In predicate logic, an existential quantification is the predication^[1] of a property or relation to at least one member of the domain. It is denoted by the logical operator symbol ∃ (pronounced "there exists" or "for some"), which is called the existential quantifier. Existential quantification is distinct from universal quantification ("for all"), which asserts that the property or relation holds for any members of the domain.

Symbols are encoded U+2203 ∃ there exists (HTML: ∃ ∃ as a mathematical symbol) and U+2204 ∄ there does not exist (HTML: ∄).

Basics[link]

Consider a formula that states that some natural number multiplied by itself is 25.

0·0 = 25, or 1·1 = 25, or 2·2 = 25, or 3·3 = 25, and so on.

This would seem to be a logical disjunction because of the repeated use of "or". However, the "and so on" makes this impossible to integrate and to interpret as a disjunction in formal logic. Instead, the statement could be rephrased more formally as

For some natural number n, n·n = 25.

This is a single statement using existential quantification.

This statement is more precise than the original one, as the phrase "and so on" does not necessarily include all natural numbers, and nothing more. Since the domain was not stated explicitly, the phrase could not be interpreted formally. In the quantified statement, on the other hand, the natural numbers are mentioned explicitly.

This particular example is true, because 5 is a natural number, and when we substitute 5 for n, we produce "5·5 = 25", which is true. It does not matter that "n·n = 25" is only true for a single natural number, 5; even the existence of a single solution is enough to prove the existential quantification true. In contrast, "For some even number n, n·n = 25" is false, because there are no even solutions.

The domain of discourse, which specifies which values the variable n is allowed to take, is therefore of critical importance in a statement's trueness or falseness. Logical conjunctions are used to restrict the domain of discourse to fulfill a given predicate. For example:

For some positive odd number n, n·n = 25

is logically equivalent to

For some natural number n, n is odd and n·n = 25.

Here, "and" is the logical conjunction.

In symbolic logic, "∃" (a backwards letter "E" in a sans-serif font) is used to indicate existential quantification.^[2] Thus, if P(a, b, c) is the predicate "a·b = c" and Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{N}

is the set of natural numbers, then

Failed to parse (Missing texvc executable; please see math/README to configure.): \exists{n}{\in}\mathbb{N}\, P(n,n,25)

is the (true) statement

For some natural number n, n·n = 25.

Similarly, if Q(n) is the predicate "n is even", then

Failed to parse (Missing texvc executable; please see math/README to configure.): \exists{n}{\in}\mathbb{N}\, \big(Q(n)\;\!\;\! {\wedge}\;\!\;\! P(n,n,25)\big)

is the (false) statement

For some natural number n, n is even and n·n = 25.

In mathematics, the proof of a "some" statement may be achieved either by a constructive proof, which exhibits an object satisfying the "some" statement, or by a nonconstructive proof which shows that there must be such an object but without exhibiting one.

Properties[link]

Negation[link]

A quantified propositional function is a statement; thus, like statements, quantified functions can be negated. The Failed to parse (Missing texvc executable; please see math/README to configure.): \lnot\   symbol is used to denote negation.

For example, if P(x) is the propositional function "x is between 0 and 1", then, for a domain of discourse X of all natural numbers, the existential quantification "There exists a natural number x which is between 0 and 1" is symbolically stated:

Failed to parse (Missing texvc executable; please see math/README to configure.): \exists{x}{\in}\mathbf{X}\, P(x)

This can be demonstrated to be irrevocably false. Truthfully, it must be said, "It is not the case that there is a natural number x that is between 0 and 1", or, symbolically:

Failed to parse (Missing texvc executable; please see math/README to configure.): \lnot\ \exists{x}{\in}\mathbf{X}\, P(x)

.

If there is no element of the domain of discourse for which the statement is true, then it must be false for all of those elements. That is, the negation of

Failed to parse (Missing texvc executable; please see math/README to configure.): \exists{x}{\in}\mathbf{X}\, P(x)

is logically equivalent to "For any natural number x, x is not between 0 and 1", or:

Failed to parse (Missing texvc executable; please see math/README to configure.): \forall{x}{\in}\mathbf{X}\, \lnot P(x)

Generally, then, the negation of a propositional function's existential quantification is a universal quantification of that propositional function's negation; symbolically,

Failed to parse (Missing texvc executable; please see math/README to configure.): \lnot\ \exists{x}{\in}\mathbf{X}\, P(x) \equiv\ \forall{x}{\in}\mathbf{X}\, \lnot P(x)

A common error is stating "all persons are not married" (i.e. "there exists no person who is married") when "not all persons are married" (i.e. "there exists a person who is not married") is intended:

