A mathematical picture paints a thousand words: the Pythagorean theorem.

In mathematical logic, a theorem is a type of abstract object, one token of which is a formula of a formal language which can be derived from the rules of the formal system that is applied to the formal language; another token of which is a statement in natural language, that can be proved on the basis of explicitly stated or previously agreed assumptions. In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ... For other uses, see Abstract It is a commonplace in philosophy that every thing or object is either abstract or concrete. ... In mathematical logic, a formula is a formal syntactic object that expresses a proposition. ... In mathematics, logic, and computer science, a formal language is a language that is defined by precise mathematical or machine processable formulas. ... In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ... In the philosophy of language, a natural language (or ordinary language) is a language that is spoken, written, or signed by humans for general-purpose communication, as distinguished from formal languages (such as computer-programming languages or the languages used in the study of formal logic, especially mathematical logic) and... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...

In all settings, an essential property of theorems is that they are derivable using a fixed set of inference rules and axioms without any additional assumptions. This is not a matter of the semantics of the language: the expression that results from a derivation is a syntactic consequence of all the expressions that precede it. In mathematics, the derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other interpretations, depending on the meanings of the derivation rules. In logic, especially in mathematical logic, a rule of inference is a scheme for constructing valid inferences. ... This article is about a logical statement. ... Semantics (Ancient σημαντικός semantikos significant, from semainein to signify, mean, from sema sign, token), is the study of meaning in communication. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...

The proofs of theorems have two components, called the hypotheses and the conclusions. The proof of a mathematical theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions. The concept of a theorem is therefore fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical. In mathematics and logic, premises are the formulas on which a step of a logical argument depends to obtain a consequence of those premises. ... A conclusion is a final proposition, which is arrived at after the consideration of evidence, arguments or premises. ... Deductive reasoning is the process of reaching a conclusion that is guaranteed to follow, if the evidence provided is true and the reasoning used to reach the conclusion is correct. ... The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion. ... A central concept in science and the scientific method is that all evidence must be empirical, or empirically based, that is, dependent on evidence or consequences that are observable by the senses. ...

Although they can be written in a completely symbolic form, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organised and clearly worded informal arguments intended to demonstrate that a formal symbolic proof can be constructed. Such arguments are typically easier to check than purely symbolic ones — indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows to establish that certain formulas are theorems of the formal system. ...

Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgements vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's last theorem is a particularly well-known example of such a theorem. Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ...

Formal and informal notions

Logically most theorems are of the form of an indicative conditional: if A, then B. Such a theorem does not state that B is always true, only that B must be true if A is true. In this case A is called the hypothesis of the theorem (note that "hypothesis" here is something very different from a conjecture) and B the conclusion. The theorem "If n is an even natural number then n/2 is a natural number" is a typical example in which the hypothesis is that n is an even natural number and the conclusion is that n/2 is also a natural number. Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ... The indicative conditional is the logical operation given by statements of the form If A then B in ordinary English (or similar natural languages). ... Look up Hypothesis in Wiktionary, the free dictionary. ... In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

In order to be proven, a theorem must be expressible as a precise, formal statement. Nevertheless, theorems are usually expressed in natural language rather than in a completely symbolic form, with the intention that the reader will be able to produce a formal statement from the informal one. In addition, there are often hypotheses which are understood in context, rather than explicitly stated.

It is common in mathematics to choose a number of hypotheses that are assumed to be true within a given theory, and then declare that the theory consists of all theorems provable using those hypotheses as assumptions. In this case the hypotheses that form the foundational basis are called the axioms (or postulates) of the theory. The field of mathematics known as proof theory studies formal axiom systems and the proofs that can be performed within them. For the algebra software named Axiom, see Axiom computer algebra system. ... Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ...

A planar map with five colors such that no two regions with the same color meet. It can actually be colored in this way with only four colors. The four color theorem states that such colorings are possible for any planar map, but every known proof involves a computational search that is too long to check by hand.

