Logic
The formal mathematical study of the methods, structure, and validity of mathematical deduction and proof. Formal logic seeks todevise a complete, consistent formulation of mathematics such that propositions can be formally stated and proved using a smallnumber of symbols with well-defined meanings. While this sounds like an admirable pursuit in principle, in practice the study ofmathematical logic can rapidly become bogged down in pages of dense and unilluminating mathematical symbols, of which Whiteheadand Russell's Principia Mathematica (1925) is perhaps the best (or worst) example.
A very simple form of logic is the study of ``Truth Tables'' and digital logic circuits in which one or more outputs depend on acombination of circuit elements (AND, NAND, OR, XOR, etc.; ``gates'') and the input values. In such a circuit, values at eachpoint can take on values of only True (1) or False (0). de Morgan's Duality Law is a useful principle forthe analysis and simplification of such circuits.
A generalization of this simple type of logic in which possible values are True, False, and ``undecided'' is calledThree-Valued Logic. A further generalization called Fuzzy Logic treats ``truth'' as a continuous quantity rangingfrom 0 to 1.
See also Absorption Law, Alethic, Boolean Algebra, Boolean Connective, Bound, CalibanPuzzle, Contradiction Law, de Morgan's Duality Law, de Morgan's Laws, Deducible, ExcludedMiddle Law, Free, Fuzzy Logic, Gödel's Incompleteness Theorem,Khovanski's Theorem, Logical Paradox, Logos, Löwenheim-Skolem Theorem, Metamathematics, Model Theory, Quantifier, Sentence, Tarski's Theorem,Tautology, Three-Valued Logic, Topos, Truth Table, Turing Machine, Universal Statement,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.com/do_you_know/falsity.html. 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. 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. Whitehead, A. N. and Russell, B. Principia Mathematica, 2nd ed. Cambridge, England: Cambridge University Press, 1962.
Logic
- Polyiamond
- Polyking
- Polylogarithm
- Polymorph
- Polynomial
- Polynomial Bar Norm
- Polynomial Bracket Norm
- Polynomial Curve
- Polynomial Factor
- Polynomial Norm
- Polynomial Remainder Theorem
- Polynomial Ring
- Polynomial Root
- Polynomial Series
- Polyomino
- Polyomino Tiling
- Polyplet
- Polytope
- Poncelet's Closure Theorem
- Poncelet's Continuity Principle
- Poncelet-Steiner Theorem
- Poncelet's Theorem
- Poncelet Transform
- Poncelet Transverse
- Pong Hau K'i