Matrix Logic. Theory and Applications - download pdf or read online

By A. Stern

During this pioneering paintings, the writer develops a basic formula of good judgment by way of concept of matrices and vector areas. the invention of matrix common sense represents a landmark within the extra formalization of good judgment. For the 1st time the ability of direct mathematical computation is utilized to the complete set of good judgment operations, permitting the derivation of either the classical and modal logics from a similar formal base. the hot formalism permits the writer to amplify the alphabet of the truth-values with unfavourable common sense antivalues and to hyperlink matrix common sense descriptions with the Dirac formula of quantum concept - a outcome having basic implications and repercussions for technology as a complete. As a unified language which allows a logical exam of the underlying phenomena of quantum box idea and vice versa, matrix good judgment opens new avenues for the learn of basic interactions and provides upward thrust to a innovative end that physics as such may be seen and studied as a good judgment within the basic experience. eventually, modelling itself on particular sciences, matrix good judgment doesn't refute the classical good judgment yet as a substitute comprises it as a distinct deterministic restrict

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Example text

The contradiction law x ΛΧ = 0 is dual to the law of the excluded middle: xv x= 1 . The duality principle, together with the algebraic transformations of Boolean algebra, enable us to construct new equalities without referring to the truth-tables. And, given a basis set of equalities, any equality of the system can be deduced from it by a series of equivalent transformations. The equivalent transformations of Boolean algebra apparently change the formula but not the logic function realized by this formula.

All such ordered pairs constitute the range of the function. When the values of the pair are specified, a logic function yields one of two Boolean values. With each logic function we associate a converse function, in which the order of arguments is reversed. A logic function for which L(x,y) = L(y,x) is called symmetric; otherwise it is asymmetric. Under these criteria the set of 16 logic functions is divided into an octet of symmetric and an octet of asymmetric functions: Conventional Logic 47 {<->, Θ, A, v, Î , I, 1, 0} - symmetric, {—», <-, \ /, ±, T, -I, h } - asymmetric.

Birkhoff and J. van Neumann in 1936, quantum logic denies the universal validity of the commutative and distributive laws of classical logic.

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