Forgive me. "Overtly premised on things we know to be false." Though obviously any argument immediately predicts its own premises. ;)
From https://en.wikipedia.org/wiki/Tychonic_system :
> Tycho admired aspects of Copernicus's heliocentric model of the solar system, but felt that it had problems as concerned physics, astronomical observations of stars, and religion. Regarding the Copernican system Tycho wrote,
>> This innovation expertly and completely circumvents all that is superfluous or discordant in the system of Ptolemy. On no point does it offend the principle of mathematics. Yet it ascribes to the Earth, that hulking, lazy body, unfit for motion, a motion as quick as that of the aethereal torches, and a triple motion at that.
> In regard to physics, Tycho held that the Earth was just too sluggish and heavy to be continuously in motion. According to the accepted Aristotelian physics of the time, the heavens (whose motions and cycles were continuous and unending) were made of "Aether" or "Quintessence"; this substance, not found on Earth, was light, strong, and unchanging, and its natural state was circular motion. By contrast, the Earth (where objects seem to have motion only when moved) and things on it were composed of substances that were heavy and whose natural state was rest
So, the Copernican system was acknowledged to be mathematically superior, and the Ptolemaic system was acknowledged to be a particularly obvious case of special pleading. The Earth was considered exceptional under the Tychonian system because it was too heavy to be constantly in motion -- as a large terrestrial object, its natural state was rest.
Interestingly enough, I remember learning about a Renaissance scientist who did several experiments to the effect that there is no such thing as a "natural state of rest" for terrestrial objects...
Furthermore, the idea that the Earth is exceptional is severely undermined by the moons of Jupiter. From them, you would see exactly what you see from Earth, and you could readily fit a Tychonian model with a sun going around Jupiter -- but why would you? (This is what I was referring to originally, but this particular argument is slightly weaker than I was thinking of.)
I'll also note that, as far as Wikipedia is concerned, the Tychonic system wasn't exactly well-established at the time:
> It was chiefly through the influence of the Jesuit scientists that the Roman Catholic Church adopted the Tychonic system, over a period of nine years (from 1611 to 1620), in a process directly prompted by the Galilean telescopic discoveries.