggested by the largest number of nonprimitive recursive functions in the conjecture. Since both of these are likely, we will choose arbitrarily. We will induct according to the following scheme:. Linear arithmetic, the lemma COUNT-NUMBERP, the definition of ZEROP inform us that the measure decreases according to the well-founded relation LESSP in each induction step of the scheme; the above induction scheme produces the following two new conjectures: Case 2.. and after winding itself through a number of induction proofs concludes that Case 1..
This simplifies, expanding the definitions of ZEROP, TIMES, PLUS, EQUAL, to: T. That finishes the proof of *1.1, which finishes the proof of *1. Q. E. D. COMMUTATIVITY-OF-TIMES Many proofs have been done or confirmed with the system list concatenation insertion sort a binary adder an expression compiler for a stack machine uniqueness of prime factorizations invertibility of the RSA encryption algorithm unsolvability of the halting problem for Pure Lisp FM8501 microprocessor Gödel's incompleteness theorem CLI Stack Gauss' law of quadratic reciprocity Byzantine Generals and Clock Synchronization A compiler for a subset of the Nqthm language bi-phase mark asynchronous communications protocol Motorola MC68020 and Berkeley C String Library Paris–Harrington Ramsey theorem The equivalence of NFSA and DFSA A more powerful version, called PC-Nqthm was developed by Matt Kaufmann; this gave the proof tools that the system uses automatically to the user, so that more guidance can be given to the proof. This is a great help, as the system has an unproductive tendency to wander down infinite chains of inductive proofs.
A Computational Logic Handbook, R. S. Boyer and J S. Moore, Academic Press, 1997; the Boyer-Moore Theorem Prover and Its Interactive Enhancement, with M. Kaufmann and R. S. Boyer and Mathematics with Applications, 29, 1995, pp. 27–62. In 2005 Robert S. Boyer, Matt Kaufmann, J Strother Moore received the ACM Software System Award for their work on the Nqthm theorem prover; the Automated Reasoning System, Nqthm The Boyer-Moore Theorem Prover Even though the system is no longer being supported, it is still available at Runnable version on GitHub
In computer science, Coq is an interactive theorem prover. It allows the expression of mathematical assertions, mechanically checks proofs of these assertions, helps to find formal proofs, extracts a certified program from the constructive proof of its formal specification. Coq works within the theory of the calculus of inductive constructions, a derivative of the calculus of constructions. Coq is not an automated theorem prover but includes automatic theorem proving tactics and various decision procedures; the Association for Computing Machinery rewarded Thierry Coquand, Gérard Pierre Huet, Christine Paulin-Mohring, Bruno Barras, Jean-Christophe Filliâtre, Hugo Herbelin, Chetan Murthy, Yves Bertot and Pierre Castéran with the 2013 ACM Software System Award for Coq. Seen as a programming language, Coq implements a dependently typed functional programming language, while seen as a logical system, it implements a higher-order type theory; the development of Coq has been supported since 1984 by INRIA, now in collaboration with École Polytechnique, University of Paris-Sud, Paris Diderot University and CNRS.
In the 1990s, École Normale Supérieure de Lyon was part of the project. The development of Coq was initiated by Gérard Pierre Huet and Thierry Coquand, after which more than 40 people researchers, contributed features of the core system; the implementation team was successively coordinated by Gérard Pierre Huet, Christine Paulin-Mohring and Hugo Herbelin. Coq is for the most part implemented in OCaml with a bit of C; the core system can be extended due to a mechanism of plug-ins. The word coq means "rooster" in French, stems from a local tradition of naming French research development tools with animal names. Up to 1991, Coquand was implementing a language called the Calculus of Constructions and it was called CoC at this time. In 1991, a new implementation based on the extended Calculus of Inductive Constructions was started and the name changed from CoC to Coq an indirect reference to Thierry Coquand who developed the Calculus of Constructions along with Gérard Pierre Huet and the Calculus of Inductive Constructions along with Christine Paulin-Mohring.
Coq provides. Programs written in Gallina have the weak normalization property – they always terminate; this is one way. This may be surprising. Georges Gonthier and Benjamin Werner used Coq to create a surveyable proof of the four color theorem, completed in September 2004. Based on this work, a significant extension to Coq was developed called Ssreflect. Despite the name, most of the new features added to Coq by Ssreflect are general-purpose features, useful not for the computational reflection style of proof; these include: Additional convenient notations for irrefutable and refutable pattern matching, on inductive types with one or two constructors Implicit arguments for functions applied to zero arguments –, useful when programming with higher-order functions Concise anonymous arguments An improved set tactic with more powerful matching Support for reflectionSsreflect 1.4 is available dual-licensed under the open source CeCILL-B or CeCILL-2.0 license, is compatible with Coq 8.4. CompCert: an optimizing compiler for all of the C programming language, programmed and proved in Coq.
