It depends on what kind of logic you're working in. You need to choose a mathematical universe in which to work i.e. an axiomatic system as you allude to in the question; mathematicians have realised this a long time ago when they bumped into paradoxes like the liar paradox, Russel's paradox etc. So let's explore a few logics.
Classical propositional calculus: https://en.wikipedia.org/wiki/Propositional_calculus. In this logic, the only thing you're allowed to talk about, at the most atomic level, is propositions. You can connect them up with AND and OR operators, and the unary NOT operator can be applied, but that's about it. The "universe" you're choosing here is analogous to the innards of a computer: a proposition is a "bit" which can be either "on" or "off" and all sentences are "circuits" using AND-gates or OR-gates. In this logic you're not allowed to talk about ice cream, tastiness etc. They simply aren't part of the formalism. Now, of course you could label one of these atomic true/false bits with whatever label you want e.g. "This ice cream is tasty", but as far as the logic is concerned, this is just a label, nothing more. We could equally have just called the proposition $P$ or $Q$. The true underlying entity is still an atomic bit.
So, propositional logic is clearly not powerful enough to encode your sentence. What's next? Well, the next step up is first-order logic: https://en.wikipedia.org/wiki/First-order_logic#:~:text=First%2Dorder%20logic%E2%80%94also%20known,%2C%20linguistics%2C%20and%20computer%20science. Here, we are allowed to quantify over non-logical entities in order to build up logical sentences. However, as you allude to in the question, things get a bit murky at this point and we run the risk of running into paradoxes if you just let those sentences freely range over anything whatsoever. For example, you can form sentences such as "Does the set of all sets contain itself?" In order to put a safety harness on the theory and protect it from such malicious paradoxes, and in order to cross the boundary from philosophy to well-defined mathematics, you need to choose a domain of discourse: https://en.wikipedia.org/wiki/Domain_of_discourse. The domain of discourse is some well-defined universe with an axiomatic characterisation.
So to say "This ice cream is tasty" is a philosophical question, not a mathematical one. However, if you first define a mathematical universe with a set of "ice cream" objects and a "tastiness" predicate, of course you can form such a sentence which, in your custom defined world, will have a perfectly well defined true/false value. I.e. it will be a proposition in that world.
It's also worth saying that propositional calculus and first order logic aren't the only kinds of logics out there. They were enough to highlight the main issues brought up in this question, but they certainly aren't the only logics that we could have discussed. You may want to explore the truth/falsehood of that sentence in other logics. But again, you'd still have to rigorously define what ice cream and tastiness are, mathematically speaking, and you'd still have to be explicit about the logic in which you're working.