Some theorems are "trivial," in the sense that they follow from definitions, axioms, and other theorems in obvious ways and do not contain any surprising insights. Some, on the other hand, may be called "deep": their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics.^[1] A theorem might be simple to state and yet be deep. An excellent example is Fermat's Last Theorem, and there are many other examples of simple yet deep theorems in number theory and combinatorics, among other areas. Image File history File links 4CT_Non-Counterexample_1. ... This article is about the mathematical construct. ... Example of a four-colored map The four color theorem (also known as the four color map theorem) states that given any plane separated into regions, such as a political map of the states of a country, the regions may be colored using no more than four colors in such... Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ...

There are other theorems for which a proof is known, but the proof cannot easily be written down. The most prominent examples are the Four color theorem and the Kepler conjecture. Both of these theorems are only known to be true by reducing them to a computational search which is then verified by a computer program. Initially, many mathematicians did not accept this form of proof, but it has become more widely accepted in recent years. The mathematician Doron Zeilberger has even gone so far as to claim that these are possibly the only nontrivial results that mathematicians have ever proved.[1] Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities and hypergeometric identities.^[2]obiectul of theorem Example of a four-colored map The four color theorem (also known as the four color map theorem) states that given any plane separated into regions, such as a political map of the states of a country, the regions may be colored using no more than four colors in such... In mathematics, the Kepler conjecture is a conjecture about sphere packing in three dimensional Euclidean space. ... Doron Zeilberger (דורון ציילברגר, born July 2, 1950 in Israel) is an Israeli mathematician, known for his work in combinatorics. ...

Relation to proof

The notion of a theorem is deeply intertwined with the concept of proof. Indeed, theorems are true precisely in the sense that they possess proofs. Therefore, to establish a mathematical statement as a theorem, the existence of a line of reasoning from axioms in the system (and other, already established theorems) to the given statement must be demonstrated.

Although the proof is necessary to produce a theorem, it is not usually considered part of the theorem. And even though more than one proof may be known for a single theorem, only one proof is required to establish the theorem's validity. The Pythagorean theorem and the law of quadratic reciprocity are contenders for the title of theorem with the greatest number of distinct proofs.obiectul In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... In number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic. ...

Theorems in logic

Logic, especially in the field of proof theory, considers theorems as statements (called formulas or well formed formulas) of a formal language. A set of deduction rules, also called transformation rules or a formal grammar, must be provided. These deduction rules tell exactly when a formula can be derived from a set of premises. Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... In logic, WFF is an abbreviation for well-formed formula. ... In mathematics, logic, and computer science, a formal language is a language that is defined by precise mathematical or machine processable formulas. ... In computer science and linguistics, a formal grammar, or sometimes simply grammar, is a precise description of a formal language — that is, of a set of strings. ...

Different sets of derivation rules give rise to different interpretations of what it means for an expression to be a theorem. Some derivation rules and formal languages are intended to capture mathematical reasoning; the most common examples use first-order logic. Other deductive systems describe term rewriting, such as the reduction rules for λ calculus. First-order logic (FOL) is a formal deductive system used by mathematicians, philosophers, linguists, and computer scientists. ... Rewriting in mathematics, computer science and logic covers a wide range of methods of transforming strings, written in some fixed alphabet, that are not deterministic but are governed by explicit rules. ... The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ...

The definition of theorems as elements of a formal language allows for results in proof theory that study the structure of formal proofs and the structure of provable formulas. The most famous result is Gödel's incompleteness theorem; by representing theorems about basic number theory as expressions in a formal language, and then representing this language within number theory itself, Gödel constructed examples of statements that are neither provable nor disprovable from axiomatizations of number theory. In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ...

Relation with scientific theories

Theorems in mathematics and theories in science are fundamentally different in their epistemology. A scientific theory cannot be proven; its key attribute is that it is falsifiable, that is, it makes predictions about the natural world that are testable by experiments. Any disagreement between prediction and experiment demonstrates the incorrectness of the scientific theory, or at least limits its accuracy or domain of validity. Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in the same way such evidence is used to support scientific theories. The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion. ... Theory of knowledge redirects here: for other uses, see theory of knowledge (disambiguation) According to Plato, knowledge is a subset of that which is both true and believed Epistemology or theory of knowledge is the branch of philosophy that studies the nature, methods, limitations, and validity of knowledge and belief. ... This page discusses how a theory or assertion is falsifiable (disprovable opp: verifiable), rather than the non-philosophical use of falsification, meaning counterfeiting. ... In the scientific method, an experiment (Latin: ex- periri, of (or from) trying) is a set of observations performed in the context of solving a particular problem or question, to retain or falsify a hypothesis or research concerning phenomena. ...