Disjoint-set data structure: correctness proof in Coq was published in 2007. Feit–Thompson theorem: formal proof using Coq was completed in September 2012. Four color theorem: formal proof using Coq was completed in 2005. Agda Calculus of constructions Curry–Howard correspondence Isabelle – similar/competing software Intuitionistic type theory HOL The Coq proof assistant – the official English website coq/coq – the project's source code repository on GitHub JsCoq Interactive Online System – allows Coq to be run in a web browser, without the need for any software installation Coq Wiki Mathematical Components library – used library of mathematical structures, part of, the Ssreflect proof language Constructive Coq Repository at Nijmegen Math Classes Coq at Open HubTextbooksThe Coq'Art – a book on Coq by Yves Bertot and Pierre Castéran Certified Programming with Dependent Types – online and printed textbook by Adam Chlipala Software Foundations – online textbook by Benjamin C. Pierce et al. An introduction to small scale reflection in Coq – a tutorial on SSreflect by Georges Gonthier and Assia MahboubiTutorialsIntroduction to the Coq Proof Assistant – video lecture by Andrew Appel at Institute for Advanced Study Video tutorials for the Coq proof assistant by Andrej Bauer
Inductive reasoning is a method of reasoning in which the premises are viewed as supplying some evidence for the truth of the conclusion, this is in contrast to deductive reasoning. While the conclusion of a deductive argument is certain, the truth of the conclusion of an inductive argument may be probable, based upon the evidence given. Many dictionaries define inductive reasoning as the derivation of general principles from specific observations, though some sources find this usage "outdated". Inductive reasoning is a form of argument that—in contrast to deductive reasoning—allows for the possibility that a conclusion can be false if all of the premises are true. Instead of being valid or invalid, inductive arguments are either strong or weak, according to how probable it is that the conclusion is true. We may call an inductive argument plausible, reasonable, justified or strong, but never certain or necessary. Logic affords no bridge from the probable to the certain; the futility of attaining certainty through some critical mass of probability can be illustrated with a coin-toss exercise.
Suppose someone tests to see if the coin is either a fair one or two-headed. She flips it ten times, ten times it comes up heads. At this point, there is a strong reason to believe. After all, the chance of ten heads in a row is.000976: less than one in one thousand. After 100 flips, every toss has come up heads. Now there is "virtual" certainty. Still, one can neither empirically rule out that the next toss will produce tails. No matter how many times in a row it comes up heads this remains the case. If one programmed a machine to flip a coin over and over continuously at some point the result would be a string of 100 heads. In the fullness of time, all combinations will appear; as for the slim prospect of getting ten out of ten heads from a fair coin—the outcome that made the coin appear biased—many may be surprised to learn that the chance of any sequence of heads or tails is unlikely and yet it occurs in every trial of ten tosses. That means all results for ten tosses have the same probability as getting ten out of ten heads, 0.000976.
If one records the heads-tails sequences, for whatever result, that exact sequence had a chance of 0.000976. An argument is deductive; that is, the conclusion cannot be false if the premises are true. If a deductive conclusion follows duly from its premises it is valid. An examination of the following examples will show that the relationship between premises and conclusion is such that the truth of the conclusion is implicit in the premises. Bachelors are unmarried. Socrates is mortal; the conclusion for a valid deductive argument is contained in the premises since because its truth is a matter of logical relations. It cannot say more than its premises. Inductive premises, on the other hand, draw their substance from fact and evidence, the conclusion accordingly makes a factual claim or prediction, its reliability varies proportionally with the evidence. Induction wants to reveal something new about the world. One could say. To better see the difference between inductive and deductive arguments, consider that it would not make sense to say: "all rectangles so far examined have four right angles, so the next one I see will have four right angles."
This would treat logical relations as something factual and discoverable, thus variable and uncertain. Speaking deductively we may permissibly say. "All unicorns can fly. This deductive argument is valid. Inductive reasoning is inherently uncertain, it only deals in the extent to which, given the premises, the conclusion is credible according to some theory of evidence. Examples include a many-valued logic, Dempster–Shafer theory, or probability theory with rules for inference such as Bayes' rule. Unlike deductive reasoning, it does not rely on universals holding over a closed domain of discourse to draw conclusions, so it can be applicable in cases of epistemic uncertainty. Another crucial difference between these two types of argument is that deductive certainty is impossible in non-axiomatic systems such as reality, leaving inductive reasoning as the primary route to knowledge of such systems. Given that "if A is true that would cause B, C, D to be true", an example of deduction would be "A is true therefore we can deduce that B, C, D are true".
An example of induction would be "B, C, D are observed to be true therefore A might be true". A is a reasonable explanation for B, C, D being true. For example: A large enough asteroid impact would create a large crater and cause a severe impact winter that could drive the non-avian dinosaurs to extinction. We observe that there is a large crater in the Gulf of Mexico dating to near the time of the extinction of the non-avian dinosaurs. Therefore, it is possible. Note, that the asteroid explanation for the mass extinction is not correct. Other events with the potential to affect global climate coincide with