On Logistic, 2 of 8; Reduction

         2. Reduction

          In logistic we can define “reduction”, a.k.a. “reciprocal addition”:

          x  {+} y    =   1 /  ( 1/x  + 1/y )

          This corresponds to Or; so let us call reduction ‘or’, just as we call addition ‘and’. I also call reduction ‘with’; “x with y”. It is useful in Kirchhoff’s law of parallel resistances; and in computing combined work-times when work-rates add.

          {+} is also denoted 1/+, and called “reciprocal addition”, “harmonic addition” and “parallel addition”.          

          These identities apply:

          The De Morgan Identities:       

              1         =      1  +   1  

           a {+} b             a      b

              1         =      1  {+}  1 

           a + b               a       b

          These follow directly from the definitions by inversion or by substituting reciprocals. I name them after the DeMorgan logic identities which distribute NOT over AND and OR, with alternation.

          They imply the Reciprocal Equations Theorem:

                    a {+} b  =  c           if and only if             1/a + 1/b  =  1/c

                    a + b  =  c    if and only if             1/a {+} 1/b  =  1/c

          Product Over Sum:

         a {+} b  =     ab  _       

                      b + a

          This follows directly from the definitions. It implies:

          Complementarity:

         (a+b)(a{+}b)  =       ab

          a + b    =         ab    _     

                           b {+} a

          Distribution:

                    c(a{+}b)  =  ca {+} cb

          Proof:

          c(a{+}b) =    cab     =     ccab      =     cacb    =  ca {+} cb

                   b + a        c(b + a)         cb + ca     

          Complementarity and Distribution imply Triads:

                    a(a+b)  {+}  b(a+b)   =   ab,             

         so if   a+b = c   then   ac{+}bc = ab

          Therefore, these sums:

                    1+3 = 4;   2+3 = 5;   7+3 = 10    

          yield these reductions:

                    4 {+} 12  =   3   ;    10 {+} 15   =   6  ;    70{+}30   =   21

          which yield these sums of reciprocals:

                    1/4 + 1/12  =  1/3 ;   1/10 + 1/15 = 1/6  ;   1/70 + 1/30  =  1/21 

         

          Field Laws:

         a {+} 0  =  0

         a {+} ∞  =  a

         a {+} –a  =   ∞

         a {+} b  =  b {+} a

         a {+} (b {+} c)  =  (a {+} b) {+} c

         a (b {+} c)  =  ab {+} ac

         (a {+} b)(c {+} d)  =  ac {+} ad {+} bc {+} bd

          a  {+}  c   =    ad {+} bc _ 

          b       d           bd

          Note that {+} resembles + , except that its identity element is  infinity, not zero. {+} and + are mirror images of each other, but not identical. For instance, x{+}x = x/2 ,  whereas  x+x = 2x.

          Any rational number can be constructed from 1, + and {+}; for instance:

          2/3 = (1+1){+}(1+1){+}(1+1)

          Therefore any rational proportion can be defined from + and {+}; for instance:

          (2/3)x  = (x+x){+}(x+x){+}(x+x)   = (x{+}x{+}x)+ (x{+}x{+}x)

          The infinite harmonic series

                    1 + 1/2 + 1/3 + …  =   1/0

          translates to these infinite reductions:

                    1 {+} 2 {+} 3 {+} … =  0

                    1 {+} (1+1) {+} (1+1+1) {+} … =  0

          Therefore in logistic:

                    Liar or (Liar and Liar) or (Liar and Liar and Liar) or…   

                    equals True.

          Consider these familiar-looking laws:

          Common Denominators:

           a  +  b   =    a + b_    

           c     c          c

           a  {+}  b   =    a {+} b_    

           c       c           c

          General Denominators (or, Rabbit Rule):

           a  +  c   =   ad + bc                 \  /

           b     d          bd                    \/  

                                                  /\ 

           a  {+}  c   =   ad {+} bc             /  \

           b       d          bd                  

          They have these less-familiar looking counterparts:

          Common Numerators:

           c  +  c   =       c    _  

           a     b        a {+} b

           c  {+}  c   =      c   _        

           a       b        a + b

          General Numerators (or, Stool Rule):

           a  +  c   =      ac                       

           b     d       ad {+} bc               \  /

                                                  \/

           a  {+}  c   =     ac                   /\

           b       d       ad + bc               /  \

          The last four rules are the same as the rules for addition, with the roles of numerator and denominator reversed, and addition alternating with reduction. Fractions stand on their heads when they pass through the reciprocal looking-glass.