Failed to parse (Missing texvc executable; please see math/README to configure.): \lnot\ \exists{x}{\in}\mathbf{X}\, P(x) \equiv\ \forall{x}{\in}\mathbf{X}\, \lnot P(x) \not\equiv\ \lnot\ \forall{x}{\in}\mathbf{X}\, P(x) \equiv\ \exists{x}{\in}\mathbf{X}\, \lnot P(x)

Negation is also expressible through a statement of "for no", as opposed to "for some":

Failed to parse (Missing texvc executable; please see math/README to configure.): \nexists{x}{\in}\mathbf{X}\, P(x) \equiv \lnot\ \exists{x}{\in}\mathbf{X}\, P(x)

Unlike the universal quantifier, the existential quantifier distributes over logical disjunctions:

Failed to parse (Missing texvc executable; please see math/README to configure.): \exists{x}{\in}\mathbf{X}\, P(x) \or Q(x) \to\ (\exists{x}{\in}\mathbf{X}\, P(x) \or \exists{x}{\in}\mathbf{X}\, Q(x))

Rules of Inference[link]

A rule of inference is a rule justifying a logical step from hypothesis to conclusion. There are several rules of inference which utilize the existential quantifier.

Existential introduction (∃I) concludes that, if the propositional function is known to be true for a particular element of the domain of discourse, then it must be true that there exists an element for which the proposition function is true. Symbolically,

Failed to parse (Missing texvc executable; please see math/README to configure.): P(a) \to\ \exists{x}{\in}\mathbf{X}\, P(x)

The reasoning behind existential elimination (∃E) is as follows: If it is given that there exists an element for which the proposition function is true, and if a conclusion can be reached by giving that element an arbitrary name, that conclusion is necessarily true, as long as it does not contain the name. Symbolically, for an arbitrary c and for a proposition Q in which c does not appear:

Failed to parse (Missing texvc executable; please see math/README to configure.): \exists{x}{\in}\mathbf{X}\, P(x) \to\ ((P(c) \to\ Q) \to\ Q)

Failed to parse (Missing texvc executable; please see math/README to configure.): P(c) \to\ Q

must be true for all values of c over the same domain X; else, the logic does not follow: If c is not arbitrary, and is instead a specific element of the domain of discourse, then stating P(c)  might unjustifiably give more information about that object.

The empty set[link]

The formula Failed to parse (Missing texvc executable; please see math/README to configure.): \exists {x}{\in}\emptyset \, P(x)

is always false, regardless of P(x). This is because Failed to parse (Missing texvc executable; please see math/README to configure.): \emptyset
denotes the empty set, and no x of any description – let alone an x fulfilling a given predicate P(x) – exist in the empty set. See also vacuous truth.

See also[link]

• Uniqueness quantification
• Quantifiers
• First-order logic

Notes[link]

1. ^ The term "predication" in grammar means the predicate of a sentence which refers to subject and is an adverb or adjective, or equivalent, that describes an attribute of the subject. In logic, "predication" is a declaration (or assertion) that is claimed to be self-evident and can be assumed as the basis for argument.
2. ^ This symbol is also known as the existential operator. It is sometimes represented with V.

References[link]

• Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-56881-262-0. 

Robert Tepper is an American singer, best known for his hit song "No Easy Way Out" from the Rocky IV motion picture soundtrack.

Born in Bayonne, New Jersey, Tepper moved to New York where he became a staff writer. He paired up with fellow singer Benny Mardones, and together they wrote Mardones' hit ballad "Into the Night," which earned them a Grammy Awards nomination.

In 1985, Tepper signed with Scotti Brothers and moved to Los Angeles. Actor/director Sylvester Stallone was taken with Tepper's song "No Easy Way Out," which subsequently led to its inclusion in the movie Rocky IV. "No Easy Way Out" climbed into the Top 40, reaching #22 on Billboard's Hot 100 in 1986, and momentarily putting Tepper in the public eye. Several European labels, including Ministry of Sound, released "No Easy Way Out" with their own dance versions, and the hard rock group Bullet For My Valentine also recorded the song in 2008. In 1986, one year after "Rocky IV" another song from the album "no easy way out" called "angel of the city" became the main song of Stallone´s film "Cobra". With "Angel of the city" Tepper was not as successful as with "no easy way out"; the movie "Cobra" was not a hit. Tepper released two solo albums for Scotti Bros. Records but both albums received little promotion from the label. Also in 1986 he co-wrote the single "Le Bel Age" for Pat Benatar with peaked at #54 on the Billboard charts. He went on to join the classic hard rock group "Iron Butterfly" for a few years. His third solo album No Rest For The Wounded Heart was released exclusively in Europe in 1996 on the MTM Music label.

