By N. S. Hellerstein
This publication is ready "diamond", a good judgment of paradox. In diamond, a press release could be actual but fake; an "imaginary" kingdom, halfway among being and non-being. Diamond's imaginary values clear up many logical paradoxes unsolvable in two-valued boolean common sense. Diamond is a brand new strategy to resolve the dilemmas of upper arithmetic. during this quantity, paradoxes through Russell, Cantor, Berry and Zeno are all resolved. This publication comprises sections: effortless; which covers the vintage paradoxes of mathematical good judgment and indicates how they are often resolved during this new method; and complex, which relates diamond to Boolean good judgment, three-valued common sense, Gödelian meta-mathematics and predicament video games.
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Additional info for Diamond: A Paradox Logic
The Heap arises at the onset of uncertainty . In practice, the Heap contains a boring number of sand grains ; and the smallest Heap contains the smallest boring number of sand grains! 18 Diamond, A Paradox Logic Finitude. Finite is the opposite of infinite; but in paradox-land, that's no excuse! In fact the concept of finiteness is highly paradoxical; for though finite numbers are finite individually and in finite groups, yet they form an infinity. Let us attempt to evaluate finiteness. Let F ='finitude', or'finity; the generic finite expression .
These normal forms are just like their counterparts in boolean logic, except that they allow differential terms. 46 Diamond, A Paradox Logic Theorem: The Primary Normal Forms F(x) = (A and x) or (B and not(x)) or (C and dx) or D F(x) = (a or not(x)) and (b or x) and (c or Dx) and d where A,B,C,D,a,b,c,d are all free of variable x, and: AorD = F(t) = aandd B or D = F(f) = b and d A or B or C or D = F(i) or F(j) = d D = F(i) and F(j) = a and b and c and d Proof: We get the first two equations from the Disjunctive and Conjunctive Normal Forms by collecting like terms with respect to the variable x.
Students of feasibility will recognize this as a variant of the Boolean Consistency Problem, and therefore NP-complete. Diamond logic's completeness suggest this: Conjecture. Diamond is a "categorical" DeMorgan algebra: Any De Morgan algebra is a subalgebra of images of products of diamond . These De Morgan algebras need not have the Interference axiom; they are subalgebras of ones that do. Thus diamond is to De Morgan algebras as two-valued logic is to Boolean algebra. I consider diamond to be a 2-dimensional extension of twovalued logic that solves paradox, just as the complex numbers are a 2dimensional extension of the real line that solves x2=- 1 .