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In mathematical logic, a **witness** is a specific value *t* to be substituted for variable *x* of an existential statement of the form ∃*x* φ(*x*) such that φ(*t*) is true.

For example, a theory *T* of arithmetic is said to be inconsistent if there exists a proof in *T* of the formula "0=1". The formula I(*T*), which says that *T* is inconsistent, is thus an existential formula. A witness for the inconsistency of *T* is a particular proof of "0 = 1" in *T*.

Boolos, Burgess, and Jeffrey (2002:81) define the notion of a witness with the example, in which *S* is an *n*-place relation on natural numbers, *R* is an *n*-place recursive relation, and ↔ indicates logical equivalence (if and only if):

In predicate calculus, a **Henkin witness** for a sentence in a theory *T* is a term *c* such that *T* proves φ(*c*) (Hinman 2005:196). The use of such witnesses is a key technique in the proof of Gödel's completeness theorem presented by Leon Henkin in 1949.

The notion of witness leads to the more general idea of game semantics. In the case of sentence the winning strategy for the verifier is to pick a witness for . For more complex formulas involving universal quantifiers, the existence of a winning strategy for the verifier depends on the existence of appropriate Skolem functions. For example, if *S* denotes then an equisatisfiable statement for *S* is . The Skolem function *f* (if it exists) actually codifies a winning strategy for the verifier of *S* by returning a witness for the existential sub-formula for every choice of *x* the falsifier might make.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Witness_(mathematics)

**Mathematics** (from Greek μάθημα *máthēma*, “knowledge, study, learning”) is the study of topics such as quantity (numbers),structure,space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.

Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's *Elements*. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Mathematics

**Mathematics**, also known as **Allah Mathematics**, (born: **Ronald Maurice Bean**) is a hip hop producer and DJ for the Wu-Tang Clan and its solo and affiliate projects.

Born and raised in Jamaica, Queens, New York, Mathematics was introduced to hip hop by his brother who used to bring home recordings of the genre's pioneers like Grandmaster Flash & The Furious Five, Treacherous Three and Cold Crush Brothers. He began his career in 1987 DJing block parties and park jams in Baisley Projects, going by the name Supreme Cut Master. In 1988, he became the full-time DJ for experienced rapper Victor C, doing countless shows in clubs and colleges in New York City.

In 1990, Mathematics linked up with GZA/Genius; he soon became one of the Wu-Tang Clan's founding members, but at the time GZA was struggling to build a career on the Cold Chillin' label. This partnership earned Mathematics a spot on his first official tour, The Cold Chillin Blizzard Tour (with popular acts such as Biz Markie, Big Daddy Kane, Kool G. Rap & DJ Polo and Marley Marl).

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Mathematics_(producer)

**"Mathematics"** is a b-side single from Mos Def's solo debut album, *Black on Both Sides*. It contains lyrics about various social issues and asks the listener to add them up and come to conclusions about them. Many references to numbers are found in this song and at times, Mos Def rhymes statistics in numerical order. The song highlights the differences between the White and African-American citizens of the US and uses the lyrics "Do your math..." (from Erykah Badu's "On & On") telling young African-American's to 'do their maths' so they can avoid being part of the numerous degrading statistics he raps about in the opening and third verses of the song. The song is produced by DJ Premier whose famous scratch samples make up the song's bridge. Premier has called it one of his favorite beats.

It can be found on the soundtrack of Madden NFL 2002. The song is also played briefly in the CSI: Crime Scene Investigation episode "Crate 'n' Burial".

The bridge of "Mathematics" contains DJ Premier's signature scratched vocals from various hip hop songs. The lyrics of those samples as well as information about their origin can be found below:

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Mathematics_(song)

* Witness* is a 1985 American crime thriller film directed by Peter Weir and starring Harrison Ford and Kelly McGillis. The screenplay by William Kelley, Pamela Wallace, and Earl W. Wallace focuses on a detective protecting a young Amish boy who becomes a target after he witnesses a murder in Philadelphia.

The film was nominated for eight Academy Awards and won two, for Best Original Screenplay and Best Film Editing. It was also nominated for seven BAFTA Awards, winning one for Maurice Jarre's score, and was also nominated for six Golden Globe Awards. William Kelley and Earl W. Wallace won the Writers Guild of America Award for Best Original Screenplay and the 1986 Edgar Award for Best Motion Picture Screenplay presented by the Mystery Writers of America.

Rachel Lapp (McGillis), a young Amish widow, and her 8-year-old son Samuel (Haas) are traveling by train to visit Rachel's sister. Samuel is amazed by the sights in the big city, but at 30th Street Station in Philadelphia, he witnesses two men attack and murder a third (Carhart). Detective John Book (Ford) is assigned to the case and he and his partner, Sergeant Elton Carter (Jennings), question Samuel. The victim was an undercover police officer. Samuel is unable to identify the perpetrator from mug shots or a police lineup, but notices a newspaper clipping with a picture of narcotics officer James McFee (Glover) and recognizes him as one of the killers. John remembers that McFee was previously responsible for a drug raid on expensive chemicals used to make amphetamines, but the evidence had mysteriously disappeared.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Witness_(1985_film)

**Witness** was the name of an evangelical newspaper established in 1840 by the Scottish geologist and writer, Hugh Miller. He continued to edit the paper at an office on the Royal Mile in Edinburgh until his suicide in December 1856. He was the principal contributor to the publication, averaging over 10,000 words a week.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Witness_(religious_newspaper)

* Witness* is the 18th album by trumpeter Dave Douglas. It was released on the RCA label in 2001 and features performances by Douglas, Chris Speed, Joe Daley, Mark Feldman, Erik Friedlander, Drew Gress, Bryan Carrott, Michael Sarin, Ikue Mori, Joshua Roseman and Yuka Honda with Tom Waits providing vocals on one track.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Witness_(Dave_Douglas_album)

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