Logic (from the Greek λογική , logikē ) [ 1 ] refers to both the study of modes of reasoning (which are valid , and which are fallacious ) [ 2 ] [ 3 ] and the use of valid reasoning. In the latter sense, logic is used in most intellectual activities, including philosophy and science, but in the first sense, is primarily studied in the disciplines of philosophy , mathematics , semantics , and computer science . It examines general forms that arguments may take. In mathematics, it is the study of valid inferences within some formal language . [ 4 ] Logic is also studied in argumentation theory . [ 5 ]
Laws of Form weblog
ATTENTION: Best viewed with Mozilla ™ or Netscape ™, 1280 x 1024 resolution. 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. Multiple Form Logic: An extension of George Spencer-Brown's LAWS of FORM
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 . 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 ).
In mathematics , Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems . 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. [ edit ] Hilbert's problem and its interpretation Hilbert's second problem