Don't get me wrong; you seem to have your logic stuff together. But in the proposition you posited, isn't it still possible for ALL the claims to be false? I think so. Where, if at all, did I go wrong on this?
No, it is not possible that there isn’t “truth”.
But notice that I used the term “finite”.
Any “finite collection of things that are said to be “truth”” is by definition lacking adequate diversity to account for all possibilities. A finite list may all be false, but we can no more say that based on logic alone than we can say this one must be true.
To see truth, as a matter of belief, at least requires one to bet the farm, as it were, to look at what is claimed and the evidences given and then go all in. To stay on the sideline, as politically correct universalists must, is to risk being cut off from ever knowing any truth, even a bad approximation of it (as indeed many false religions have morals and wisdom simply because they were usually invented when and where men still possessed some memory, or even just an inherited cultural bias, against things that are flagrantly wrong).
Now, I will say what I wrote earlier had this flaw: is isn’t merely the number of options but their scope. It is actually possible to, by ignoring all aspects of there being “truth” and focus merely that there is truth at all, to construct a list of options so absent of content, mere philosophical / ontological statements without burden of details, that you can say, logically, one must be true.
These are useless.
The one I know of is this, to be presented with two options:
There is a supernature beyond the realm of dimensions and time, on which creation depends but which is distinct from creation.
There is no such supernature at all.
We may recognize the second as atheism, if the more manly philosophical type onwards to the wishy washy Golden Compass sort it hardly matters. With either (and what may be construed in between) the whole system is all and there is nothing besides it upon which it is predicated.
The former may be loosely recognized as some form of theism with an actual creator.
Between these two there are no other options, one must be true and one must be false; however, I say they are useless because in lacking detail at all, in being so barren, they serve little more purpose with respect to truth than a sign warning to observe warning signs does.