Is it possible to define something in terms of what is true about it?
Suppose for example, the case where in mathematics a definition is given in terms of formal symbols and rules for which strings involving them are designated as 'true'.
Physical systems, more precisely the state of a physical system, is also defined "in terms of what is true about it". In this case, the "true statements" are yes/no propositions that can be operationally tested by constructible apparatus (very much a logical-positivist-like approach, but arising from strictly physical considerations).
Then, measuring the yes/no-outcomes for a maximal set of consistent propositions completely determines the measured system's state. Here, "consistent" means simultaneously measurable, i.e., no measurement disturbs the already-determined outcome of any other. And "maximal" means that any additional measurement necessarily distrurbs one or more already-determined outcomes.
And, in fact, there's no other way to define a physical system's state. So, not only "can you define...", but you've got no choice "but to define...".
... too-long-for-a-comment answer to @Gordon's comment below
The remarks above are just a standard result. It's typically derived as a quantum result since classical propositions are always consistent (also called "compatible"). But the same idea holds for everyday measurements of macroscopic systems, as follows.
Firstly, as I parenthetically remarked above vis-a-vis logical positivism, particularly the verifiability theory of meaning, a proposition is associated with the procedure or apparatus used to verify/measure it. And then the general idea here is that whenever two or more propositions are compatible, you can construct a single apparatus that simultaneously measures them both (or them all).
You can see that exactly stated in https://books.google.com/books?id=dVs8PcZ0Hd8C&pg=PA115 (just above the commutator, Eq.6.25). I'm not sure why Isham calls that "trivially compatible" (I'm not sure how they'd commute otherwise). So let's just overlook that wrinkle (unless somebody can followup with a counterexample).
The ultimate upshot of this is that any observable can be decomposed into a maximal set of yes/no-propositions (which are just observables with exactly two possible outcomes/eigenvalues). And that's precisely stated and derived somewhere in Chapter 5 or Chapter 6 of https://books.google.com/books/about/Foundations_of_quantum_mechanics.html?id=FwpRAAAAMAAJ But I'm not immediately finding it in my printed copy, which I studied carefully some years (or decades) ago. And google seems reluctant to display entire pages, anyway (though I did notice an easy-to-find pirated pdf you can download, but won't explicitly give out the url).
Perhaps the easiest-to-follow discussion comprises the first six pages (right, just six) of https://books.google.com/books?id=WWYbAQAAIAAJ at which point Schwinger concludes "...the symbol of this compound measurement is...[math elided]...which then describes a complete measurement, such that the system possesses definite values for the maximum number of attributes; any attempt to determine the value of still another independent physical quantity will produce uncontrollable changes in one or more of the previously assigned values." (he didn't seem to have much use for periods)
Anyway, the above quote's from pages 5-6 (and you might likely want to read through page 12), but google also seems reluctant to display entire pages of this book, and I'm not seeing any pdf's. Schwinger's subsequent https://books.google.com/books?id=fDX6CAAAQBAJ&pg=PA001 develops that discussion much more formally, and is a correspondingly harder read, but google seems willing to display pages.
Finally, re Boltzmann, no. These are complete measurements determining pure states, not density matrices determining canonical ensembles. That wouldn't really address the OP's question regarding "things" (I'm taking his "something" to explicitly mean thing rather than ensemble). Whatever philosophical argument you want to have about the definition of "thing", a pure state is as close to "thing" as you're going to physically get.
You give the example yourself: Yes, algebraic structures, such as groups, rings, fields, or lattices are defined in exactly this way. The definition of, say, a group does not specify what the set of elements and the operations of a group are, but it specifies what kind of formulas these elements and operations must necessarily satisfy – not more, and not less.