newtonian mechanics - Do we all actually just weigh half of what our bathroom scales tell us?

Here's the reasoning: Newton's third law tells us that every force has an equal but opposite reaction force. So besides my weight pushing down on the scale from above, there's also an equal force pushing up on the scale from below, so the spring or pressure-sensor-thingy inside the scale is being squeezed from two directions by double the force! That's tipping the scales, and therefore everybody is actually just half as heavy as it says on the readout.