As a point on interest, when discussing dB, you have to state whether you're dealing with voltage or power.
The formula for calculating power dB is .. 10 (log(P1/P2)
The formula for calculating voltage dB is .. 20 (log(V1/V2)
So, doubling power results in a 3 dB gain, and doubling voltage results in a 6dB gain.
When you're dealing with transmitters, you're dealing with power to the antenna. Receivers deal with voltage from the antenna.
So, a power increase of 3dB from the transmitting antenna will result in a 6dB voltage increase at the receiving antenna. (3dB is a doubling of power, 6dB is a doubling of voltage).
As TITANO points out, S meter measurement gets kind of dodgy. Different manufacturers did use their own system. Plus, S meters are notoriously imprecise.
But, the popular 'industry standard' is to use 50 microVolts on the antenna = 0dB. This is measured as S9.. Cut that in half (25 mV) and you have a 6 dB decrease which should be S8. Cut that in half again down to 12.5 microVolts and you have S7 and so on. These are theoretical values.
As I indicated in my previous post, doubling the power out (3dB) will provide one 'theoretical' S unit at the receiver (6dB).