The Collatz conjecture: one way to illustrate its complexity is to extend the iteration from the natural numbers to the complex numbers. The result is a fractal, which (in accordance with universality) resembles the Mandelbrot set.

Nonetheless, there is some degree of empiricism and data collection involved in the discovery of mathematical theorems. By establishing a pattern, sometimes with the use of a powerful computer, mathematicians may have an idea of what to prove, and in some cases even a plan for how to set about doing the proof. For example, the Collatz conjecture has been verified for start values up to about 2.88 × 10¹⁸. The Riemann hypothesis has been verified for the first 10 trillion zeroes of the zeta function. Neither of these statements is considered to be proven. Image File history File links Download high resolution version (996x597, 390 KB) Summary Fractal of the Collatz map. ... Image File history File links Download high resolution version (996x597, 390 KB) Summary Fractal of the Collatz map. ... In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. ... Initial image of a Mandelbrot set zoom sequence with continuously coloured environment The Mandelbrot set is a set of points in the complex plane, the boundary of which forms a fractal. ... The Collatz conjecture is an unsolved conjecture in mathematics. ... There is also the Riemann hypothesis for curves over finite fields. ... In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ...

Such evidence does not constitute proof. For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample (i.e., a natural number n for which the Mertens function M(n) equals or exceeds the square root of n) is known: all numbers less than 10¹⁴ have the Mertens property, and the smallest number which does not have this property is only known to be less than the exponential of 1.59 × 10⁴⁰, which is approximately 10 to the power 4.3 × 10³⁹. Since the number of particles in the universe is generally considered to be less than 10 to the power 100 (a googol), there is no hope to find an explicit counterexample by exhaustive search at present. The Mertens conjecture is a statement about the behaviour of a certain function as its argument increases. ... The exponential function is one of the most important functions in mathematics. ... For the Internet company, see Google. ... In computer science, brute-force search is a trivial but very general problem-solving technique, that consists of systematically enumerating all possible candidates for the solution and checking whether each candidate satisfies the problems statement. ...

Note that the word "theory" also exists in mathematics, to denote a body of mathematical axioms, definitions and theorems, as in, for example, group theory. There are also "theorems" in science, particularly physics, and in engineering, but they often have statements and proofs in which physical assumptions and intuition play an important role; the physical axioms on which such "theorems" are based are themselves falsifiable. Group theory is that branch of mathematics concerned with the study of groups. ...

Terminology

Theorems are often indicated by several other terms: the actual label "theorem" is reserved for the most important results, whereas results which are less important, or distinguished in other ways, are named by different terminology.

• A proposition is a statement not associated with any particular theorem. This term sometimes connotes a statement with a simple proof.
• A lemma is a "pre-theorem", a statement that forms part of the proof of a larger theorem. The distinction between theorems and lemmas is rather arbitrary, since one mathematician's major result is another's minor claim. Gauss's lemma and Zorn's lemma, for example, are interesting enough that some authors present the nominal lemma without going on to use it in the proof of a theorem.
• A corollary is a proposition that follows with little or no proof from one other theorem or definition. That is, proposition B is a corollary of a proposition A if B can readily be deduced from A.
• A claim is a necessary or independently interesting result that may be part of the proof of another statement. Despite the name, claims must be proved.

There are other terms, less commonly used, which are conventionally attached to proven statements, so that certain theorems are referred to by historical or customary names. For examples: This article is about the word proposition as it is used in logic, philosophy, and linguistics. ... In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than an independent statement, in and of itself. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... In the theory of polynomials, Gausss lemma, named after Carl Friedrich Gauss, refers to two related statements: The first result states that the product of two primitive polynomials is also primitive (a polynomial is primitive if the greatest common divisor of its coefficients is 1). ... Zorns lemma, also known as the Kuratowski-Zorn lemma, is a proposition of set theory that states: Every non-empty partially ordered set in which every chain (i. ... A theorem is a statement which can be proven true within some logical framework. ...

