Landau 1961 begins with a brief presentation of special relativity. This question is about the validity of a certain argument that they use in building up the foundations of the subject from scratch, so I think I should first sketch what they consider foundational.
They define an inertial frame as one in which Newton's first law holds. They telescope Einstein's two 1905 postulates into a single principle of relativity, which is that the laws of physics are form-invariant w.r.t. changes of inertial frame. They assert (based on unspecified experiments) that forces are propagated at a finite speed. They also seem to assume (presumably also based on experiment) that this speed is a maximum and universal for all interactions.
Given a maximum velocity, they argue that time is not absolute. The argument is that in Galilean relativity, velocities combine linearly, which is incompatible with a maximum velocity and with the Michelson-Morley experiment. Also, Galilean relativity posits absolute time. Therefore absolute time must be wrong. This seems like a clear logical fallacy to me. (A implies B, and A also implies C. B is false, so C must be false.)
[EDIT] Oops, crucial mistake in my original statement of the question. The parenthetical originally ended with "C must be true." Should have ended with "C must be false."
Is there any way of salvaging this argument? Am I misunderstanding it?
Landau and Lifshitz, The classical theory of fields, 2nd ed., 1961