Is Galilei's reasoning on free fall valid? Galileo Galilei discovered by experiments that all bodies tend to fall with the same rapidity (I use it in an intuitive sense, you can replace it by 'acceleration' used in today's physics language), independently of their weight. He also provided the following rationale, based on the proof by contradiction (I do not have the original wording at hand, but I believe I can paraphrase the idea, which I am interested in.)
Let's imagine one heavy body and one light body. Suppose that heavy bodies fall faster than light bodies, as almost everybody believes. Connect the two bodies so that they form another body. This resulting body is heavier than the original heavy body, so according to the assumption it should fall faster. But on the other hand, the original heavy body is being inhibited in its fall by the connected light body, since this "wants" to fall with less rapidity. Due to this inhibition, the heavy body part should fall with less rapidity than it falls normally alone. So we arrive at contradiction, and the only way to resolve it is to reject the assumption. Instead, all bodies fall with the same rapidity (acceleration).
What do you think of this argument - is it valid, or no? For what reasons? It is very compelling at first, but on the other hand, should not the law of free fall be experimental law, rather than logical necessity? If so, where is the problem with the reasoning?
 A: What happens if we instead consider a pair of charged bodies with different charges and replace the constant gravitational field with a constant electric field in the vertical direction? Suppose that both charged bodies are attracted by the ground and that no gravitational field is present.
In Galileo's reasoning no description of the gravitational interaction is actually provided, so gravitational field could be replaced by the electric one.
Along Galileo's reasoning  should we conclude that the charged bodies will reach the ground simultaneously? It seems so.
It would be generally false, obviously, also because what happens also depends on the inertial masses of the bodies, that play a role but are not mentioned. So, in my opinion Galileo's reasoning is untenable.
A: Rapidity is usually taken to be synonymous with velocity. But using your definition, this is a very logical deduction. However it almost presupposes what is already known in order to arrive at it's conclusion. Because when you connect the two bodies, it is plausible that whatever mysterious quality that makes heavier bodies fall faster then flows into the lighter body, informing it of the total mass of the system. For things falling in air, this mysterious quality would be the ratio of the drag force to the gravitational force, and is transmitted through internal forces. For things in orbit, it is tidal forces. It is not self-evident that there should be no such mysterious quality in a vacuum, and is only determined by experiment.
A: Galileo's reasoning is basically :
Given the "heavier objects fall faster postulation" (presuming no other parameters), where $G(m_i)>G(m_j)$ whenever $m_i>m_j$: 
$$\mathbb{I}:x_1(t)=G(t,m_1);x_2(t)=G(t,m_2)$$ 
Setting the ground as zero, as well as upward is positive. 
The consequence of postulation $\mathbb{I}$ (Connect the two bodies):
$$x_{(1+2),\mathbb{I}}(t)=G(t,m_1+m_2)\tag{1}$$ 
Where the position of connected objects $x_{(1+2),\mathbb{I}}(t)<x_1(t)$.
 (heavier object falls faster).
Then Galileo postulates "connecting two - one light and one heavy - the connected object 'wants' to fall with less rapidity" ($x_1(t)$ represent the position of lighter disconnected object) :
$$\mathbb{II}:x_{(1+2),\mathbb{II}}(t)=\lambda[x_1(t)-x_2(t)]+x_2(t) $$
Where $\lambda$ is some scaling parameter, satisifying : $0<\lambda <1$, such that the connected object's $x_{{1+2},\mathbb{II}}(t)$ always somewhere between $x_1(t)$ & $x_2(t)$. Since, $x_2$(the disconnected heavier object) is the upper bound and the lower bound for $x_{(1+2)\mathbb{I}}$ and $x_{(1+2)\mathbb{II}}$respectively; Galileo claims therefore there is a contradiction.
Galileo's argument is kind of "valid" (but kind of a cheat to me), since the postulation $\mathbb{II}$ is actually a premise - You can't derive postulation $\mathbb{II}$ from postulation $\mathbb{I}^\dagger$, as well as there is no particular strong reason for either postulations at that time(both gives us contradicting consequences).
Now what's left is how much we can justify Galileo's postulation $\mathbb{II}$: 
If we further assume: A.) $G$ has no other parameters (such as charges) B.) $G$ is linear over mass $m$: $G(t,m)=m*G(t)$: then simply by adding $x_1$ and $x_2$ from the "heavier object fall faster postulation $\mathbb{I}$", we have : $$x_1+x_2=(m_1+m_2)G(t)\tag{2}$$
While based on $(1):$
$$x_{(1+2),\mathbb{I}}(t)=G(t,m_1+m_2)=(m_1+m_2)G(t)<x_1+x_2\tag{3}$$
plug $(2)$ into $(3)$, then we have:
$$(m_1+m_2)G(t)<(m_1+m_2)G(t), \forall t>t_{releasing}$$
Therefore, contradiction. Since postulation $\mathbb{II}$ plays no roles here, then postulation $\mathbb{I}$ is wrong (or assumption $A$, $B$).
Therefore, Galileo's reasoning is valid under a few presumptions. Where $G$ (and $\lambda$) exactly has to be determined by experiment. 

$\dagger :$ At least I can't think of one. If you happen to know one, please do tell me.