• Identity, used for theorems which state an equality between two mathematical expressions. Examples include Euler's identity and Vandermonde's identity.
• Rule, used for certain theorems such as Bayes' rule and Cramer's rule, that establish useful formulas.
• Law. Examples include the law of large numbers, the law of cosines, and Kolmogorov's zero-one law.^[3]
• Principle. Examples include Harnack's principle, the least upper bound principle, and the pigeonhole principle.
• A Converse is a reverse theorem. For example, If a theorem states that A is a related to B, its converse would state that B is related to A. The converse of a theorem need not be always true.

A few well-known theorems have even more idiosyncratic names. The division algorithm a theorem expressing the outcome of division in the natural numbers and more general rings. The Banach–Tarski paradox is a theorem in measure theory that is paradoxical in the sense that it contradicts common intuitions about volume in three-dimensional space. In mathematics, the term identity has several important uses: An identity is an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an equality which is true under more particular conditions. ... For other uses, see List of topics named after Leonhard Euler. ... In combinatorial mathematics, Vandermondes identity, named after Alexandre-Théophile Vandermonde, states that This may be proved by simple algebra relying on the fact that (see factorial) but it also admits a more combinatorics-flavored bijective proof, as follows. ... Look up rule, ruling in Wiktionary, the free dictionary. ... Bayes theorem is a result in probability theory, which gives the conditional probability distribution of a random variable A given B in terms of the conditional probability distribution of variable B given A and the marginal probability distribution of A alone. ... Cramers rule is a theorem in linear algebra, which gives the solution of a system of linear equations in terms of determinants. ... // The law of large numbers (LLN) is any of several theorems in probability. ... Fig. ... In probability theory, Kolmogorovs zero-one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, called a tail event, will either almost surely happen or almost surely not happen; that is, the probability of such an event occurring is zero or one. ... A principle (not principal) is something, usually a rule or norm, that is part of the basis for something else. ... In complex analysis, Harnacks principle is a theorem about the behavior of sequences of harmonic functions. ... This article or section does not adequately cite its references or sources. ... The inspiration for the name of the principle: pigeons in holes. ... For other uses, see Converse (disambiguation). ... The division algorithm is a theorem in mathematics which precisely expresses the outcome of the usual process of division of integers. ... A ball can be decomposed into a finite number of pieces and reassembled into two balls identical to the original. ... In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ... Look up paradox in Wiktionary, the free dictionary. ...

An unproven statement that is believed to be true is called a conjecture (or sometimes a hypothesis, but with a different meaning from the one discussed above). To be considered a conjecture, a statement must usually be proposed publicly, at which point the name of the proponent may be attached to the conjecture, as with Goldbach's conjecture. Other famous conjectures include the Collatz conjecture and the Riemann hypothesis. In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ... Look up Hypothesis in Wiktionary, the free dictionary. ... Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ... The Collatz conjecture is an unsolved conjecture in mathematics. ... There is also the Riemann hypothesis for curves over finite fields. ...

Layout

A theorem and its proof are typically laid out as follows:

Theorem (name of person who proved it and year of discovery, proof or publication).

Statement of theorem.

Proof.

Description of proof.

The end of the proof may be signalled by the letters Q.E.D. meaning "quod erat demonstrandum" or by one of the tombstone marks "□" or "∎" meaning "End of Proof", introduced by Paul Halmos following their usage in magazine articles. This article is about Latin phrase Q.E.D., as used in proofs. ... For other meanings of the abbreviation QED, see QED. Q. E. D. is an abbreviation of the Latin phrase quod erat demonstrandum (literally, that which was to be demonstrated). This is a translation of the Greek oper edei deixai which was used by many early mathematicians including Euclid and Archimedes. ... The tombstone, or halmos, symbol — (Unicode U+220E) — is used in mathematics to denote the end of a proof. ... Paul Halmos Paul Richard Halmos (March 3, 1916 — October 2, 2006) was a Hungarian-born American mathematician who wrote on probability theory, statistics, operator theory, ergodic theory, functional analysis (in particular, Hilbert spaces), and mathematical logic. ...

