Download Diamond: A Paradox Logic by N. S. Hellerstein PDF

By N. S. Hellerstein

This publication is set "diamond", a good judgment of paradox. In diamond, an announcement may be real but fake; an "imaginary" nation, halfway among being and non-being. Diamond's imaginary values remedy many logical paradoxes unsolvable in two-valued boolean good judgment. Diamond is a brand new technique to clear up the dilemmas of upper arithmetic. during this quantity, paradoxes by way of Russell, Cantor, Berry and Zeno are all resolved. This ebook includes sections: uncomplicated; which covers the vintage paradoxes of mathematical good judgment and indicates how they are often resolved during this new process; and complex, which relates diamond to Boolean common sense, three-valued good judgment, Gödelian meta-mathematics and drawback video games.

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Call this "t/f', or "i". ft ft ft .... ; call this "Ft", or 'J". f f f f f f .... ; call this "f/f", or "f'. "/" is pronouced "but"; thus i is "true but false" and j is "false but true". These four values form a diamond-shaped lattice: true = t/t i = t/f j = f/t false = f/f This is "diamond logic"; a wave logic with two components and four truth values. It describes the logic waves of period 2. The values i and j can be interpreted as "underdetermined" and "overdetermined " states; where "underdetermined " means "insufficient data for definite answer", and "overdetermined " means "contradictory data".

And tm„(x„) ) where each %(x) is one of these functions: { x, not x, dx, t, f, i, j } Conjunctive Normal Form: F(x) _ (t11(x1) or t12(x2) or ... or t 1n(x1J ) and (t21(x1) or t22(x) or ... or t2n(xn) ) and ... and (tm,(xl) or tm2(x2) or ... ) ) where each tu(x) is one of these functions: ( x, notx, Dx, t,f,i,j ) We do this by distributing negations downwards, canceling doublenegations, and distributing enough times. These normal forms are just like their counterparts in boolean logic, except that they allow differential terms.

At any point on the boundary, is the statement true, or is it false? ) This page is intentionally left blank Chapter 2 Diamond The buzzer; wave logic Diamond values; tiff; under/over-determined Harmonic functions; not, and, or, but, others Diamond circuits; phased-delay, dual-rail Brownian forms and laws Boundary logic A. The Buzzer Chapter I posed the problem of paradox but left it undecided . Is the Liar true or false? Boolean logic cannot answer . What bold expedient would decide the question?

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