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I'm interested in applying logic to day-to-day reasoning. The problem is that formal logic seems really restrictive to limit inductive arguments to be only universal ("all swans are white"). Few things in the world are true across all instances.
My question is if it's fine to make statements like "all swans are things that are probably white". Because it also seems like this is the same thing as saying "some swans are white".
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.
My question is if it's fine to make statements like "all swans are things that are probably white". Because it also seems like this is the same thing as saying "some swans are white".
All swans are things that are probably white" and "some swans are white" differ in quantity -- "all" vs "some." Moreover, the predicate-terms "things that are probably white" and "white [things]" differ greatly: "things that are probably white" denotes a thing whose quality of whiteness depends on an undefined "probability" (whose lack of definition makes this an unacceptable proposition) -- which we impute to all swans; Whereas "white [things]" denotes things that are white -- that is, we are imputing whiteness to some swans.
