Loop Invariants as Tautologies
A Tautology is a formula (in a certain logic) that is true under every model of that logic. That is, it is equivalent to the formula "$True$".

A loop invariant, however, is a certain claim that is usually true under some models, and false under others (a model in this case is an algorithm). Then, you prove that the invariant is true under your specific model.

If you add axioms to your logic that forces then only model to be the one of your specific program, then indeed this would be a tautology. But such a process (adding axioms and proving that something is a tautology), is what is more commonly known as "proof". (For clarification: even if you add enough axioms, you may not be able to prove your claim, even if it's a tautology, in case of incomplete systems).

For example, consider a loop that increases a variable $i$ by $1$. An invariant of the loop may be that if before the loop $i&gt;0$, then after the loop $i&gt;0$. Indeed, this loop satisfies it. But it is not a tautology, since we can come up with other loops that do not satisfy it.