By A. Stern
During this pioneering paintings, the writer develops a basic formula of common sense by way of thought of matrices and vector areas. the invention of matrix good judgment represents a landmark within the additional formalization of common sense. For the 1st time the facility 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 magnify the alphabet of the truth-values with unfavorable good judgment antivalues and to hyperlink matrix common sense descriptions with the Dirac formula of quantum conception - a consequence having primary implications and repercussions for technological know-how as a complete. As a unified language which allows a logical exam of the underlying phenomena of quantum box thought and vice versa, matrix common sense opens new avenues for the research of basic interactions and offers upward thrust to a innovative end that physics as such will be considered and studied as a common sense within the basic experience. ultimately, modelling itself on targeted sciences, matrix common sense doesn't refute the classical good judgment yet as a substitute accommodates it as a different deterministic restrict
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Extra info for Matrix Logic. Theory and Applications
Example text
Now we shall present the complete set of elementary Boolean functions and underline some of their properties relevant to our discussion. Boolean functions of one argument There are four Boolean functions of one argument: identity I(x), negation l(x), and the logic constants l(x) and 0(x). The chart below shows that the structure of these functions is identical with the structure of the truth-tables for the unary connectives we considered above, except that the truth-values t and f are now replaced by the numerical values 1 and 0 respectively: Γ\ΜΧ) x ^Χ.
In a more general case, when the standard basis set is used, brackets can be eliminated if we introduce the following ordering of connectives: <-»,->, v, Λ , Ι . This convention avoids brackets, since it is possible to restore these in the inverse order 1, A , V , ->,<->. In the expression l x v y Λ z —»y <-»x brackets will be restored as follows: ((((Tx)v(yAz))->y)<->x). However, not every Boolean expression can be written without brackets. For example, further elimination is not possible in the expressions x -> (y -> z), (x —> y) Λ z.
The fundamental theorem of Boolean algebra states that any logic formula can be expressed in the disjunctive normal form or conjunctive normal form, defined as follows: An expression is said to be in disjunctive normal form if it represents a disjunction of one or more disjuncts, each of which is a continued conjunction of atomic variables or their negations: m n v (Λ x1J ) i=l j=l Any expression which is not a contradiction is logically equivalent to some disjunctive normal form. ) j=l »' Any expression which is not a tautology is logically equivalent to some conjunctive normal form.
