Why conductivity increases in semiconductor when doped?
Conductivity can be dominated by holes (in P-type material), or by electrons (in N-type material), but the generation and recombination of hole/electron pairs must be equal (or it's not in equilibrium, and will
soon have different carrier concentrations).

The generation rate (set by temperature) isn't a variable, but the
recombination depends on the concentration of the (relatively rare)
electrons and holes.   It is proportional to both.   So, we have
an equation
$$N_e \times N_{hole}  = constant $$
One cannot manipulate the electron concentration without affecting the hole concentration.

Conductivity is proportional to a weighted sum of the hole and electron concentrations.
Holes, in silicon, are less effective, but we'll ignore that.

Now, visualize a rectangle, where the vertical sides are the hole concentration, and the horizontal sides are the electron concentration. 
The product (the rectangle area) is fixed because the product of concentrations is a constant.

The perimeter length of that rectangle is the conductivity; either extreme P doping (hole concentration) or N doping (electron concentration) yields a large-perimeter rectangle.   Similar electron and hole concentrations make the rectangle a square, with minimum perimeter for the given area. 

The formulae insist that high conductivity results from either extreme of doping.