You have asked about using the term 'probably' in this post. Yes, you can use any word at all in a proposition if it's meaningful. There are different types of logic such as syllogistic, sentential, FOPC, and modal for starters, and some are more sophisticated than others. Ultimately, when analyzing language, one chooses a logic depending on what is important. Necessity, probability, possibility, and obligation are different aspects of language related to linguistic ideas such as tense, evidentiality, conditionals, and so on, and basic logics often lack symbols to represent them.
Additionally, it seems what you are looking at is how to make logic fit natural language and ordinary practice. If you're interested in improving inductive logic, start with that article. There's a big difference between purely Aristotelian syllogisms of Logic 101 and informal logic. Conifold has pointed you towards two entries: logic and probability and generalized quantifiers. There is no shortage of logics, so another interesting variant is fuzzy logic; as well, explore the nature of concepts themselves. Traditional logics presume a nature of definition (generally necessity and sufficiency) that some considered a special case of how concepts function. Wittgenstein's passage on family resemblance moved a lot of thinkers in a different direction. Lastly, if you're interested on improving propositional structure, look at grammar. Since Chomsky's work, the notion of surface and deep structure has provoked philosophers of language to explore issues of syntax and semantics from a number of perspectives that give insight on what goes on in ordinary language use with a philosophical bent.