In 2009, Sony/BMG released his first two albums digitally.

In recent years he ventured into writing music for television and film, and collaborated on a musical. There are plans for his fourth solo album to be released in 2011.^{[dated info]} The music will be a departure from the big rock sound of the 1980s and has been described as "acoustic, singer/songwriter with an electric tinge."

In 2012 he released his fourth solo album New Life Story.

Discography[link]

Albums[link]

• No Easy Way Out (Scotti Bros., 1986)
• Modern Madness (Scotti Bros., 1988)
• No Rest For The Wounded Heart (MTM Music, 1996)
• New Life Story (2012)

External links[link]

• Robert Tepper home page
• Robert Tepper Official MySpace page
• Robert Tepper Fan Site
• Robert Tepper fan page on MySpace

References[link]

Robert Norman "Bob" Ross (October 29, 1942 – July 4, 1995) was an American painter, art instructor, and television host.^[1] He is best known as the creator and host of The Joy of Painting, a television program that ran on PBS in the United States.

Personal life[link]

Ross was born in Daytona Beach, Florida, and attended school until the ninth grade.^[2] Raised in Orlando, Florida,^[3] Ross enlisted in the U.S. Air Force at age 18^[2] and was living in Florida early in his military career when the Air Force transferred him to Eielson AFB (in Alaska), where he first saw the snow and mountains that later became recurring themes in his artwork; he developed his quick-painting technique in order to be able to create art for sale in brief daily work breaks.^[4] Having held military positions that required him to be, in his own words, "mean" and "tough," "the guy who makes you scrub the latrine, the guy who makes you make your bed, the guy who screams at you for being late to work," Ross decided that if he ever moved on from the military, "it wasn't going to be that way any more," "vowing never to scream again".^[4] Ross discovered after beginning his sideline in painting that he was soon able to earn more from selling his work than from his Air Force position. After leaving the Air Force, he studied with Bill Alexander before becoming famous worldwide with his own television program, The Joy of Painting.^[1]

Ross had a son, Steven, from his first marriage to Lynda Brown. Steven occasionally appeared on The Joy of Painting and is a Bob Ross–certified instructor.^[1] Ross and Lynda's marriage ended in divorce in 1981. Ross married again, this time to Jane. Jane died of cancer in 1993,^[1] and Ross himself suffered from lymphoma in his later years. In early 1994, Ross cancelled The Joy of Painting to continue battling the disease, with his final show airing on May 17, 1994. On July 4, 1995, Ross died at home and was survived by his ex-wife Lynda, his son Steve, a half-brother, and a full brother.^[5][6] He is buried at Woodlawn Memorial Park in Gotha, Florida.^[7]

Ross had a brother Jim, whom he mentioned in passing on his show.^[8]

Television show[link]

Ross was the host of the public television series The Joy of Painting. The show ran from January 11, 1983 to May 17, 1994 and still^[update] appears in reruns in many broadcast areas and countries, including the PBS oriented Create. During each half-hour segment, Ross would instruct viewers in the art of oil painting using a quick-study technique that kept colors to a minimum and broke paintings down into simple steps that virtually anyone could follow. Art critic Mira Schor compared him to another PBS television host, noting that the softness of Ross's voice and the slow pace of his speech was similar to Fred Rogers.^[9]

Ross later founded his own successful line of art supplies and how-to books, and also offered painting classes taught by instructors trained in the "Bob Ross method," building a $15 million business.^[2] In a 1990 interview, Ross mentioned that all his programs were donated free of charge to PBS stations and that his earnings came instead from sales of his 20 books and 100 videotapes (the total to that date), as well as profits from some 150 Bob Ross-ATE teachers and a line of art materials sold through a national supplier.^[4] Ross also mentioned on the show "Towering Glacier" (#2341) that he donated all the paintings made on the show to PBS stations around the country to "help them out."^[10]

Ross also filmed wildlife footage, squirrels in particular, usually from his own garden. Small animals often appeared on his show, even during some of his trickier works, as he would often take in injured or abandoned squirrels and other assorted wildlife and look after them.^[4]

Painting[link]

Ross utilized the wet-on-wet oil painting technique, in which the painter continues adding paint on top of still wet paint rather than waiting a lengthy amount of time to allow each layer of paint to dry. Combining this method with the use of two inch and other types of brushes as well as painting knives allowed Ross to paint trees, water, clouds and mountains in a matter of seconds. Each painting would start with simple strokes that appeared to be nothing more than colored smudges. As he added more and more strokes, the blotches transformed into intricate landscapes.^[11][12] Ross dedicated the first episode of the second season of "The Joy of Painting with Bob Ross" to William Alexander, explaining that "years ago, Bill taught me this fantastic [wet-on-wet] technique, and I feel as though he gave me a precious gift, and I'd like to share that gift with you [the viewer]".^[13]