The exact style will depend on the author or publication. Many publications provide instructions or macros for typesetting in the house style. For other uses, see Macro (disambiguation) A macro in computer science is a rule or pattern that specifies how a certain input sequence (often a sequence of characters) should be mapped to an output sequence (also often a sequence of characters) according to a defined procedure. ... A publishing companys or periodicals house style is the collection of conventions in its manual of style. ...

It is common for a theorem to be preceded by definitions describing the exact meaning of the terms used in the theorem. It is also common for a theorem to be preceded by a number of propositions or lemmas which are then used in the proof. However, lemmas are sometimes embedded in the proof of a theorem, either with nested proofs, or with their proofs presented after the proof of the theorem. For other uses, see Definition (disambiguation). ... This article is about the word proposition as it is used in logic, philosophy, and linguistics. ... In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than an independent statement, in and of itself. ...

Corollaries to a theorem are either presented between the theorem and the proof, or directly after the proof. Sometimes corollaries have proofs of their own which explain why they follow from the theorem ; theorem of simona popp; A theorem is a statement which can be proven true within some logical framework. ...

Lore

It has been estimated that over a quarter of a million theorems are proved every year.^[4]

The well-known aphorism, "A mathematician is a device for turning coffee into theorems", is probably due to Alfréd Rényi, although it is often attributed to Rényi's colleague Paul Erdős (and Rényi may have been thinking of Erdős), who was famous for the many theorems he produced, the number of his collaborations, and his coffee drinking.^[5] An aphorism (literally distinction or definition, from Greek αφοριζειν to define) expresses a general truth in a pithy sentence. ... Alfréd Rényi (March 20, 1921 – February 1, 1970) was a Hungarian mathematician who made contributions in combinatorics and graph theory but mostly in probability theory. ... Paul ErdÅ‘s (Hungarian: ErdÅ‘s Pál, in English occasionally Paul Erdos or Paul Erdös, March 26, 1913 – September 20, 1996), was an immensely prolific (and famously eccentric) Hungarian-born mathematician. ... The title given to this article is incorrect due to technical limitations. ...

The classification of finite simple groups is regarded by some to be the longest proof of a theorem; it comprises tens of thousands of pages in 500 journal articles by some 100 authors. These papers are together believed to give a complete proof, and there are several ongoing projects to shorten and simplify this proof.^[6] The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. ...

Look up theorem in Wiktionary, the free dictionary.

Template:Logic und Theorem Wikipedia does not have an article with this exact name. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 151 languages. ...

See also

• Metatheorem
• List of theorems
• Gödel's incompleteness theorem, that establishes very general conditions under which a formal system will contain a true statement for which there exists no proof within the system.
• Inference
• Toy theorem
• Kepler
• Tomotika and Tamada
• caius iacob

In logic, a metatheorem is a statement about theorems or about some axiomatic theory. ... This is a list of theorems, by Wikipedia page. ... In mathematical logic, G̦dels incompleteness theorems are two celebrated theorems proven by Kurt G̦del in 1931. ... Inference is the act or process of deriving a conclusion based solely on what one already knows. ... In mathematics, a toy theorem is a simplified version of a more general theorem. ... Johannes Kepler Johannes Kepler (December 27, 1571 РNovember 15, 1630), a key figure in the scientific revolution, was a German astronomer, mathematician and astrologer. ...

Notes

1. ^ See Deep Theorem, cited below.
2. ^ Petkovsek et al. 1996.
3. ^ The word law can also refer to an axiom, a rule of inference, or, in probability theory, a probability distribution.
4. ^ Hoffman 1998, p. 204.
5. ^ Hoffman 1998, p. 7.
6. ^ An enormous theorem: the classification of finite simple groups, Richard Elwes, Plus Magazine, Issue 41 December 2006.

This article is about a logical statement. ... In logic, especially in mathematical logic, a rule of inference is a scheme for constructing valid inferences. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... A probability distribution describes the values and probabilities that a random event can take place. ...

References

• Hoffman, P. (1998). The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. Hyperion, New York. 
• Petkovsek, Marko; Wilf, Herbert; Zeilberger, Doron (1996). "A = B". A.K. Peters, Wellesley, Massachusetts. 

External links

Logic Portal

• Eric W. Weisstein, Deep Theorem at MathWorld.
• Eric W. Weisstein, Theorem at MathWorld.