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.

Mathematical logic

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order logic, and definability. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see Logic in computer science for those.

Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics) rather than trying to find theories in which all of mathematics can be developed.

An Introduction to .....

An Introduction to ..... is a 1994 compilation album by Roy Harper.

History

The album features 13 Harper songs from a 25 year period and is "...a collection of various styles and periods...purely intended for people who may not know where to start (with Harpers music)." Roy Harper

Track listing

All tracks credited to Roy Harper

Personnel

Crash Course

Crash Course (also known as Driving Academy) is a 1988 made for television teen film directed by Oz Scott.

Plot

Crash Course centers on a group of high schoolers in a driver’s education class; many for the second or third time. The recently divorced teacher, super-passive Larry Pearl, is on thin ice with the football fanatic principal, Principal Paulson, who is being pressured by the district superintendent to raise driver’s education completion rates or lose his coveted football program. With this in mind, Principal Paulson and his assistant, with a secret desire for his job, Abner Frasier, hire an outside driver’s education instructor with a very tough reputation, Edna Savage, aka E.W. Savage, who quickly takes control of the class.

The plot focuses mostly on the students and their interactions with their teachers and each other. In the beginning, Rico is the loner with just a few friends, Chadley is the bookish nerd with few friends who longs to be cool and also longs to be a part of Vanessa’s life who is the young, friendly and attractive girl who had to fake her mother’s signature on her driver’s education permission slip. Kichi is the hip-hop Asian kid who often raps what he has to say and constantly flirts with Maria, the rich foreign girl who thinks that the right-of-way on the roadways always goes to (insert awesomely fake foreign Latino accent) “my father’s limo”. Finally you have stereotypical football meathead J.J., who needs to pass his English exam to keep his eligibility and constantly asks out and gets rejected by Alice, the tomboy whose father owns “Santini & Son” Concrete Company. Alice is portrayed as being the “son” her father wanted.