Ross noted that the landscapes he painted — typically mountains, lakes, snow, and log cabin scenes — were strongly influenced by his years living in Alaska, where he was stationed for the majority of his Air Force career. He repeatedly stated on the show his belief that everyone had inherent artistic talent and could become an accomplished artist given time, practice, and encouragement, and to this end was often fond of saying, "We don't have mistakes here, we just have happy accidents."^[14] Ross was well known for other catchphrases he used while painting as he crafted "happy little trees".^[15] In most episodes of The Joy of Painting, he noted that one of his favorite parts of painting was cleaning the brush, specifically his method of drying off a brush, which he had dipped in odorless thinner, by striking it against the thinner can and easel. He would smile and often laugh aloud as he "beat the devil out of it." He also used a palette which had been lightly sanded down which was necessary to avoid catching the reflections of strong studio lighting. At the end of each episode, Ross was known for saying, "so from all of us here, I'd like to wish you happy painting, and God bless my friend."

When asked about his laid-back approach to painting and eternally calm and contented demeanor, he once commented: "I got a letter from somebody here a while back, and they said, 'Bob, everything in your world seems to be happy.' That's for sure. That's why I paint. It's because I can create the kind of world that I want, and I can make this world as happy as I want it. Shoot, if you want bad stuff, watch the news."^[16]

Rivalry with William Alexander[link]

Bob Ross was a student of William Alexander, who hosted the public television series The Magic World of Oil Painting from the early 1970s until 1982. On his show, Alexander highlighted his mastery of the alla prima or wet-on-wet style of oil painting in order to promote his paint supply business, Alexander Art and his painting classes. Ross later became a Bill Alexander instructor. His series, The Joy of Painting, was picked up by many of the PBS stations that carried The Magic World of Oil Painting.^[17]

At the beginning of The Joy of Painting's second season in 1984, Ross dedicated the show to Alexander and Alexander filmed a promo for his former student: "I hand off my mighty brush to a mighty man, and that is Bob Ross."^[13][17] In 1987 someone from Alexander Art told Ross that they could not keep up with the demand generated by the The Joy of Painting and suggested that Ross start his own line of art supplies.^[17] As Bob Ross Incorporated grew into a $15 million a year business Alexander told the New York Times that he felt "betrayed" by his one-time student: "He betrayed me. I invented 'wet on wet.' I trained him and he is copying me -- what bothers me is not just that he betrayed me, but that he thinks he can do it better." ^[18]

In popular culture[link]

This "In popular culture" section may contain minor or trivial references. Please reorganize this content to explain the subject's impact on popular culture rather than simply listing appearances, and remove trivial references. (May 2012)

In "Derbez en Cuando" and "De Nuez en Cuando", two mexican comedy shows by Eugenio Dervez, who parodied this painter as "Bob Atroz"

Bob Ross was parodied in the Comedy Central program Tosh.0 in an episode where Tosh dressed up as Ross while conducting an interview with an artist from the web.

In 2006 a character loosely based off of Ross was featured in the popular comic based cartoon The Boondocks. In the episode an art teacher, Ross, takes Riley under his wing to help him improve his Graffitti.

Family Guy included a parody of Ross in the beginning of an episode.

An easter egg on homestarrunner.com features Marzipan dressed up as and painting like Ross.

He was also parodied by Benny Hill in a short gag, where his character is surrounded by wildlife and some animals end up attacking him. On a similar gag, parodied Ross is painting, then when he turns back to the camera a lady comes into scene and distracted Ross paints her buttocks in error, then she slaps him in the face.

During the Powerpuff Girls episode "Roughing It Up", Professor Utonium pays homage to Ross while he and the girls are painting in the woods, even using Ross's "happy little trees" catchphrase.

In season 2, episode 3 of Dilbert, "S02E03 - Art", a character sharing many traits of Ross named "Rusty Shanks" is shown demonstrating bizarre takes on abstract art, including painting using ones own head as the brush. The character is quickly hushed off-stage by men in black suits apparently sent as enforcers to collect outstanding debts from Rusty.

See also[link]

• Wet-on-wet

References[link]

External links[link]

Wikiquote has a collection of quotations related to: Bob Ross

• Bob Ross Incorporated
• Bob Ross at the Internet Movie Database