Logic

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Logic. Logic (from the Ancient Greek: λογική, logike)[1] has two meanings: first, it describes the use of valid reasoning in some activity; second, it names the normative study of reasoning or a branch thereof.[2][3] In the latter sense, it features most prominently in the subjects of philosophy, mathematics, and computer science.

Logic

Logic was studied in several ancient civilizations, including India,[4] China,[5] Persia and Greece. 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. Laws of Form weblog. Multiple Form Logic: An extension of George Spencer-Brown's LAWS of FORM. ATTENTION: Best viewed with Mozilla ™ or Netscape ™, 1280 x 1024 resolution.

Multiple Form Logic: An extension of George Spencer-Brown's LAWS of FORM

An enhanced generalisation of George Spencer Brown 's " Laws of Form ", simplifying & elucidating Propositional Logic and the Philosophy of Logic. This site also includes some Automatic Theorem Proving Software (you can download) written in Visual Prolog ™ and LPA Win-Prolog ™ Last update: 5 October 2007 (version 1.8) CONTENTS: (1) "Laws of Form" and the unknown history of Multiple Form Logic ™ George Spencer-Brown's " Laws of Form " is a revolutionary book about Logic, which influenced many researchers and artists in the world, for about three decades.

Gödel's incompleteness theorems. Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic.

Gödel's incompleteness theorems

The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem.

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). Hilbert's second problem. In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems.

Hilbert's second problem

It asks for a proof that arithmetic is consistent – free of any internal contradictions. In the 1930s, Kurt Gödel and Gerhard Gentzen proved results that cast new light on the problem. Some feel that these results resolved the problem, while others feel that the problem is still open. Hilbert's problem and its interpretation[edit]