Logic
The formal mathematical study of the methods, structure, and validity of mathematical deduction and proof.
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 
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
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.
Referenced on Wolfram|Alpha:
Logic
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