Lemma (logic)

In informal logic and argument mapping, a lemma is simultaneously a contention for premises below it and a premise for a contention above it. Transitivity: If one has proof that B follows from A and proof of A, then one has proof of B.

See also

Lemma

Lemma may refer to:

See also

Lemma (botany)

Lemma is a phytomorphological term used in botany referring to a part of the spikelet of grasses (Poaceae). It is the lowermost of two chaff-like bracts enclosing the grass floret. It often bears a long bristle called an awn, and may be similar in form to the glumes - chaffy bracts at the base of each spikelet. It is usually interpreted as a bract but it has also been interpreted as one remnant (the abaxial) of the three members of outer perianth whorl (the palea may represent the other two members, having been joined together).

A lemma's shape, their number of veins, whether they are awned or not, and the presence or absence of hairs are particularly important characters in grass taxonomy.

Lemma (mathematics)

In mathematics, a "helping theorem" or lemma (plural lemmata or lemmas) from the Ancient Greek λῆμμα (lemma, "anything which is received, such as a gift, profit, or a bribe”) is a proven proposition which is used as a stepping stone to a larger result rather than as a statement of interest by itself.

Comparison with theorem

There is no formal distinction between a lemma and a theorem, only one of intention – see Theorem terminology. However, a lemma can be considered a minor result whose sole purpose is to help prove a theorem – a step in the direction of proof, so to speak.

Well-known lemmas

A good stepping stone can lead to many others. Some powerful results in mathematics are known as lemmata, such as Bézout's lemma, Dehn's lemma, Euclid's lemma, Farkas' lemma, Fatou's lemma, Gauss's lemma, Greendlinger's lemma, Itō's lemma, Jordan's lemma, Nakayama's lemma, Poincaré's lemma, Riesz's lemma, Schur's lemma, Schwarz's lemma, Urysohn's lemma, Yoneda's lemma and Zorn's lemma. While these results originally seemed too simple or too technical to warrant independent interest, they have turned out to be central to the theories in which they occur.

Logic

Logic (from the Ancient Greek: λογική, logike) is the branch of philosophy concerned with the use and study of valid reasoning. The study of logic also features prominently in mathematics and computer science.

Logic was studied in several ancient civilizations, including Greece, India, and China. In the West, logic was established as a formal discipline by Aristotle, who gave it a fundamental place in philosophy. The study of logic was part of the classical trivium, which also included grammar and rhetoric. Logic was further extended by Al-Farabi who categorized it into two separate groups (idea and proof). Later, Avicenna revived the study of logic and developed relationship between temporalis and the implication. In the East, logic was developed by Hindus, Buddhists and Jains.

Logic is often divided into three parts: inductive reasoning, abductive reasoning, and deductive reasoning.

The study of logic

The concept of logical form is central to logic. The validity of an argument is determined by its logical form, not by its content. Traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic.

Ω-logic

In set theory, Ω-logic is an infinitary logic and deductive system proposed by W. Hugh Woodin (1999) as part of an attempt to generalize the theory of determinacy of pointclasses to cover the structure . Just as the axiom of projective determinacy yields a canonical theory of , he sought to find axioms that would give a canonical theory for the larger structure. The theory he developed involves a controversial argument that the continuum hypothesis is false.

Woodin's Ω-conjecture asserts that if there is a proper class of Woodin cardinals (for technical reasons, most results in the theory are most easily stated under this assumption), then Ω-logic satisfies an analogue of the completeness theorem. From this conjecture, it can be shown that, if there is any single axiom which is comprehensive over (in Ω-logic), it must imply that the continuum is not . Woodin also isolated a specific axiom, a variation of Martin's maximum, which states that any Ω-consistent (over ) sentence is true; this axiom implies that the continuum is .

Logic (rapper)

HIDDEN ERROR: Usage of "spouse?? ??Jess Andrea??" is not recognizedHIDDEN ERROR: Usage of "past names and nick names" is not recognized

Sir Robert Bryson Hall II (born January 22, 1990), known by his stage name Logic, is an American rapper and singer. He was born and raised in Gaithersburg, Maryland. He has amassed a large Internet following and is a member of the RattPack, his group of friends that he works with. Logic has released four official mixtapes and two studio albums, the most recent being The Incredible True Story.

He is currently signed with Visionary Music Group and Def Jam and released his debut album Under Pressure on October 21, 2014. The album debuted at number four on the Billboard 200 and sold 73,000 copies in its first week. Logic released his sophomore album The Incredible True Story on November 13, 2015.

Early life

Robert Bryson Hall II grew up in the West Deer Park public housing projects, a low-income crime-ridden neighborhood. Throughout his childhood, both his African-American father and White mother suffered from cocaine addiction and alcoholism. His father was not a major part of his childhood, but they are in contact with each other now. Growing up, his brothers sold crack cocaine and even sold the drug to his father. He attended Gaithersburg High School but did not graduate. He began skipping classes in the 10th grade. "I started doing badly and failed every class but English, so they kicked me out of school, they gave up on me." Logic was expelled and used his free time to dive into pursuing his music career.