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Logic


Logic is the formal mathematical study of the methods, structure, and validity of mathematical deduction and proof.

According to Wolfram (2002, p. 860), logic is the most widely discussed formal system since antiquity.

In Hilbert's day, formal logic sought to devise a complete, consistent formulation of mathematics such that propositions could be formally stated and proved using a small number of symbols with well-defined meanings. The difficulty of formal logic was demonstrated in the monumental Principia Mathematica (1925) of Whitehead and Russell's, in which hundreds of pages of symbols were required before the statement 1+1=2 could be deduced.

The foundations of this program were obliterated in the mid 1930s when Gödel unexpectedly proved a result now known as Gödel's second incompleteness theorem. This theorem not only showed Hilbert's goal to be impossible, but also proved to be only the first in a series of deep and counterintuitive statements about rigor and provability in mathematics.

A very simple form of logic is the study of "truth tables" and digital logic circuits in which one or more outputs depend on a combination of circuit elements (AND, OR, NAND, NOR, NOT, XOR, etc.; "gates") and the input values. In such a circuit, values at each point can take on values of only true (1) or false (0). de Morgan's duality law is a useful principle for the analysis and simplification of such circuits.

A generalization of this simple type of logic in which possible values are true, false, and "undecided" is called three-valued logic. A further generalization called fuzzy logic treats "truth" as a continuous quantity ranging from 0 to 1.


See also

Absorption Law, Alethic, Boolean Algebra, Boolean Connective, Bound, Caliban Puzzle, Contradiction Law, de Morgan's Duality Law, de Morgan's Laws, Deducible, Free, Fuzzy Logic, Gödel's First Incompleteness Theorem, Gödel's Second Incompleteness Theorem, Khovanski's Theorem, Law of the Excluded Middle, Logos, Löwenheim-Skolem Theorem, Metamathematics, Model Theory, Paradox, Quantifier, Sentence, Tarski's Theorem, Tautology, Three-Valued Logic, Topos, Truth Table, Turing Machine, Universal Turing Machine, Venn Diagram, Wilkie's Theorem Explore this topic in the MathWorld classroom

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References

Adamowicz, Z. and Zbierski, P. Logic of Mathematics: A Modern Course of Classical Logic. New York: Wiley, 1997.Bogomolny, A. "Falsity Implies Anything." http://www.cut-the-knot.org/do_you_know/falsity.shtml.Carnap, R. Introduction to Symbolic Logic and Its Applications. New York: Dover, 1958.Church, A. Introduction to Mathematical Logic, Vol. 1. Princeton, NJ: Princeton University Press, 1996.Enderton, H. B. A Mathematical Introduction to Logic. New York: Academic Press, 1972.Enderton, H. B. Elements of Set Theory. New York: Academic Press, 1977.Heijenoort, J. van. From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879-1931. Cambridge, MA: Cambridge University Press, 1967.Gödel, K. On Formally Undecidable Propositions of Principia Mathematica and Related Systems. New York: Dover, 1992.Jeffrey, R. C. Formal Logic: Its Scope and Limits. New York: McGraw-Hill, 1967.Kac, M. and Ulam, S. M. Mathematics and Logic: Retrospect and Prospects. New York: Dover, 1992.Kleene, S. C. Introduction to Metamathematics. Princeton, NJ: Van Nostrand, 1971.Smullyan, R. M. First-Order Logic. New York: Dover, 1995.Weisstein, E. W. "Books about Logic." http://www.ericweisstein.com/encyclopedias/books/Logic.html.Whitehead, A. N. and Russell, B. Principia Mathematica, 2nd ed. Cambridge, England: Cambridge University Press, 1962.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. 860-861, 2002.

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Logic

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Weisstein, Eric W. "Logic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Logic.html

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